Jekyll2021-04-28T21:12:21+00:00https://howshouldithinkabout.com/feed.xmlHow Should I Think AboutA place for experimental essaysThe Buffett-Munger system in their own words2021-03-03T00:00:00+00:002021-03-03T00:00:00+00:00https://howshouldithinkabout.com/investing/buffett-and-munger-on-investing<p class="small"><em>This article is not investment advice. It is for educational purposes only.</em></p>
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<p>Here is my compilation of lessons on investing by Warren Buffett and Charlie Munger.</p>
<p>I divided their lessons into 19 sections — use the Contents menu to navigate across them. Go deeper into the context of each quote by clicking and expanding them.</p>
<p>My main source are decades of Q&A sessions during Berkshire’s annual meetings. <a href="https://buffett.cnbc.com/annual-meetings">Their video recordings (with searchable transcripts) have been made available online by CNBC</a>.</p>
<p>The other primary sources are, of course, the famous <a href="https://berkshirehathaway.com/letters/letters.html">Buffett’s Annual Letters to Berkshire Shareholders</a>.</p>
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<h2 id="the-overarching-principles">The overarching principles</h2>
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“The first rule of investing is don’t lose. And the second rule of investment is don’t forget the first rule”
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<div>
<p>From <a href="https://www.youtube.com/watch?v=8OcegOGAGIs">his first TV interview, aired in 1985</a>:</p>
<blockquote>
<p>The first rule of investing is don’t lose. And the second rule of investment is don’t forget the first rule.</p>
<p>And that’s all the rules there are. I mean that. If you buy things for far below what they’re worth, and you buy a group of them, you basically don’t lose money.</p>
</blockquote>
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<summary>
“What we’re doing in investment — and what everybody is doing in investment — is they’re laying out money now to get more money back later on”
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<div>
<p>From the <a href="https://buffett.cnbc.com/video/2008/05/03/morning-session---2008-berkshire-hathaway-annual-meeting.html">2008 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Well, what we’re doing in investment — and what everybody is doing in investment — is they’re laying out money now to get more money back later on.</p>
</blockquote>
<p>From the <a href="https://buffett.cnbc.com/video/2002/05/04/afternoon-session---2002-berkshire-hathaway-annual-meeting.html">2002 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>All investment is, is laying out some money now to get more money back in the future. Now, there’s two ways of looking at the getting the money back. One is from what the asset itself will produce. That’s investment.</p>
<p>One is from what somebody else will pay you for it later on, irrespective of what the asset produces, and I call that speculation.</p>
<p>So, if you are looking to the asset itself, you don’t care about the quote because the asset is going to produce the money for you. And that’s how — that’s what society, as a whole, is going to get from investing in that asset.</p>
<p>Then there’s the other way of looking at it, is what somebody will pay you tomorrow for it, even if it’s valueless. And that’s speculation. And of course, society gets nothing out of that eventually, but one group profits at the expense of another.</p>
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“Your goal as an investor should simply be to purchase, at a rational price, a part interest in an easily-understandable business whose earnings are virtually certain to be materially higher 5, 10 and 20 years from now”
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<div>
<p>From his <a href="https://berkshirehathaway.com/letters/1996.html">1996 Annual Letter</a>:</p>
<blockquote>
<p>Your goal as an investor should simply be to purchase, at a rational price, a part interest in an easily-understandable business whose earnings are virtually certain to be materially higher 5, 10 and 20 years from now.</p>
<p>Over time, you will find only a few companies that meet these standards — so when you see one that qualifies, you should buy a meaningful amount of stock.</p>
<p>You must also resist the temptation to stray from your guidelines: If you aren’t willing to own a stock for 10 years, don’t even think about owning it for 10 minutes. Put together a portfolio of companies whose aggregate earnings march upward over the years, and so also will the portfolio’s market value.</p>
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Valuation is about “identifying the key variables in a particular business, and evaluating how predictable they are”
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<p>From <a href="https://www.youtube.com/watch?v=MAo5I5saSfo">1998 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>If I were teaching of course on investments, there would be simply one valuation study after another. With the students trying to identify the key variables in that particular business, and evaluating how predictable they were. Because that is the first step. If something is not very predictable, forget it.</p>
<p>On the final exam, I would take an Internet company and I would say: “The question is, how much is this worth?” And anybody that gave me an answer I would flunk!</p>
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“Accounting is the language of business, and it is an imperfect language, but unless you are willing to put in the effort to learn accounting, you really shouldn’t select stocks yourself”
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<p>Buffett quoted in the book <a href="https://www.goodreads.com/work/quotes/4475839-warren-buffett-and-the-interpretation-of-financial-statements-the-searc">Warren Buffett and the Interpretation of Financial Statements</a>:</p>
<blockquote>
<p>You have to understand accounting and you have to understand the nuances of accounting. It’s the language of business and it’s an imperfect language, but unless you are willing to put in the effort to learn accounting — how to read and interpret financial statements — you really shouldn’t select stocks yourself.</p>
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“I don’t look at the primary message of Graham as [having] anything to do with formulas. There [are] three important aspects to it: (1) your attitude toward the stock market; (2) the margin of safety; (3) looking at stocks as businesses”
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<div>
<p>During the <a href="https://buffett.cnbc.com/video/1995/05/01/morning-session---1995-berkshire-hathaway-annual-meeting.html">1995 Berkshire Hathaway Annual Meeting</a>, Buffett summarized the key principles of investing, as taught by <a href="https://en.wikipedia.org/wiki/Benjamin_Graham">Benjamin Graham</a>, as follows:</p>
<blockquote>
<p>I don’t look at the primary message, from our standpoint, of Graham, really, as [having] anything to do with formulas.</p>
<p>There [are] three important aspects to it:</p>
<ol>
<li>
<p>One is your attitude toward the stock market. That’s covered in Chapter 8 of “<a href="https://en.wikipedia.org/wiki/The_Intelligent_Investor">The Intelligent Investor</a>.” I mean, if you’ve got that attitude toward the market, you start ahead of 99% of all people who are operating in the market. So, you have an enormous advantage.</p>
</li>
<li>
<p>Second principle is the margin of safety, which again, gives you an enormous edge, and actually has applicability far beyond just the investment world.</p>
</li>
<li>
<p>And then the third is just looking at stocks as businesses, which gives you an entirely different view than most people that are in the market.</p>
</li>
</ol>
<p>And with those three sort of philosophical benchmarks, the exact evaluation technique you use is not really that important.</p>
<p>Because you’re not going to go way off the track, whether you use <a href="https://en.wikipedia.org/wiki/Walter_Schloss">Walter Schloss</a>’s approach — or mine, or whatever. <a href="https://en.wikipedia.org/wiki/Philip_L._Carret">Phil Carret</a> has a slightly different approach. But it’s got those three cornerstones to it, I will guarantee. And believe me, he’s done very well.</p>
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<p>Over the long term, it is the level of returns on capital of a business that really matter (i.e., that is what dominates investment outcomes):</p>
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“If you buy a stock at a sufficiently low price, there will usually be some hiccup in the fortunes of the business that gives you a chance to unload at a decent profit. [But] time is the enemy of the mediocre [businesses].” “[This] cigar-butt strategy worked very well while I was managing small sums. With large sums, it would never work well”
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<div>
<p>From the <a href="https://berkshirehathaway.com/letters/1989.html">1989 Annual Letter</a>:</p>
<blockquote>
<p>If you buy a stock at a sufficiently low price, there will usually be some hiccup in the fortunes of the business that gives you a chance to unload at a decent profit, even though the long-term performance of the business may be terrible. I call this the “cigar butt” approach to investing. A cigar butt found on the street that has only one puff left in it may not offer much of a smoke, but the “bargain purchase” will make that puff all profit.</p>
<p>Unless you are a liquidator, that kind of approach to buying businesses is foolish:</p>
<ol>
<li>
<p>First, the original “bargain” price probably will not turn out to be such a steal after all. In a difficult business, no sooner is one problem solved than another surfaces — never is there just one cockroach in the kitchen.</p>
</li>
<li>
<p>Second, any initial advantage you secure will be quickly eroded by the low return that the business earns.</p>
</li>
</ol>
<p>For example, if you buy a business for $8 million that can be sold or liquidated for $10 million and promptly take either course, you can realize a high return. But the investment will disappoint if the business is sold for $10 million in ten years and in the interim has annually earned and distributed only a few percent on cost.</p>
<p>Time is the friend of the wonderful business, the enemy of the mediocre.</p>
</blockquote>
<p>From the <a href="https://www.berkshirehathaway.com/letters/2014ltr.pdf">2014 Annual Letter</a>:</p>
<blockquote>
<p>My cigar-butt strategy worked very well while I was managing small sums. Indeed, the many dozens of free puffs I obtained in the 1950s made that decade by far the best of my life for both relative and absolute investment performance.</p>
<p>Even then, however, I made a few exceptions to cigar butts, the most important being GEICO. Thanks to a 1951 conversation I had with Lorimer Davidson, a wonderful man who later became CEO of the company, I learned that GEICO was a terrific business and promptly put 65% of my $9,800 net worth into its shares. Most of my gains in those early years, though, came from investments in mediocre companies that traded at bargain prices. Ben Graham had taught me that technique, and it worked.</p>
<p>But a major weakness in this approach gradually became apparent: Cigar-butt investing was scalable only to a point. With large sums, it would never work well.</p>
<p>In addition, though marginal businesses purchased at cheap prices may be attractive as short-term investments, they are the wrong foundation on which to build a large and enduring enterprise. Selecting a marriage partner clearly requires more demanding criteria than does dating. (Berkshire, it should be noted, would have been a highly satisfactory “date”: If we had taken Seabury Stanton’s $11.375 offer for our shares, BPL’s weighted annual return on its Berkshire investment would have been about 40%.)</p>
<p>It took Charlie Munger to break my cigar-butt habits and set the course for building a business that could combine huge size with satisfactory profits. […] From my perspective, though, Charlie’s most important architectural feat was the design of today’s Berkshire. The blueprint he gave me was simple: Forget what you know about buying fair businesses at wonderful prices; instead, buy wonderful businesses at fair prices.</p>
<p>The year 1972 was a turning point for Berkshire (though not without occasional backsliding on my part — remember my 1975 purchase of Waumbec). We had the opportunity then to buy See’s Candy for Blue Chip Stamps, a company in which Charlie, I and Berkshire had major stakes, and which was later merged into Berkshire.</p>
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<p class="small center muted">· · ·</p>
<h2 id="intrinsic-value-is-the-discounted-value-of-the-cash-that-can-be-taken-out-of-a-business-during-its-remaining-life-and-its-an-estimate">Intrinsic value is the discounted value of the cash that can be taken out of a business during its remaining life (and it’s an estimate)</h2>
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What is intrinsic value?
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<p>From <a href="https://berkshirehathaway.com/owners.html">Berkshire Hathaway’s An Owner’s Manual</a>:</p>
<blockquote>
<p>Intrinsic value can be defined simply: It is the discounted value of the cash that can be taken out of a business during its remaining life.</p>
</blockquote>
<p>Here is another Buffett writing on intrinsic value that clarifies the meaning of <em>discount</em>. From his <a href="https://berkshirehathaway.com/letters/1992.html">1992 Annual Letter</a>:</p>
<blockquote>
<p>In [the book] The Theory of Investment Value, written over 50 years ago, John Burr Williams set forth the equation for [intrinsic] value, which we condense here:</p>
<p>“The value of any stock, bond or business today is determined by the cash inflows and outflows — discounted at an appropriate interest rate — that can be expected to occur during the remaining life of the asset.”</p>
<p>Note that the formula is the same for stocks as for bonds.</p>
<p>Even so, there is an important, and difficult to deal with, difference between the two: A bond has a coupon and maturity date that define future cash flows; but in the case of equities, the investment analyst must himself estimate the future “coupons.”</p>
</blockquote>
<p>In other words, the role of discount rates is to find the “present” value of “concealed-value coupons” that a given stock (or business) is expected to pay over future years.</p>
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<summary>
“Intrinsic value is an estimate rather than a precise figure.” “We are trying to figure out what businesses are going to be worth in 10 or 20 years”
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<div>
<p>From the <a href="https://berkshirehathaway.com/owners.html">Berkshire Hathaway’s An Owner’s Manual</a>:</p>
<blockquote>
<p>The calculation of intrinsic value, though, is not so simple. As our definition suggests, intrinsic value is an estimate rather than a precise figure, and it is additionally an estimate that must be changed if interest rates move or forecasts of future cash flows are revised.</p>
</blockquote>
<p>From <a href="https://www.youtube.com/watch?v=5ioeNrmn4eY">1997 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>As we have said, in high tech businesses or something like that, we don’t have the faintest idea of what the “coupons” are going to be. When we get into businesses where we think we can understand them reasonably well, we are trying to print the “coupons”. We are trying to figure out what businesses are going to be worth in 10 or 20 years.</p>
<p>They are a number filters which say to us, we don’t know what that business is gonna be worth in 10 or 20 years, and we can’t even make an educated guess. Obviously, we don’t think we know to three decimal places (or two decimal places, or anything like that) what precisely what’s going to be produced. But we have a high degree of confidence that we’re in the ballpark with certain kinds of businesses.</p>
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“We really like the decision to be obvious enough to us that it doesn’t require making a detailed calculation”
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<div>
<p>From the <a href="https://www.youtube.com/watch?v=UWvOK-_EtDM">1995 Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: When you’re valuing the companies and you discount back the future earnings that you talked about. How many years out you generally go? If you don’t go out a general number of years, how do you arrive at that time period?</em></p>
<p>Warren: Despite the fact that we can define that in a very kind of simple and direct equation, we’ve never actually sat down and and written out a set of numbers that to relate that equation. We do it in our heads in a way, obviously. I mean, that’s what it’s all about.</p>
<p>But there is no piece of paper. And there never was a piece of paper that shows what our calculation on Helzberg Diamonds, See’s Candy or the Buffalo News was in that respect. So it would be attaching a little more scientific quality to our analysis than there really is. If I could give you some gobbledygook, like “We do it for 18 years, and stick a terminal value on it, and do all of this”.</p>
<p>We are sitting in the office, thinking about that question with each business, each investment. And we have discount rates, in a general way, in mind, but we really like the decision to be obvious enough to us that it doesn’t require making a detailed calculation.</p>
<p>It’s the framework, but it’s not applied in the sense that we actually fill in all the variables.</p>
<p>Charlie: Somebody once subpoenaed our staffing papers on some acquisition. And of course not only did we not have any staffing papers, we didn’t have any staff.</p>
</blockquote>
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<summary>
Ideally, estimate the intrinsic value first, and only then look at price
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<p>From a <a href="https://www.youtube.com/watch?v=8OcegOGAGIs">TV interview in 1984</a>:</p>
<blockquote>
<p>I would rather value a stock (or a business) first, and not even know the price — so that I’m not influenced by the price in establishing my valuation.</p>
<p>And then look at the price later to see whether it’s way out of line with what my value is.</p>
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Is “a bird in the hand is worth two in the bush”? You can only answer if you know when you would get those birds, and what the interest rates are
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<div>
<p>From the <a href="https://buffett.cnbc.com/video/2000/04/29/birds-hands-and-bushes-the-first-investment-primer.html">2000 Annual Meeting</a> (<a href="https://www.youtube.com/watch?v=vo_TWaV6Xy8">YouTube</a>):</p>
<blockquote>
<p>When would you guess it was written the first investment primer that I know of? It was pretty good advice, and it was delivered about 600 B.C. by <a href="https://en.wikipedia.org/wiki/Aesop">Aesop</a>. Aesop, you’ll remember, said “a bird in the hand is worth two in the bush”.</p>
<p>Now, Aesop was on to something, but he didn’t finish it. Because there’s a couple of other questions that go along with that:</p>
<ol>
<li>He forgot to say exactly when you were going to get the two birds from the bush, and</li>
<li>He forgot to say what interest rates were that you had to measure this against</li>
</ol>
<p>But if he’d given those two factors, he would have defined investment for the next 2600 years.</p>
<p>Will you trade a bird in the hand? [In] investing, [the equivalent is to] lay out cash today.</p>
<p>The question is, as an investment decision, you have to evaluate how many birds are in the bush. You may think there are 2 birds in the bush or 3 birds in the bush. And you have to decide when they’re going to come out (i.e., when you’re going to “acquire” them).</p>
<p>If interest rates are 5% and you’re going to get 2 birds from the bush in 5 years <em>vs.</em> [just] 1 now, 2 birds in the bush are much better than 1 bird in the hand now. You would want to trade your bird in the hand. I’ll take 2 birds in the bush because if we’re going to get them in 5 years that’s roughly 14% compounded, [given that] interest rates are only 5% percent.</p>
<p>But if interest rates were 20%, you would decline to take 2 birds in the bush 5 years from now. You would say, that’s not good enough. Because out of 20% percent, if I just keep this bird in my hand and compound it, I’ll have more birds than 2 birds in the bush in 5 years.</p>
<p>Now, what’s all that got to do with growth?</p>
<p>Well, usually people associate growth with a lot more birds in the bush but you still have to decide when you’re going to get them. And you have to measure that against interest rates and you have to measure it against other bushes and other equations. And that’s all investing is. It’s a value decision based on what it is worth, how many birds are in that bush, when you’re going to get them, and what interest rates are.</p>
</blockquote>
<p>He then went on explaining the challenges with high stock prices in the context the <a href="https://en.wikipedia.org/wiki/Dot-com_bubble">Dot-com bubble</a>:</p>
<blockquote>
<p>Let’s just take a company that has marvelous prospects. It is paying you nothing now, and you buy it at a valuation of $500 billion. Now, if you feel that 10% is the appropriate [annual] rate of return (and you can pick your figure). That means that if it pays you nothing this year, but it starts paying next year, it has to be able to pay you $55 billion in perpetuity [more precisely, for the next 26 years], each year. But if it’s not gonna pay until the 3rd year, then it has to pay you $65 billion in perpetuity [more precisely, for the next 29 years] to justify the present price.</p>
<p>Every year that you wait to take a bird out of the bush means that you have to take out more birds. It’s that simple. I question in my mind whether sometimes whether people who buy stocks at $500 billion dollars are really thinking of the mathematics implicit in what they are doing.</p>
<p>Let’s assume there’s only going to be a one-year delay before the business starts paying out to you. And you want to get a 10% return. If you buy at $500 billion, that means $55 billion of cash that they have to be able to disgorge to you year after year, after year. To do that they have to make perhaps $80 billion dollars (or close) pre-tax. Now you might look around at the universe of businesses in this world and see how many are earning $80 billion pre-tax (or 70 or 60 or 50 or 40 or 30). And you won’t find any.</p>
<p>So it requires a rather extraordinary change in profitability to give you enough birds out of that particular bush to make it worthwhile to give up the one that you have in your hand.</p>
</blockquote>
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<summary>
“It gets very dangerous to project out high growth rates because you get into this paradox. If you say the growth rate of a company is going to be 9% between now and judgment day and you use a 7% discount rate, it goes off [and] you get into infinity. And that’s where people get in a lot of trouble”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2004/05/01/morning-session---2004-berkshire-hathaway-annual-meeting.html">2004 Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: Mr. Buffett, my question is on business valuation and growth. In one of your letters, you mentioned the discounting formula on earnings divided by the difference between the discount and the growth rate.</em></p>
<p><em>But if the growth rate is larger than the discount rate and if we use this formula, then we get a negative number. And one way around this — let’s call it method A — is to have two growth stages, one with a high growth and the second stage with a low growth.</em></p>
<p><em>And the second way, method B, would be to estimate how much the earnings is on the third year for the company and then multiply this by the average price-to-earning ratio to get the price in the tenth year.</em></p>
<p><em>I don’t know if you use the method A or method B, but if not, I would like to ask, Mr. Buffett, how do you estimate how much a company is worth if the growth rate is larger than the discount rate?</em></p>
<p>Buffett: Well, you put your finger on an interesting mathematical relationship. Because if you’re using a present value discount formula and you put in a growth rate that is higher than the discount rate, as you have postulated, the answer, of course, will be infinity.</p>
<p>And there are a lot of managements around who like to think their stocks are worth infinity, but we haven’t found one yet.</p>
<p>That precise subject was covered in <a href="https://www.jstor.org/stable/2976852">a paper called ‘Growth Stocks and The Petersburg Paradox’ by a fellow named David Durand</a> probably 30 years ago. And somewhere, we probably have a copy at our office.</p>
<p>[Note: see also <a href="https://www.turtletrader.com/outliers.pdf">‘Integrating the Outliers: Two Lessons from the St. Petersburg Paradox’ by Michael J. Mauboussin and Kristen Bartholdson</a>]</p>
<p>It gets very dangerous to project out high growth rates because you get into this paradox. If you say the growth rate of a company is going to be 9% between now and judgment day and you use a 7% discount rate, it goes off, you know, you get into infinity. And that’s where people get in a lot of trouble.</p>
<p>The idea of projecting out extremely high growth rates for very long periods of time has caused investors to lose very, very large sums of money.</p>
<p>There aren’t many companies [that can grow significantly over long periods of time]. Just take a look at the Fortune 500, go back 50 years and look at the companies that were there and how many have really maintained rates much above 10%. It’s not an easy hurdle. And when you get up to 15%, you know, you’re in the atmosphere and rarified atmosphere.</p>
<p>So there’s a real danger in projecting out high growth rates. Charlie and I will very seldom — virtually never — get up into high digits. You can lose a lot of money doing that.</p>
<p>You may miss an opportunity some time, but I haven’t seen people who have been consistently successful doing that. And you do run into this paradox you mentioned.</p>
<p>Charlie: Well, you’re obviously right, when you get a mathematical result that is infinity, to back off and realize that can’t happen. And, of course, what people do is they project that the growth rate will reduce and, indeed, eventually stop. And then you get more realistic numbers. What else could anyone do?</p>
</blockquote>
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<summary>
“As a practical matter, if you estimated for 20 years or so, the terminal values get less important”
</summary>
<div>
<p>From the <a href="https://www.youtube.com/watch?v=qQIfkVjUeTI">2000 Annual Meeting</a>:</p>
<blockquote>
<p>As a practical matter, if you estimated for 20 years or so, the terminal values get less important.</p>
<p>You do want to have in your mind a stream of cash that will be thrown off over, say a 20 year period, that makes sense discounted at a proper interest rate compared to what you’re paying today, and that’s what investment’s all about.</p>
</blockquote>
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<p class="small center muted">· · ·</p>
<h2 id="use-a-common-discount-rate-when-comparing-between-investment-alternatives">Use a common discount rate when comparing between investment alternatives</h2>
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<summary>
Long-term government bonds are a common, ‘risk-free’ yardstick
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=5ioeNrmn4eY">1997 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>We basically use long-term risk-free government bond-type interest rates to think back in terms of what we should discount at.</p>
</blockquote>
<p>Also from the <a href="https://www.youtube.com/watch?v=0HSZRWg9Ne0">same meeting</a>:</p>
<blockquote>
<p><em>Question: Why would you use the risk-free rate?</em></p>
<p>Warren: The risk-free rate is used merely to equate one item to another. In other words, we’re looking for whatever is the most attractive, but, in terms of present valuing anything, we’re going to use a number.</p>
<p>Obviously, we can always buy the government bonds, so that becomes the yardstick rate.</p>
<p>It doesn’t mean we want to buy government bonds. It doesn’t mean we want to buy government bonds if the best thing we can find is only has a present value that works out at a 0.5% a year better than the government bond. But [gov’t bonds are] the appropriate yardstick in our view to simply use and compare across all kinds of investment opportunities (oil wells, farms, whatever it may be).</p>
<p>[Of course, the comparison among investments then] gets into degree of certainty. But [the risk-free rate is the] yardstick, and it does serve to make that a constant throughout the valuation process.</p>
</blockquote>
<p>From <a href="https://www.youtube.com/watch?v=5ioeNrmn4eY">1999 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>It doesn’t mean we would pay that figure once we use that discount number. But we would use that to establish comparability across investment alternatives.</p>
<p>So, if we were looking at 50 companies, and making the sort of calculation that you just talked about, we would probably use the long-term government rate to discount it back. But we wouldn’t pay that number after we discounted it back.</p>
<p>We would look for appropriate [‘margins of safety’] from that figure. [In fact,] it doesn’t really make any difference whether you use a higher figure and then look across them, or use our figure and look for the biggest [‘margin of safety’].</p>
</blockquote>
</div>
</details>
<details>
<summary>
The discount rate accounts for the <em>value of cash flows over time</em>, it is <em>not</em> about their ‘risk’ — “The purity of the idea is that you’re discounting future cash, and it doesn’t make any difference whether cash comes from a risky business or a ‘so-called’ safe business”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=JsbVonhF-q4">1996 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: I have a question in determining a company’s intrinsic value. You seem to write or indicate that you project out a company’s owner earnings for a number of years and then discount that back by prevailing rates. My question is, How much of a premium, if any, to prevailing risk-free rates do you demand when you discount back the owner earnings of a company? Or stated differently, for example, today with long rates at about 7%, if you did the same exercise with Coca-Cola, at what rate of interest would you discount back their owner earnings?</em></p>
</blockquote>
<blockquote>
<p>We get asked that question a lot, and we’ve answered to some extent in past annual reports about what discount rate to use. We basically think in terms of the long-term government rate.</p>
<p>We don’t think we’re any good at predicting interest rates, but probably there may be times of, what would seem like, very low rates, we might use a little higher rate, but we don’t put the risk factor in it per se.</p>
<p>Because essentially the purity of the idea is that you’re discounting future cash, and it doesn’t make any difference whether cash comes from a risky business or a “so-called” safe business. So, the value of the cash delivered by a water company, which is going to be around for a hundred years, is not different than the value of the cash derived from some high-tech company — if any.</p>
<p>It may be harder for you to make the estimate. And you may, therefore, want a bigger discount when you get all through with the calculation. But up to the point where you decide what you’re willing to pay, you may decide you can’t estimate it at all. I mean, that’s what happens with us for most companies.</p>
<p>We believe in using the a government bond type interest rate. We believe in trying to stick with businesses that where we think we can see the future reasonably well. You never see it perfectly obviously.</p>
</blockquote>
</div>
</details>
<details>
<summary>
Of course interest rates affect how you value a business
</summary>
<div>
<p>Here’s a <a href="https://junto.investments/daily-journal-2021-transcript/">recent remark</a> by <a href="https://en.wikipedia.org/wiki/Charlie_Munger">Charlie Munger</a> on the record low interest rates of present day and its effects on stock prices:</p>
<blockquote>
<p><em>Question: Previously, you have said; It takes character to sit with all that cash and do nothing. I didn’t get to where I am by going after mediocre opportunities. In the past few years, equity prices have increased significantly and cash has arguably become riskier due to central banking policy. Have you considered amending this quote or lowering your standards?</em></p>
<p>Charlie Munger: I think everybody is willing to hold stocks at higher price-earnings multiples when interest rates are as low as they are now. And so I don’t think it’s necessarily crazy that good companies sell at way higher multiples than they used to.</p>
<p>On the other hand, as you say, I didn’t get rich by buying stocks at high price-earnings multiples in the midst of crazy speculative booms. I’m not going to change.</p>
<p>I am more willing to hold stocks at high multiples than I would be if interest rates were a lot lower. Everybody is.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“We are affected in that valuation process to a considerable degree by interest rates, but not by whether they’re 7.3% or 7.0% or 7.5%. We will be thinking much differently if their long-term rates are 11% or 5%, but we don’t have any magic multiples in mind”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=7x484sx70kg">1995 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>We are affected in that valuation process to a considerable degree by interest rates, but not by whether they’re 7.3% or 7.0% or 7.5%. I mean, we will be thinking much differently if their long-term rates are 11% or 5%, but we don’t have any magic multiples in mind.</p>
<p>We’re thinking we want to be in the business the 10 years from now is earning a whole lot more money than it is now. And that we will still feel good about the prospects of the business at that time. That’s the kind of business we’re trying to buy all of [in private transactions], and that’s the kind of business that we try and buy part of [via the stock market]. And then sometimes we buy others, too.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“How much more [than the government bond rates] do we want?” “We want enough so that we feel very comfortable [that] (1) if they close on the stock market for a couple of years, [and] (2) if interest rates go up another 100 basis points or 200 basis points, we’re still happy with what we bought”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=176DGSK3GWc">2007 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>How much more [than the government bond rates] do we want?</p>
<p>Well, if government bond rates were 2%, we’re not gonna buy a business to earn 3 or 3.5% expectancy over the years. We just don’t want to commit our money that way. We’d rather sit around a wait a little while.</p>
<p>If they’re 4.75%, what do we hope to get over time? Well, we want to get a fair amount more than that [4.75%].</p>
<p>But I can’t tell you that we sit on every morning and I call Charlie in Los Angeles and say what’s our hurdle rate today? I mean, we have never used the term, you know.</p>
<p>We want enough so that we feel very comfortable [that]:</p>
<ol>
<li>if they close on the stock market for a couple of years,</li>
<li>if interest rates go up another 100 basis points or 200 basis points,</li>
</ol>
<p>We’re still happy with what we bought.</p>
<p>I know it sounds kind of fuzzy, but it is fuzzy.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“If we thought we were getting a stream of cash over the next 30 years that we felt extremely certain about, we would use a discount rate that would be somewhat less than if it was one where we thought we might get some surprises in 5-10 years”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=-blp9SREK4g">1994 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>I would say that in a world of 7% long-term bond rates that we would certainly want to think we were discounting future after-tax streams of cash at least a 10% rate. But that will depend on the certainty we feel about the business. The more certain we feel about a business, the closer we are willing to play it.</p>
<p>We have to feel pretty certain about any business before we’re even interested at all. But there are still degrees of certainty.</p>
<p>If we thought we were getting a stream of cash over the next 30 years that we felt extremely certain about, we would use a discount rate that would be somewhat less than if it was one where we thought we might get some surprises in 5 or 10 years — [if we thought that that] possibility [of surprises] existed.</p>
</blockquote>
</div>
</details>
<details>
<summary>
You don’t need an opinion on <em>future</em> interest rates to make a judgement on a business today
</summary>
<div>
<p>From <a href="https://buffett.cnbc.com/video/1994/04/25/morning-session---1994-berkshire-hathaway-annual-meeting.html">1994 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: I’m interested in that many of the holdings of Berkshire are in industries that are perceived as interest rate-sensitive industries, including Wells Fargo, Salomon, Freddie Mac, even GEICO. And yet you have an admitted sort of ambivalence towards interest rates or changes in interest rates. And it therefore seems that you don’t feel that those changes affect the fundamental attractiveness of those businesses. I thought maybe you could share your thoughts on what you see in these businesses that the investment community as a whole is ignoring.</em></p>
<p>Well, the value of every business, the value of a farm, the value of an apartment house, the value of any economic asset, is 100% sensitive to interest rates, because all you are doing in investing is transferring some money to somebody now in exchange for what you expect the stream of money to come in, over a period of time. And the higher interest rates are, the less that present value is going to be.</p>
<p>So every business, by its nature, whether it’s Coca-Cola or Gillette or Wells Fargo, is in its intrinsic valuation, is a 100% sensitive to interest rates.</p>
<p>Now, the question as to whether a Wells Fargo or a Freddie Mac or whatever it may be, whether their business gets better or worse internally, as opposed to the valuation process, because of higher interest rates, that is not easy to figure.</p>
<p>I mean, GEICO, if they write their insurance business at the same underwriting ratio — in other words they have the same loss and expense experience relative to premiums — they benefit by higher interest rates, obviously, over time, because they’re a float business, and the float is worth more to them.</p>
<p>Now, externally, getting back to the valuation part, the present value of those earnings also becomes less then. But the present value of Coke’s earnings becomes less in a higher interest rate environment.</p>
<p>Wells Fargo, whether they earn more or less money under any given interest rate scenario is hard to figure. There may be one short-term effect and there may be another long-term effect.</p>
<p>So I do not have to have a view on interest rates — and I don’t have a view on interest rates — to make a decision as to an insurance business, or a mortgage guarantor business, or a banking business, or something of the sort, relative to making a judgment about Coke or Gillette.</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="always-mind-your-opportunity-cost">Always mind your opportunity cost</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<details>
<summary>
“At any given time, when we consider an investment, we have to compare it to the best alternative investment we have at that time”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=3yu3UYlI2Y4">2001 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Charlie: Obviously considerations of cost are important in business. And obviously opportunity costs, which is a doctrine of economics (really, a doctrine of lifesmanship), are also very important.</p>
<p>We’ve always had that kind of basic thinking. Of course, capital isn’t free.</p>
<p>You have perfectly good old-fashioned doctrines, like opportunity cost: at any given time, when we consider an investment, we have to compare it to the best alternative investment we have at that time.</p>
</blockquote>
</div>
</details>
<details>
<summary>
Hurdle rates are no substitute for actually thinking things through — and properly comparing among the best opportunities available
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=176DGSK3GWc">2007 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Charlie: The concept of a hurdle rate makes nothing but sense. And yet, a lot of terrible errors are made by people who are talking about hurdle rates.</p>
<p>Just because you can measure something and guess it, it doesn’t mean that it’s the controlling variable of what you’re dealing with in a messy world.</p>
<p>I don’t see any substitute for thinking about a whole lot of investment options. And thinking about why one is better than another, and what the likely returns are from each, etc, etc.</p>
<p>The trouble of hurdle rate concept — not that we don’t have one in a sense — is that it doesn’t work as well as a system of comparing things.</p>
<p>In other words, if I have something available that I think will give me 8% for sure, and I can buy all I want of it. And you’ve got a perfectly good investment that I think will earn 7%, I don’t have to waste five minutes with you.</p>
<p>In the real world, your opportunity costs are what you want to make your decisions based.</p>
<p>Warren: And even if you had something you were really familiar with and very sure on the 8%, 8.5% wouldn’t tempt you if somebody came along. That’s the practical matter of it.</p>
<p>I’ve been on 19 corporate boards. I would say that, of the presentations I’ve seen (and I’ve seen a lot of them), every one had a calculation of internal rate of return (IRR). If they burned them all, the boards would have been better off.</p>
<p>I mean, there is so much nonsense presented. Because the presenters essentially know what the listeners desires of hearing, and what is needed in order to get through something that the CEO wants to do anyway that. You just get nonsense figures.</p>
<p>Charlie: I have a young friend who sells private partnership interests to investors. He’s in a really tough field where it’s hard to get decent returns. And I said, “What return do you tell them you’re aiming for?” And he said 20%. I asked, “How did you pick that number?” He said, if I chose any lower number they wouldn’t get me the money.</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="against-the-academic-models-of-modern-finance">Against the academic models of modern finance</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<details>
<summary>
“We define risk as the possibility of harm or injury. Now, it became very fashionable in the academic world, and then that spilled over into the financial markets, to define risk in terms of volatility, of which beta became a measure. Using [such] measures of risk, something whose return varies from year to year between 20% and 80% is riskier, as defined, than something whose return is 5% a year, every year. We just think the financial world has gone haywire in terms of measures of risk”
</summary>
<div>
<p>From <a href="https://buffett.cnbc.com/video/1994/04/25/morning-session---1994-berkshire-hathaway-annual-meeting.html">1994 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: I’d like to ask you to expound on your view of risk in the financial world, and I ask that against the background of what appear to be a number of inconsistencies between your view of risk and the conventional view of risk.</em></p>
<p><em>I mention that in a recent article you pointed out inconsistency in the use of beta as a measure of risk, which is a common standard.</em></p>
<p>Warren: Well, we do define risk as the possibility of harm or injury.</p>
<p>And in that respect we think [risk] inextricably wound up in your time horizon for holding an asset. If you intend to buy XYZ Corporation at 11:30 this morning and sell it out before the close today, that is — in our view — a very risky transaction. Because we think 50% of the time you’re going to suffer some harm or injury.</p>
<p>If you have a time horizon on a business, we think the risk of buying something like Coca-Cola at the price we bought it at a few years ago is essentially, is so close to nil in terms of our perspective holding period. But if you asked me the risk of buying Coca-Cola this morning and you’re going to sell it tomorrow morning, I say that is a very risky transaction.</p>
<p>Now, as I pointed out in the annual report, it became very fashionable in the academic world, and then that spilled over into the financial markets, to define risk in terms of volatility, of which beta became a measure. But that is no measure of risk to us.</p>
<p>Interesting thing is that using [such] conventional measures of risk, something whose return varies from year to year between 20% and 80% is riskier, as defined, than something whose return is 5% a year, every year. We just think the financial world has gone haywire in terms of measures of risk.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“Stick an extra 6% on the interest rate to allow for [the ‘risk’ of an investment]. I tend to think that’s kind of nonsense.”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=JsbVonhF-q4">1996 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>[We don’t] try to have the whole panoply with all different kinds of risk rates. Because, frankly, we think that just be playing games with numbers. I don’t think you can stick numbers on a highly speculative business where the whole industry is going to change in five years and have that mean anything.</p>
</blockquote>
<blockquote>
<p>If you say, “I’m going to stick an extra 6% on the interest rate to allow for the fact…” I tend to think that’s kind of nonsense. I mean, it may look mathematical, but it’s mathematical gibberish in my view. You better just stick with businesses that you can understand and use the government bond rate.</p>
</blockquote>
<p>From <a href="https://buffett.cnbc.com/video/1998/05/04/afternoon-session---1998-berkshire-hathaway-annual-meeting.html">1998 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Warren: All the capital asset pricing model-type reasoning with different rates of risk-adjusted return and all that, we think it is nonsense.</p>
<p>We do think it’s also nonsense to get into situations, or to try and evaluate situations, where we don’t have any conviction to speak of as to what the future is going to look like. And we don’t think you can compensate for that by having a higher discount rate and saying it’s riskier. So, “I don’t really know what’s going to happen and I’ll have a higher discount rate.” That just is not our way of approaching things.</p>
<p>Charlie: Yeah. This great emphasis on volatility in corporate finance we just regard as nonsense.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“I think you’d have to believe in the tooth fairy to believe that you could easily outperform the market by 7% per annum just by investing in high volatility stocks”
</summary>
<div>
<p><a href="https://fs.blog/2015/03/charlie-munger-academic-economics/">Charlie Munger’s 2003 Herb Kay Memorial Lecture — “Academic Economics: Strengths and Weaknesses, after Considering Interdisciplinary Needs” — at the University of California at Santa Barbara</a>:</p>
<blockquote>
<p>Berkshire’s whole record has been achieved without paying one ounce of attention to the efficient market theory in its hard form. And not one ounce of attention to the descendants of that idea, which came out of academic economics and went into corporate finance and morphed into such obscenities as the capital asset pricing model, which we also paid no attention to.</p>
<p>I think you’d have to believe in the tooth fairy to believe that you could easily outperform the market by seven-percentage points per annum just by investing in high volatility stocks.</p>
</blockquote>
</div>
</details>
<p>Using the <a href="https://en.wikipedia.org/wiki/Capital_asset_pricing_model">capital asset pricing model</a> (CAPM) is ‘inanity’:</p>
<details>
<summary>
“They invented all this ridiculous mathematics, which concluded that the companies that made the most money had the highest cost of capital; well, all I can say is it’s not for us”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=3yu3UYlI2Y4">2001 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Charlie: Well, cost of capital. First, obviously considerations of cost are important in business. And obviously opportunity costs, which is a doctrine of economics (really, a doctrine of lifesmanship), are also very important.</p>
<p>We’ve always had that kind of basic thinking. Of course, capital isn’t free and, of course, you incurr in cost of capital when you’re borrowing money, or at least you configure cost of loans. But the theorists had to develop some theory for what <em>equity</em> costs, and there they just went bonkers.</p>
<p>They said, if you earned 100% on capital because you had some marvelous business, your cost of capital was a 100%. And therefore you shouldn’t look at any opportunity that delivered a lousy 80%. That is the kind of thinking which came out of the <a href="https://en.wikipedia.org/wiki/Capital_asset_pricing_model">capital asset pricing model (CAPM)</a>, and so forth. That I’ve always considered inanity.</p>
<p>What is Berkshire’s cost of capital? We have this damn capital. It just keeps multiplying and multiplying. What does this cost?</p>
<p>If you have perfectly good old-fashioned doctrines, like opportunity cost, at any given time, when we consider an investment, we have to compare it to the best alternative investment we have at that time. We had perfectly good old-fashioned ideas that are very basic to use, but they weren’t good enough for these modern theaters. So they invented all this ridiculous mathematics, which concluded that the companies that made the most money had the highest cost of capital. Well, all I can say is it’s not for us.</p>
<p><em>[He is likely referring to the fact the model assumes a linear relationship between the return of an asset and its beta, which determines the so-called cost of equity.]</em></p>
<p>Warren: What you find, of course, is that the cost of capital is about a 25% below the return promised by any deal that the CEO wants to do.</p>
<p>I have listened to cost of capital discussions at all kinds of corporate board meetings. And, you know, I’ve never found anything that made very much sense in it. Except for the fact that is what they learned in business school, and that’s what the consultants talked about. And most of the board members would nod their head without knowing what the hell was going on. And that’s been my history with the cost of capital.</p>
</blockquote>
</div>
</details>
<p>Buffett does not use <a href="https://en.wikipedia.org/wiki/Modern_portfolio_theory">modern portfolio theory</a> (MPT) models to account for the time distribution of future cash flows or its ‘correlation’ with the general market. Nor does he try to “smooth out” returns. In fact, he takes in ‘lumpiness’ when properly rewarded for it (via their insurance business):</p>
<details>
<summary>
“We’re intelligently making these guesses as best we can based on our own circumstances and our own abilities. I think it’s crazy to do it based on somebody else’s circumstances and somebody else’s abilities [as it is the case with academic models]”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=fCt0sqnLpjQ">2003 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: You have just said that you actually apply the same discount rate across stocks. I’m sure you know that modern finance (e.g. <a href="https://en.wikipedia.org/wiki/Modern_portfolio_theory">modern portfolio theory</a>) actually suggests that you should not do that. [It recommends] that you should be thinking about the timing of cash flows and, in particular, the covariance with the general market.</em></p>
<p><em>Now, you’ve made a point of emphasizing that when you think of risk, you think of risk primarily in terms of, will you get the cash flows that you predict you will get over time? Sort of numerator risk if you think in terms of discounted cash flow. I think everyone here will have to acknowledge that your results speak for themselves — [so this] has probably been a very effective way of thinking about risk.</em></p>
<p><em>But there is a true economic cost to think about the timing of cash flows as well. And it may be a much smaller cost but it is still a real cost. I might, for example, suggest you think about somebody deciding between two jobs. The jobs are completely identical, and the person expects to make the same amount of money from each job. But there’s one difference, and the difference is one job will pay him more when the economy’s in the tank, and the other job will pay him more when the economy is going gangbusters. Now, if he asked you which job was actually worth more, my guess is you would tell him that the one that would pay him more when the economy’s in the tank. And the reason is, if he wanted to make more money by moonlighting or doing something else, it’d be much easier when the economy is doing better. That’s the essential logic behind the idea that you look at the covariance of when cash flows come in with the overall market.</em></p>
<p><em>It’s a real cost, even though it is difficult to measure. And even if it is a smaller risk than the numerator risk, the risk of getting the actual cash flows, since it’s a real cost I imagine you must think about it. My question to you would be, how do you think about it? And if you decide not to, why?</em></p>
<p>Charlie: Years ago, when Warren ran a partnership, and to some extent the partnership that I ran was the operated in the same way, we implicitly did what you’re suggesting. Part of the partnership funds were in so-called “event arbitrage investments”. And those tended to generate returns occasionally when the market generally was in the tank. And “alternative investments” would more mimic the general market. So we were doing what this academic theory prescribes, you know, 40 years ago. But we didn’t use the modern lingo.</p>
<p>Warren: We’ve got some preference for having a lot of money coming in all the time. But we do go into into insurance transactions with huge volatility, which could mean that a big chunk of money could go out at one time (or in a very short period of time). But we won’t give up a lot in expectable return for smoothness.</p>
<p>[In other words], if you give us a choice of having money come in every week at the same present value of money coming in very lumpy ways that we wouldn’t know about, we would choose the smooth.</p>
<p>But if you give us a choice of a higher present value for the lumpiness, we will take the lumpiness. And usually we get offered that choice. Other people value smoothness so highly that we do get a spread, in our view, for lumpy returns.</p>
<p>Maybe you’ve heard it already that Pepsi Cola is having a contest <em>[called <a href="https://en.wikipedia.org/wiki/Play_for_a_Billion">Play for a Billion</a>]</em>. They’re going to have a drawing in September. The contest goes through a lot of little phases, but in the end there’s going to be one person who’s going to have 1 chance in 1000 of winning a 1 billion dollars. That billion dollar will have a present value of maybe 250 million.</p>
<p><em>[The $1 billion prize was to be paid in 40 annual payments as follows: $5 million each year for the first 20 years, $10 million each year for years 21-39, and a balloon payment of $710 million at year 40. A immediate cash option of $250 million was available as well.]</em></p>
<p>If whoever gets to that position hits the number, we will pay it and we don’t mind paying out 250 million dollars, as long as we got paid appropriately for us. And that would create bad cash flow that particular week (or maybe even for two weeks).</p>
<p>We’re willing assume that for a payment. And very, very few people in the world are. Even those that can afford it. We would even assume it for $2.5 billion in present value. We’d want more proportionally to assume it for that, but Charlie and I think would agree that we would take that on if we got paid well enough for it. We wouldn’t do it for $25 billion <em>[presumably because it would be too big of a check for Berkshire’s size at the time]</em>.</p>
<p>We would we will do things and therefore, you know, we get the calls on that sort of thing. And that is more profitable business over time than bread-and-butter business. It also could lead to you having an intense interest in watching the television show when the drawing takes place, and making sure who draws the number, too.</p>
<p>Charlie: Once you’re talking about opportunity costs that are personal to yourself and your own situation, you’ve departed from modern finance. Totally. And that’s what we’ve done.</p>
<p>We’re intelligently making these guesses as best we can based on our own circumstances and our own abilities. I think it’s crazy to do it based on somebody else’s circumstances and somebody else’s abilities.</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="against-diversification-if-you-are-a-professional">Against diversification if you are a ‘professional’</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<details>
<summary>
“Diversification is protection against ignorance. It makes little sense if you know what you are doing”
</summary>
<div>
<p><a href="https://www.goodreads.com/quotes/616685-diversification-is-protection-against-ignorance-it-makes-little-sense-if">Warren Buffett once stated</a> that “Diversification is protection against ignorance. It makes little sense if you know what you are doing.”</p>
<p>From the <a href="https://buffett.cnbc.com/video/2003/05/03/afternoon-session---2003-berkshire-hathaway-annual-meeting.html">2003 Annual Meeting</a>:</p>
<blockquote>
<p>Charlie: What’s interesting is that at least 90 percent of the professional investment management operations don’t think the way we do at all. They just think, if they hire enough people, they can be better at determining whether Pfizer or Merck is going to do better over the next 20 years.</p>
<p>And they can do that, stock by stock, all through the 500 and have wide diversification. And at the end of 10 years they’ll be way ahead of other people, and, of course, they won’t.</p>
<p>Very few people have this idea of searching for just a few opportunities.</p>
</blockquote>
<p>From <a href="https://archive.is/h7H8C#selection-5801.0-5805.339">2008 talks at Emory’s Goizueta Business School and McCombs School of Business at UT Austin</a>:</p>
<blockquote>
<p>I have 2 views on diversification:</p>
<ol>
<li>
<p>If you are a professional and have confidence, then I would advocate lots of concentration.</p>
</li>
<li>
<p>For everyone else, if it’s not your game, participate in total diversification. The economy will do fine over time. Make sure you don’t buy at the wrong price or the wrong time. That’s what most people should do, buy a cheap index fund, and slowly dollar cost average into it. If you try to be just a little bit smart, spending an hour a week investing, you’re liable to be really dumb.</p>
</li>
</ol>
<p>If it’s your game, diversification doesn’t make sense. It’s crazy to put money into your 20th choice rather than your 1st choice. “Lebron James” analogy. If you have Lebron James on your team, don’t take him out of the game just to make room for someone else. If you have a harem of 40 women, you never really get to know any of them well.</p>
</blockquote>
<p>From <a href="https://buffett.cnbc.com/video/2008/05/03/morning-session---2008-berkshire-hathaway-annual-meeting.html">2008 Berkshire Hathaway Annual Meeeting</a>:</p>
<blockquote>
<p>Charlie: Yeah. You, students of America, go to these elite business schools and law schools [to] learn corporate finance the way it’s now taught, and investment management the way it’s now taught.</p>
<p>And some of these people write articles in the newspaper and other places and they say, “Well, the whole secret of investment is diversification.” That’s the mantra.</p>
<p>They’ve got it exactly back-ass-ward. The whole secret of investment is to find places where it’s safe and wise to non-diversify. It’s just that simple.</p>
<p>Diversification is for the know-nothing investor; it’s not for the professional.</p>
<p>Warren: And there’s nothing wrong with the know-nothing investor practicing it. It’s exactly what they should practice. It’s exactly what a good professional investor should not practice. There’s no contradiction in that.</p>
<p>A know-nothing investor will get decent results as long as they know they’re a know-nothing investor, diversify as to time they purchase their equities, and as to the equities they purchase. That’s crazy for somebody that really knows what they’re doing.</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="handle-risk-through-a-subjective-threshold-of-certainty">Handle risk through a (subjective) threshold of certainty</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<details>
<summary>
“We look at riskiness as being sort of a go/no-go valve. If we think we don’t know what’s going to happen in the future [we simply skip that business]. That doesn’t mean [its future is] necessarily [bad], it just means we don’t know, [and that] is risky for us.”
</summary>
<div>
<p>From <a href="https://buffett.cnbc.com/video/1998/05/04/afternoon-session---1998-berkshire-hathaway-annual-meeting.html">1998 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: Could you please explain how you differentiate between types of businesses in your cash flow valuation process, given that you use the same discount rates across companies? For example, in valuing Coke and GEICO, how do you account for the difference in the riskiness of their cash flows?</em></p>
<p>Warren: We don’t worry about risk in the traditional — the way you’re taught at Wharton. But it’s a good question, believe me.</p>
<p>If we could see the future of every business perfectly, it wouldn’t make any difference whether the money came from running streetcars or from selling software, because all the cash that came out, which is all we’re measuring between now and judgment day, would spend the same to us. The industry that it’s earned in means nothing except to the extent that it may tell you something about the ability to develop the cash. But it has no meaning on the quality of the cash once it becomes distributable.</p>
<p>We look at riskiness, essentially, as being sort of a go/no-go valve in terms of looking at the future businesses. In other words, if we think we simply don’t know what’s going to happen in the future — that doesn’t mean it’s necessarily risky, it just means we don’t know. It means it’s risky for us. It might not be risky for someone else who understands the business. In that case, we just give up. We don’t try to predict those things.</p>
<p>And we don’t say, “Well, we don’t know what’s going to happen, so therefore we’ll discount it at 9% instead of 7%,” some number that we don’t even know. That is not our way to approach it.</p>
<p>We feel that once it passes a threshold test of being something about which we feel quite certain, that the same discount factor tends to apply to everything. And we try to do only things about which we are quite certain when we buy into the businesses.</p>
</blockquote>
<p>From <a href="https://www.youtube.com/watch?v=YfYKcJBYkv8">2003 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>As we pointed out many times in the past, intrinsic value is terribly important and very fuzzy. We do our best to work in the kind of businesses where our predictions are of fairly highly probable nature. And that leaves out all kinds of companies.</p>
</blockquote>
<p><a href="https://www.youtube.com/watch?v=JsbVonhF-q4&lc=UgyHfHkfKaX6beS2TQB4AaABAg.9JZPzRyffYt9K4OizP51M_">YouTube user Infinite Rings put it succintly on a comment</a>:</p>
<blockquote>
<p>There is no sense in fiddling with discount rates based on how speculative an investment is. You either understand what you are investing in, and can made reasonable assumptions about cash flows, or you can’t, in which case your underlying assumptions are the issue, not the discount rate at the end.</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="use-future-owner-earnings-in-business-valuations-avoid-ebitda">Use <em>future</em> owner earnings in business valuations, avoid EBITDA</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<p>In 1987, Buffett defined owner earnings as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.9em"><mtext>(a)</mtext></mstyle><mstyle mathsize="0.9em"><mo>+</mo><mstyle mathsize="0.9em"><mtext>(b)</mtext></mstyle><mstyle mathsize="0.9em"><mo>−</mo><mstyle mathsize="0.9em"><mtext>(c)</mtext></mstyle></mstyle></mstyle></mrow><annotation encoding="application/x-tex">\text{\small (a)} \small + \text{\small (b)} \small - \text{\small (c)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5">(a)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin sizing reset-size6 size5">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">(b)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin sizing reset-size6 size5">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">(c)</span></span></span></span></span>, where:</p>
<ul>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.9em"><mtext>(a)</mtext></mstyle><mstyle mathsize="0.9em"><mo>=</mo></mstyle></mrow><annotation encoding="application/x-tex">\text{\small (a)} \small =</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5">(a)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel sizing reset-size6 size5">=</span></span></span></span> <a href="https://en.wikipedia.org/wiki/Generally_Accepted_Accounting_Principles_(United_States)">GAAP</a> reported earnings</li>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.9em"><mtext>(b)</mtext></mstyle><mstyle mathsize="0.9em"><mo>=</mo></mstyle></mrow><annotation encoding="application/x-tex">\text{\small (b)} \small =</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5">(b)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel sizing reset-size6 size5">=</span></span></span></span> depreciation, depletion, goodwill amortization, and certain other non-cash charges (such as non-cash inventory costs, deferred income taxes, and employee stock compensation)</li>
<li><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.9em"><mtext>(c)</mtext></mstyle><mstyle mathsize="0.9em"><mo>=</mo></mstyle></mrow><annotation encoding="application/x-tex">\text{\small (c)} \small =</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5">(c)</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel sizing reset-size6 size5">=</span></span></span></span> the average annual amount of capitalized expenditures (for plant and equipment, etc) and any incremental working capital that the business requires to fully maintain its long-term competitive position and its unit volume</li>
</ul>
<p>At the time, companies were not required to report cash flow statements. Nowadays it is possible to unpack Buffett’s terms and get most of them directly from those statements.</p>
<p><a href="https://www.oldschoolvalue.com/stock-valuation/what-is-owner-earnings/">Old School Value has a great article</a> (<a href="https://archive.is/l69Rw">archive</a>) that breaks Buffett’s formula down to more up-to-date terms:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.1600em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mstyle mathsize="0.9em"><mtext>Owner earnings</mtext></mstyle><mstyle mathsize="0.9em"><mo>=</mo><mspace width="0.5em"/></mstyle></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mstyle mathsize="0.9em"><mtext>reported earnings (a)</mtext></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mstyle mathsize="0.9em"><mo>+</mo><mstyle mathsize="0.9em"><mtext>depreciation, goodwill amortization (b)</mtext></mstyle></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mstyle mathsize="0.9em"><mo>±</mo><mstyle mathsize="0.9em"><mtext>other non-cash charges (b)</mtext></mstyle></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mstyle mathsize="0.9em"><mo>−</mo><mstyle mathsize="0.9em"><mtext>average annual maintenance capex (c)</mtext></mstyle></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mstyle mathsize="0.9em"><mo>±</mo><mstyle mathsize="0.9em"><mtext>changes in working capital (c)</mtext></mstyle></mstyle></mrow></mstyle></mtd></mtr></mtable></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{equation*}
\begin{split}
\text{\small Owner earnings} \small = \enspace &\text{\small reported earnings (a)} \\
&\small+ \text{\small depreciation, goodwill amortization (b)} \\
&\small\pm \text{\small other non-cash charges (b)} \\
&\small- \text{\small average annual maintenance capex (c)} \\
&\small\pm \text{\small changes in working capital (c)} \\
\end{split}
\end{equation*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:7.500000000000002em;vertical-align:-3.500000000000001em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4.000000000000001em;"><span style="top:-6.000000000000001em;"><span class="pstrut" style="height:6em;"></span><span class="mord"><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4em;"><span style="top:-6.0600000000000005em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"><span class="mord text"><span class="mord sizing reset-size6 size5">Owner earnings</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel sizing reset-size6 size5">=</span><span class="mspace sizing reset-size6 size5" style="margin-right:0.5em;"></span></span></span><span style="top:-4.559999999999999em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"></span></span><span style="top:-3.0599999999999983em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"></span></span><span style="top:-1.5599999999999983em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"></span></span><span style="top:-0.05999999999999828em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.5000000000000018em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:4em;"><span style="top:-6.0600000000000005em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"><span class="mord"></span><span class="mord text"><span class="mord sizing reset-size6 size5">reported earnings (a)</span></span></span></span><span style="top:-4.559999999999999em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin sizing reset-size6 size5">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">depreciation, goodwill amortization (b)</span></span></span></span><span style="top:-3.0599999999999983em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin sizing reset-size6 size5">±</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">other non-cash charges (b)</span></span></span></span><span style="top:-1.5599999999999983em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin sizing reset-size6 size5">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">average annual maintenance capex (c)</span></span></span></span><span style="top:-0.05999999999999828em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin sizing reset-size6 size5">±</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">changes in working capital (c)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.5000000000000018em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:3.500000000000001em;"><span></span></span></span></span></span></span></span></span></span></span></span>
<p>Total capital expenditures are available in present-day cash flow statements, but <em>maintenance</em> capex is not. It’s up to us to figure out how much of the total capex goes towards maintaining the business <em>vs.</em> how much is being used to (eventually) grow it.</p>
<p>Working capital is usually obtained from the balance sheet, and it is defined as <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mstyle mathsize="0.9em"><mtext>(current assets)</mtext></mstyle><mstyle mathsize="0.9em"><mo>−</mo><mrow><mtext>(</mtext><mstyle mathsize="0.9em"><mtext>current liabilities)</mtext></mstyle></mrow></mstyle></mrow><annotation encoding="application/x-tex">\text{\small (current assets)} \small - \text{(\small current liabilities)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5">(current assets)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin sizing reset-size6 size5">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">(current liabilities)</span></span></span></span></span>. But <em>changes</em> in working capital is different. It is a cash flow item, and it is easier to get it directly from the cash flow statement.</p>
<p>Crucially, let’s not lose sight that the key quantity in Buffett’s system is the <em>future</em> owner earnings, not past ones. But, for <em>some</em> business in <em>some</em> contexts, previous owner earnings can be useful for forecasting future ones.</p>
<details>
<summary>
“We consider the owner earnings figure, not the GAAP figure, to be the relevant item for valuation purposes”
</summary>
<div>
<p>When an acquisition happens, all sorts of accounting acrobatics can take place and lead to confusion about the standing of a business.</p>
<p>In 1987, Warren Buffett explained some of those difficulties using their own acquisition of the <a href="https://en.wikipedia.org/wiki/Scott_Fetzer_Company">Scott Fetzer Company</a> — a diversified manufacturer — as a practical example.</p>
<p>From Buffett’s <a href="https://berkshirehathaway.com/letters/1986.html">1986 Annual Letter</a>:</p>
<blockquote>
<p>What does all this mean for owners? Did the shareholders of Berkshire buy a business that earned $40.2 million in 1986 or did they buy one earning $28.6 million? Were those $11.6 million of new charges a real economic cost to us? Should investors pay more for the stock of Company O [i.e., Scott Fetzer right before acquisition] than of Company N [the same company, right after acquisition]? And, if a business is worth some given multiple of earnings, was Scott Fetzer worth considerably more the day before we bought it than it was worth the following day?</p>
<p>If we think through these questions, we can gain some insights about what may be called ‘owner earnings.’ These represent:</p>
<p>Owner earnings = (reported earnings) + (depreciation, depletion, amortization, and certain other non-cash charges) − (the average annual amount of capitalized expenditures for plant and equipment, etc. that the business requires to fully maintain its long-term competitive position and its unit volume)</p>
<p>If the business requires additional working capital to maintain its competitive position and unit volume, the increment also should be included. However, businesses following the <a href="https://en.wikipedia.org/wiki/Inventory_valuation">LIFO inventory method</a> usually do not require additional working capital if unit volume does not change.</p>
<p>Our owner-earnings equation does not yield the deceptively precise figures provided by GAAP, since the third factor must be a guess — and one sometimes very difficult to make. Despite this problem, we consider the owner earnings figure, not the GAAP figure, to be the relevant item for valuation purposes — both for investors in buying stocks and for managers in buying entire businesses. We agree with Keynes’s observation: “I would rather be vaguely right than precisely wrong.”</p>
<p>[…] Questioning GAAP figures may seem impious to some. After all, what are we paying the accountants for if it is not to deliver us the “truth” about our business. But the accountants’ job is to record, not to evaluate. The evaluation job falls to investors and managers.</p>
<p>Accounting numbers, of course, are the language of business and as such are of enormous help to anyone evaluating the worth of a business and tracking its progress. Charlie and I would be lost without these numbers: they invariably are the starting point for us in evaluating our own businesses and those of others. Managers and owners need to remember, however, that accounting is but an aid to business thinking, never a substitute for it.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“Most managers probably will acknowledge that they need to spend something more than (b) on their businesses over the longer term just to hold their ground in terms of both unit volume and competitive position. When this imperative exists — that is, when (c) exceeds (b) — GAAP earnings overstate owner earnings. Frequently this overstatement is substantial”
</summary>
<div>
<p>Now, from the same <a href="https://berkshirehathaway.com/letters/1986.html">1986 Annual Letter</a>:</p>
<blockquote>
<p>Most managers probably will acknowledge that they need to spend something more than (b) on their businesses over the longer term just to hold their ground in terms of both unit volume and competitive position. When this imperative exists — that is, when (c) exceeds (b) — GAAP earnings overstate owner earnings. Frequently this overstatement is substantial. The oil industry has in recent years provided a conspicuous example of this phenomenon. Had most major oil companies spent only (b) each year, they would have guaranteed their shrinkage in real terms.</p>
<p>All of this points up the absurdity of the “cash flow” numbers that are often set forth in Wall Street reports. These numbers routinely include (a) plus (b) — but do not subtract (c).</p>
<p>Most sales brochures of investment bankers also feature deceptive presentations of this kind. These imply that the business being offered is the commercial counterpart of the Pyramids — forever state-of-the-art, never needing to be replaced, improved or refurbished.</p>
<p>“Cash Flow”, true, may serve as a shorthand of some utility in descriptions of certain real estate businesses or other enterprises that make huge initial outlays and only tiny outlays thereafter. A company whose only holding is a bridge or an extremely long-lived gas field would be an example. But “cash flow” is meaningless in such businesses as manufacturing, retailing, extractive companies, and utilities because, for them, (c) is always significant. To be sure, businesses of this kind may in a given year be able to defer capital spending. But over a five- or ten-year period, they must make the investment — or the business decays.</p>
<p>Why, then, are “cash flow” numbers so popular today? In answer, we confess our cynicism: we believe these numbers are frequently used by marketers of businesses and securities in attempts to justify the unjustifiable (and thereby to sell what should be the unsalable).</p>
<p>When (a) — that is, GAAP earnings — looks by itself inadequate to service debt of a junk bond or justify a foolish stock price, how convenient it becomes for salesmen to focus on (a) + (b). But you shouldn’t add (b) without subtracting (c): though dentists correctly claim that if you ignore your teeth they’ll go away, the same is not true for (c). The company or investor believing that the debt-servicing ability or the equity valuation of an enterprise can be measured by totaling (a) and (b) while ignoring (c) is headed for certain trouble.</p>
</blockquote>
<p>From <a href="https://berkshirehathaway.com/owners.html">Berkshire Hathaway’s An Owner’s Manual</a>:</p>
<blockquote>
<p>We do not think so-called EBITDA (earnings before interest, taxes, depreciation and amortization) is a meaningful measure of performance. Managements that dismiss the importance of depreciation — and emphasize “cash flow” or EBITDA — are apt to make faulty decisions, and you should keep that in mind as you make your own investment decisions.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“We do not think so-called EBITDA (earnings before interest, taxes, depreciation and amortization) is a meaningful measure of performance. Managements that dismiss the importance of depreciation — and emphasize ‘cash flow’ or EBITDA — are apt to make faulty decisions”
</summary>
<div>
<p>From <a href="https://berkshirehathaway.com/owners.html">Berkshire Hathaway’s An Owner’s Manual</a>:</p>
<blockquote>
<p>We do not think so-called EBITDA (earnings before interest, taxes, depreciation and amortization) is a meaningful measure of performance. Managements that dismiss the importance of depreciation — and emphasize “cash flow” or EBITDA — are apt to make faulty decisions, and you should keep that in mind as you make your own investment decisions.</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="glimpses-into-their-thinking-about-returns-on-capital">Glimpses into their thinking about returns on capital</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<p>It is all about capital.</p>
<p>Capital <em>in</em> as incremental investment, capital <em>out</em> as future owner earnings.</p>
<p>From the <a href="https://berkshirehathaway.com/letters/1992.html">1992 Annual Letter</a>:</p>
<blockquote>
<p>Leaving the question of price aside, the best business to own is one that over an extended period can employ large amounts of incremental capital at very high rates of return.</p>
<p>The worst business to own is one that must, or will, do the opposite — that is, consistently employ ever-greater amounts of capital at very low rates of return.</p>
<p>Unfortunately, [not only] the first type of business is very hard to find, most high-return businesses need relatively little capital [amounts.]</p>
</blockquote>
<p>From the <a href="https://buffett.cnbc.com/video/2000/04/29/morning-session---2000-berkshire-hathaway-annual-meeting.html">2000 Annual Meeting</a>:</p>
<blockquote>
<p>The very best businesses, the really wonderful businesses, require no book value. We want to buy businesses, really, that will deliver more and more cash — and not need to retain cash, which is what builds up book value over time.</p>
<p>Book value is no starting point at all of any kind in — whether it’s The Washington Post or Coca-Cola or Gillette. It’s a factor we ignore.</p>
<p>We do look at what a company is able to earn on invested assets and what it can earn on incremental invested assets. But the book value, we do not give a thought to.</p>
</blockquote>
<p>Here is Buffett, in his <a href="https://berkshirehathaway.com/letters/1994.html">1994 Annual Letter</a>, describing Scott Fetzer’s exceptional returns:</p>
<details>
<summary>
“We’ve tried to put in the [1994] annual report pretty much how we approach securities. The best businesses, by definition, are going to be businesses that earn very high returns on capital employed over time. So, by nature, if we want to own good businesses, we’re going to own things that have relatively little capital employed compared to our purchase price”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/1995/05/01/morning-session---1995-berkshire-hathaway-annual-meeting.html">1995 Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: In the book, “Warren Buffett Way,” the author describes the capital growth model that you’ve used to evaluate intrinsic value in common stock purchases. My question is, do you also still use the formula Ben Graham described in “The Intelligent Investor,” that uses evaluating anticipated growth, but also book value? It seems to me that fair value is always a bit higher when using Mr. Graham’s formula than the stream of cash discounted back to present value that is in “Warren Buffett Way” and also that you’ve alluded to in annual reports.</em></p>
<p>Warren: We’ve tried to put in the [1994] annual report pretty much how we approach securities. And book value is, virtually, not a consideration at all.</p>
<p>The best businesses, by definition, are going to be businesses that earn very high returns on capital employed over time. So, by nature, if we want to own good businesses, we’re going to own things that have relatively little capital employed compared to our purchase price.</p>
<p>That would not have been Ben Graham’s approach. But Ben Graham was not working with very large sums of money. And he would not have argued with this approach, he just would’ve said his was easier. And it is easier, perhaps, when you’re working with small amounts of money.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“Scott Fetzer’s earnings have increased steadily since we bought it, but book value has not grown commensurately. Consequently, return on equity, which was exceptional at the time of our purchase, has now become truly extraordinary”
</summary>
<div>
<p>From the <a href="https://berkshirehathaway.com/letters/1994.html">1994 Annual Letter</a>:</p>
<blockquote>
<p>We paid $315.2 million for Scott Fetzer, which at the time had $172.6 million of book value. The $142.6 million premium we handed over indicated our belief that the company’s intrinsic value was close to double its book value. In the table below we trace the book value of Scott Fetzer, as well as its earnings and dividends, since our purchase:</p>
</blockquote>
<table>
<thead>
<tr>
<th>(in US$ mi)</th>
<th style="text-align: right">1986</th>
<th style="text-align: right">1987</th>
<th style="text-align: right">1988</th>
<th style="text-align: right">1989</th>
<th style="text-align: right">1990</th>
<th style="text-align: right">1991</th>
<th style="text-align: right">1992</th>
<th style="text-align: right">1993</th>
<th style="text-align: right">1994</th>
</tr>
</thead>
<tbody>
<tr>
<td>Book Value <code class="language-plaintext highlighter-rouge">(BOY, 1)</code></td>
<td style="text-align: right">172.6</td>
<td style="text-align: right">87.9</td>
<td style="text-align: right">95.5</td>
<td style="text-align: right">118.6</td>
<td style="text-align: right">105.5</td>
<td style="text-align: right">133.3</td>
<td style="text-align: right">120.7</td>
<td style="text-align: right">111.2</td>
<td style="text-align: right">90.7</td>
</tr>
<tr>
<td>Earnings <code class="language-plaintext highlighter-rouge">(2)</code></td>
<td style="text-align: right">40.3</td>
<td style="text-align: right">48.6</td>
<td style="text-align: right">58.0</td>
<td style="text-align: right">58.5</td>
<td style="text-align: right">61.3</td>
<td style="text-align: right">61.4</td>
<td style="text-align: right">70.5</td>
<td style="text-align: right">77.5</td>
<td style="text-align: right">79.3</td>
</tr>
<tr>
<td>Dividends <code class="language-plaintext highlighter-rouge">(3)</code></td>
<td style="text-align: right">125.0</td>
<td style="text-align: right">41.0</td>
<td style="text-align: right">35.0</td>
<td style="text-align: right">71.5</td>
<td style="text-align: right">33.5</td>
<td style="text-align: right">74.0</td>
<td style="text-align: right">80.0</td>
<td style="text-align: right">98.0</td>
<td style="text-align: right">76.0</td>
</tr>
<tr>
<td>Book Value <code class="language-plaintext highlighter-rouge">(EOY, 4)</code></td>
<td style="text-align: right">87.9</td>
<td style="text-align: right">95.5</td>
<td style="text-align: right">118.6</td>
<td style="text-align: right">105.5</td>
<td style="text-align: right">133.3</td>
<td style="text-align: right">120.7</td>
<td style="text-align: right">111.2</td>
<td style="text-align: right">90.7</td>
<td style="text-align: right">94.0</td>
</tr>
<tr>
<td>Return on Average Equity</td>
<td style="text-align: right">126%</td>
<td style="text-align: right">97%</td>
<td style="text-align: right">87%</td>
<td style="text-align: right">116%</td>
<td style="text-align: right">79%</td>
<td style="text-align: right">107%</td>
<td style="text-align: right">130%</td>
<td style="text-align: right">174%</td>
<td style="text-align: right">168%</td>
</tr>
</tbody>
</table>
<p>[BOY stands for beginning of the year. EOY stands for ending of the year. Note, therefore, that <code class="language-plaintext highlighter-rouge">(4) = (1)+(2)-(3)</code>. The ROE line was added and calculated by me, not Buffett. I took ROE to be <code class="language-plaintext highlighter-rouge">(2+3) / ((4+1)/2)</code>. ]</p>
<blockquote>
<p>Because it had excess cash when our deal was made, Scott Fetzer was able to pay Berkshire dividends of $125 million in 1986, though it earned only $40.3 million. I should mention that we have not introduced leverage into Scott Fetzer’s balance sheet. In fact, the company has gone from very modest debt when we purchased it to virtually no debt at all (except for debt used by its finance subsidiary). Similarly, we have not sold plants and leased them back, nor sold receivables, nor the like. Throughout our years of ownership, Scott Fetzer has operated as a conservatively-financed and liquid enterprise.</p>
<p>As you can see, Scott Fetzer’s earnings have increased steadily since we bought it, but book value has not grown commensurately. Consequently, return on equity, which was exceptional at the time of our purchase, has now become truly extraordinary.</p>
<p>You might expect that Scott Fetzer’s success could only be explained by a cyclical peak in earnings, a monopolistic position, or leverage. But no such circumstances apply. Rather, the company’s success comes from the managerial expertise of CEO Ralph Schey, of whom I’ll tell you more later.</p>
</blockquote>
</div>
</details>
<p>In the excerpts below, Buffett talks about returns on net tangible assets. Net tangible assets are <a href="https://www.investopedia.com/terms/n/nettangibleassets.asp">defined as</a> the total assets of a company, minus any intangible assets such as goodwill, patents, and trademarks, less all liabilities and the par value of preferred stock.</p>
<p>Or, in slightly shorter and simplified forms:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable rowspacing="0.1600em" columnspacing="1em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mtable rowspacing="0.2500em" columnalign="right left" columnspacing="0em"><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mstyle mathsize="0.9em"><mtext>Net tangible assets</mtext></mstyle><mstyle mathsize="0.9em"></mstyle></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mo>=</mo><mstyle mathsize="0.9em"><mtext>(Total assets</mtext></mstyle><mstyle mathsize="0.9em"><mo>−</mo><mstyle mathsize="0.9em"><mtext>Total liabilities)</mtext></mstyle><mstyle mathsize="0.9em"><mo>−</mo><mstyle mathsize="0.9em"><mtext>(Intangibles)</mtext></mstyle></mstyle></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mstyle mathsize="0.9em"><mo>=</mo><mstyle mathsize="0.9em"><mtext>(Equity)</mtext></mstyle><mstyle mathsize="0.9em"><mo>−</mo><mstyle mathsize="0.9em"><mtext>(Intangibles)</mtext></mstyle></mstyle></mstyle></mrow></mstyle></mtd></mtr><mtr><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow></mrow></mstyle></mtd><mtd><mstyle scriptlevel="0" displaystyle="true"><mrow><mrow></mrow><mstyle mathsize="0.9em"><mo>=</mo><mstyle mathsize="0.9em"><mtext>(Tangible equity)</mtext></mstyle></mstyle></mrow></mstyle></mtd></mtr></mtable></mstyle></mtd></mtr></mtable><annotation encoding="application/x-tex">\begin{equation*}
\begin{split}
\text{\small Net tangible assets} \small &= \text{\small (Total assets} \small - \text{\small Total liabilities)} \small - \text{\small (Intangibles)} \\
&\small = \text{\small (Equity)} \small - \text{\small (Intangibles)} \\
&\small = \text{\small (Tangible equity)} \\
\end{split}
\end{equation*}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:4.500000000000002em;vertical-align:-2.0000000000000004em;"></span><span class="mord"><span class="mtable"><span class="col-align-c"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5000000000000013em;"><span style="top:-4.5em;"><span class="pstrut" style="height:4.5em;"></span><span class="mord"><span class="mord"><span class="mtable"><span class="col-align-r"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5000000000000004em;"><span style="top:-4.5600000000000005em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"><span class="mord text"><span class="mord sizing reset-size6 size5">Net tangible assets</span></span></span></span><span style="top:-3.06em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"></span></span><span style="top:-1.5599999999999992em;"><span class="pstrut" style="height:2.9em;"></span><span class="mord"></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.000000000000001em;"><span></span></span></span></span></span><span class="col-align-l"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:2.5000000000000004em;"><span style="top:-4.66em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5">(Total assets</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin sizing reset-size6 size5">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">Total liabilities)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin sizing reset-size6 size5">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">(Intangibles)</span></span></span></span><span style="top:-3.16em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel sizing reset-size6 size5">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">(Equity)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin sizing reset-size6 size5">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">(Intangibles)</span></span></span></span><span style="top:-1.6599999999999993em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord"></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel sizing reset-size6 size5">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">(Tangible equity)</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.000000000000001em;"><span></span></span></span></span></span></span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:2.0000000000000004em;"><span></span></span></span></span></span></span></span></span></span></span></span>
<p>Below, Buffett also writes about <em>unleveraged</em> net tangible assets, which can be calculated as:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mstyle mathsize="0.9em"><mtext>Unleveraged net tangible assets</mtext></mstyle><mstyle mathsize="0.9em"><mo>=</mo><mstyle mathsize="0.9em"><mtext>(Equity</mtext></mstyle><mstyle mathsize="0.9em"><mo>+</mo><mstyle mathsize="0.9em"><mtext>Long-term debt)</mtext></mstyle><mstyle mathsize="0.9em"><mo>−</mo><mstyle mathsize="0.9em"><mtext>(Intangibles)</mtext></mstyle></mstyle></mstyle></mstyle></mrow><annotation encoding="application/x-tex">\text{\small Unleveraged net tangible assets} \small = \text{\small (Equity} \small + \text{\small Long-term debt)} \small - \text{\small (Intangibles)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.799992em;vertical-align:-0.174996em;"></span><span class="mord text"><span class="mord sizing reset-size6 size5">Unleveraged net tangible assets</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel sizing reset-size6 size5">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">(Equity</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin sizing reset-size6 size5">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">Long-term debt)</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin sizing reset-size6 size5">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.9em;vertical-align:-0.225em;"></span><span class="mord text sizing reset-size6 size5"><span class="mord">(Intangibles)</span></span></span></span></span></span>
<p>As explained by <a href="https://theoraclesclassroom.com">Adam J. Mead</a> in <a href="https://www.youtube.com/watch?v=M0QNqZ3zgoA">this YouTube video</a>, Buffett prefers to evaluate <em>unleveraged</em> net tangible assets because he looks for businesses that earn a good return on <em>total</em> capital: “It is as if the business were <em>entirely</em> equity financed. Because returns on equity can be enhanced with the use of debt, he’s looking [instead] for a business that stands alone as a good business without any kind of financial engineering.”</p>
<p>Now onto Buffett’s own words:</p>
<details>
<summary>
‘Manufacturing, Service and Retailing’ businesses should be evaluated on their earnings on unleveraged net tangible assets. After-tax returns in 12-20% range are good, anything above 25% after-tax is terrific
</summary>
<div>
<p>From the <a href="https://berkshirehathaway.com/letters/2013ltr.pdf">2013 Annual Letter</a>:</p>
<blockquote>
<p>The crowd of companies in this section sells products ranging from lollipops to jet airplanes. Some of these businesses, measured by <em>earnings on unleveraged net tangible assets</em>, enjoy terrific economics, producing profits that run from 25% after-tax to far more than 100%. Others generate good returns in the area of 12% to 20%. A few, however, have very poor returns, a result of some serious mistakes I made in my job of capital allocation. I was not misled: I simply was wrong in my evaluation of the economic dynamics of the company or the industry inwhich it operated.</p>
<p>Fortunately, my blunders usually involved relatively small acquisitions. Our large buys have generall yworked out well and, in a few cases, more than well. I have not, however, made my last mistake in purchasing either businesses or stocks. Not everything works out as planned.</p>
<p>Viewed as a single entity, the companies in this group are an excellent business. They employed an <em>average</em> of $25 billion of <em>net tangible assets</em> during 2013 and, with large quantities of excess cash and little leverage, earned 16.7% after-tax on that capital.</p>
<p>Of course, a business with terrific economics can be a bad investment if the purchase price is excessive. We have paid substantial premiums to <em>net tangible assets</em> for most of our businesses, a cost that is reflected in the large figure we show for goodwill. Overall, however, we are getting a decent return on the capital we have deployed in this sector. Furthermore, the intrinsic value of these businesses, in aggregate, exceeds their carrying value by a good margin. Even so, the difference between intrinsic value and carrying value in the insurance and regulated-industry segments is far greater. It is there that the truly big winners reside.</p>
</blockquote>
</div>
</details>
<p>Above, Buffett is using after-tax numbers, but in many other occasions, when comparing between different businesses across periods of time, he has used pre-tax figures. Why? <a href="https://www.youtube.com/watch?v=M0QNqZ3zgoA&lc=UgxwpDh73wMKEQE2YrR4AaABAg.9JUdtj3vlNa9JYWh1Xw8WK">According to Adam</a>, it’s because pre-tax returns removes distortions over time due to the interest expense write-off, as well as any tax changes:</p>
<blockquote>
<p>I believe Buffett uses pre-tax figures to eliminate the “noise” and focus on the underlying profitability of the business. I talk about it in my <a href="https://www.amazon.com/Complete-Financial-History-Berkshire-Hathaway/dp/0857199129">upcoming book</a>, too. Since BRK buys on a 100% equity basis, looking at pre-tax operating results eliminates the effects of financing. Taxes are important, and should be considered, just in the proper context.</p>
</blockquote>
<p>Back to Buffett:</p>
<details>
<summary>
See’s Candy is “the prototype of a dream business” — “1972 pre-tax earnings were less than $5 million, and the capital then required to conduct the business was $8 million. [So] it was earning 60% pre-tax on invested capital”
</summary>
<div>
<p>From the <a href="https://berkshirehathaway.com/letters/2007ltr.pdf">2007 Annual Letter</a>:</p>
<blockquote>
<p>Let’s look at the prototype of a dream business, our own See’s Candy.</p>
<p>[…] We bought See’s for $25 million when its sales were $30 million and pre-tax earnings were less than $5 million. The capital then required to conduct the business was $8 million. (Modest seasonal debt was also needed for a few months each year.) Consequently, the company was earning 60% pre-tax on invested capital [that is, 5/8 = 0.625].</p>
<p>Two factors helped to minimize the funds required for operations. First, the product was sold for cash, and that eliminated accounts receivable. Second, the production and distribution cycle was short, which minimized inventories.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“FlightSafety is an excellent but not extraordinary business. It requires a significant reinvestment of earnings if it is to grow” — $270 million pre-tax earnings in 2007 on net investment in fixed assets of $1,079 million, and a pre-tax return on capital of 25%
</summary>
<div>
<p>From the <a href="https://berkshirehathaway.com/letters/2007ltr.pdf">2007 Annual Letter</a>:</p>
<blockquote>
<p>One example of good, but far from sensational, business economics is our own FlightSafety. [… The] business requires a significant reinvestment of earnings if it is to grow.</p>
<p>When we purchased FlightSafety in 1996, its pre-tax operating earnings were $111 million, and its net investment in fixed assets was $570 million. Since our purchase, depreciation charges have totaled $923 million. But capital expenditures have totaled $1.635 billion, most of that for simulators to match the new airplane models that are constantly being introduced. (A simulator can cost us more than $12 million, and we have 273 of them.) Our fixed assets, after depreciation, now amount to $1.079 billion. Pre-tax operating earnings in 2007 were $270 million, a gain of 159 million since 1996. That gain gave us a good, but far from See’s-like, return on our incremental investment of $509 million.</p>
<p>Consequently, if measured only by economic returns, FlightSafety is an excellent but not extraordinary business. Its put-up-more-to-earn-more experience is that faced by most corporations. For example, our large investment in regulated utilities falls squarely in this category. We will earn considerably more money in this business ten years from now, but we will invest many billions to make it.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“We tend to prefer the business which drowns in cash”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2008/05/03/morning-session---2008-berkshire-hathaway-annual-meeting.html">2008 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Charlie: We tend to prefer the business which drowns in cash. It just makes so much money that one of the main principles of owning it is you have all this cash coming in.</p>
<p>There are other businesses, like the construction equipment business of my old friend John Anderson. He used to say about his business: “You work hard all year, and at the end of the year there’s your profit sitting in the yard.” There was never any cash. Just more used construction equipment. We tend to hate businesses like that.</p>
<p>Warren: Yeah. It’s a lot easier to understand a business that’s mailing you a check every month.</p>
<p>That’s what an apartment house [could be] if you own [a good one]. [In fact,] you can probably value an apartment property pretty well if you know anything about the city in which it’s in. And if you have the financial statements, you could make a reasonable guess as to what the future earnings are likely to be. But that’s because it is a business that gives you cash. Now, [of course] there can be surprises in that arena as well.</p>
<p>There are a lot of businesses I wouldn’t buy if I thought the management was the most wonderful in the world because, if they were in the wrong business, it really doesn’t make much difference.</p>
</blockquote>
</div>
</details>
<p>The following return figures from See’s Candy are staggering per se, but I’d say the most important fact is to realize they are the returns on <em>incremental</em> investments. This is the <em>ideal</em> kind of metric in one’s mind when making an investment decision.</p>
<p>It is fun to imagine that we were in Buffett and Munger’s shoes in 1972, when they were buying See’s and thinking about future owner earnings and capital needs of the business over the next decades:</p>
<details>
<summary>
See’s Candy earned $1.35 billion pre-tax on $32 million of <em>incremental</em> investment from 1972 to 2007, bringing in an return on <em>incremental</em> capital of 155% [averaged out over 35 years]; from 2008 to 2014, profits were $550 million on $8 million of added investment for an average return on <em>incremental</em> capital of 436%
</summary>
<div>
<p>From the <a href="https://berkshirehathaway.com/letters/2007ltr.pdf">2007 Annual Letter</a>:</p>
<blockquote>
<p>Last year See’s sales were $383 million, and pre-tax profits were $82 million. The capital now required to run the business is $40 million. This means we have had to reinvest only $32 million since 1972 to handle the modest physical growth — and somewhat immodest financial growth — of the business. In the meantime pre-tax earnings have totaled $1.35 billion. All of that, except for the $32 million, has been sent to Berkshire (or, in the early years, to Blue Chip). After paying corporate taxes on the profits, we have used the rest to buy other attractive businesses. Just as Adam and Eve kick-started an activity that led to six billion humans, See’s has given birth to multiple new streams of cash for us. (The biblical command to “be fruitful and multiply” is one we take seriously at Berkshire.)</p>
<p>There aren’t many See’s in Corporate America. Typically, companies that increase their earnings from $5 million to $82 million require, say, $400 million or so of capital investment to finance their growth. That’s because growing businesses have both working capital needs that increase in proportion to sales growth and significant requirements for fixed asset investments.</p>
<p>[…] It’s far better to have an ever-increasing stream of earnings with virtually no major capital requirements. Ask Microsoft or Google.</p>
</blockquote>
<p>From the <a href="https://www.berkshirehathaway.com/letters/2014ltr.pdf">2014 Annual Letter</a>:</p>
<blockquote>
<p>See’s was a legendary West Coast manufacturer and retailer of boxed chocolates, then annually earning about $4 million pre-tax while utilizing only $8 million of net tangible assets. Moreover, the company had a huge asset that did not appear on its balance sheet: a broad and durable competitive advantage that gave it significant pricing power. That strength was virtually certain to give See’s major gains in earnings over time. Better yet, these would materialize with only minor amounts of incremental investment. In other words, See’s could be expected to gush cash for decades to come. […]</p>
<p>To date, See’s has earned $1.9 billion pre-tax, with its growth having required added investment of only $40 million.</p>
</blockquote>
</div>
</details>
<p>Although returns on equity can be manipulated (e.g. through stock repurchases), Buffett explains that there are limits to how much an industry as whole can earn. For instance, 30% return on tangible equity in banking seemed unsustainable to him if the GDP is only growing 3% on real terms:</p>
<details>
<summary>
“Let’s just say every company retained all of its earnings and they earned 20% on equity. You could not have corporate profits growing at 20% as a part of the economy year after year.” “Warren is right. You can’t have massive accumulations of earnings that are retained and keep earning these rates of return on them”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/1998/05/04/morning-session---1998-berkshire-hathaway-annual-meeting.html">1998 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: I’ve watched returns on equity for the banking sector in the U.S. go up a good bit over the last few years. And returns on tangible equity for some of the major banks that have led to consolidation have gone up a good bit more. Leads me to wonder whether these returns are sustainable over the near-term or the longer-term, 5, 10 years out.</em></p>
<p>Buffett: Well, that’s the $64 question, because the returns on equity — and particularly tangible equity, and [even more] particularly tangible equity in the banking sector — have hit numbers that are unprecedented. And then the question is, if they’re unprecedented, are they unsustainable?</p>
<p>Charlie and I would not base our actions on the premise that they are sustainable. Twenty percent-plus returns on book equity — and much higher returns on tangible equity. In the banking field, you have a number of enterprises that on tangible equity are getting up close to the 30% range.</p>
<p>Now, can a system where the GDP in real terms is growing, maybe, 3% — where in nominal terms it grows 4 to 5% — can businesses consistently earn 20% on equity?</p>
<p>They certainly can if they retain most of their earnings, because you would have corporate profits rising as a percentage of GDP, to the point that would get ludicrous.</p>
<p>So under those conditions [of 3% GDP real growth], you’d either have to have huge payouts — either by repurchases of shares or by dividends or by takeovers, actually — that would keep the level of capital reasonably consistent among industries.</p>
<p>Let’s just say every company retained all of its earnings and they earned 20% on equity. You could not have corporate profits growing at 20% as a part of the economy year after year.</p>
<p>This has been a better world than we foresaw in terms of returns, so we’ve been wrong before. And we’re not making a prediction now, but we would not want to buy things on the basis that these returns would be sustained.</p>
<p>We told you last year, if these returns are sustained and interest rates stayed at these levels or fell lower, that stock prices, in aggregate, are justified. And we still believe that. But those are two big ifs.</p>
<p>And a particularly big if, in my view, is the one about returns on equity and on tangible assets. It certainly goes against classic economic theory to believe that they can be sustained.</p>
<p>Charlie, how do you feel about it?</p>
<p>Charlie: Well, I think a lot of the increase in return on equity has been caused by the increasing popularity of <a href="https://en.wikipedia.org/wiki/Jack_Welch">Jack Welch</a>’s idea that “if you can’t be a leader in a line of business, get out of it”. And if you have fewer people in the business, returns on equity can go up.</p>
<p>Then it’s got more and more popular to buy in shares, even at very high prices per share. And if you keep the equity low enough by buying shares back, well, you could make return on equity whatever you want.</p>
<p>We had, to some extent, a slow revolution in corporate attitudes. But Warren is right. You can’t have massive accumulations of earnings that are retained and keep earning these rates of return on them.</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="their-expectations-about-returns-on-capital-have-evolved-from-high-tens-to-low-tens-as-berkshire-grew-in-size">Their expectations about returns on capital have evolved from high tens to low tens (as Berkshire grew in size)</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<p>In 1994, wholly-owned subsidiaries were charged 14-20% for any incremental capital they requested. In other words, this was the range of returns they “implicitly” expected at the time:</p>
<details>
<summary>
“When capital invested in an operation is significant, we also both charge managers a high rate for incremental capital they employ and credit them at an equally high rate for capital they release. [The rates] range between 14% and 20%”
</summary>
<div>
<p>From <a href="https://berkshirehathaway.com/letters/1994.html">Warren Buffett’s 1994 Annual Letter</a>:</p>
<blockquote>
<p>In setting compensation, we like to hold out the promise of large carrots, but make sure their delivery is tied directly to results in the area that a manager controls. When capital invested in an operation is significant, we also both charge managers a high rate for incremental capital they employ and credit them at an equally high rate for capital they release.</p>
<p>The product of this money’s-not-free approach is definitely visible at <a href="https://en.wikipedia.org/wiki/Scott_Fetzer_Company">Scott Fetzer</a>. If Ralph Schey [its CEO] can employ incremental funds at good returns, it pays him to do so: His bonus increases when earnings on additional capital exceed a meaningful hurdle charge.</p>
<p>But our bonus calculation is symmetrical: If incremental investment yields sub-standard returns, the shortfall is costly to Ralph as well as to Berkshire. The consequence of this two-way arrangement is that it pays Ralph — and pays him well — to send to Omaha any cash he can’t advantageously use in his business.</p>
</blockquote>
<p>From the <a href="https://www.youtube.com/watch?v=6sdAaxdvQvU">1995 Annual Meeting</a>, that took place just few months after the letter was published:</p>
<blockquote>
<p><em>Question: In describing your allocation of capital to your wholly-owned subsidiaries, you wrote in the annual report [above] that “you charge manages a high rate for incremental capital they employ, and credit them at an equally high rate for capital they release.” How do you determine this high rate? And how do they determine how much capital they can release?</em></p>
<p>The question’s about incentive arrangements we have with managers (or other situations) where we either advance capital to a wholly-owned subsidiary or withdrawal it. Usually that ties in with the compensation plan. We want our managers to understand just how highly we do value capital, and we feel there’s there’s nothing that creates a better understanding than to charge them for it.</p>
<p>We have different arrangements. Sometimes it’s based a little on the history of the the company. It may be based, a little bit, on the industry. It may be based on interest rates at the time that we first draw it up. We have arrangements depending on those variables, and perhaps some others, and perhaps just how we felt the day we drew it up.</p>
<p>They range between 14% and 20% in terms of capital advanced.</p>
<p>Sometimes we have an arrangement where if it’s a seasonal business (where for a few months of the year they have a seasonal requirement), we give it to them very cheap at LIBOR. But if they use more capital over beyond that, we start saying, “Well, that’s permanent capital”, so we charge them considerably more.</p>
<p>Now, if we buy a business that’s using a couple hundred million of capital, and we work out a bonus arrangement, and the manager figures out a way to do the business with less capital, we may credit him at a very high rate. The same rate we would use in charging him in terms of his bonus arrangement.</p>
<p>We believe in managers knowing that money costs money. I would say that just generally my experience in business is that most managers, when using their own money, understand that money costs money. But sometimes managers, when using other people’s money, start thinking of it a little bit like free money. And that’s a habit we don’t want to encourage around Berkshire.</p>
<p>By sticking these rates on capital, we are telling the people who run our business how much capital is worth to us. And I think that’s a useful guideline in terms of the decisions they’re making. Because we [Charlie and I] don’t make very many decisions about our operating business. We make very, very few. I don’t see budgets in most cases from our 100%-owned subsidiaries. And if I don’t see them, no one else sees them. I mean, we have no staff at headquarters looking at this kind of thing. We give them great responsibility, but we do want them to know how we calibrate the use of capital.</p>
<p>So far, I would say it’s really worked quite well. Our managers don’t mind being measured. And I think they enjoy seeing a batting average posted — and a batting average that does not include a cost of capital as a phony batting average.</p>
</blockquote>
</div>
</details>
<p>By 2003 their expectations had changed dramatically:</p>
<details>
<summary>
“Unless, however, we see a very high probability of at least 10% pre-tax returns, we will sit on the sidelines. With short-term money returning less than 1% after-tax, sitting it out is no fun”
</summary>
<div>
<p>From Warren Buffett’s <a href="https://berkshirehathaway.com/letters/2002pdf.pdf">2002 Annual Letter</a>:</p>
<blockquote>
<p>Despite three years of falling prices, which have significantly improved the attractiveness of common stocks, we still find very few that even mildly interest us. That dismal fact is testimony to the insanity of valuations reached during The Great Bubble. Unfortunately, the hangover may prove to be proportional to the binge.</p>
<p>The aversion to equities that Charlie and I exhibit today is far from congenital. We love owning common stocks — if they can be purchased at attractive prices. In my 61 years of investing, 50 or so years have offered that kind of opportunity. There will be years like that again. Unless, however, we see a very high probability of at least 10% pre-tax returns (which translate to 6.5-7% after corporate tax), we will sit on the sidelines. With short-term money returning less than 1% after-tax, sitting it out is no fun. But occasionally successful investing requires inactivity.</p>
</blockquote>
<p>From <a href="https://www.youtube.com/watch?v=fCt0sqnLpjQ">2003 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: [Give what you have written in this year’s letter], my question would be: How do you adjust that required rate of return across periods of time? So, for example, when interest rates are higher, do you look for a different equity premium return over different periods of time?</em></p>
<p>Warren: The 10% you mentioned, basically that’s the figure we quit on. We quit on buying, we don’t want to buy equities where our real expectancy is below 10%. Now, that’s true whether short rates are 6%, or whether short rates are 1%. We just feel that that it would get very sloppy to start dipping below that.</p>
<p>We have over time gotten very partial to the businesses where we think the predictability is high, but we still want a threshold return of 10%, which is not that great after tax anyway.</p>
</blockquote>
<p>In May 2020, <a href="https://twitter.com/chrisbloomstran">Christopher Bloomstran</a> of <a href="https://www.semperaugustus.com">Semper Augustus</a> commented on Berkshire’s implicit hurdle rates during <a href="https://www.youtube.com/watch?v=yFIriGnO-Oc">interview on The Acquirers Podcast</a>:</p>
<blockquote>
<p>We think Berkshires hurdle rate for investing capital is still around 10%.</p>
<p>When they did the <a href="https://www.cnbc.com/2019/04/30/buffetts-berkshire-hathaway-to-invest-10-billion-in-occidental-petroleum-for-anadarko-takeover.html">Occidental Petroleum (OXY) deal</a>, they did it at yields that they thought would be north of 10%.</p>
<p>When <a href="https://markets.businessinsider.com/news/stocks/warren-buffett-invested-3-billion-general-electric-ge-2008-crisis-2020-6-1029327040?op=1">they did the GE</a> and <a href="https://finance.yahoo.com/news/warren-buffett-invested-5b-goldman-195420885.html">they did the Goldman Sachs</a> deals [after the 2008 Financial Crisis], those were 10% preferreds. They were callable at a premium over a period of time, so yield on that paper was probably 13%. And then they got warrants underneath. So, if you do the math on what the warrants wind up being overtime, those were high-teens returns on capital.</p>
<p>We don’t think Berkshires is in the business of making investments in businesses that earn 7% returns [on capital].</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="examples-of-comfortable-margins-of-safety-and-some-other-obvious-investment-decisions">Examples of ‘comfortable’ margins of safety and some other ‘obvious’ investment decisions</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<p>A “comfortable” margin of safety — per Benjamin Graham — would be buying at 40% of your estimated intrinsic value:</p>
<details>
<summary>
“I’ve bought stocks many times [that] were in businesses that I thought I understood — where, if I knew enough about the financial past, it would tell me enough about the financial future that I could buy. I couldn’t say [precisely if] the stock was worth X or 105% of X or 95% of X, but if I could buy it at 40% of X, I would feel that I had this margin of safety that Graham would talk about, and I could make a decision”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2008/05/03/morning-session---2008-berkshire-hathaway-annual-meeting.html">2008 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Warren: The first [point] you’d have to [look at] is, do I understand enough about this business so that the financial statements will tell me the information that’s useful to me in making a judgment about what the <em>future</em> financial statements are going to look like? And in many cases, the answer would be no. Probably in a great majority of the cases it would be no.</p>
<p>But I’ve bought stocks many times [that] were in businesses that I thought I understood — where, if I knew enough about the financial past, it would tell me enough about the financial future that I could buy.</p>
<p>I couldn’t say [precisely if] the stock was worth X or 105% of X or 95% of X, but if I could buy it at 40% of X, I would feel that I had this margin of safety that Graham would talk about, and I could make a decision.</p>
</blockquote>
</div>
</details>
<p>Here is a real-world example of “comfortable” margin of safety. It features Buffett and their investment on <a href="https://en.wikipedia.org/wiki/PetroChina">PetroChina</a> in 2002 (maybe 2003).</p>
<p>At the time he compared PetroChina vs. <a href="https://en.wikipedia.org/wiki/Yukos">Yokus</a> (from Russia) vs. Western oil companies, and it was way too cheap:</p>
<details>
<summary>
“If you can buy [a business] at what turned out to be 3x earnings, and you get 45% [due to their dividend policy] of the 33% [earnings yield], you’re getting a [0.45 * 1/3 =] 15% cash yield on your investment.” “It was a very attractive price. We would have bought more but the price jumped up”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2005/04/30/morning-session---2005-berkshire-hathaway-annual-meeting.html">2005 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>We bought PetroChina a few years ago, again, after reading [just] the annual report.</p>
<p>Fortunately it was in English. It was [our] first Chinese stock — and really the last. I mean, it won’t necessarily be the last, but it’s the only one that we’ve owned so far.</p>
<p>We put about $400 million into it. At the time, and still, it produces about 3% of the world’s oil, which is a lot of oil. It produces, probably, 80% or so as much as Exxon Mobil will produce. And it’s a huge company.</p>
<p>Last year it earned $12 billion. Now, if you look at the Fortune 500 list, my guess is you won’t find more than about five companies in the United States that earn $12 billion or more. So it’s a major company.</p>
<p>At the time we bought it, the total market value was $35 billion. So we bought it at about 3x what it earned last year. It does not have unusual amounts of leverage.</p>
<p>In the annual report, they say something which very, very few companies do say, but which I think is actually fairly important. They say they will pay out about 45% of the amount they earn. So, if you can buy it at what turned out to be 3x earnings, and you get 45% of 33%, you’re getting a 15% cash yield on your investment.</p>
<p>It’s a very good annual report. Chinese government owns 90% of the company. We own 1.3%. If we vote with them, the two of us control the business. It’s a thought that hasn’t occurred to them, but I’ll keep pointing it out.</p>
<p>But, you know, it’s a very major business — at what was a very attractive price.</p>
<p>Unfortunately, the government shares and our shares have the same economic interest but they are classified differently, so that the government’s 90% are called one thing and the 10% with the public are called A-shares. We have to report in Hong Kong when we own 10% of a company. Unfortunately, the 10% [threshold was] applied only to the 10% of A-shares, so we had to reveal our ownership when we only had 1% of the economic interest in the company. We would have bought more but the price jumped up, and we are happy to have our 1.3% (or whatever it is), and we think that they’ve done a good job in running the business.</p>
<p>They’ve got large gas reserves, which they’re starting to develop now. It’s a very major enterprise. Employs almost 500,000 people.</p>
<p>And the interesting thing was, a few years ago relatively few people in the investment world probably even thought about the fact that PetroChina was over there and was a much larger business than just about any oil company in the world, except for BP and Exxon Mobil.</p>
<p>But I should emphasize. The annual report of PetroChina [is] easy to read. Understandable. They declare their policies. Anybody could get it. You can read it.</p>
<p>We did not go over and — we never had any contact with the management before we bought the stock. We’d never attended an investor presentation or anything of the sort. I mean, it’s right there in black and white, in a report that anybody can get. And we just sit in the office and read those things, and we were able to put $400 million out that’s now worth about $2 billions (or something like that)
.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“If it had been selling at the same multiple as a U.S. domestic company, would I have regarded it as more attractive? No. There’s [always] some disadvantages to being in a culture that you don’t perfectly understand, or where tax laws can change, your ownership rules can change. But the discount at which PetroChina was selling, compared to other international oil companies, was, in my view at the time, ridiculous. So, that’s why we bought it”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2005/04/30/morning-session---2005-berkshire-hathaway-annual-meeting.html">2005 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>I think I’m right on this [but] at the time, Yukos, which is the big oil company in Russia, was probably far better known among the investment community in the United States than PetroChina. And I compared the two. At the time, thought to myself, would I rather have the money in Russia or in China? PetroChina, in my view, was far cheaper. And I felt that the economic climate was likely to be better in China, you know.</p>
<p>If it had been selling at the same multiple as a U.S. domestic company, would I have regarded it as more attractive? No. I mean, there’s some disadvantages, always, to being in a culture that you don’t perfectly understand, or where tax laws can change, your ownership rules can change.</p>
<p>But the discount at which PetroChina was selling, compared to other international oil companies, was, in my view at the time, ridiculous. So, that’s why we bought it.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“It was bought simply because it was very cheap in relation to earnings, in relation to reserves, in relation to daily oil production, and relation to refining capacity. Whatever metric you wanted to use, it was far cheaper than Exxon, or BP, or Shell, or companies like that. [Of course you should] factor in about whether there could be some huge disruption in Chinese-American relationships or something of the sort, where you would lose for reasons other than what happened in terms of world oil prices, [etc]. But we’re happy with it”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2004/05/01/afternoon-session---2004-berkshire-hathaway-annual-meeting.html">2004 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Warren: The company is very similar to big oil companies in the world. PetroChina may have been the fourth largest — fourth most profitable — oil company in the world last year. I may be wrong on that. But they produce 80 or 85% as much crude daily as Exxon does, as I remember. And it’s a big, big company. And it’s not complicated. You know, obviously, a company with half a million employees, and all of that. But [it is] a big integrated oil company, it’s fairly easy to get your mind around the economic characteristics that will exist in the business.</p>
<p>It was bought not because it was in China, but it was bought simply because it was very, very cheap in relation to earnings, in relation to reserves, in relation to daily oil production, and relation to refining capacity. Whatever metric you wanted to use, it was far cheaper than Exxon, or BP, or Shell, or companies like that.</p>
<p>Now, you can say it should be cheaper, because you don’t what’ll happen with it 90% owned by the government in China, and that’s obviously a factor that you [would] stick in valuation. But I did not think that was a factor that accounted for the huge differential in the price at which it could be bought.</p>
<p>[Of course you should] factor in your own thinking about whether there could be some huge disruption in Chinese-American relationships or something of the sort, where you would lose for reasons other than what happened in terms of world oil prices, and that sort of thing. But we’re happy with it.</p>
<p>Charlie: If a thing is cheap enough, obviously you can afford a little more country risk, or regulatory risk, or whatever. This is not complicated.</p>
<p>Warren: If you’re buying something like that at well under half what — or maybe a third — of what comparable oil companies are selling for, that’s not high-level stuff.</p>
</blockquote>
</div>
</details>
<p>A few years later, Buffett even shared how much he had valued PetroChina at:</p>
<details>
<summary>
“I came to the conclusion it was worth $100 billion, and then I checked the price and it was selling for $35 billion, roughly”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2008/05/03/afternoon-session---2008-berkshire-hathaway-annual-meeting.html">2008 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>It was in 2002 and 2003, and the report came out in the spring, and I read it. And that’s the only thing I ever did. I never contacted any management. I never got a brokerage report. I never asked for anybody’s opinion.</p>
<p>But what I did do is I came to the conclusion that the company — and it’s not hard to understand crude oil production and refining and marketing and the chemical operation they have. I mean, you can do the same thing with Exxon or BP or any of them, and I look at them [all].</p>
<p>I came to the conclusion it was worth $100 billion, and then I checked the price and it was selling for $35 billion, roughly.</p>
<p>What’s the sense of talking to management? Basically, if you talk to management of almost every company, they’ll say they think their stock is a wonderful buy, and they’ll give you all the good stuff and skip over things that [are not that good]. It just doesn’t make any difference.</p>
<p>Now, if I thought the company was worth 40 billion and had been selling for $35 billion, then at that point you have to start trying to refine your analysis more. But there’s no reason to refine your analysis. I mean, I didn’t need to know whether it was worth $97 billion or $103 billion if I was buying it at $35 billion.</p>
<p>So any further refining of analysis would be a waste of time when what I should be doing is buying the stock. We really like things that you don’t have to carry out to three decimal places, you know.</p>
</blockquote>
</div>
</details>
<p>There are other famous examples of mispriced opportunities that Buffett and Munger took advantage of over the years.</p>
<p>For instance, here are they describing their “high-conviction” (and somewhat “contrarian”) bets on American Express in 1964, The Washington Post in 1973, GEICO in 1976:</p>
<details>
<summary>
“A business with something glorious underneath disguised by terrible numbers that cause cutoff points in other people’s minds is ideal for us if we can figure it out. And we’ve had a couple of those in our history that have made us a lot of money”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2000/04/29/morning-session---2000-berkshire-hathaway-annual-meeting.html">2000 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Warren: One of the best buys we ever made was in 1976 when we bought a significant percentage that became, through repurchases, 50% of GEICO. [It took place] at a time when the company was losing a lot of money, and was destined to lose a lot of money in the immediate future.</p>
<p>The fact that they were losing money was not lost on us. But we thought we saw a future there that was significantly different than the current situation. So it would not bother us, in the least, to buy into a business that currently was losing money for some reason that we understood, and where we thought that the future was going to be significantly different.</p>
<p>There are all kinds of decisions that involve the future looking different in some important way than the present. Most of our decisions relate to things where we expect the future <em>not</em> to change much, but you [do] get [all kinds of situations].</p>
<p>American Express was a good example. When we bought it in 1964, a fellow named Tino de Angelis had caused them an incredible trouble. It was one of those decisions that looked at time as if it could break the company. So we knew, if you’ve been charging for what Tino had stolen from the company against the income account that year (or the legal cost), you were looking at a significant loss.</p>
<p>But the question was “what was American Express gonna look like 10 or 20 years later?” And we felt very good about that.</p>
<p>Charlie: A business with something glorious underneath disguised by terrible numbers that cause cutoff points in other people’s minds is ideal for us if we can figure it out. And we’ve had a couple of those in our history that have made us a lot of money.</p>
</blockquote>
</div>
</details>
<p>This is pure gold and worth repeating: “Most of our decisions relate to things where we expect the future <em>not</em> to change much, but you [do] get [all kinds of situations]. There are [some] decisions that [do] involve the future looking different in some important way than the present.”</p>
<details>
<summary>
“When we bought The Washington Post, it went down 50% in a matter of a few months. Best thing that could’ve happened! I mean, doesn’t get any better than that. Business was fundamentally very nonvolatile in nature, but it was a volatile stock. And, you know, that is a great combination from our standpoint”
</summary>
<div>
<p>From <a href="https://buffett.cnbc.com/video/1998/05/04/afternoon-session---1998-berkshire-hathaway-annual-meeting.html">1998 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>If we have a business about which we’re extremely confident as to the business result, we would prefer that it have high volatility than low volatility. We will make more money out of a business where we know where the endgame is going to be if it bounces around a lot.</p>
<p>For example, if people reacted to the monthly earnings of See’s Candy, which might lose money eight months out of the year and makes a fortune in November and December. If people reacted to that and therefore made its stock as an independent company very volatile, that would be terrific for us because we would know it was all nonsense. And we would buy in July and sell in January! Well, obviously, things don’t behave that way.</p>
<p>But when we see a business about which we’re very certain, but the world thinks that its fortunes are going up and down, and therefore it behaves with great volatility — we love it. That’s way better than having a lower beta. So we actually would prefer what other people would call risk.</p>
<p>When we bought The Washington Post, it went down 50% in a matter of a few months. Best thing that could’ve happened! I mean, doesn’t get any better than that. Business was fundamentally very nonvolatile in nature — TV stations and a strong, dominant newspaper, that’s a nonvolatile business — but it was a volatile stock. And, you know, that is a great combination from our standpoint.</p>
</blockquote>
</div>
</details>
<p>In other words, price “volatility” in a business whose fundamentals are good and <em>not</em> volatile is a great buying opportunity.</p>
<p>Sometimes, despite going-in returns looking attractive and “safe”, things go wrong.</p>
<p>Take for example Buffet’s stakes in the four major U.S. airlines that he bought in 2016. He has explained that the underlying earnings yield he foresaw for this position was [1/8 to 1/7, that is] 12.5 to 14.3%. But then COVID-19 hit the world, and those attractive returns were suddenly gone:</p>
<details>
<summary>
“We paid $7 or $8 billion to own 10% of the four large companies in the [U.S.] airline business. And we felt for that, we were getting $1 billion roughly of earnings. Now, we weren’t getting a billion dollars of dividends, but we felt our share of the underlying earnings was $1 billion”
</summary>
<div>
<p>From the <a href="https://www.rev.com/blog/transcripts/warren-buffett-berkshire-hathaway-annual-meeting-transcript-2020">2020 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>I want to be sure that if I’m talking to you about investments and stocks more than I usually have, I want you to know what Berkshire’s actually doing. Now, you’ll see in the month of April that we net sold $6 billion or so of securities.</p>
<p>And that isn’t because we thought the stock market was going to go down or anything of this order ([like] somebody [changing] their target price or this year’s earnings forecast).</p>
<p>[In selling due to COVID-19,] I just decided that I’d made a mistake in evaluating. That was an understandable mistake. It was a probability-weighted decision. When we bought that, we were getting an attractive amount for our money when investing across the airlines business.</p>
<p>We bought roughly 10% of the four largest airlines. We probably paid $7 or $8 billion to own 10% of the four large companies in the airline business. And we felt for that, we were getting a billion dollars roughly of earnings. Now, we weren’t getting a billion dollars of dividends, but we felt our share of the underlying earnings was a billion dollars.</p>
<p>We felt that that number was more likely to go up than down over a period of time. It would be cyclical obviously, but it was as if we bought the whole company.</p>
<p>We bought it through the New York Stock Exchange, and we can only effectively buy 10% roughly of the four [to avoid the <a href="https://www.sec.gov/smallbusiness/goingpublic/officersanddirectors">SEC rule of 2-day transaction reporting</a>]. And we treat it mentally exactly as if we were buying a business.</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="if-you-have-found-a-truly-wonderful-business-it-is-ok-to-demand-less-of-a-margin-of-safety">If you have found a truly wonderful business, it is OK to demand less of a margin of safety</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<p>There is no better way to put the topic of ‘quality’ than in Charlie Munger’s own words.</p>
<p>In his famous 1994 talk at USC Business School, ‘<a href="https://fs.blog/great-talks/a-lesson-on-worldly-wisdom/">A Lesson on Elementary Worldly Wisdom As It Relates To Investment Management & Business</a>’, he said:</p>
<blockquote>
<p>We’ve really made the money out of high-quality businesses. In some cases, we bought the whole business. And in some cases, we just bought a big block of stock. But when you analyze what happened, the big money’s been made in high-quality businesses. And most of the other people who’ve made a lot of money have done so in high-quality businesses.</p>
</blockquote>
<details>
<summary>
“The investment game always involves considering both quality and price. And the trick is to get more quality than you’re paying for in the price. It’s just that simple.” “But not easy”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/1998/05/04/afternoon-session---1998-berkshire-hathaway-annual-meeting.html">1998 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: my question has to deal with kind of quality versus price.</em></p>
<p><em>I’ve been to three annual meetings and I’ve heard great things about Coke every year. But as far as I’m aware, you have not bought any additional shares of Coke over the last three years even though the stock has done just fine.</em></p>
<p><em>If an investor has a relatively short timeframe, say three to five years, how much weight do you think one should give to quality versus price?</em></p>
<p>Warren: Well, if your timeframe is three to five years, I wouldn’t advise it being that way. Because I think if you think you’re going to get out then, it gets more leaning toward the bigger fool theory. The best way to look at any investment is, how will I feel if I own it forever, you know, and put all my family’s net worth in it?</p>
<p>If you talk about quality meaning the certainty that the business will perform as you expect it to perform over a period of time, so the range of possible performance is fairly narrow — that’s the kind of business we like to buy.</p>
<p>And all I can say is that we like to pay a comfortable price, and that depends to some extent on what interest rates are.</p>
<p>We haven’t found comfortable prices for the kind of businesses we like in the last year. We don’t find them uncomfortable, in the sense that we want to sell them. But they’re not prices at which we [would buy again]. We added to Coke one time about five years ago or thereabouts, and it’s conceivable we would add again. It’s a lot more conceivable we would add than subtract.</p>
<p>But that’s the way we feel about most of the businesses. We did make a decision last year that we thought bonds were relatively attractive, and we trimmed certain holdings and eliminated certain small holdings in order to make a bigger commitment in bonds.</p>
<p>Charlie: Yeah. You talk about quality versus price. The investment game always involves considering both quality and price. And the trick is to get more quality than you’re paying for in the price. It’s just that simple.</p>
<p>Warren: But not easy.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“I think the main thing to do is find wonderful businesses.” “If you happen to come into some added money at some time when something dramatic has happened, [swing hard!]. But you don’t want to spend your life waiting around for them”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/1996/05/06/afternoon-session---1996-berkshire-hathaway-annual-meeting.html">1996 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: You said [that] if you have three great companies, wonderful businesses, they could last you a lifetime.</em></p>
<p><em>One thing that struck me in a way that great businesses get pounded down. And then you bet big on them, like American Express and Disney at one time.</em></p>
<p><em>And my question is, I have capital to invest, but I haven’t yet invested it. I have three great companies, which I’ve identified: Coca-Cola, Gillette, and McDonald’s.</em></p>
<p><em>And my question is, if I have a lifetime ahead of me, where I want to keep an investment for more than 20 or 30 years, is it better to wait a year or two to see if one of those companies stumble, or to get in now and just stay with it over a long time horizon?</em></p>
<p>Warren: I won’t comment on the three companies that you’ve named. But in general terms, unless you find the prices of a great company really offensive. If you feel you’ve identified [a great company] — and, by definition, a great company is one that’s going to remain great for 30 years. (If it’s gonna be a great company for 3 years, it ain’t a great company.)</p>
<p>[Here is an analogy about what it means to be a great company:] If you were going to take a trip for 20 years, you wouldn’t feel bad leaving the money with no orders with your broker, and no power of attorney or anything, and you just go on the trip. And when you come back, it’s gonna be a terribly strong company.</p>
<p>[If you have identified companies like those,] I think it’s better just to own them. We could attempt to buy and sell some of the things that we own that we think are fine businesses, but they’re too hard to find. I mean, we found See’s Candy in 1972. Here and there we get the opportunity to do something, but they’re too hard to find.</p>
<p>To sit there and hope that you buy them in the throes of some panic — that you sort of take the attitude of a mortician waiting for a flu epidemic (or something) —, I’m not sure that will be a great technique.</p>
<p>I mean, it may be great if you inherit it. <a href="https://en.wikipedia.org/wiki/J._Paul_Getty">J. Paul Getty</a> inherited [his fater] money at the bottom in 1932. He didn’t inherit exactly, he talked to his mother out of it. He benefited enormously by having access to a lot of cash in the early 30s, that he didn’t have access to in the late 20s. So, you can get some accidents like that. But that’s a lot to count on.</p>
<p>If you start with the Dow at X, and you think it’s too high. When it goes to 90% of X, do you buy? And what if it goes [down] to 50% of X?</p>
<p>You [almost?] “never” get the benefit of those extremes anyway, unless you just come into some accidental sum of money at some times. So I think the main thing to do is find wonderful businesses.</p>
<p>[…] If you happen to come into some added money at some time when something dramatic has happened. I mean, we did well back in 1964 because American Express ran into a crook. We did well in 1976 because GEICO’s managers and auditors didn’t know what the loss reserves should have been the previous couple of years.</p>
<p>So, we’ve had our share of flu epidemics [!], but you don’t want to spend your life waiting around for them.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“Let’s just say there was no stock market and the owner of the best business in whatever your hometown is came to you and said, ‘Look, my brother just died. He owned 20% of the business. I want somebody to go in with me to buy that 20%. Price looks a little high maybe, but this is what I think I can get for it. Do you want to buy in?’” “I think probably the thing to do is to take it”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/1996/05/06/afternoon-session---1996-berkshire-hathaway-annual-meeting.html">1996 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Is <a href="https://en.wikipedia.org/wiki/Philip_L._Carret">Phil Carret</a> here? We’ve got the world’s hero of investing. Phil, would you stand up? Phil is 99. He wrote a book on investing in 1924 [“[Buying a Bond](https://www.amazon.com/Buying-Bond-Commemorative-Philip-Carret/dp/0870341243”]. Phil has done awfully well by finding businesses he likes, and sticking with them, and not worrying too much about what they do day to day.</p>
<p>Let’s just say there was no stock market and the owner of the best business in whatever your hometown is came to you and said, “Look, my brother just died. He owned 20% of the business. I want somebody to go in with me to buy that 20%. Price looks a little high maybe, but this is what I think I can get for it. Do you want to buy in?”</p>
<p>I think that (1) if you like the business, and (2) you like the person that’s coming to you, and (3) the price sounds reasonable, and (4) you really know the business, I think probably the thing to do is to take it. And don’t worry about how its quoted.</p>
<p>It won’t be quoted tomorrow, or next week, or next month. I think people’s investment would be more intelligent if it’s you know stocks were quoted about once a year, but it isn’t going to happen that way.</p>
</blockquote>
</div>
</details>
<p>In fact, when purchasing new subsidiaries, they say they look for “fair” prices, not “cheap” ones:</p>
<details>
<summary>
“In buying a new subsidiary, Berkshire would seek to pay a fair price for a good business that the Chairman could pretty well understand”
</summary>
<div>
<p>From the <a href="https://www.berkshirehathaway.com/letters/2014ltr.pdf">2014 Annual Letter</a>:</p>
<blockquote>
<p>In buying a new subsidiary, Berkshire would seek to pay a fair price for a good business that the Chairman could pretty well understand. Berkshire would also want a good CEO in place, one expected to remain for a long time and to manage well without need for help from headquarters.</p>
</blockquote>
</div>
</details>
<p>Incidentally, Charlie Munger is <a href="https://25iq.com/2015/11/07/ama-on-charlie-munger-what-did-charlie-munger-learn-from-phil-fisher/">also quoted</a> as having said:</p>
<blockquote>
<p>If you can buy the best companies, over time the pricing takes care of itself.</p>
</blockquote>
<p>Yet on the topic of margin of safety <em>vs</em>. quality, an interesting test is to observe the prices they pay when repurchasing Berkshire’s own stock. Arguably, the gap between their estimated intrinsic value and the buyback price should be one of the <em>thinnest</em> margins of safety that they are OK with.</p>
<p class="small center muted">· · ·</p>
<h2 id="cash-is-needed-against-worst-case-scenarios-and-it-has-optionality">Cash is needed against worst-case scenarios (and it has optionality)</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<details>
<summary>
“We’re not really ever positioning ourselves [to time either interest rates or a crash in equity prices]. We’re simply trying to do the smartest thing we can every day when we come to the office. And if there’s nothing smart to do, cash is the default option”
</summary>
<div>
<p>From <a href="https://buffett.cnbc.com/video/2003/05/03/afternoon-session---2003-berkshire-hathaway-annual-meeting.html">2003 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Warren: In terms of how we’re positioned, you know, we have 16 billion of cash, not because we want 16 billion of cash, or because we expect interest rates to go up, or because we expect equities to go down. We have 16 billion in cash because we don’t see anything that makes us want to part with that cash where we feel we’re getting enough for our money.</p>
<p>But we would spend it Monday morning on the right sort of business, or even if we could find equities that we liked, or if we could find some junk bonds we liked. We’re not finding [junk bonds] this year at all, because prices have changed dramatically.</p>
<p>So, we’re not really ever positioning ourselves. We’re simply trying to do the smartest thing we can every day when we come to the office. And if there’s nothing smart to do, cash is the default option.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“In terms of future opportunities, the issue is, is it at all likely that there’ll be an opportunity like 1973-4, or 1982, even, when equities generally are just mouthwatering? I think there’s a very [high] chance that neither Warren or I will live to see either of those occasions again”
</summary>
<div>
<p>From <a href="https://buffett.cnbc.com/video/2003/05/03/afternoon-session---2003-berkshire-hathaway-annual-meeting.html">2003 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Charlie: In terms of future opportunities, the issue is, is it at all likely that there’ll be an opportunity like 1973-4, or 1982, even, when equities generally are just mouthwatering?</p>
<p>I think there’s a very excellent chance that neither Warren or I will live to see either of those occasions again.</p>
<p>If so, Berkshire’s not going to have a lot of no-brainer opportunities. We’re going to have to grind ahead the way we’ve been doing it recently, which is not all bad.</p>
<p>Warren: It’s not impossible, though, we’ll get some mouthwatering opportunities. I mean you just don’t know in markets. It’s unbelievable what markets do over time.</p>
</blockquote>
<p>(And yet, just five years later they got the 2008 Great Recession.)</p>
</div>
</details>
<details>
<summary>
“We are guessing at our <em>future</em> opportunity cost. Warren is basically saying that he’s guessing that [we should have opportunities], in due course, to put our money at pretty attractive rates of return. And therefore he’s not gonna waste a lot of firepower now at lower returns”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=fCt0sqnLpjQ">2003 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Warren: We feel also, obviously, that we will get opportunities (in the future) that are at least at that level [of 10%], and perhaps substantially above. So there’s just a point at which we drop out of the game. And it’s arbitrary. There’s no scientific studies or anything. But I will bet you that a lot of years in the future we will be able to find equities that we understand, and that have a probability of returns at 10% or greater.</p>
<p>Charlie: I think that in the last analysis, everything we do comes back to opportunity cost. To some considerable extent, we are guessing at our future opportunity cost. Warren is basically saying that he’s guessing that [we should have opportunities], in due course, to put our money at pretty attractive rates of return. And therefore he’s not gonna waste a lot of firepower now at lower returns. But that’s an opportunity cost calculation.</p>
<p>And if interest rates were to more or less permanently settle at 1% or something like that, Warren were to reappraise his notions of future opportunity cost, he would change the numbers. Like Keynes said, “When the facts change, I change my mind. What do you do?”</p>
<p>Warren: With our $16 billion that’s getting 1.25% pre-tax, that’s 200 million dollars a year. We could very easily buy government’s [bonds] doing in 20 years and get roughly 5%. So we could change that 200 million a year to 800 million year of income. And we’re making a decision, as Charlie says, is that it’s better to take 200 million for a while on the theory that we’ll find something that gives us 10% or better than to commit to the 800 million a year. And then find that in a year (or there abouts), when the better opportunities came along, that what we’ve committed to had a big principal loss in it.</p>
<p>It’s not terribly scientific but I can tell you is that in practice it seems to work pretty well.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“Cash is to a business as oxygen is to an individual: never thought about when it is present, the only thing in mind when it is absent”
</summary>
<div>
<p>From Warren Buffett’s <a href="https://www.berkshirehathaway.com/letters/2014ltr.pdf">2014 Annual Letter</a>:</p>
<blockquote>
<p>At a healthy business, cash is sometimes thought of as something to be minimized — as an unproductive asset that acts as a drag on such markers as return on equity. Cash, though, is to a business as oxygen is to an individual: never thought about when it is present, the only thing in mind when it is absent.</p>
<p>American business provided a case study of that in 2008. In September of that year, many long-prosperous companies suddenly wondered whether their checks would bounce in the days ahead. Overnight, their financial oxygen disappeared. At Berkshire, our “breathing” went uninterrupted. Indeed, in a three-week period spanning late September and early October, we supplied $15.6 billion of fresh money to American businesses.</p>
<p>We could do that because we always maintain at least $20 billion – and usually far more – in cash equivalents. And by that we mean U.S. Treasury bills, not other substitutes for cash that are claimed to deliver liquidity and actually do so, except when it is truly needed. When bills come due, only cash is legal tender. Don’t leave home without it.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“We’re operating on the basis that we will get chances to deploy capital. They will come in clumps in all likelihood. And they will come when other people don’t want to allocate capital”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=sb5j4SzKzjI">2019 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: Warren, you are a big advocate of index investing, and of not trying to time the market. But by your having Berkshire hold such a large amount of cash in T-bills, it seems to me you don’t practice what you preach. I’m thinking that a good alternative would be for you to invest most of Berkshire’s excess cash in a well-diversified index fund until you find an attractive acquisition or buy back stocks.</em></p>
<p><em>Had you done that over the past 15 years, all the time keeping the $20 billion cash cushion you want, I estimate that at the end of 2018, the company’s 112 billion balance in cash, cash equivalents, in short-term investments and T-bills, would’ve instead been worth about 155 billion.</em></p>
<p><em>The difference between the two figures is an opportunity cost equal to more than 12 percent of Berkshire’s current book value. What is your response to what I say?</em></p>
<p>Warren: That’s a perfectly decent question, and I wouldn’t quarrel with the numbers. And I would say that that is an alternative, for example, that my successor may wish to employ. Because, on balance, I would rather own an index fund than carry Treasury bills.</p>
<p>I would say that if we’d instituted that policy in 2007 or 2008, we might have been in a different position in terms of our ability to move late in 2008 or 2009.</p>
<p>It has certain execution problems with hundreds of billions of dollars than it does if you were having a similar policy with a billion or 2 billion or something of the sort.</p>
<p>But it’s a perfectly rational observation. And certainly, looking back on ten years of a bull market, it really jumps out at you.</p>
<p>But I would argue that if you were working with smaller numbers, it would make a lot of sense. And if you’re working with large numbers, it might well make sense in the future at Berkshire to operate that way.</p>
<p>You know, we committed 10 billion a week ago. And there are conditions under which we could spend a hundred billion dollars very, very quickly. And if we did — if those conditions existed — it would be capital very well deployed, and much better than in an index fund.</p>
<p>We’re operating on the basis that we will get chances to deploy capital. They will come in clumps in all likelihood. And they will come when other people don’t want to allocate capital.</p>
<p>Charlie: Well, I plead guilty to being a little more conservative with the cash than other people. But I think that’s all right.</p>
<p>We could have put all the money into a lot of securities that would’ve done better than the S&P with 20/20 hindsight. Remember, we had all that extra cash all that period, if something had come along in the way of opportunities and so on.</p>
<p>I don’t think it’s a sin to be a little strong on cash when you’re as a big a company as we are.</p>
<p>I watched Harvard [Endowment Fund] use the last ounce of their cash, including all their prepaid tuition from the parents, and plunge it into the market at exactly the wrong moment and make a lot of forward commitments to private equity. And they suffered, like, two or three years of absolute agony. We don’t want to be like Harvard.</p>
<p>Warren: We do like having a lot of money to be able to operate very fast and very big. We know we won’t get those opportunities frequently.</p>
<p>I don’t think, you know, [that] in the next 20 or 30 years there’ll be two or three times when it’ll be raining gold and all you have to do is go outside. But we don’t know when they will happen. And we have a lot of money to commit.</p>
<p>And I would say that if you told me I had to either carry short-term Treasury bills or have index funds and just let that money be invested in America generally, I would take the index funds.</p>
<p>But we still have hopes.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“We don’t prepare ourselves for a single problem. We prepare ourselves for problems that sometimes create their own momentum. In 2008-2009 you didn’t see all the problems the first day.” “We take a worst-case scenario into mind that probably is a considerably worse case than most people do. So I don’t look at [our current cash position] as [that] huge”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=A5lQbkqlJro">2020 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: Why are you recommending listeners to buy now, yet you’re not comfortable buying now as evidenced by your huge cash position?</em></p>
<p>Well, as I just explained the position isn’t that huge when I look at worst-case possibilities. I would say that that there are things that I think are quite improbable — and I hope they don’t happen — but that doesn’t mean they won’t happen.</p>
<p>For example in our insurance business, we could have the world’s (or the country’s) number one hurricane that it’s ever had it, but that doesn’t preclude the fact we [could get] the biggest earthquake a month later.</p>
<p>So we don’t prepare ourselves for a single problem. We prepare ourselves for problems that sometimes create their own momentum. In 2008-2009 you didn’t see all the problems the first day. What really kicked it off was the Freddie and Fannie, the <a href="https://en.wikipedia.org/wiki/Government-sponsored_enterprise">GSEs</a>, went into conservatorship in early September. And then, when money market funds broke the buck…</p>
<p>There are things that trip other things. We take a worst-case scenario into mind that probably is a considerably worse case than most people do. So I don’t look at [it] as [that] huge.</p>
<p>And I’m not recommending that people buy stocks today or tomorrow or next week or next month. I think it all depends on your circumstances, but you shouldn’t buy stocks unless you expect, in my view, you expect to hold them for a very extended period and you are prepared financially and psychologically to hold them, the same way you would hold a farm and never look at a quote and never pay…</p>
</blockquote>
</div>
</details>
<details>
<summary>
“Blanche DuBois [from the famous play] said that she had ‘always depended on the kindness of strangers’. We [at Berkshire] don’t want to be dependent on the kindness of friends even, because there are times when money almost stops”
</summary>
<div>
<p>From <a href="https://www.rev.com/blog/transcripts/warren-buffett-berkshire-hathaway-annual-meeting-transcript-2020">2020 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>I show our cash and Treasury bill position on March 31st. And you might look at that and say, “Well, you’ve got $125 billion or so in cash and Treasury bills. And you’ve got…” At least at that point, we had about, I don’t know, $180 billion or so in equities.</p>
<p>And you can say, “Well, that’s a huge position to have in Treasury bills versus just $180 billion in equities.” But we really have far more than that in equities because we own a lot of businesses. We own 100% of the stock of a great many businesses, which to us are very similar to the marketable stocks we own. We just don’t own them all. They don’t have a quote on them.</p>
<p>But we have hundreds of billions of wholly owned businesses. So our $124 billion is not some 40% or so cash positions, it’s far less than that.</p>
<p>And we will always keep plenty of cash on hand, and for any circumstances, with a 9/11 comes along, if the stock market is closed, as it was in World War I—it’s not going to be, but I didn’t think we were going to be having a pandemic when I watched that <a href="https://www.espn.com/mens-college-basketball/game?gameId=401168325">Creighton-Villanova game in January</a> either.</p>
<p>So we want to be in a position at Berkshire where… Well, you remember <a href="https://en.wikipedia.org/wiki/Blanche_DuBois">Blanche DuBois in A Streetcar Named Desire</a>? That goes back before many of you. In Blanche’s case, she said that she had ‘always depended on the kindness of strangers’. We don’t want to be dependent on the kindness of friends even, because there are times when money almost stops. And we had one of those, interestingly enough. We had it, of course, in 2008 and ’09.</p>
<p>But right around in the day or two leading up to March 23rd, we came very close but fortunately we had a Federal Reserve that knew what to do, but investment-grade companies were essentially going to be frozen out of the market.</p>
<p>CFOs all over the country had been taught to sort of maximize returns on equity capital, so they financed themselves to some extent through commercial paper because that was very cheap and it was backed up by bank lines and all of that. And they let the debt creep up quite a bit in many companies.</p>
<p>And then of course they had the hell scared out of them by what was happening in markets, particularly the equity markets. And so they rushed to draw down lines of credit. And that surprised the people who had extended those lines of credit; they got very nervous. And the capacity of Wall Street to absorb a rush to liquidity that was taking place in mid-March was strained to the limit to the point where the Federal Reserve, observing these markets, decided they had to move in a very big way.</p>
<p>We got to the point where the U.S. Treasury market, the deepest of all markets, got somewhat disorganized. And when that happens, believe me, every bank and CFO in the country knows is, and they react with fear. And fear is the most contagious disease you can imagine; it makes the virus look like a piker.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“In late September of 2008, we had committed to put $6.6 billion in Wrigley. And then Goldman Sachs needed $5 billion, GE needed $3 billion. I sold a couple billion dollars’ worth of J&J just because I didn’t like getting our cash level down below a certain point, under the circumstances that existed then”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2009/05/02/morning-session---2009--berkshire-hathaway-annual-meeting.html">2009 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Normally we have plenty of money around. But in late September of last year [2008], we had committed to put $6.6 billion in Wrigley. And then Goldman Sachs needed $5 billion, GE needed $3 billion.</p>
<p>I sold a couple billion dollars’ worth of J&J just because I didn’t like getting our cash level down below a certain point, under the circumstances that existed then.</p>
<p>That not was a negative decision on J&J. It just meant that I wanted a couple billion more around. And I saw an opportunity to do something that I probably wouldn’t see too much later. Whereas, I could always buy J&J back at a later date. But that’s an unusual situation.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“We want to have all our money working in decent businesses; but sometimes we can’t find them, or sometimes cash comes in unexpectedly, or sometimes we sell something, and we have more cash around than we would like. And frankly, I think these asset allocation things that tacticians in Wall Street put out, you know, about 60% stocks and 30% [something] — we think that’s total nonsense”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2001/04/28/morning-session---2001-berkshire-hathaway-annual-meeting.html">2001 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Warren: Well, there are times when we’re awash in cash. And there have been plenty of times when we didn’t have enough cash.</p>
<p>I remember in the late ’60s, when bank credit was very difficult, Charlie and I were looking for money over in the Middle East. The guy that wanted us to repay him in dinars was also the guy that determined the value of those things. So, we were not terribly excited about meeting up with him on payday and having him decide the exchange rate on that date.</p>
<p>But we, obviously, are looking every day for ways to deploy cash.</p>
<p>We would never have cash around just to have cash. I mean, we would never think that we should have a cash position of X percent. And frankly, I think these asset allocation things that tacticians in Wall Street put out, you know, about 60% stocks and 30 — we think that’s total nonsense.</p>
<p>We want to have all our money working in decent businesses. But sometimes we can’t find them, or sometimes cash comes in unexpectedly, or sometimes we sell something, and we have more cash around than we would like.</p>
<p>And more cash around than we would like means that we have 10 or 15 cents around. Because we want money employed, but we’ll never employ it just to employ it. And in recent years, we’ve tended to be cash heavy, but not because we wanted cash per se.</p>
<p>In the mid-’70s, you know, we were scraping around for every dime we could find to buy things. We don’t like lots of leverage, and we never will. We’ll never borrow lots of money at Berkshire. It’s just not our style.</p>
<p>But you will find us quite unhappy over time if cash just keeps building up. And I think, one way or another, we’ll find ways to use it.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“We consider a portion of that stash to be untouchable, having pledged to always hold at least $20 billion in cash equivalents to guard against external calamities. We have also promised to avoid any activities that could threaten our maintaining that buffer”
</summary>
<div>
<p>From the <a href="https://berkshirehathaway.com/letters/2018ltr.pdf">2018 Annual Letter</a>:</p>
<blockquote>
<p>Berkshire held $112 billion at yearend in U.S. Treasury bills and other cash equivalents, and another $20 billion in miscellaneous fixed-income instruments. We consider a portion of that stash to be untouchable, having pledged to always hold at least $20 billion in cash equivalents to guard against external calamities. We have also promised to avoid any activities that could threaten our maintaining that buffer.</p>
<p>Berkshire will forever remain a financial fortress. In managing, I will make expensive mistakes of commission and will also miss many opportunities, some of which should have been obvious to me. At times, our stock will tumble as investors flee from equities. But I will never risk getting caught short of cash.</p>
</blockquote>
<p>As of May 2020, <a href="https://twitter.com/chrisbloomstran">Christopher Bloomstran</a> of <a href="https://www.semperaugustus.com">Semper Augustus</a> put Berkshire’s cash in perspective during an <a href="https://www.youtube.com/watch?v=yFIriGnO-Oc">interview on The Acquirers Podcast</a>:</p>
<blockquote>
<p>To put the cash in perspective. Berkshire has taken a lot of heat over the last three or four years for sitting on this giant cash pile, right? That’s now north of $130 billion dollars. With the sales of the airlines and incremental profits coming out some of the businesses, they might have $140 billion dollars in cash today.</p>
<p>If you look at the last 20 years, the cash in the business has averaged about 12% of Berkshire’s total assets. At [2019] year-end it was 16%. It’s probably crept up now to where, you know, you’re pushing back on $800 billion dollars in assets. It’s not that large.</p>
<p>And when you think about the insurance operation itself. It had statutory capital of $215 or $220 billion dollars at [2019] year-end. [And you think about their] stock portfolio, as big as it was, pushing $250 billion dollars. [It] got down to $160-165 at the lows on March 23rd [of 2020]. [And it is] probably back up to $190 billion dollars today [in May 2020].</p>
<p>[Also,] all of the cash is not available for investment. Mr. Buffett’s always talked about the $20 billion dollars, that would just always be the fortress, and would always be a permanent cash reserve. I look at that [$20 billion] number, really, as more likely approximating one year’s worth of insurance losses paid as cash. Which at the current run rate would be about $37 billion dollars.</p>
<p>And then you’ve got cash throughout the operation — in the rail and in the energy businesses. Although you can’t tell where it is now, because we’re seeing less transparency of the MSR [Berkshire’s Manufacturing, Services and Retailing business line] businesses themselves in the last three years. There is cash that’s held in those operations. And the cash the MSR businesses [have] probably offsets [their] debt.</p>
<p>[All in all] I think there’s probably, call it, $60 billion dollars of cash that really is <em>not</em> available for long duration investment in businesses or in common stocks [out of the $130-140]. I think it’s the permanent cash reserve for the insurance operations plus working capital that’s required in the other businesses.</p>
<p>So, in today’s world, there’s the balance somewhere between $60 and $140. It’s still a good chunk of money. It’s still $80 billion dollars [but it is <em>not</em> $140 billion].</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="how-do-they-size-their-bets">How do they size their bets?</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<details>
<summary>
“If we were only running our own net worth — I’m certain a very significant number of times, if you go over 50 years, there have been a lot of times when you would have put at least 75% of your net worth into an idea”
</summary>
<div>
<p>From <a href="https://buffett.cnbc.com/video/2008/05/03/morning-session---2008-berkshire-hathaway-annual-meeting.html">2008 Berkshire Hathaway Annual Meeeting</a>:</p>
<blockquote>
<p><em>Question: I’ve read that there were several times in your investing career when you were confident enough in one idea to put a lot of your money into it — say, 25% or more.</em></p>
<p><em>I believe a couple of those cases were American Express and the Washington Post in the ’70s. And I’ve heard you discuss your thinking on those.</em></p>
<p><em>But could you talk about any of the other times you’ve been confident enough to make such a big investment and what your thinking was in those cases?</em></p>
<p>Warren: If we were only running our own net worth — I’m certain a very significant number of times, if you go over 50 years, there have been a lot of times when you would have put at least 75% of your net worth into an idea. Wouldn’t there, Charlie?</p>
<p>Let’s just assume you didn’t have Berkshire in the picture. We’ve seen all kinds of ideas we would have put 75% of our net worth in.</p>
<p>Charlie: Warren, there have been times in my life when I’ve had more than a 100% of my net worth invested in things.</p>
<p>Warren: That’s because you had a friendly banker; I didn’t.</p>
<p>Several times I had 75% percent of my net worth in one situation.</p>
<p>Over a long period of time, you will see things that it would be a mistake — if you’re working with smaller sums — not to have half your net worth in.</p>
<p>You really do see things that are lead-pipe cinches. And you’re not going to see them often and they’re not going to be talking about them on television or anything of the sort, but there will be some extraordinary things happen in a lifetime where you can put 75% of your net worth or something like that in a given situation.</p>
<p>The problem has been the guys that have put 500% of their net worth in. If you just take <a href="https://en.wikipedia.org/wiki/Long-Term_Capital_Management">LTCM</a>. Very smart guys. Very decent guys. Some friends of mine. High grade. Knew their business.</p>
<p>But they put, you know, maybe 25 times their net worth into things that were a cinch, if they hadn’t have gone in that heavily. I mean, they were in things that had to converge, but they didn’t get to play out the hand.</p>
<p>But if they’d have had a 100% of their net worth in them, it would have worked out fine. If they would have had 200 percent of their net worth in it, it would have worked out fine. But they instead went to, you know, maybe 2500% or something like that.</p>
<p>I mean, actually, there’s quite a few people in this room that have close to a hundred percent of their net worth in Berkshire, and some of them have had it for 40 or more years. Berkshire was not in a cinch category. It was in the strong probability category, I think.</p>
<p>But I saw things in 2002 in the junk bond field. I saw things in the equity markets.</p>
<p>If you could have bought Cap Cities with Tom Murphy running it in 1974, it was selling at a third or a fourth what the properties were worth and you had the best manager in the world running the place and you had a business that was pretty damn good even if the manager wasn’t. You could have put a hundred percent of your net worth in there and not worry.</p>
<p>You could put a hundred percent of your net worth in Coca-Cola, earlier than when we bought it, but certainly around the time we bought it, and that would not have been a dangerous position.</p>
<p>It would be far more dangerous to do a whole bunch of other things that brokers were recommending to people.</p>
<p>And you will find opportunities that, if you put 20% of your net worth in it, you’ll have wasted the opportunity of a lifetime, you know, in terms of not really loading up.</p>
<p>And we’ve had the chance to do that, way, way in our past, when we were working with small sums of money. We’ll never get a chance to do that working with the kinds of money that Berkshire does.</p>
<p>We try to load up on things. And there will be markets when we get a chance to from time to time, but very seldom do we get to buy as much of any good idea as we would like to.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“If I were running $50, $100, $200 million, I would have 80% in 5 positions, with 25% for the largest. [But] if it’s your game and you really know your business, you can load up. There were various times I would have gone up to 75%, even in the past few years”
</summary>
<div>
<p>From <a href="https://archive.is/h7H8C#selection-5801.0-5805.339">2008 talks at Emory’s Goizueta Business School and McCombs School of Business at UT Austin</a>:</p>
<blockquote>
<p>Charlie and I operated mostly with 5 positions. If I were running $50, $100, $200 million, I would have 80% in 5 positions, with 25% for the largest.</p>
<p>In 1964 I found a position I was willing to go heavier into, up to 40%. I told investors they could pull their money out. None did. The position was American Express after the <a href="https://en.wikipedia.org/wiki/Salad_Oil_scandal">Salad Oil Scandal</a>.</p>
<p>In 1951 I put the bulk of my net worth into GEICO.</p>
<p>Later in 1998, <a href="https://en.wikipedia.org/wiki/Long-Term_Capital_Management">LTCM</a> was in trouble. With the spread between the on-the-run versus off-the-run 30-year Treasury bonds, I would have been willing to put 75% of my portfolio into it. There were various times I would have gone up to 75%, even in the past few years. If it’s your game and you really know your business, you can load up.</p>
<p>Over the past 50-60 years, Charlie and I have never permanently lost more than 2% of our personal worth on a position. We’ve suffered quotational loss, 50% movements. That’s why you should never borrow money. We don’t want to get into situations where anyone can pull the rug out from under our feet.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“I would say for anyone working with normal capital who really knows the businesses they have gone into, six is plenty, and I probably have half of what I like best”
</summary>
<div>
<p>From a <a href="https://archive.is/h7H8C#selection-5845.692-5849.724">1998 lecture at the University of Florida Business School</a>:</p>
<blockquote>
<p>Once you are in the businesses of evaluating businesses and you decide that you are going to bring the effort and intensity and time involved to get that job done, then I think diversification is a terrible mistake to any degree.</p>
<p>If you really know businesses, you probably shouldn’t own more than six of them. If you can identify six wonderful businesses, that is all the diversification you need. And you will make a lot of money.</p>
<p>And I can guarantee that going into a seventh one instead of putting more money into your first one is gotta be a terrible mistake. Very few people have gotten rich on their seventh best idea. But a lot of people have gotten rich with their best idea.</p>
<p>So I would say for anyone working with normal capital who really knows the businesses they have gone into, six is plenty, and I probably have half of what I like best. I don’t diversify personally. All the people I’ve known that have done well with the exception of Walter Schloss, Walter diversifies a lot. I call him Noah, he has two of everything.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“As long as the odds are in our favor, and we’re not risking the whole company on one throw or anything close to it, we don’t mind volatility in results; what we want is the favorable odds”
</summary>
<div>
<p>From <a href="https://buffett.cnbc.com/video/1998/05/04/afternoon-session---1998-berkshire-hathaway-annual-meeting.html">1998 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Charlie: [Let me] put it this way: as long as the odds are in our favor, and we’re not risking the whole company on one throw or anything close to it, we don’t mind volatility in results. What we want is the favorable odds. We figure the volatility, over time, will take care of itself at Berkshire.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“We’ve never had a lot of things that pulled us way back. So we never went two steps forward and one step back. We probably went two steps forward and a fraction of a step back. But avoiding the catastrophes is a very important thing. And it will be important in the future. You will have your chance to participate in catastrophes”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=KAF6bG6HbAA">2007 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>The biggest thing to is to have something in the way you’re programmed so that you don’t ever do anything where you can lose a lot.</p>
<p>I mean, our best ideas have not been better than other people’s best ideas, but we’ve never had a lot of things that pulled us way back. So we never went two steps forward and one step back. We probably went two steps forward and a fraction of a step back. But avoiding the catastrophes is a very important thing. And it will be important in the future. You will have your chance to participate in catastrophes.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“Charlie and I made a dozen or so very big decisions relative to net worth, but not as big as they should have been. And we’ve known we were right on those going in. I mean, they just weren’t that complicated, and we knew we were focusing on the right variables and that they were dominant”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=MAo5I5saSfo">1998 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>You have to make a few good decisions in your lifetime. The important thing is to know when you find [a company] where you really do know the key variables, which ones are important, and you do think you’ve got a fix on them.</p>
<p>What we’ve done well. Charlie and I made a dozen or so very big decisions relative to net worth, but not as big as they should have been.</p>
<p>And we’ve known we were right on those going in. I mean, they just weren’t that complicated, and we knew we were focusing on the right variables and that they were dominant. And we knew that even though we couldn’t take it out to five decimal places or anything like, we knew that, in a general way, we were right about them. And that’s what we look for, the fat pitch.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“If we knew it was an honest coin, and someone wanted to give us 7-to-5 (or something of the sort) on one flip, how much of Berkshire’s net worth would we put on that flip? Well, it would sound like a big number to you. It would not be a huge percentage of the net worth, but it would be a significant number. We will do things when probabilities favor us”
</summary>
<div>
<p>From <a href="https://buffett.cnbc.com/video/1994/04/25/morning-session---1994-berkshire-hathaway-annual-meeting.html">1994 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: I’d like to ask you to expound on your view of risk in the financial world, and I ask that against the background of what appear to be a number of inconsistencies between your view of risk and the conventional view of risk.</em></p>
<p><em>And I mention that derivatives are dangerous, and yet you feel comfortable playing at derivatives through Salomon Brothers. And betting on hurricanes is dangerous, and yet you feel comfortable playing with hurricanes through insurance companies.</em></p>
</blockquote>
<blockquote>
<p>Warren: The risk, in terms of our super-cat [super catastrophic insurance] business, is not that we lose money in any given year. We know we’re going to lose money in some given day, that is for certain. And we’re extremely likely to lose money in a given year [if a contract went on for a long enough period]. Our time horizon of writing that business would be at least a decade. And we think the probability of losing money over a decade is low. So we feel that, in terms of our horizon of investment, that is <em>not</em> a risky business.</p>
<p>[In fact,] it’s a whole lot less risky than writing something that’s much more predictable.</p>
<p>We are perfectly willing to lose money on a given transaction — arbitrage being an example, any given insurance policy being another example. We are perfectly willing to lose money on any given transaction.</p>
<p>We are <em>not</em> willing to enter into transactions in which we think the probability of doing a number of mutually independent events, but of a similar type, has an expectancy of loss. And we hope that we are entering into our transactions where our calculations of those probabilities have validity. And to do so, we try to narrow it down. [And] there are a whole bunch of things we just won’t do because we don’t think we can write the equation on them. […]</p>
<p>We, Charlie and I, by nature are pretty risk-averse. But we are very willing to enter into transactions [with the odds are in our favor].</p>
<p>If we knew it was an honest coin, and someone wanted to give us 7-to-5 (or something of the sort) on one flip, how much of Berkshire’s net worth would we put on that flip? Well, it would sound like a big number to you. It would not be a huge percentage of the net worth, but it would be a significant number. We will do things when probabilities favor us.</p>
<p>Charlie: Yeah, I would say we try and think like Fermat and Pascal as if they’d never heard of modern finance theory. I really think that a lot of modern finance theory can only be described as disgusting.</p>
</blockquote>
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</details>
<p class="small center muted">· · ·</p>
<h2 id="assessing-managements">Assessing managements</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<details>
<summary>
“There’s a lot to be said for a sort of candid, simple, coherent prose.” “We give very little advice to our managers, but one thing we always do say is to tell us the bad news immediately. And I don’t see why that isn’t good advice for the manager of a public company”
</summary>
<div>
<p>From the <a href="https://www.youtube.com/watch?v=duxPgfFuKds">1998 Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: I’d like your advice on how to understand annual reports. What you look for, what’s important, what’s not important, and what you’ve learned over the years from reading thousands of reports? Thank you.</em></p>
<p>Warren: Well, we’ve read a lot of reports, I will tell you that.</p>
<p>We start by looking at the reports of companies that we think we can understand. We hope to be reading reports of businesses (I do read hundreds of them every year) that are understandable to us.</p>
<p>And then we see from that report whether the management is telling us about the things that we would want to know about if we owned a 100% of the company.</p>
<p>And when we find a management that does tell us about those things, and that is candid in the same way that a manager of a subsidiary would be candid with us, and talks in language that we can understand, it definitely improves our feeling about investing in such a business.</p>
<p>And the reverse turns us off, to some extent. So if we read a bunch of public relations gobbledygook, you know, and we see lots of pictures and no facts, it has some effect on our attitude toward a business.</p>
<p>We want to understand the business better when we get through with the annual report than when we picked it up. And that is not difficult for a management to do if they want to do it.</p>
<p>If they don’t want to do it, you know, we think that is a factor in whether we want to be their partners over a 10-year period or so.</p>
<p>But we’ve learned a lot from annual reports. For example, I would say that the Coca-Cola annual report over the last good many years is an enormously informative document. I mean, I can’t think of any way if I’d have a conversation with Roberto Goizueta, or now Doug Ivester, and they were telling me about the business, they would not be telling me more than I get from reading that annual report.</p>
<p>We bought that stock based on an annual report. We did not buy it based on any conversation of any kind with the top management of Coca-Cola before we bought our interest. We simply bought it based on reading the annual report, plus our knowledge of how the business worked.</p>
<p>Charlie: Yeah. I do think that if you’ve got a standardized bunch of popular jargon that looks like it came out of the same consulting firm, I do think it’s a big turnoff. That’s not to say that some of the consulting mantras aren’t right. But I think there’s a lot to be said for a sort of candid, simple, coherent prose.</p>
<p>Warren: Almost every business has problems, and we’d just assume that the manager would tell us about them. We would like that in the businesses we run. In fact, we give very little advice to our managers, but one thing we always do say is to tell us the bad news immediately. And I don’t see why that isn’t good advice for the manager of a public company.</p>
<p>Over time, you know, I’m positive it’s the best policy. But a lot of companies, for example, have investor relations people, and they are dying just to pump out what they think is good news all the time. And they have this attitude that, you know, you’ve got a bunch of animals out there to be fed. And that they’re going to feed them what they want to eat all the time.</p>
<p>And over time the animals learn. So we’ve tried to stay away from businesses like that.</p>
<p>Charlie: What you seldom see in an annual report is a sentence like this: “This is a very serious problem and we haven’t quite figured out yet how to handle it.” But believe me, that is an accurate statement much of the time.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“I think you can learn a lot by reading the annual letters. For one thing, if it’s clearly the product of some Investor Relations Department, or outside consultant, or something of the sort that tells you something about the individual [CEO and/or chair]. If he’s not willing to talk once a year through a few pages to the people who gave them their money to invest. I’ve really got some questions about people like that”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=zS-95ZsXxD8">2007 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: Warren and Charlie you spend a lot of time evaluating the management quality and integrity of the companies that you may invest in. In my current job I do not have the opportunity to do that. As I read through annual reports and financial statements, what do you suggest I focus on to help me to determine the quality and integrity of management?</em></p>
<p>Warren: We’ve spent many, many years — and we bought many things — without meeting managements at all, without having any entrée to them.</p>
<p>The $5 billion of stocks that we may have bought in the first quarter, most of those were companies that I’ve never met the management, never talked to them. We read a lot. We read annual reports. We read about competitors. We read about the industries they’re in.</p>
<p>Obviously, if we’re buying the whole business, that’s a different question. Because we’re gonna buy it, be in bed with them, they’re gonna run them. So we care very much about whether they’re going to behave in the future as they have in the past, once we own the business. We’ve had very good luck on that.</p>
<p>But in terms of marketable securities, we read the reports. Charlie and I were just talking about one the other day. We read an annual report of a large oil company — hundred pages, public relations people, lots of pictures, spent a fortune on it. Yet you can’t find in that report what they’re finding cost per thousand of cubic feet (or per barrel of oil) was last year. That’s the most important figure in an oil & gas company over a period of years, but every year counts.</p>
<p>The fact that wouldn’t even be discussed. (And the reason it wasn’t discussed, it was [because the number was] absolutely terrible) But the fact that wouldn’t even be discussed — and to the extent it was touched on it wasn’t done on in a dishonest manner. When we read things where we basically are getting dishonest messages from the management, it makes a difference to us.</p>
<p>In marketable securities we can solve that by selling the stock. It’s not the same thing as buying the entire business, but I think you can learn a lot by reading the annual letters. For one thing, if it’s clearly the product of some Investor Relations Department, or outside consultant, or something of the sort that tells you something about the individual. If he’s not willing to talk once a year through a few pages to the people who gave them their money to invest. I’ve really got some questions about people like that.</p>
<p>I like that feeling that the I’m hearing directly from somebody who regards me as a partner. And you may not get it all the way, but when I get a zero percent of the way, I don’t like it.</p>
<p>In marketable securities, we’ve still bought into some extremely good businesses where we thought they were run by people who didn’t really like very well. [And so far it] didn’t feel like they screwed up.</p>
<p>Charlie: There are two things. The quality of the business and the quality of the management. If the business is good enough, it will carry a lousy manager. In the converse case, a really good manager gets in a really lousy business, you’ll ordinarily have a very imperfect record.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“Read the proxy statements, see what they think of. See how they treat themselves <em>vs.</em> how they treat the shareholders. Look at what they have accomplished, considering what the hand was that they were dealt when they took over compared to what is going on in the industry”
</summary>
<div>
<p>From <a href="https://buffett.cnbc.com/video/1994/04/25/morning-session---1994-berkshire-hathaway-annual-meeting.html">1994 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: You talk about good management with corporations, and that you try and buy companies with good management. I feel that I have about as much chance of meeting good managers, other than yourself, as I do bringing Richard Nixon back to life. How do I, as an average investor, find out what good management is?</em></p>
<p>Well, I think you judge management by two yardsticks. One is how well they run the business. I think you can learn a lot about that by reading about both what they’ve accomplished and what their competitors have accomplished, and seeing how they have allocated capital over time.</p>
<p>You have to have some understanding of the hand they were dealt when they themselves got a chance to play the hand. But, if you understand something about the business they’re in — and you can’t understand it in every business, but you can find industries or companies where you can understand it — then you simply want to look at how well they have been doing in playing the hand, essentially, that’s been dealt with them.</p>
<p>And then the second thing you want to figure out is how well that they treat their owners. And I think you can get a handle on that, oftentimes. A lot of times you can’t. I mean it — they’re many companies that obviously fall somewhere in that 20th to 80th percentile and it’s a little hard to pick out where they do fall.</p>
<p>But, I think you can usually figure out. It’s not hard to figure out that, say, Bill Gates, or Tom Murphy, or Don Keough, or people like that, are really outstanding managers. And it’s not hard to figure out who they’re working for.</p>
<p>And I can give you some cases on the other end of the spectrum, too. It’s interesting how often the ones that, in my view, are the poor managers also turn out to be the ones that really don’t think that much about the shareholders, too. The two often go hand in hand.</p>
<p>But, I think reading of reports — and reading of competitors’ reports — I think you’ll get a fix on that in some cases. You don’t have to make a hundred correct judgments in this business or 50 correct judgments. You only have to make a few. And that’s all we try to do.</p>
<p>And, generally speaking, the conclusions I’ve come to about managers have really come about the same way you can make yours. I mean they come about by reading reports rather than any intimate personal knowledge or — and knowing them personally at all.</p>
<p>So, read the proxy statements, see what they think of. See how they treat themselves <em>vs.</em> how they treat the shareholders. Look at what they have accomplished, considering what the hand was that they were dealt when they took over compared to what is going on in the industry.</p>
<p>And I think you can figure it out sometimes. You don’t have to figure out very often.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“I frequently ask CEOs of companies what they would do differently if they owned the whole place themselves”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2009/05/02/morning-session---2009--berkshire-hathaway-annual-meeting.html">2009 Annual Meeting</a>:</p>
<blockquote>
<p>I frequently ask CEOs of companies what they would do differently if they owned the whole place themselves.</p>
<p>You know, when I’m talking to, either companies where we’ve invested in or other companies [that] friends of mine run. [I often ask:] “What would you do different if this was a hundred percent owned by you and your family?”</p>
<p>And they give me a list of things.</p>
<p>There is no list at Berkshire. You know, we basically run this place the same way we’d run if we owned a 100% of it.</p>
<p>And that is a difference in terms of people joining in with us. They don’t have to adjust their lives to a bunch of rules that are kind of self-imposed, in terms of how people think about public companies, in terms of earnings, predictions, and all of that sort of thing.</p>
<p>There are certain people that would prefer to be associated with an enterprise like that. [They would] know that they’ve made a one decision on where that business — that they built up over decades, and cherish, and everything — is going to go, and they’re not going to get surprised later on.</p>
<p>They’re not going to get some management consultant come in and say, “You ought to have a pure player, Wall Street’s saying, so you ought to spin this off or sell it,” or anything like that. And they know we’re not going to leverage it up.</p>
<p>So they know they’re really going to get to do what they love the most, which is to continue to run their business, not bothered by bankers or lawyers or public expectancies or anything of the sort.</p>
<p>And, like I said earlier, that’s a real advantage.</p>
<p>Charlie: Yeah, in the show business, they say the show has legs if it’s going to last a long time. I think Berkshire Hathaway’s system has legs.</p>
</blockquote>
</div>
</details>
<p>Crooks give off a lot of the same messages, time after time:</p>
<details>
<summary>
“They tell you things that are too good to be true”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=c_fierpQAtk">2002 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Charlie and I have found that — at least to us — many of the crooks look like crooks. We haven’t shorted them, but we have spotted a lot of frauds over the years in public companies. Years before the roof fell in.</p>
<p>They usually are people that tell you things that are too good to be true for one thing. They tell you very mediocre businesses are wonderful businesses for one reason or another. They just have a smell about them, you know.</p>
<p>Charlie [and I have] a hobby of keeping track of the [crooks] of the world. There’s a syndrome. They give off a lot of the same messages, time after time.</p>
<p>Wall Street has no filter against them. Wall Street loves them as long as you know as long as they’re pushing out securities and the commissions.</p>
</blockquote>
<blockquote>
<p>Charlie: We didn’t stop Normandy America. With <a href="https://archive.is/MGcbY">Lou Simpson</a> (<a href="https://www.bloomberg.com/news/articles/2019-07-05/ex-berkshire-manager-lou-simpson-converts-firm-to-family-office">source</a>), Warren Buffett, and Charlie Munger on the board of <a href="https://en.wikipedia.org/wiki/Salomon_Brothers">Salomon Brothers</a>.</p>
<p>Warren: <a href="https://archive.is/yKywP">Normandy America was a case of some guy that manufactured a record out in California</a> (<a href="https://www.nytimes.com/1995/08/17/business/a-double-blow-to-salomon-offering-ended-rating-cut.html">source</a>). He was promoting a bunch of securities including, Berkshire Hathaway. He was going to go public and Salomon was courting him. The record was total baloney. I think they actually went public for a day or so, and then the SEC pulled it back.</p>
<p>Charlie: It was a very embarrassing episode, and when we remonstrated against this obvious insanity. They told us the underwriting committee had approved it.</p>
<p>Warren: I don’t think they changed underwriting committees either.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“I think management should tell it like it is at all times and not be a big promoter of its own stock”
</summary>
<div>
<p>Charlie Munger at the <a href="https://junto.investments/daily-journal-2021-transcript/">2021 Daily Journal AGM</a> [<a href="https://www.yahoo.com/entertainment/charlie-munger-speaks-daily-journal-162005167.html">video</a>]:</p>
<blockquote>
<p><em>Question: Does management, in your opinion, have a moral responsibility to have their shares trade as close to fair value as possible?</em></p>
<p>Charlie Munger: I don’t think you can make that a moral responsibility. Because if you do that, I’m a moral leper. The Daily Journal stock sells way above the price I would pay if I was buying new stock. So, no I don’t think it’s the responsibility of management to assure where the stock sells.</p>
<p>I think management should tell it like it is at all times and not be a big promoter of its own stock.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“We have <em>never</em> invested in companies that are hell-bent on issuing shares. That behavior is one of the surest indicators of a promotion-minded management, weak accounting, a stock that is overpriced and — all too often — outright dishonesty”
</summary>
<div>
<p>From the <a href="https://www.berkshirehathaway.com/letters/2014ltr.pdf">2014 Annual Letter</a>:</p>
<blockquote>
<p>At both BPL [i.e. the old Buffet Partnerships] and Berkshire, we have <em>never</em> invested in companies that are hell-bent on issuing shares. That behavior is one of the surest indicators of a promotion-minded management, weak accounting, a stock that is overpriced and — all too often — outright dishonesty.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“I see more predictions of future earnings growth at a high rate, not less.” “It’s what the investor relations departments want the managements to say. It makes their life easier, you know. But they don’t have to be there five years from now — or 10 years from now — doing [that same kind of] thing”
</summary>
<div>
<p>From <a href="https://buffett.cnbc.com/video/2001/04/28/morning-session---2001-berkshire-hathaway-annual-meeting.html">2001 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: My question concerns Gillette. Do you think their goal of trying to grow earnings at 15-plus percent kind of got them into their current inventory problems at the trade? And as well, the Duracell acquisition. I know at the time, neither one of you were the biggest fans of the deal. I just want to know how you feel about it now.</em></p>
<p>Warren: Well, I think it’s a mistake for any company to predict 15% a year growth. But plenty of them do. For one thing, unless the U.S. economy grows at 15% a year, eventually any 15% number catches up with you. It just, it doesn’t make sense. Very, very few large companies can compound their earnings at 15%. It isn’t going to happen.</p>
<p>You can look at the Fortune 500. If you pick any company on there that currently has record earnings, and you want to pick out 10 of them that over the next 20 years will average 15% or greater, I will bet you that more than half of your list will not make it.</p>
<p>So, I think it’s a mistake, and as I’ve said in the annual report, I think it leads people to stretch on accounting. I think it tends to make them change trade practices.</p>
<p>And you know, I’m not singling out Gillette in the least, but I can tell you that if you look at the companies that have done it, you will find plenty of examples of people who have made those sort of mistakes.</p>
<p>Charlie: I think that kind of stuff happens all the time. It will continue to happen. It’s just built into the system.</p>
<p>I see more predictions of future earnings growth at a high rate, not less. A few people have sort of taken an abstinence pledge, but it’s very few. It’s what the analysts want to hear.</p>
<p>Warren: It’s what the investor relations departments want the managements to say. It makes their life easier, you know. But they don’t have to be there five years from now — or 10 years from now — doing [that same kind of] thing.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“I would say the number of times we’re going to buy into a company (whether it’s through stocks or through the entire company) where people are talking about EBITDA is going to be about zero”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=c_fierpQAtk">2002 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: I would like to direct a question to Mr. Munger pertaining to cash flow analysis. Given the practices of the numerous corporations of deliberately fabricating cash flow numbers — which occurred in some of the telcos, where they characterize like kind exchanges as the product sales — how do you ferret out this type of fraud? What do you recommend an individual investor to do, short of obtaining a degree in forensic accounting to uncover this type of fraud?</em></p>
<p>Charlie: I think you’re asking for a lot if you want some simple way of not being taken in by the frauds of the world. I don’t think there is any short answer. [But] I think there are whole fields that you can just quitclaim because it looks like there’s too much fraud in it.</p>
<p>Warren: You raise the question about cash flow. I would say the number of times we’re going to buy into a company whether it’s through stocks or through the entire company where people are talking about EBITDA is going to be about zero. If we take all the people in the world have talked about EBITDA, and all the people in the world who haven’t talked about EBITDA, there are more frauds in the first group percentage by a substantial margin, very substantial.</p>
<p>It’s very interesting to me that if you look at some enormously successful companies — Walmart, General Electric, Microsoft — I don’t think that term has ever appeared in their annual reports. So when people are talking about that sort of thing either they’re trying to con you in some way, or they con themselves.</p>
<p>Charlie: Or both.</p>
<p>Warren: Or both, that often happens. If you set out to con somebody, after a while, you con yourself.</p>
<p>If they think you’re focusing on EBITDA, they may arrange things so that that number looks bigger than it really is.</p>
<p>You take telecoms. They’re spending every dime that comes in. It isn’t cash flow. I mean, the cash is flowing out. But you can look at their statement, and there’s billions of dollars supposedly, and depreciation, and so on.</p>
<p>You know, interest is an expense. And, actually, taxes are going to be expensive [too]! Anybody that tells you that they are making a lot of money before taxes (or in terms of EBITDA) [and that that money] is meaningful… [It is an absurd!]</p>
<p>You get depreciation by laying out money ahead of time. It’s the worst kind of expense. We [at Berkshire] look for float, where we get the money and then pay out later on. But depreciation occurs because you buy an asset first, and then you get the deduction later on. It’s the worst kind of expense there is.</p>
<p>And you start paying taxes when you actually make money, and when the depreciation runs out at some point.</p>
<p>It just amazes me how widespread the usage of EBITDA has become. I would say there have been people who have tried to dress up financial statements in a way to appeal the people who were impressed by such a number.</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="when-do-they-sell-stocks">When do they sell stocks?</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<details>
<summary>
“We would sell [stocks] if we needed money for something else. But that has not been the problem the last 10 or 15 years. We [would also] sell when we’re reevaluating the economic characteristics of the business. [When we sell,] that doesn’t mean we think that the company is going into some disastrous period or anything remotely like that. [It means instead that] we don’t think that their competitive advantage is as strong as we thought it was when we initially made the decision”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2002/05/04/morning-session---2002-berkshire-hathaway-annual-meeting.html">2002 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: You have said that your favorite time to own a stock is forever. Yet, you sold McDonald’s and Disney after not owning them for long. How do you do decide when to hold forever and when to sell?</em></p>
<p>Well, it’s a very good question about selling. It’s not our natural inclination to sell.</p>
<p>We have held the Washington Post stock since 1973. I’ve never sold a share of Berkshire, having bought the first shares in 1962.</p>
<p>We’ve held Coke stock since 1988. We’ve held Gillette stock since 1989. Held American Express stock since 1991. We had actually previously been in American Express in the ’60s, and in Disney.</p>
<p>So, there are companies we are familiar with.</p>
<p>[First of all,] we would sell if we needed money for something else. But that has not been the problem the last 10 or 15 years.</p>
<p>Forty years ago my sales were all because I found something that I liked even better. I hated to sell what I sold, but I also didn’t want to borrow money. So I would reluctantly sell something that I thought was terribly cheap to buy something that was even cheaper.Those were the times when I had more ideas than money. Now I’ve got more money than ideas, and that’s a different equation.</p>
<p>So now we sell when we’re reevaluating the economic characteristics of the business.</p>
<p>Don’t want to name names — but take a stock we’ve sold, of some sort. We probably had one view of the long-term competitive advantage of the company at the time we bought it, and we may have modified that. That doesn’t mean we think that the company is going into some disastrous period (or anything remotely like that). We think McDonald’s has a fine future. We think Disney has a fine future. And there are others. But we probably don’t think that their competitive advantage is as strong as we thought it was when we initially made the decision.</p>
<p>That may mean that we were wrong when we made the decision originally. It may mean that we’re wrong now, and that their strengths are every bit as what they were before. But, for one reason or another, we think that the strengths may have been eroded to some degree.</p>
<p>A classic case on that would be the newspaper industry, generally, for example. I mean, in 1970, Charlie and I were looking at the newspaper business. We felt it was about as impregnable a franchise as could be found. We still think it’s quite a business, but we do not think the franchise in 2002 is the same as it was in 1970. We do not think the franchise of a network television station in 2002 is the same as it was in 1965.</p>
<p>And those beliefs change quite gradually. And who knows whether they’re precise — whether they’re right, even. But that is the reason, in general, that we sell now.</p>
<p>If we got into some terribly cheap market, we might sell some things that we thought were cheap to buy something even cheaper, after we’d bought lots and lots of equities. But that’s not the occasion right now.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“We don’t [hold stocks forever]. If we lose confidence in the management, if we lose confidence in the durability of the competitive advantage, if we recognize we made a mistake when we went into it — we [have sold] plenty of times”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2009/05/02/morning-session---2009--berkshire-hathaway-annual-meeting.html">2009 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: How do you justify holding stocks forever when the fundamentals have permanently changed?</em></p>
<p>Warren: Well, the answer is we don’t [hold stocks forever]. If we lose confidence in the management, if we lose confidence in the durability of the competitive advantage, if we recognize we made a mistake [in the original analysis] when we went into it — we sell plenty of times. So it’s not unheard of.</p>
<p>On the other hand, if you really get a wonderful business with outstanding management — but mostly the wonderful business part of it — when in doubt, keep holding. But it’s no inviolable rule.</p>
<p>Now, among the businesses we own [100%] we have an attitude that when we buy a business, it’s for keeps. And we make only two exceptions: when they promise to start losing money indefinitely or if we have major labor problems. But otherwise, we are not going to sell something just ’cause we get offered more money for it, even than it’s worth.</p>
<p>And that’s a peculiarity we have. And we want our partners to know about that.</p>
<p>We do think it probably helps us in terms of buying businesses over time. It’s also the way we want to run our business.</p>
<p>But with stocks, bonds, we sell them. But we’re more reluctant to sell them than most people. I mean, if we made the right decision going on, we like to ride that a very long time. And we’ve owned many — we’ve owned some stocks for decades.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“We like those airlines, but the world has changed for the airlines, and I don’t know how it’s changed. I don’t know whether the Americans will change their habits because of an extended period that we’re semi shut down in the economy”
</summary>
<div>
<p>From the <a href="https://www.youtube.com/watch?v=A5lQbkqlJro">2020 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: Please, did I understand correctly Mr. Buffett, to say that Berkshire Hathaway sold its interest in four different airlines and if so, can you name them? Can the names of those airlines be identified?</em></p>
<p>I wouldn’t normally talk about it, but I think it requires an explanation, and it requires an explanation that means we were not disappointed at all in the businesses that were being run and the management, but we did come to a different opinion on it.</p>
<p>They’re the four largest US airlines, that’s American Airlines, and Delta Airlines, and Southwest Airlines, and United Continental. And I think collectively they probably, or at least 80% of the revenue passenger miles that is flowing in the United States. And they have significant international flying too, excluding Southwest.</p>
<p>We like those airlines, but the world has changed for the airlines, and I don’t know how it’s changed, and I hope it corrects itself in a reasonably prompt way. I don’t know whether the Americans will have now changed their habits (or will change their habits) because of an extended period (if it happens) that we’re semi shut down in the economy.</p>
<p>I don’t know whether the trends toward what people have been doing by phone. I mean, it’s been seven weeks since I’ve had a haircut. It’s been more than seven weeks since I put on attire — [for me it’s been] just a question of which sweatsuit I wear. So who knows how we come out of this? I think that there are certain industries, and unfortunately I think the airline industry among others, that are really hurt by a [de facto] shut down, by events that are far beyond their control.</p>
</blockquote>
<p>Also from the <a href="https://www.rev.com/blog/transcripts/warren-buffett-berkshire-hathaway-annual-meeting-transcript-2020">2020 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>And it turned out I was wrong about that business because of something that was not in any way the fault of four excellent CEOs.</p>
<p>I mean, believe me, [it is] no joy being a CEO of an airline. But the companies we bought are well managed, they did a lot of things right. It’s a very, very, very difficult business because you’re dealing with millions of people every day, and if something goes wrong for 1% of them, they are very unhappy. So I don’t envy anybody the job of being CEO of an airline, but I particularly don’t enjoy being it in a period like this, where people have been told basically not to fly.</p>
<p>The airline business, and I may be wrong (and I hope I’m wrong), has changed in a very major way. It’s obviously changed in the fact that there’re four companies are each going to borrow perhaps an average of at least $10 or $12 billion <em>each</em>.</p>
<p>You have to pay that back out of earnings over some period of time. I mean, you’re $10 or $12 billion worse off, if that happens. And, of course, in some cases they’re having to sell stock or sell the right to buy a stock at these prices. And that takes away from the upside down.</p>
<p>And I don’t know whether it’s 2 or 3 years from now that as many people will fly as many passenger miles as they did last year. They may and they may not, but the future is much less clear to me.</p>
</blockquote>
</div>
</details>
<p>Keep your winners:</p>
<details>
<summary>
“[If an] investor followed a policy of purchasing an interest in, say, 20% of the future earnings of a number of outstanding college basketball stars. A handful of these would go on to achieve NBA stardom, and the investor’s take from them would soon dominate his royalty stream. To suggest that this investor should sell off portions of his most successful investments simply because they have come to dominate his portfolio is akin to suggesting that the Bulls trade Michael Jordan because he has become so important to the team”
</summary>
<div>
<p>From Robert G. Hagstrom’s book <a href="https://www.amazon.com/Warren-Buffett-Portfolio-Mastering-Investment/dp/0471392642">The Warren Buffett Portfolio: Mastering the Power of the Focus Investment Strategy</a>:</p>
<blockquote>
<p>But if you own a superior company, the last thing you want to do is to sell it. “When carried out capably, a [low-turnover] investment strategy will often result in its practitioner owning a few securities that will come to represent a very large portion of his portfolio,” explains Buffett.</p>
<p>“This investor would get a similar result if he followed a policy of purchasing an interest in, say, 20% of the future earnings of a number of outstanding college basketball stars. A handful of these would go on to achieve NBA stardom, and the investor’s take from them would soon dominate his royalty stream. To suggest that this investor should sell off portions of his most successful investments simply because they have come to dominate his portfolio is akin to suggesting that the Bulls trade Michael Jordan because he has become so important to the team.”</p>
</blockquote>
<p>The original source is Outstanding Investor Digest (August 8, 1997, page 15).</p>
</div>
</details>
<details>
<summary>
“You <em>may</em> sell if you believe the valuations between different kinds of markets are somewhat out of whack. We have done a little trimming last year in that manner. But that could well be a mistake. The real thing to do with a <em>great</em> business is just hang on for dear life”
</summary>
<div>
<p>From the <a href="https://www.youtube.com/watch?v=PTe3UFR8fMA">1998 Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: What criteria do you use to sell stock? I kind of understand how you buy it, but I’m not sure how you sell.</em></p>
<p>Warren: The best thing to do is buy a stock that you don’t ever want to sell. And that’s what we’re trying to do.</p>
<p>That’s true when we buy an entire business. I mean, we bought all of GEICO. We bought all of See’s Candy or The Buffalo News. We’re not buying those to resell. What we’re trying to do is buy a business that we will be happy with if we own it the rest of our lives, and we expect to with those.</p>
<p>It’s the same principle applies to marketable securities. You get extra options with marketable securities. You can add to holdings. We can never own more than a 100% of a business, but if we own 2% of a business and we like it at a given price, we can add and have 4% or 5%. So that’s an advantage.</p>
<p>Sometimes, if we need money to move to another sector, like we did last year, we will trim some holdings, but that doesn’t mean we’re negative on those businesses at all. I mean, we think they’re wonderful businesses or we wouldn’t own them. [But] we would sell if we needed money for other things. The GEICO stock that I bought in 1951, I sold in 1952. And it went on to be worth — before the 1976 problems — 100 or more times what I’d paid. But I didn’t have the money to do something else. So you sell if you need money for something else.</p>
<p>You <em>may</em> sell if you believe the valuations between different kinds of markets are somewhat out of whack. We have done a little trimming last year in that manner. But that could well be a mistake. I mean, the real thing to do with a great business is just hang on for dear life.</p>
<p>Charlie: [For] the sales that do happen, the ideal way is when you found something you like immensely better. Isn’t that obvious that’s the ideal way to sell?</p>
<p>Warren: And incidentally, the ideal purchase is to have something that you already liked be selling at a price where you feel like buying more of it. I mean, we probably should have done more of that in the past in some situations.</p>
<p>But that’s the beauty of marketable securities. If you’re in a wonderful business, you do get a chance, periodically, maybe to double up in it, or something of the sort.</p>
<p>If the stock market were to sell a lot cheaper than it is now, we would probably be buying more of the businesses that we already own. They would certainly be the first ones that we would think about. They’re the businesses we like the best.</p>
</blockquote>
</div>
</details>
<p>Charlie on not selling <a href="https://www.byd.com">BYD</a> despite its recent ‘nosebleed’ prices:</p>
<details>
<summary>
“We admire the company, like its position [and also its management]. We pay huge taxes when we sell something. On balance, we hold in certain of these position when, normally, we wouldn’t buy a new position.” “One of my smartest friends in VC is constantly getting huge clumps of stocks at nosebleed prices. And what he does is he sells about half of them always. That way, whatever happens, he feels smart. I don’t follow that practice but I don’t criticize it either”
</summary>
<div>
<p>From the 2021 <a href="https://en.wikipedia.org/wiki/Daily_Journal_Corporation">Daily Journal Corporation</a>’s Annual General Meeting (<a href="https://www.yahoo.com/entertainment/charlie-munger-speaks-daily-journal-162005167.html">2-hour video stream</a>, <a href="https://junto.investments/daily-journal-2021-transcript/">transcript</a>):</p>
<blockquote>
<p><em>Question: BYD is in the Daily Journal stock portfolio with a very big paper gain. The stock has gained so much this year and last year. The stock appreciated probably way more than intrinsic value. How do you decide to hold on to a stock or sell some?</em></p>
<p>Charlie: Well, that’s a very good question. BYD stock did nothing for the first five years we held it. Last year it quintupled. What happened is that BYD is very well positioned for the transfer of Chinese automobile production from gasoline-driven cars to electricity-driven cars. You can imagine it’s in a wonderful position and that excited the people in China, which has its share of crazy speculators. And so, the stock went way up.</p>
<p>We admire the company and like its position. And we pay huge taxes to a combination of the federal government and the state of California when we sell something. On balance, we hold in certain of these position when, normally, we wouldn’t buy a new position. Practically everybody does that.</p>
<p>One of my smartest friends in venture capital is constantly getting huge clumps of stocks at nosebleed prices. And what he does is he sells about half of them always. That way, whatever happens, he feels smart. I don’t follow that practice but I don’t criticize it either.</p>
</blockquote>
<p>And also:</p>
<blockquote>
<p><em>Question: Do you believe the valuations for electric car manufacturers are in bubble territory? Both Berkshire and Li Lu own BYD Company which you spoke highly of in the past. BYD sells at nearly 200 P/E. This is cheap compared to Tesla currently valued at over 1100 times P/E and 24 times sales. I know Berkshire is a long-term owner and rarely sells securities of high-quality companies it owns in its portfolio simply because it’s overvalued. For example, Coca-Cola in the past. However, is there a price too high that the company’s future profits simply cannot justify? And since we are on the subject of selling potentially overvalued security, could you provide your systems for selling securities?</em></p>
<p>Charlie Munger: Well, I so rarely hold a company like BYD that goes to a nosebleed price that I don’t think I’ve got a system yet. I’m just learning as I go along. I think you can count on the fact that if we really like the company and we like the management — and that is the way we feel about BYD — we’re likely to be a little too loyal. I don’t think we’ll change on that.</p>
</blockquote>
</div>
</details>
<p>Sometimes mistakes are made. The best is to liquidate the position immediately (and, yes, lose money):</p>
<details>
<summary>
“When we sell something, very often it’s going to be our entire stake. We don’t trim positions. If we like a business, we’re going to buy as much of it as we can and keep it as long as we can. When we change our mind, we don’t take half measures or anything of the sort. [And] we were selling them at far lower prices than we paid.”
</summary>
<div>
<p>From the <a href="https://www.youtube.com/watch?v=A5lQbkqlJro">2020 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Warren: We would have bought other airlines, too, incidentally, but those were the four big ones. Those were the ones we could put some money into and we put whatever it was, $7 or $8 billion into it. And we did not take out anything like $7 or $8 billion. And that was my mistake. But it’s always a problem if there are things on the lower levels of probabilities that happen sometimes, and it happened to the airlines. And I’m the one who made the decision.</p>
<p><em>Question: But Warren, just to clarify on his question. He asked, “Did you sell your whole stake in all four of those companies?</em></p>
<p>Oh, yeah. The answer is yes.</p>
<p>When we sell something, very often it’s going to be our entire stake. We don’t trim positions. That’s just not the way we approach it any more than if we buy a 100% of a business, we’re going to sell it down to 90% or 80%. If we like a business, we’re going to buy as much of it as we can and keep it as long as we can.</p>
<p>When we change our mind, we don’t take half measures or anything of the sort. [And] we were selling them at far lower prices than we paid.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“You don’t have to make it back the way you lost it”
</summary>
<div>
<p>From <a href="https://buffett.cnbc.com/video/1995/05/01/morning-session---1995-berkshire-hathaway-annual-meeting.html">1995 Annual Meeting</a>:</p>
<blockquote>
<p>Charlie: I’d like to repeat that [saying] about “not having to get it back the way you lost it”. You know, that’s the reason so many people are ruined by gambling. They get behind, and then they feel they have to get it back the way they lost it. It’s a deep part of the human nature.</p>
<p>And it’s very smart just to lick it by will, and little phrases like that are very useful.</p>
<p>Warren: Yeah, one of the important things in stocks is that the stock does not know that you own it. You have all these feelings about it. You remember what you paid, you know? You remember who told you about it. All these little things, you know?</p>
<p>[But the stock] doesn’t give a damn! It just sits there.</p>
<p>[If] a stock [is] at 50, [and] somebody’s paid a 100, they feel terrible. Somebody else paid 10, they feel wonderful. All these feelings, and it has no impact whatsoever.</p>
<p>So, as Charlie says, gambling is the classic example. Someone builds a business over years — <em>that</em> they know how to do. And then they go out some place, and get into a mathematically disadvantageous game. [They] start losing it, and they think they’ve got to make it back, not only the way they lost it, but that night. And it’s a great mistake.</p>
</blockquote>
</div>
</details>
<p>Their stance on selling is famously different for their wholly-owned businesses (<em>vs</em>. publicly-traded stocks). Their relationships with fully-owned businesses are more akin to marriages:</p>
<details>
<summary>
“The unwillingness to sell [our wholly-owned] businesses goes back a long way. We don’t see a reason to go around ending friendships we have with people, or contact, or relationships. It just doesn’t make any sense to us. We don’t want to get committed to that sort of activity”
</summary>
<div>
<p>From the <a href="https://www.youtube.com/watch?v=DQPDahlPi6A">1995 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Warren: The unwillingness to sell businesses, like I say, goes back a long way. If that hurts performance, it’s peanuts. That’s simply a fact — a function of the attitude Charlie and I have. We find it a rarity when we find people in the business that we want to associate with. When we do find that, we enjoy it.</p>
<p>We don’t see any reason to make an extra half a percent a year or 1% a year — don’t try us on higher numbers! We don’t see a reason to go around ending friendships we have with people, or contact, or relationships. It just doesn’t make any sense to us. We don’t want to get committed to that sort of activity.</p>
<p>We know we wouldn’t do it if we were a private company. Now, in Berkshire [a public company], we feel we’ve enunciated that position. We want to get that across to everybody who might join with us, because we don’t want them to expect us to do it.</p>
<p>We want them to expect us to work hard to get a decent result, and to make sure that the shareholders get the same result we get, and all of that sort of thing. But we don’t want to enter into any implicit contract with our fellow shareholders that will cause us to have to behave in a way that we really don’t want to behave.</p>
<p>If that’s the price of making more money, it’s a price we don’t want to pay. There’s other things we forgo also, but that is the one that people might disagree with us on. So, we want to be very sure that everybody understands that, going in. That’s part of what you buy here.</p>
<p>I don’t think it’ll hurt performance that much anyway. But to the extent that it does, it’s a limitation you get with us.</p>
<p>Charlie: I don’t think there’s any way to measure it, exactly. But my guess is that, if you could appraise something you might call the character of the people that are running the operating businesses in Berkshire, many of whom helped create the businesses in the first place, and are leading citizens in their community, like the <a href="https://en.wikipedia.org/wiki/Helzberg_Diamonds">Helzbergs</a>.</p>
<p>I don’t think there’s any other corporation in America that has done as well as we have, if you measure the human quality of the people who are in it.</p>
<p>Now, you can say we’ve collected high-grade people because we sure as hell couldn’t create them. But one way or another, this is a remarkable system. And why would we tinker with it?</p>
<p>Warren: If you want to attract high-grade people, you probably ought to try and behave pretty well yourself.</p>
<p>Besides, it wouldn’t be any fun doing the other. I was in that position, a little bit, when I ran the partnership back in the ’60s. And I really — you know, people were coming into partnership with me. And my job was to turn out the best return that we could. And I found that if I got into a business, that presented certain alternatives that I didn’t like. So, Berkshire’s much more satisfactory in that respect.</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="what-you-track-is-the-gap-between-your-estimate-of-intrinsic-value-and-price">What you track is the gap between your estimate of intrinsic value and price</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<p>One should worry less about their cost basis. Instead, one should be vigilant (and up-to-date) with their own estimates of intrinsic value:</p>
<details>
<summary>
“When Charlie and I ran funds, we didn’t worry about whether [the price of] something was up or down. We worried about what it was worth compared to what it was selling for”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2009/05/02/morning-session---2009--berkshire-hathaway-annual-meeting.html">2009 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: If either of you were starting a smaller investment fund today, let’s say a $26 million fund hypothetically. With this smaller asset base, what would you do differently, both in terms of the number of positions and frequency of turnover?</em></p>
<p><em>For example, if you owned a portfolio of 10 stocks and five of them doubled in a short time period, would it make sense to actively manage the portfolio and take profits in the five that had doubled and redeploy the proceeds into your positions, into the ones that had not moved higher, where, presumably, more upside exists and the odds are more dramatically stacked in your favor? Or would you favor the strategy of sitting on your hands in the name of long-term investing?</em></p>
<p>Warren: We would own the half of dozen or so stocks we like best. And it wouldn’t have anything to do with what our cost on them was. It would only have to do with our evaluation of their price versus value. It doesn’t make any difference what the cost is.</p>
<p>And incidentally, if they went down 50%, we would say the same thing. Using your illustration, I don’t know whether that fund has actually had something that went up or went down.</p>
<p>So, we would — our cost basis, except in rare cases — and we actually have a situation like this at Berkshire now, which I may explain a little later. But the cost basis doesn’t have anything to do the fund.</p>
</blockquote>
<blockquote>
<p>When Charlie and I ran funds, we didn’t worry about whether [the price of] something was up or down. We worried about what it was worth compared to what it was selling for.</p>
<p>And we tried to have most of our money in a relatively few — very few — positions which we thought we knew very well. We do the same thing now. We’d do the same thing a hundred years from now.</p>
<p>Charlie: Yes, [Warren is] tactfully suggesting that you [the questioner] adopt a different way of thinking.</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="pay-special-attention-to-the-competitive-dynamics-in-an-industry">Pay special attention to the competitive dynamics in an industry</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<details>
<summary>
“If we own stock in a company and there are eight other companies in the same industry, I want to [read] the reports for the other eight, because I can’t understand how my company is doing unless I understand what the other eight are doing”
</summary>
<div>
<p>From the <a href="https://www.youtube.com/watch?v=Ky_876v9GsU">1996 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Warren: I like to know as much as I can about the person that’s running it, and how they think about the business, and what’s really going on in the business.</p>
<p>In other words, I would like to have a report that would be identical to, if I owned half of a company but was away for a year — and I had a partner who owned the other half — when I came back, that he would tell me about what had taken place during the past year and what he foresaw coming up, and all of that. That is what I think the purpose of the [annual] report is.</p>
<p>Now, the SEC mandates a lot of information, and some of that is helpful. But there’s an <em>intent</em> behind the report. If it’s a sales document, I’m less interested. I don’t see any way to mandate what I’m talking about. But that’s the kind of report I’m looking for.</p>
<p>What I’m trying to do as I read reports? I like to understand just generally what’s going on in all kinds of businesses. If we own stock in a company in an industry, and there are eight other companies that are in the same industry, I want to own or be on the mailing list for the reports for the other eight, because I can’t understand how my company is doing unless I understand what the other eight are doing.</p>
<p>I want to have the perspective of, in terms of market share, what’s going on in the business. Or their margins, or the trend of margins, all kinds of things that I can’t get unless I [read the reports]. I can’t be an intelligent owner of a business unless I know what all the other businesses in that industry are doing. So, I try to get that information out of a report.</p>
<p>If I’m thinking about investing in a specific company, I try to size up their business and the people that are running it. And over the years, I have found reading a lot of reports to be quite useful in terms of making business decisions at Berkshire.</p>
<p>If we own all of a business, I want to own shares in all of the competitors just to keep track of what’s going on. And I want to be able to intelligently evaluate how our managers are doing that. And I can’t do that unless I know the industry backdrop against which they’re working.</p>
<p>It’s amazing, you know, how well you can do in investing, really, with what I would call outside information. I’m not sure how useful [inside information] is. But outside information — there’s all kinds of information around, as to businesses. And you don’t have to understand all of them. You just have to understand the ones that you’re thinking about getting in. And you can do it, if you just — nobody will do it for you.</p>
<p>In my view, you can’t read Wall Street reports and get anything out of them. You have to do it yourself and get your arms around it. I don’t think we’ve ever gotten an idea, you know, in 40 years from a Wall Street report. But we’ve gotten a lot of ideas from annual reports.</p>
<p>Charlie: What I find is that it takes a long time to read the annual report even if it’s a comparatively simple business, because if you really are trying to understand it, it’s not a bit easy.</p>
<p>Warren: Yeah. I would say that, on average, in a business we’re really interested in, even though we know what to skip, to some extent, and what to read, I mean, it’s going to be 45 minutes or an hour on a report.</p>
<p>And if there are six or eight companies in the industry, that’s going to be six or eight hours, perhaps. And then their quarterlies and a lot of other [stuff].</p>
<p>The way you learn about businesses is by absorbing information about them, thinking, deciding what counts, and what doesn’t count, relating one thing to another. And, you know, that’s the job.</p>
<p>And you can’t get that by looking at a bunch of little numbers on a chart bobbing up and down. Or reading market commentary and periodicals or anything of the sort. That just won’t do it. You’ve got to understand the businesses. That’s where it all begins and ends.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“The majority of the securities Charlie and I bought, we’ve never met the management and never talked to them, but we have primarily worked off financial statements, our general understanding of business, and some specific understanding of the industry in the business we’re buying”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2008/05/03/morning-session---2008-berkshire-hathaway-annual-meeting.html">2008 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>We’ve bought lots and lots of securities. The majority of the securities Charlie and I bought, we’ve never met the management and never talked to them, but we have primarily worked off financial statements, our general understanding of business, and some specific understanding of the industry in the business we’re buying.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“I did a lot of work in the earlier years just getting familiar with businesses. I would go out and talk to customers, suppliers, and maybe ex-employees in some cases. Everybody. I would ask every CEO, ‘If you could only buy stock in one company [in your industry] that was not your own, which one would it be and why?’ I have done that in the past on the business I felt I could understand, so I don’t have to do that anymore. You do get a database in your head”
</summary>
<div>
<p>From a <a href="https://archive.is/h7H8C#selection-3110.0-3123.407">1998 lecture at the University of Florida Business School</a>:</p>
<blockquote>
<p>Everybody has got a different circle of competence. The important thing is not how big the circle is, the important thing is the size of the circle; the important thing is staying inside the circle. And if that circle only has 30 companies in it out of 1000s on the big board, as long as you know which 30 they are, you will be OK. You should know those businesses well enough so you don’t need to read lots of work.</p>
<p>Now, I did a lot of work in the earlier years just getting familiar with businesses, and the way I would do that is use what <a href="https://en.wikipedia.org/wiki/Philip_Arthur_Fisher">Phil Fisher</a> would call, the “Scuttlebutt Approach.” I would go out and talk to customers, suppliers, and maybe ex-employees in some cases. Everybody. Everytime I was interested in an industry, say it was coal, I would go around and see every coal company. I would ask every CEO, “If you could only buy stock in one coal company that was not your own, which one would it be and why?” You piece those things together, you learn about the business after awhile.</p>
<p>Funny, you get very similar answers as long as you ask about competitors. “If you had a silver bullet and you could put it through the head of one competitor, which competitor and why?” You will find who the best guy is in the industry. So there are a lot of things you can learn about a business. I have done that in the past on the business I felt I could understand, so I don’t have to do that anymore.</p>
<p>The nice thing about investing is that you don’t have to learn anything new. You can do it if you want to, but if you learn Wrigley’s chewing gum 40 years ago, you still understand Wrigley’s chewing gum. There are not a lot of great insights to get of the sort as you go along. So you do get a database in your head.</p>
<p>I had a guy, Frank Rooney, who ran Melville for many years; his father-in-law died and had owned H.H. Brown, a shoe company. And he put it up with Goldman Sachs. But he was playing golf with a friend of mine here in Florida and he mentioned it to this friend, so my friend said “Why don’t you call Warren?” He called me after the match and in five minutes I basically had a deal.</p>
<p>But I knew Frank, and I knew the business. I sort of knew the basic economics of the shoe business, so I could buy it. Quantitatively, I have to decide what the price is. But, you know, that is either yes or no. I don’t fool a lot around with negotiations. If they name a price that makes sense to me, I buy it. If they don’t, I was happy the day before, so I will be happy the day after without owning it.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“When I made an offer for Clayton Homes, I’d never visited the business. I’d never met the people. I’d done it over the phone. I’d read Jim Clayton’s book. I looked at the 10-Ks. I knew every company in the industry. I look at competitors.” “[Nowadays] we do not find it particularly helpful to talk to managements. The figures tell us more than a management does”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2003/05/03/afternoon-session---2003-berkshire-hathaway-annual-meeting.html">2003 Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: How do you get a few excellent investment ideas to be so successful? Do you read any special newspapers or industry magazines? Or do you visit the headquarters or any subsidiaries of companies?</em></p>
<p><em>And which sources of information, like books (for example, Value Line, Standard and Poor’s, Moody’s), databases (like Reuters, Bloomberg, DataStream), annual reports, internet, and so on, do you use to get the right impression of a company?</em></p>
<p>Warren: The answer is sort of all of the above. I mean, we read a lot. And we read daily publications, we read weekly or monthly periodicals, we read annual reports, we read 10-Ks, we read 10-Qs.</p>
<p>Fortunately, the investment business is a business where knowledge cumulates. I mean, everything you learn when you’re 20 or 30 — you may tweak some as you go along, but it all kind of builds into a knowledge base that’s useful forever.</p>
<p>I read. Charlie used to read, [and he] may still read a fair amount. But do I read a lot of 10-Ks, read a lot of annual reports. 40 or 50 years ago I did a lot of talking to managements. I used to go out and take a trip every now and then and really drop in on maybe 15 or 20 companies. I haven’t done that for a long, long time.</p>
<p>Everything we do, pretty much, I find through public documents. When I made an offer for <a href="https://en.wikipedia.org/wiki/Clayton_Homes">Clayton Homes</a>, I’d never visited the business. I’d never met the people. I’d done it over the phone. I’d read Jim Clayton’s book <a href="https://www.amazon.com/First-Dream-Jim-Clayton/dp/0972638903">‘First a Dream’</a>. I looked at the 10-Ks. I knew every company in the industry. I look at competitors. And I try to understand the business and not have any preconceived notions. And there is adequate information out there to evaluate a great many businesses.</p>
<p>We do not find it particularly helpful to talk to managements. Managements frequently want to come to Omaha and talk to me, and they usually have a variety of reasons that they say they want to talk to me, but what they’re really hoping is we get interested in their stock. That never works.</p>
<p>Managements are not the best reporting parties in most cases. The figures tell us more than a management does. So we do not spend any real amount of time talking to management. When we buy a business, we look at the record to determine what the management’s like, and then we want to size them up, personally, as I said earlier, whether they will keep working. But we don’t give a hoot about anybody’s projections. We don’t even want to hear about them, in terms of what they’re going to do in the future. We’ve never found any value in anything like that.</p>
<p>But just a general business knowledge — what we’ve seen work, what we’ve seen has not worked. There’s a lot you absorb over time.</p>
<p>Charlie: Yeah. The more basic knowledge you have, I think the less new knowledge you have to get. The game is a lot like that fellow that plays chess blindfolded. He’s got a memory of the board and everything that happened before. And that enables him to do the next move in a way he never could if you just showed him the board midgame, cold. In terms of what publications, I don’t know about Warren, [but] I would hate to give up The Wall Street Journal.</p>
<p>Warren: You want to read lots of financial material as it comes along. And actually, The New York Times has a far better business section than they had 25 years ago. But you want to read Fortune, you know. You want to read lots of annual reports. You really want to have a database in your mind, so that you can tell what kind of a business you’re looking at, in general, by looking at the figures.</p>
<p>We never look at any analyst reports. If I read one it was because the funny papers weren’t available, you know. I don’t understand why people do it.</p>
</blockquote>
</div>
</details>
<p>At the same time, you must, of course, think about demand, too:</p>
<details>
<summary>
“If we own stock in a company and there are eight other companies in the same industry, I want to [read] the reports for the other eight, because I can’t understand how my company is doing unless I understand what the other eight are doing”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2005/04/30/morning-session---2005-berkshire-hathaway-annual-meeting.html">2005 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: Can you describe how you made your investment decision to invest in Anheuser-Busch and how you estimated its intrinsic value? How long did it take you to make this decision?</em></p>
<p><em>And is Budweiser inevitable, like Coca-Cola?</em></p>
<p>Warren: […] I observe, just generally, consumer habits. Currently, the beer industry sales are very flat. Wine and spirits have gained in that general category at the expense of beer. So if you look at the industry figures, they’re not going anywhere.</p>
<p>Miller’s has been rejuvenated to some degree. So Anheuser, which has had a string of earnings gains that have been quite substantial over the years and market share gains, is experiencing very flat earnings, having to spend more money to maintain share, in some cases, having promotional pricing. So they are going through a period that is certainly less fun for them than was the case a few years ago.</p>
<p>And it’s a fairly easy-to-understand product and consumer behavior is fairly easy to understand. It’s an exceptionally strong business.</p>
<p>Beer business is not going to grow significantly in the U.S. Worldwide, beer is popular in a great many places, and Anheuser will have a very strong position in it. But I would not expect the earnings to do much for some time, but that’s fine with us.</p>
<p>What we’re looking for is businesses with a durable competitive advantage. I don’t think there’s any question that Anheuser has a very, very strong consumer position. Now, as I said, Miller has been rejuvenated to some degree.</p>
<p>The other thing about it is, of course, in beer you do not see the prevalence of private labels or generic products that you see in a great many consumer products that had strong positions over the years, [but] that are being attacked. That’s a small plus.</p>
<p>But beer consumption per capita is going no place. And there’s nothing that will change that. Interestingly enough, the average person in this climate drinks about 64 ounces of liquid a year. I think it’s roughly 27% of that will be carbonated soft drinks. And, of course, of that Coca-Cola products will be 40-odd percent.</p>
<p>So, of the 64 ounces of liquid that Americans are drinking every day, you can figure something like 11 ounces of that, man, woman, and child, will be a Coca-Cola product.</p>
<p>Beer, as I remember — I could be wrong on this — but I think beer is about 10% of all liquids. So, one out of every 10 ounces that’s consumed by Americans of any kind of liquid is, I believe, is beer.</p>
<p>Coffee, incidentally, despite what you read about the popularity of Starbucks — which is very real, of course, — but coffee has just gone down and down and down over the last 30 or 40 years.</p>
</blockquote>
</div>
</details>
<p>It’s crucial to grasp the “economic forces” that are “likely determine the future of a wide variety of businesses”:</p>
<details>
<summary>
“[Ted and Todd, the investment managers that will succeed him,] have excellent ‘business minds’ that grasp the economic forces likely to determine the future of a wide variety of businesses. They are aided in their thinking by an understanding of what is predictable and what is unknowable”
</summary>
<div>
<p>From his <a href="https://www.berkshirehathaway.com/letters/2011ltr.pdf">2011 Annual Letter</a>:</p>
<blockquote>
<p>Todd Combs built a $1.75 billion portfolio (at cost) last year, and Ted Weschler will soon create one of similar size. Each of them receives 80% of his performance compensation from his own results and 20% from his partner’s. When our quarterly filings report relatively small holdings, these are not likely to be buys I made (though the media often overlook that point) but rather holdings denoting purchases by Todd or Ted.</p>
<p>One additional point about these two new arrivals. Both Ted and Todd will be helpful to the next CEO of Berkshire in making acquisitions. They have excellent “business minds” that grasp the economic forces likely to determine the future of a wide variety of businesses. They are aided in their thinking by an understanding of what is predictable and what is unknowable.</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<h2 id="remember-that-building-your-own-database-is-a-lifelong-process">Remember that building your own ‘database’ is a lifelong process</h2>
<p class="right small"><a href="#" class="muted">↑</a></p>
<details>
<summary>
“You start by learning the basic rules of bookkeeping, which are rules of addition and subtraction; and then you have to spend a lot of time before that accounting gets related to the larger reality, and that’s a lifelong process”
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2003/05/03/morning-session---2003-berkshire-hathaway-annual-meeting.html">2003 Annual Meeting</a>:</p>
<blockquote>
<p><em>Question: I would like to know the accounting book you like best.</em></p>
<p>Warren: Well, it’s been a long time since I’ve read an accounting book. I read Finney back when I was in college, I remember that. And I always liked accounting. And for any of you in business, you know, you basically can’t get enough accounting. But I am not really up to date on accounting books. Maybe Charlie’s been reading some of those lately.</p>
<p>I would hope, actually, that if you read the Berkshire reports over time that you get certain, perhaps, lessons on accounting.</p>
<p>But I think you learn more accounting — once you know the basics of it — by reading good business articles that deal with accounting issues, accounting scandals, that sort of thing.</p>
<p>And — you start with the accounting figures as the raw material of understanding a business, but you have to bring something additional to that.</p>
<p>I can’t think of any good books on that subject. I think I’ve read a lot of good magazine articles that contributed to my knowledge over the years. And I’ve just, you know, I’ve read a lot of annual reports, and seen what people can do with accounting.</p>
<p>As I’ve said before, if I don’t understand it, I figure it’s probably because the management doesn’t want me to understand it. And if the management doesn’t want me to understand it, there’s probably something wrong going on. I mean, people don’t obfuscate with numbers, usually, without a purpose. And when you run into that the best thing to do is you stay away.</p>
</blockquote>
<blockquote>
<p>Charlie: Yeah, asking Warren what good books he knows about accounting is like asking him what good books does he have about breathing.</p>
<p>What the implication of that is, is that you start by learning the basic rules of bookkeeping, which are sort of like the basic rules of addition and subtraction. And then you have to spend a lot of time before that accounting gets related to the larger reality, and that’s a lifelong process.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“[I’ve read] the annual reports of Anheuser-Busch for 25 years. I’ve [also] read the reports of Coca-Cola and Gilette (and all kinds of other companies) long before we invested in them.” “I have been reading [IBM’s] annual report for more than 50 years, but it wasn’t until a Saturday in March last year [2011] that my thinking crystallized”
</summary>
<div>
<p>From the <a href="https://www.youtube.com/watch?v=xa2B9c4FyRI">2005 Annual Meeting</a>:</p>
<blockquote>
<p>You have to look for opportunities to that fit within that framework as you go through life. and you can’t do something every day. You can <em>learn</em> every day, but you can’t <em>act</em> every day.</p>
<p>I talked about reading the annual reports of Anheuser-Busch for 25 years. I’ve [also] read the reports of Coca-Cola and Gilette (and all kinds of other companies) long before we invested in them.</p>
</blockquote>
<p>From his <a href="https://www.berkshirehathaway.com/letters/2011ltr.pdf">2011 Annual Letter</a>:</p>
<blockquote>
<p>As was the case with Coca-Cola in 1988 and the railroads in 2006, I was late to the IBM party. I have been reading the company’s annual report for more than 50 years, but it wasn’t until a Saturday in March last year [2011] that my thinking crystallized.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“I’ve never prepared a spreadsheet”
</summary>
<div>
<p>From <a href="https://www.youtube.com/watch?v=176DGSK3GWc">2007 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>We don’t <em>formally</em> have discount rates. Every time I start talking about all this stuff Charlie reminds me that I’ve never prepared a spreadsheet. [But of course,] in effect, <em>in my mind</em>, I do [use discount rates].</p>
<p>We are going to want to get a significantly higher return obviously in terms of cash produced relative to the amount we’re out laying now for a business than we are from a government bond. That has to be the yardstick at a base.</p>
</blockquote>
</div>
</details>This article is not investment advice. It is for educational purposes only.The Buffett-Munger system as I understand it2021-02-28T00:00:00+00:002021-02-28T00:00:00+00:00https://howshouldithinkabout.com/investing/the-buffett-munger-system<p class="small"><em>This article is not investment advice. It is for educational purposes only.</em></p>
<p class="small center muted">· · ·</p>
<p>Investing is making financial bets on future states of the world. You either make or lose money depending on what you had at stake, and how the state of the world turned out to be <em>vs.</em> what you had predicted.</p>
<p>If investing is betting, investing successfully over the long term <em>requires</em> an edge. Without some sort of advantange on your side, your bets will, almost inevitably, end with ruin.</p>
<p>Strategies for finding and exploiting edges abound. I am most familiar with Warren Buffett’s approach, which I call the Buffett-Munger system — to recognize the fact that Charlie Munger contributed significantly to shape it.</p>
<p class="small center muted">· · ·</p>
<h2 id="an-asset-should-pay-for-itself">An asset should pay for itself</h2>
<p>First and foremost, whenever buying a financial asset, Buffett seeks to make money from cash flows “produced” by the assets themselves — and not from a future sale of those assets at higher-than-initially-bought prices.</p>
<p>Why?</p>
<p>Because Buffett claims no insight about where stock prices are going to be in the near future. Having no insight on future movements of the market, he makes a conscious and deliberate effort to be “independent” from future prices.</p>
<p>This attitude is in stark contrast to how most people approach investing. Most people are trying to “buy low” and — crucially — “sell high”. What Buffett does is different. He also tries to “buy low”, but he wants the asset to pay itself. Selling is, therefore, <em>much</em> less important to his system.</p>
<p>There is a clear benefit in not relying on future prices. It greatly reduces the chances of being in the position of <em>having</em> to sell an asset at a potentially unfavorable price.</p>
<p>The idea seems interesting, but of course risk is not completely gone. Buffett only “shifts” it. He avoids the risk of relying on future buyers to sell at a higher price. But the risk of not earning a return because of underwhelming cash flows is still very much present.</p>
<h2 id="predict-the-future-in-specific-circumstances">Predict the future in <em>specific</em> circumstances</h2>
<p>Buffett is very aware of such a risk, and here is how he tries to minimize it:</p>
<p>For businesses in some <em>specific</em> circumstances, Buffett is extremely confident that he can predict, with a rough precision, the business owner’s earnings 10+ years out in the future</p>
<p>In other words, because Buffett doesn’t know much about where the market is going, he aims to make money <em>independently</em> of future prices of assets he owns. And the way he does that is by buying specific businesses whenever their prices seem attractive <em>vs.</em> their expected ability of generating future cash flows.</p>
<p>In many ways, this is just being conservative and applying common sense. Again, if one wants to make money while not depending on the capricious stock market, then the investment itself must pay you back in form of profits.</p>
<details>
<summary>
It is as if Buffett claimed to know <em>nearly nothing</em> about future market prices, but <em>a lot</em> about earnings of certain businesses decades ahead
</summary>
<div>
<p>From the <a href="https://buffett.cnbc.com/video/2003/05/03/afternoon-session---2003-berkshire-hathaway-annual-meeting.html">2003 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>If somebody gave me all 500 stocks in the S&P and I had to make some prediction about how they would behave relative to the market over the next couple years, I don’t know how I would do.</p>
<p>But maybe I can find one in there where I think I’m 90% in being right. This is an enormous advantage in stocks. You only have to be right on the very, very few things in your lifetime as long as you never make any big mistakes.</p>
</blockquote>
<p>In the same <a href="https://www.youtube.com/watch?v=fCt0sqnLpjQ">2003 Annual Meeting</a>, he also said:</p>
<blockquote>
<p>As a practical matter, there’s just some businesses that possess economic characteristics that make their future prospects far out far more predictable than others. There are all kinds of businesses that you just can’t remotely predict what they’ll earn, and you just have to forget about them.</p>
</blockquote>
</div>
</details>
<h2 id="make-the-future-more-amenable-to-predictions">Make the future more amenable to predictions</h2>
<p>But can he really predict the future cash flows of the kinds of businesses he buys?</p>
<p>The world is ever changing. How on earth can anyone predict decades ahead? And do that <em>consistently</em> — even if for just a few stocks?</p>
<p>Well, if we take Buffett’s astonishing track record as evidence, it does seem he is doing something quite correctly.</p>
<p>It turns out there are several tricks that Buffett employs to make such a daunting task possible. Chiefly, he is very picky about the <em>specific</em> circumstances surrounding the businesses he considers investing in.</p>
<p>What does Buffett have in mind when selecting investment candidates? An asset must pass several filters. The key ones<sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup> are:</p>
<ol>
<li>
<p>A businesses. Buffett strongly prefers to invest in businesses. He buys either a portion (via the stock market) or 100%. He shies away from gold because it is not a cash generating asset (same for other commodities). He finds real estate too hard to find mispriced opportunities and avoid it as well. The exception to business is bonds. Eventually, he takes advantage of large price dislocations in bonds, but most of the time he is out of fixed income market.</p>
</li>
<li>
<p>Businesses that he deeply understands. He invests only in businesses he thinks he is an expert on. He famously says that the subset of businesses that he understands well are within his <em>circle of competence</em>.</p>
</li>
<li>
<p>Businesses with deep and wide moats. Moats are durable competitive advantages. Buffett looks for them because they make forecasting easier. It is a trick to be more confident in extrapolating the present into the future with less chance of being too optimistic. In other words, if a business has a enduring moat, Buffett is more confident this business’ competitive dynamics are going to stay largely the same and earnings over decades will be less affected by competition.</p>
</li>
<li>
<p>Businesses ran by competent and honest people. Buffett is admittedly hands-off, in Berkshire he does capital allocation and that’s pretty much it. So, when buying an entire businesses, Buffett seeks to keep the current management in place. And when investing in the public markets, he says he looks for candid and capable management. In either case, he needs to trust the management because, again, predicting the future is hard enough. Dealing with incompetent or mischievous people would make it nearly impossible. Ideally, Buffett looks for managers who are independently wealthy (and therefore didn’t need to be there) but who deliberately chose to continue running the business because there is nothing else they would rather do (they love it).</p>
</li>
</ol>
<details>
<summary>
“Our problem — which we can’t solve by studying up — is that we have no insights into which participants in the tech field possess a truly durable competitive advantage”
</summary>
<div>
<p>From the <a href="https://www.berkshirehathaway.com/letters/final1999pdf.pdf">1999 Annual Letter</a>:</p>
<blockquote>
<p>As I mentioned earlier, several of the companies in which we have large investments had disappointing business results last year. Nevertheless, we believe these companies have important competitive advantages that will endure over time. This attribute, which makes for good long-term investment results, is one Charlie and I occasionally believe we can identify.</p>
<p>More often, however, we can’t — not at least with a high degree of conviction. This explains, by the way, why we don’t own stocks of tech companies, even though we share the general view that our society will be transformed by their products and services. Our problem — which we can’t solve by studying up — is that we have no insights into which participants in the tech field possess a truly durable competitive advantage.</p>
<p>Our lack of tech insights, we should add, does not distress us. After all, there are a great many business areas in which Charlie and I have no special capital-allocation expertise. For instance, we bring nothing to the table when itcomes to evaluating patents, manufacturing processes or geological prospects. So we simply don’t get into judgments in those fields.</p>
</blockquote>
</div>
</details>
<h2 id="design-structural-advantages">Design structural advantages</h2>
<p>Buffett’s advantages in investing stem not only from being able to predict the future of businesses in specific circumstances.</p>
<p>He thinks deeply about risks and seeks to protect himself from all sorts of adverse situations. For example, we have just learned that because he does not <em>need</em> to sell higher, he effectively avoids to be at the mercy of the market.</p>
<p>Over the years, he has also worked to mitigate several other risks that negatively affect any investor. As a matter of fact, in Berkshire Hathaway, Buffett has devised an ingenious “machine” that provides him an upper hand on several facets of investing.</p>
<p>Here is a set of structural advantages he holds when dealing with the “market”:</p>
<ol>
<li>
<p>Avoid debt, prefer float. To avoid having to deal with obligations (either roll over debt or service it) in unknown market conditions (which could snowball against him), Buffett avoids debt altogether and instead gets “leverage” from insurance float (under the terms that Berkshire itself underwrites, and therefore “controls”)</p>
</li>
<li>
<p>Avoid “timed” offerings. To avoid transactions deliberately timed by executives and investment bankers, Buffett tends to steer clear from new offerrings, and focus instead on buying stocks at open market (where he can choose the moment to act)</p>
</li>
<li>
<p>When purchasing entire businesses, buy only from sellers who <em>want</em> to sell. To avoid getting into auctions (and bidding wars) when buying private companies, Buffett never makes the open move, he instead asks for every seller to state their offering price</p>
</li>
<li>
<p>Positive selection of sellers, stemming from their stellar reputation. To attract the right kind of sellers (and avoid <em>adverse</em> selection), he does not engage in unsolicited transactions. Instead, he has built Berkshire’s reputation as a safe harbor for private businesses (i.e., they are a buyer that “never” sells, that doesn’t touch on debt, etc)</p>
</li>
</ol>
<p>The excerpts below has Buffett explaining in more detail Berkshire’s thoughtful approach to leverage — via insurance float and deferred income taxes:</p>
<details>
<summary>
“Float has some similarities to bank deposits: cash flows in and out daily to insurers, with the total they hold changing very little”
</summary>
<div>
<p>From his <a href="https://berkshirehathaway.com/letters/2020ltr.pdf">2020 Annual Letter</a>:</p>
<blockquote>
<p>Berkshire now enjoys $138 billion of insurance “float” — funds that do not belong to us, but are nevertheless ours to deploy, whether in bonds, stocks or cash equivalents such as U.S. Treasury bills.</p>
<p>Float has some similarities to bank deposits: cash flows in and out daily to insurers, with the total they hold changing very little.</p>
<p>The massive sum held by Berkshire is likely to remain near its present level for many years and, on a cumulative basis, has been <em>costless</em> to us. That happy result, of course, could change — but, over time, I like our odds.</p>
</blockquote>
<p>And from the <a href="https://berkshirehathaway.com/letters/2018ltr.pdf">2018 Annual Letter</a>:</p>
<blockquote>
<p>This collect-now, pay-later model leaves P/C [property and casualty insurance] companies holding large sums — money we call “float” — that will eventually go to others. Meanwhile, insurers get to invest this float for their own benefit. Though individual policies and claims come and go, the amount of float an insurer holds usually remains fairly stable in relation to premium volume. Consequently, as our business grows, so does our float.</p>
<p>We may in time experience a decline in float. If so, the decline will be <em>very</em> gradual — at the outside no more than 3% in any year. The nature of our insurance contracts is such that we can <em>never</em> be subject to immediate or near-term demands for sums that are of significance to our cash resources. That structure is by design and is a key component in the unequaled financial strength of our insurance companies. That strength will <em>never</em> be compromised.</p>
<p>If our premiums exceed the total of our expenses and eventual losses, our insurance operation registers an underwriting profit that adds to the investment income the float produces. When such a profit is earned, we enjoy the use of free money — and, better yet, get <em>paid</em> for holding it.</p>
</blockquote>
</div>
</details>
<details>
<summary>
“We use debt sparingly. Many managers, it should be noted, will disagree with this policy, arguing that significant debt juices the returns for equity owners. And these more venturesome CEOs will be right <em>most</em> of the time. At rare and unpredictable intervals, however, credit vanishes and debt becomes financially fatal. A Russian-roulette equation — usually win, occasionally die — may make financial sense for someone who gets a piece of a company’s upside but does not share in its downside. But that strategy would be madness for Berkshire”
</summary>
<div>
<p>From the <a href="https://berkshirehathaway.com/letters/2018ltr.pdf">2018 Annual Letter</a>:</p>
<blockquote>
<p>We use debt sparingly. Many managers, it should be noted, will disagree with this policy, arguing that significant debt juices the returns for equity owners. And these more venturesome CEOs will be right <em>most</em> of the time.</p>
<p>At rare and unpredictable intervals, however, credit vanishes and debt becomes financially fatal. A Russian-roulette equation — usually win, occasionally die — may make financial sense for someone who gets a piece of a company’s upside but does not share in its downside. But that strategy would be madness for Berkshire. Rational people don’t risk what they have and need for what they don’t have and don’t need.</p>
</blockquote>
<blockquote>
<p>Beyond using debt and equity, Berkshire has benefitted in a major way from two less-common sources of corporate funding. The larger is the float I have described. So far, those funds, though they are recorded as a huge net liability on our balance sheet, have been of more utility to us than an equivalent amount of equity. That’s because they have usually been accompanied by underwriting earnings. In effect, we have been <em>paid</em> in most years for holding and using other people’s money.</p>
<p>As I have often done before, I will emphasize that this happy outcome is far from a sure thing: Mistakes in assessing insurance risks can be huge and can take many years to surface. (Think asbestos.) A major catastrophe that will dwarf hurricanes Katrina and Michael <em>will</em> occur — perhaps tomorrow, perhaps many decades from now. “The Big One” may come from a traditional source, such as a hurricane or earthquake, or it may be a total surprise involving, say, a cyber attack having disastrous consequences beyond anything insurers now contemplate. When such a mega-catastrophe strikes, we will get our share of the losses and they will be big — <em>very</em> big. Unlike many other insurers, however, we will be looking to add business the next day.</p>
<p>The final funding source — which again Berkshire possesses to an unusual degree — is deferred income taxes. These are liabilities that we will eventually pay but that are meanwhile interest-free.</p>
<p>As I indicated earlier, about $14.7 billion of our $50.5 billion of deferred taxes arises from the unrealized gains in our equity holdings. These liabilities are accrued in our financial statements at the current 21% corporate tax rate but will be paid at the rates prevailing when our investments are sold. Between now and then, we in effect have an interest-free “loan” that allows us to have more money working for us in equities than would otherwise be the case.</p>
<p>A further $28.3 billion of deferred tax results from our being able to accelerate the depreciation of assets such as plant and equipment in calculating the tax we must currently pay. The front-ended savings in taxes that we record gradually reverse in future years. We regularly purchase additional assets, however. As long as the present tax law prevails, this source of funding should trend upward.</p>
</blockquote>
<p>(Here is a <a href="https://quickbooks.intuit.com/r/taxes/making-sense-of-deferred-tax-assets-and-liabilities/">short article by QuickBooks</a> with a few examples of deferred taxes.)</p>
</div>
</details>
<p>And here’s <a href="https://berkshirehathaway.com/letters/2014ltr.pdf">how Buffett articulates</a> the value proposition of Berkshire for prospective sellers (connected to items #3 and #4 above):</p>
<blockquote>
<p>Berkshire offers to the business owner who wishes to sell: a permanent home, in which the company’s people and culture will be retained (though, occasionally, management changes will be needed). Beyond that, any business we acquire dramatically increases its financial strength and ability to grow. Its days of dealing with banks and Wall Street analysts are also forever ended.</p>
</blockquote>
<p>Another important consequence of their approach to risk minimization is that Berkshire has been able to move decisively when others simply can’t.</p>
<details>
<summary>
“I guarantee you that people will do some exceptionally stupid things in equity markets sometime in the next 20 years. And then the question is, are we in a position to do something about that when that happens?”
</summary>
<div>
<p>From <a href="https://buffett.cnbc.com/video/2001/04/28/morning-session---2001-berkshire-hathaway-annual-meeting.html">2001 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>We have some significant advantages in buying businesses over time. We would be the preferred purchaser, I think, for a reasonable number of private companies and public companies as well.</p>
<p>Our checks clear. We will always have the money. People know that when we make a deal, it will get done, and it will get done as fast as anybody can do it. It won’t be subject to any kind of second thoughts or financing difficulties. And we bought, as you know, we bought <a href="https://en.wikipedia.org/wiki/Johns_Manville">Johns Manville</a> because the other group had financing difficulties.</p>
<p>People know they will get to run their businesses as they’ve run them before, if they care about that, and a lot of people do. Others don’t.</p>
<p>We have an ownership structure that is probably more stable than any company our size, or anywhere near our size, in the country. And that’s attractive to people.</p>
<p>And we are under no pressure to do anything dumb. You know, if we do things dumb, it’s because we do things dumb. It’s not because anybody’s making us do it.</p>
<p>So those are significant advantages. And the disadvantage, the biggest disadvantage we have is size. I mean, it is harder to double the market value of a $100 billion company than a $1 billion company, using what we have in our arsenal.</p>
<p>Charlie: Yeah. This is not a hog heaven period for Berkshire. The investment game is getting more and more competitive. And I see no sign that that is going to change.</p>
<p>Warren: But people will do stupid things in the future. There’s no question. I mean, I will guarantee you sometime in the next 20 years that people will do some exceptionally stupid things in equity markets.</p>
<p>And then the question is, are we in a position to do something about that when that happens? But we do continue to prefer to buy businesses, though. That’s what we really enjoy.</p>
</blockquote>
</div>
</details>
<p>What is the net result of having less exposure to risks?</p>
<p>Having less “things that could go wrong” means that you have actually increased your chances of having “things that could well.”</p>
<p>But that’s not all with Buffett.</p>
<h2 id="be-disciplined-with-price-and-alternatives">Be disciplined with price and alternatives</h2>
<p>When it comes to actually pulling the trigger and making an investment, Buffett does two <em>additional</em> things very well.</p>
<p>Firstly, Buffett treats every investment as a capital allocation decision that bears opportunity cost.</p>
<p>Before making any investment Buffett always consider all competing alternatives, including doing nothing and simply holding cash until something more certain appears.</p>
<details>
<summary>
“It’s crazy not to compare it to things that you’re already very certain of”
</summary>
<div>
<p>From the <a href="https://www.youtube.com/watch?v=5ioeNrmn4eY">1997 Berkshire Hathaway Annual Meeting</a>:</p>
<blockquote>
<p>Charlie: I would argue that one filter that’s useful in investing is the simple idea of opportunity cost. If you have one opportunity that you already have available in large quantity and you like it better than 98% of the other things you see, you can just screen out the other 98% because you already know something better.</p>
<p>Warren: If somebody shows us a business, the first thing goes through our head is: Would we rather own this business or more Coca-Cola? Would we rather own it than more Gillette? It’s crazy not to compare it to things that you’re [already] very certain of. There are very few businesses that you will find that we’re a certain of the future about as company such as that. And therefore we will want [new] companies where the certainty gets close to that.</p>
<p>Charlie: With this attitude you get a concentrated portfolio, which we don’t mind. This practice of ours, which is so simple, is not widely copied. I do not know why.</p>
</blockquote>
</div>
</details>
<p>In fact, because Berkshire is a diversified conglomerate able to play both in the private and public markets, he has a much broader universe to pick from (<em>vs</em>. most other insurers, <em>vs</em>. other professional money managers, etc).</p>
<p>And secondly and crucially, he is particularly disciplined on the price he buys anything. He seeks to buy only when he is <em>indeed</em> highly confident that the opportunity at hand meets his criteria.</p>
<p>This is yet another act to mitigate the risk of losing money. To reduce his chances of losing money due to predictions that might turn out to be too optimistic, Buffett seeks to buy only when there is a significant difference between his estimate of an asset’s intrinsic value <em>vs.</em> its current selling price.</p>
<p>Whenever he discusses this point, Buffett often refers to <a href="https://en.wikipedia.org/wiki/Benjamin_Graham">Ben Graham</a>’s concept of <a href="https://en.wikipedia.org/wiki/Margin_of_safety_(financial)">margin of safety</a>. He also draws an analogy between the stock market and baseball.</p>
<details>
<summary>
“You only swing when you are really confident things at your sweet spot”
</summary>
<div>
<p>From the <a href="https://www.youtube.com/watch?v=P305CTi8_FQ">2017 documentary Becoming Warren Buffett</a>:</p>
<blockquote>
<p>I was genetically blessed with a certain wiring that’s very useful in a highly developed market system, where there’s lots of chips on the table. I happened to be good at that game.</p>
<p><a href="https://en.wikipedia.org/wiki/Ted_Williams">Ted Williams</a> wrote a book called <a href="https://en.wikipedia.org/wiki/The_Science_of_Hitting">The Science of Hitting</a>. In that, he had a picture of himself at bat, and the strike zone broken into, I think, 77 squares. He said that if he waited for the pitch that was really in the sweet spot, he would bat 0.400. And if he had to swing at something on the lower corner, he would probably bat 0.235.</p>
</blockquote>
<p><img src="/assets/img/ted-williams-medium.jpg" style="width: 350px" class="center-block responsive" /></p>
<blockquote>
<p>In investing, I’m in a “no-called strike” business, which is the best business you can be. I can look at a thousand different companies, and I don’t have to be right on every one of them. Or even 50 of them. So I can pick the ball I want to hit.</p>
<p>The trick in investing is just to sit there and watch pitch after pitch go by — wait for the one right in your sweet spot.</p>
<p>[As for] people yelling “Swing, you bump”, [simply] ignore them. There’s a temptation for people to act far too frequently in stocks simply because they’re so liquid.</p>
</blockquote>
</div>
</details>
<p class="small center muted">· · ·</p>
<p>There you have it — my current understanding of Warren Buffett and Charlie Munger’s remarkable investing system.</p>
<p>Now, read how Buffett and Munger themselves have described their approach to investing over the decades.</p>
<p>Up next: <a href="/investing/buffett-and-munger-on-investing/">The Buffett-Munger system in their own words</a>.</p>
<p class="small center muted">· · ·</p>
<div class="footnotes" role="doc-endnotes">
<ol>
<li id="fn:1" role="doc-endnote">
<p>Clearly, the four filters that I have listed are somewhat intertwined. And all of them, in a way or another, tie back to the question of predictability. For instance, what is a business he “understands”? It is, of course, one that he can confidently predict their owner’s earnings decades ahead! <a href="#fnref:1" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
</ol>
</div>This article is not investment advice. It is for educational purposes only.A guided tour to the Gompertz law of mortality2021-02-03T00:00:00+00:002021-02-03T00:00:00+00:00https://howshouldithinkabout.com/aging/gompertz-law-of-mortality-from-scratch<p>Here is a question to you:</p>
<blockquote>
<p>What are your chances of dying this year?</p>
</blockquote>
<p>This is not an easy question, and you may want to avoid it altogether. But regardless of how we might feel about it, our death is (at the minimum) consequential to us and our loved ones. So let’s be fearless and explore the subject with a cool head.</p>
<p>Fortunately, a sensible answer (mathematically speaking) to our chances of dying this year isn’t that hard to guess.</p>
<p>Choose a sample of people born in the same year, and — as time goes by — take note of their age at death. Some people will unfortunately die during infancy. Most will face death much older. There is an expression, known as the Gompertz law of mortality, that give us a reasonable estimate of how many persons of the sample will die each year. The law is usually stated as</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>B</mi><msup><mi>e</mi><mrow><mi>C</mi><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">h(x) = B e^{C x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8913309999999999em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span></span>
<p>where <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(x)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span></span></span></span> is, loosely speaking, the chance of death at age <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span>, and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span> are constants obtained from experience.</p>
<p>This essay reviews how Benjamin Gompertz discovered this expression. Our tour goes back in history, examines the original papers and data, steps into his shoes, and rederives things from scratch.</p>
<p>Click on the subheaders ▶ below to expand or collapse each section. If you are short on time and would rather know your chances of death right now, scroll down directly to the interactive charts and grab the estimates there.</p>
<details>
<summary>
The first tabulations of human longevity were published in late 1600s in Europe
</summary>
<div>
<p>Mortality has bewildered humankind for aeons, but it was only in the 18th century that mortality data by age was tabulated for the first time.</p>
<p>At the time, the so-called tables of mortality were built to calculate the price of <a href="https://en.wikipedia.org/wiki/Life_annuity">life annuities</a>.</p>
<p>(For those of you that, like me, don’t know what a life annuity is: it is an insurance product in which the purchaser pays in advance for a sequence of future payments while he or she is still alive.)</p>
<p>When a insurance company sells a life annuity to someone, it must come up with an expectation for how long the purchaser will live. Insurers are then <em>very</em> interested in predicting their customers’ lifespans as accurately as possible. And that’s why insurers spurred the creation of mortality tables.</p>
<p>Early mortality tables consisted in births and deaths grouped by age for some European localities during the 1600, 1700, and 1800s. Here is a famous one, published by <a href="https://en.wikipedia.org/wiki/Joshua_Milne">Joshua Milne</a> in 1815:</p>
<p><img src="/assets/img/milne-page_405-small-tinypng.png" style="width: 350px" class="center-block responsive" /></p>
<blockquote>
<p><small>Milne, Joshua. 1815. <i><a href="https://books.google.com/books?id=8afeAAAAMAAJ">A Treatise on the Valuation of Annuities and Assurances on Lives and Survivorships: On the Construction of Tables of Mortality and on the Probabilities and Expectations of Life, Volume 1</a></i>, <a href="https://books.google.com/books?id=8afeAAAAMAAJ&pg=PA405">405</a>. London: Longman, Hurst, Rees, Orme, and Brown.</small></p>
</blockquote>
<p><a href="https://doi.org/10.1002/9780470012505.tae001">Data from several early tables has been nicely summarized by David Forfar</a>. I plot four of them in the chart below.</p>
<p>For each following data points <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo separator="true">,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, y)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mpunct">,</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mclose">)</span></span></span></span>, the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>-value represents the number of survivors aged <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span> in years. This kind of series is also known as survival curve. To make ours easier to compare, each dataset was rescaled (i.e., divided by the number of people in it) such that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">y(0) = 1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord">0</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> for all of them:</p>
<div class="warning-narrow-screen">
<p class="muted small">⚠️ Your screen is almost too narrow for the interactive chart below. Please rotate your
device for a better experience.</p>
</div>
<div class="charts">
<canvas id="canvas-18th-century-survival"></canvas>
</div>
<p class="right"><small>— Download <a href="/assets/data/18th-century-survival.csv">chart data in .csv</a></small></p>
<p>A quick look at the chart reveals a few common features among these series:</p>
<ul>
<li>A sharp decline between <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>5</mn></mrow><annotation encoding="application/x-tex">5</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">5</span></span></span></span> years,</li>
<li>A roughly linear decay from <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">10</span></span></span></span> to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>55</mn></mrow><annotation encoding="application/x-tex">55</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">55</span></span></span></span> y.o., and</li>
<li>A somewhat accelerated decay afterwards</li>
</ul>
<p>But beware of jumping into conclusions too fast!</p>
<p>These series were among the first ever tabulated, but they had serious issues. Their sample size, especially for older ages, was quite small. There were methodological and data collection issues as well.<sup id="fnref:1" role="doc-noteref"><a href="#fn:1" class="footnote" rel="footnote">1</a></sup></p>
<p>So the early data was rough. But, as soon as the first mortality tables were published, work related to them blossomed among European mathematicians. Besides its practical use in life annuities, the prospects of finding mathematical regularities on human mortality was — and still is — too alluring!</p>
</div>
</details>
<details>
<summary>
de Moivre investigated two basic mortality models in 1725
</summary>
<div>
<p>In 1725, the mathematician <a href="https://en.wikipedia.org/wiki/Abraham_de_Moivre">Abraham de Moivre</a> published a book — <a href="https://books.google.com/books?id=ed5bAAAAQAAJ">Annuities Upon Lives</a> — with his incursions into the problem of estimating human lifespans.<sup id="fnref:2" role="doc-noteref"><a href="#fn:2" class="footnote" rel="footnote">2</a></sup></p>
<p>In his book, de Moivre deliberately converted real lives into <em>fictitious</em> ones. By abstracting from real-world data, he was free to investigate models that were within reach of the Mathematics of his day. Fictitious lives are more amenable to simpler Mathematics!</p>
<p>He applied two mortality models to his fictitious populations. They had either:</p>
<ol>
<li>An equal number of deaths per year, or</li>
<li>Equal probabilities of death at each year of age</li>
</ol>
<p>The first hypothesis is known as <a href="https://en.wikipedia.org/wiki/De_Moivre%27s_law">de Moivre’s law</a>. An equal number of deaths per year results in an <a href="https://en.wikipedia.org/wiki/Arithmetic_progression">arithmetic progression</a>, and in survival curves following a line. From the previous chart, this simple model seems to fit parts of the early data. In fact, we have just observed a roughly linear decay (from <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">10</span></span></span></span> to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>55</mn></mrow><annotation encoding="application/x-tex">55</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">55</span></span></span></span> y.o.).</p>
<p>But you should also recall that the early data was inaccurate. So at the end of the day, his first model has not been very fruitful.</p>
<p>de Moivre’s second model did not reflect real human mortality either. We have later discovered though that it does describe several other phenomena found in nature. So it is useful to explore it further and sharpen our intuition about models.</p>
<p>What does “equal probabilities of death at each year of age” actually mean?</p>
<p>Here’s a way of thinking about it. This hypothesis is as if death were completely independent of age. In such a model, both the young and old die at the same rate. It does not matter if you are young (and usually healthy) or old (and typically fralty), your chances of dying in any giving day are always the same.</p>
<p>In such a hypothetical world, how does the human survival data look like?</p>
<p>They decrease at a constant and age-independent rate. Something very similar to <a href="https://en.wikipedia.org/wiki/Exponential_decay#Applications_and_examples">the geometrical progression that happens to atoms in radioactive decay</a>.</p>
<p>To make it visual, let’s consider two example curves of geometrical progressions, with constant rates <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mi>x</mi><mn>30</mn></mfrac></mrow></msup></mrow><annotation encoding="application/x-tex">e^{- \frac {x} {30}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84708em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.84708em;"><span style="top:-3.363em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915428571428572em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">30</span></span></span></span><span style="top:-3.2255000000000003em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><msup><mi>e</mi><mrow><mo>−</mo><mfrac><mi>x</mi><mn>50</mn></mfrac></mrow></msup></mrow><annotation encoding="application/x-tex">e^{- \frac {x} {50}}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.84708em;vertical-align:0em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.84708em;"><span style="top:-3.363em;margin-right:0.05em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mtight"><span class="mopen nulldelimiter sizing reset-size3 size6"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.6915428571428572em;"><span style="top:-2.656em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mtight">50</span></span></span></span><span style="top:-3.2255000000000003em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line mtight" style="border-bottom-width:0.049em;"></span></span><span style="top:-3.384em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.344em;"><span></span></span></span></span></span><span class="mclose nulldelimiter sizing reset-size3 size6"></span></span></span></span></span></span></span></span></span></span></span></span></span>, and plot them over the data points that we have seen before:</p>
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<p>What to make out of this?</p>
<p>Well, de Moivre did invite Mathematics to the party. But it took a century before the data a bit more acceptable and could be used as a basis for better models.</p>
</div>
</details>
<details>
<summary>
Gompertz advanced foundational ideas in the 1820s
</summary>
<div>
<p>One hundred years after de Moivre’s work, <a href="https://en.wikipedia.org/wiki/Benjamin_Gompertz">Benjamin Gompertz</a> made the founding contributions to the quest of a formula for mortality.</p>
<p>Gompertz was a British self-educated mathematician that worked for an insurance company. He published seminal work about a “law of human mortality”. <a href="https://royalsocietypublishing.org/doi/10.1098/rstl.1825.0026">The most famous of his papers came out in 1825</a>. Because of those contributions, he is generally recognized as one of the pioneers of Actuarial science.<sup id="fnref:3" role="doc-noteref"><a href="#fn:3" class="footnote" rel="footnote">3</a></sup></p>
<p>His papers touch on different aspects of pricing life annuities. In a way, Gompertz intended to make his job easier. Because calculations were labor intensive at the time, he tried to find general formulae that could fit the data from early mortality tables well enough.</p>
<p>Did he succeed? Not really. Most likely because the data at hand was still of low quality in the 1820s.</p>
<p>Nonetheless, Gompertz ended up making two foundational contributions to the quest of modeling mortality mathematically:</p>
<ul>
<li>a conceptual framework for the causes of death, and</li>
<li>a conjecture for how to account for the effects of aging in mortality</li>
</ul>
<p>For the framework, here’s what he wrote in 1825:</p>
<blockquote>
<p>It is possible that death may be the consequence of two generally co-existing causes; the one, chance, without previous disposition to death or deterioration; the other, a deterioration, or an increased inability to withstand destruction.</p>
</blockquote>
<p>He didn’t elucidate what exactly he meant by “chance”, but he did put foward a working model for “deterioration”:</p>
<blockquote>
<p>If mankind be continually gaining seeds of indisposition, or in other words, an increased liability to death […] it would follow that the number of living out of a given number of persons at a given age, at equal successive increments of age, would decrease in a greater ratio than the geometrical progression.</p>
</blockquote>
<p>In other words, deterioration would increase the susceptibility of humans to death and because of that, survival curves should decline faster at older ages. That rings as true to our present-day intuition — the old does seem to die more frequently than the young.</p>
<p>Gompertz went further and proposed a quantitative model to account for his conjecture of an increasing rate of death as we grow older:</p>
<blockquote>
<p>If the average exhaustions of a man’s power to avoid death were such that at the end of equal infinitely small intervals of time, he lost equal portions of his remaining power to oppose destruction which he had at the commencement of those intervals, then at the age <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span> his power to avoid death, or the intensity of his mortality might be denoted by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">Be^{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span>, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> being a constant.</p>
</blockquote>
<p>It is fair to say that this is the paragraph that made Gompertz famous. So it surely bears closer examination!</p>
<p>In Gompertz’s model of deterioration, humans hold two opposing characteristics: the “power to avoid death” and its inverse, the “intensity of mortality.” That is, as one decreases, the other increases. Moreover, the “power to avoid death” would decay in “average exhaustions” over “infinitely small intervals of time.”</p>
<p>The way he mentions these terms suggests that Gompertz intended to cast his concept of “power to avoid death” into what is now known as a <a href="https://en.wikipedia.org/wiki/Random_variable#Continuous_random_variable">continuous random variable</a>. <a href="https://en.wikipedia.org/wiki/Probability_theory">Probability theory</a> was yet to be fully formalized at his time, but he surely was familiar with <a href="https://en.wikipedia.org/wiki/Calculus">Calculus</a> and its uses to study rates of change. In fact, in his 1825 paper, Gompertz made extensive use of <a href="https://en.wikipedia.org/wiki/Fluxion">fluxions</a> — <a href="https://en.wikipedia.org/wiki/Isaac_Newton">Isaac Newton</a>’s notation for the time derivative — to analyze the rate of change of survival curves.</p>
<p>Gompertz’s paragraph then ends with the punchline. He (seemingly out of nothing!) conjectured that intensity of mortality could be expressed by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">Be^{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span>. Later in this essay, we will see that further progress showed that taking <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">Be^{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span> for the intensity of mortality works remarkably well. But where did Gompertz got this nice and tidy closed-form expression from?</p>
<p>Because he offered no explanation, it’s up to us to step into his shoes (and mind), fill in the gaps and rederive it. Let’s do it now!</p>
</div>
</details>
<details>
<summary>
Recreating the steps that Gompertz might have followed
</summary>
<div>
<p>Gompertz was into something with his analyses of rates of change of survival curves. But, as I have just remarked, there were gaps in his exposition. He did not provide a clear definition for what he meant by “intensity of mortality”. Neither did he justified why he picked <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">Be^{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span> for its analytic form.</p>
<p>If I were to fill the gaps and recreate step by step what Gompertz might have thought, I would reason as follows.</p>
<p>(I will think out aloud step by step. Anyone with familiarity of basic Calculus should follow easily, but bear in mind that I am no mathematician, so I won’t be 100% precise.)</p>
<p>First, let <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span> be the number of persons alive at age <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span> (similarly to the plots of survival curves that we have visited before).</p>
<p>To investigate the rate of change of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>, we start by computing the absolute decrease of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span> over the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">10</span></span></span></span>-year intervals:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi><mo>=</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mn>10</mn><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Delta y = y(k \mathclose + 10) - y(k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8777699999999999em;vertical-align:-0.19444em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">10</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span></span>
<p>To get a sense of the rate of change on a “more natural” timescale than <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">10</span></span></span></span>-year intervals, we then compute the average <em>annual</em> decrease over <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">10</span></span></span></span>-year intervals:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>k</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mn>10</mn><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mn>10</mn><mo stretchy="false">)</mo><mo>−</mo><mi>k</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mn>10</mn><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mn>10</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac {\Delta y} {\Delta k} = \frac {y(k \mathclose + 10) - y(k)} {(k \mathclose + 10) - k} = \frac {y(k \mathclose + 10) - y(k)} {10}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0463299999999998em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603299999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">10</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">10</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">10</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">10</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>
<p>Having realized that <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">10</span></span></span></span> years is too coarse of an interval (there certainly are tons of “action” happening in-between), we decide to compute the average over intervals of, say, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> year:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>k</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>k</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mn>1</mn></mfrac></mrow><annotation encoding="application/x-tex">\frac {\Delta y} {\Delta k} = \frac {y(k \mathclose + 1) - y(k)} {(k \mathclose + 1) - k} = \frac {y(k \mathclose + 1) - y(k)} {1}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0463299999999998em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603299999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>
<p>We then recognize that it would be even better if we could generalize this average ratio from <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1</span></span></span></span> year to any <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi mathvariant="normal">Δ</mi><mi>k</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\Delta k > 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.73354em;vertical-align:-0.0391em;"></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>. So we write:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>k</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mi mathvariant="normal">Δ</mi><mi>k</mi><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mi mathvariant="normal">Δ</mi><mi>k</mi><mo stretchy="false">)</mo><mo>−</mo><mi>k</mi></mrow></mfrac><mo>=</mo><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mi mathvariant="normal">Δ</mi><mi>k</mi><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>k</mi></mrow></mfrac></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>eq. 1</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\tag*{eq. 1} \frac {\Delta y} {\Delta k} = \frac {y(k \mathclose + \Delta k) - y(k)} {(k \mathclose + \Delta k) - k} = \frac {y(k \mathclose + \Delta k) - y(k)} {\Delta k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.0463299999999998em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603299999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="tag"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord text"><span class="mord">eq. 1</span></span></span></span></span></span>
<p>(To make it easier to refer to it later on, let’s call this expression rate of mortality.)</p>
<p>We now pause and think for a few moments about what rate of mortality actually means in plain English. (Do pause and think for yourself!)</p>
<p>We arrive at the following: it represents “the amount of deaths per unit of time.” Does it sound like something useful?</p>
<p>It surely does, but then we realize that this metric is not that great for studying human mortality over time. It still has a serious flaw.</p>
<p>As less and less people survive to older ages, we might be misled to think that there are less deaths per unit of time and therefore things are “improving” — while in fact the opposite is true. If there are only <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">10</span></span></span></span> persons alive and all <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">10</span></span></span></span> die in a year, things seem <em>relatively</em> much worse than if <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn></mrow><annotation encoding="application/x-tex">10</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">10</span></span></span></span> out of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>1000</mn></mrow><annotation encoding="application/x-tex">1000</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">1000</span></span></span></span> persons alive die in a year.</p>
<p>So we reflect a bit more and realize that “the amount of deaths per unit of time considering only the ones that are still alive at the beginning of each unit of time” is a good fix for the flaw we have identified.</p>
<p>We then try to stress test this new expression. Could this metric have gone through Gompertz’s mind back in the day?</p>
<p>If we were Gompertz, working at a insurance company, we would be concerned about life annuities. But, for the point of view of a insurer, what are annuities if not: “Given the information that I have about a purchaser (like their age), how much longer should we expect them to live and thus demand to be paid?”</p>
<p>Looking at the problem of mortality through this metric sounds like a win-win. Not only it would fix the flaw we have identified, but it would fit greatly to the problem of life annuities because it “contains” the important information of how long the purchaser have already lived.</p>
<p>So we wonder about how to translate this new quantity from words to Mathematics. Shortly after, we notice that a division by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span> seems to expresses well the conditioning of “considering only the ones that are still alive at the beginning of each unit of time”.</p>
<p>So, while there are persons alive (that is, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>≠</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">y \not = 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel"><span class="mord vbox"><span class="thinbox"><span class="rlap"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="inner"><span class="mord"><span class="mrel"></span></span></span><span class="fix"></span></span></span></span></span></span><span class="base"><span class="strut" style="height:0.36687em;vertical-align:0em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>) we have:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mn>1</mn><mi>y</mi></mfrac><mo>⋅</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>k</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mi mathvariant="normal">Δ</mi><mi>k</mi><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>k</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac {1} {y} \cdot \frac {\Delta y} {\Delta k} = \frac {1} {y(k)} \cdot \frac {y(k \mathclose + \Delta k) - y(k)} {\Delta k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.20188em;vertical-align:-0.8804400000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.0463299999999998em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603299999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>
<p>Now, because by definition <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mi mathvariant="normal">Δ</mi><mi>k</mi><mo stretchy="false">)</mo><mo>≤</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">{y(k \mathclose + \Delta k) \leq y(k)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≤</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span></span> for any survival curve, we suddenly become aware of the fact that (for any <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.69444em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span></span>) the above expression assumes only non-positive values.</p>
<p>So we decide (for convenience) to multiply it by <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span></span></span></span> so that it only assumes either positive values or zero (which are more intuitive for most uses):</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo stretchy="false">(</mo><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⋅</mo><mfrac><mn>1</mn><mi>y</mi></mfrac><mo>⋅</mo><mfrac><mrow><mi mathvariant="normal">Δ</mi><mi>y</mi></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>k</mi></mrow></mfrac><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mi mathvariant="normal">Δ</mi><mi>k</mi><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>k</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">(-1) \cdot \frac {1} {y} \cdot \frac {\Delta y} {\Delta k} = - \frac {1} {y(k)} \cdot \frac {y(k \mathclose + \Delta k) - y(k)} {\Delta k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mopen">(</span><span class="mord">−</span><span class="mord">1</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.20188em;vertical-align:-0.8804400000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.0463299999999998em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3603299999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>
<p>Having reached a seemingly useful expression, we medidate on a name for it, and settle on <em>discrete</em> hazard rate <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span>:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mi mathvariant="normal">Δ</mi><mi>k</mi><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>k</mi></mrow></mfrac></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>eq. 2</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\tag*{eq. 2} h(k) = - \frac {1} {y(k)} \cdot \frac {y(k \mathclose + \Delta k) - y(k)} {\Delta k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="tag"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord text"><span class="mord">eq. 2</span></span></span></span></span></span>
<p>Delighted by this result we decide to go even further.</p>
<p>We have already noted that Gompertz mentioned “infinitely small intervals of time” in his conjecture. So we leave the discrete and jump into the realm of limits and differentials, Calculus after all.</p>
<p>Thus, in present-day notation:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><munder><mrow><mi>lim</mi><mo></mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi><mo>→</mo><mn>0</mn></mrow></munder><mrow><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">(</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo>+</mo><mi mathvariant="normal">Δ</mi><mi>t</mi><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>t</mi></mrow></mfrac><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">)</mo></mrow></mrow><annotation encoding="application/x-tex">h(t) = \lim_{\Delta t\to 0} {\Bigg( - \frac {1} {y(t)} \cdot \frac {y(t + \Delta t) - y(t)} {\Delta t} \Bigg)}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="mop op-limits"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:0.69444em;"><span style="top:-2.355669em;margin-left:0em;"><span class="pstrut" style="height:3em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">Δ</span><span class="mord mathnormal mtight">t</span><span class="mrel mtight">→</span><span class="mord mtight">0</span></span></span></span><span style="top:-3em;"><span class="pstrut" style="height:3em;"></span><span><span class="mop">lim</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.744331em;"><span></span></span></span></span></span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="delimsizing size4">(</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">Δ</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="delimsizing size4">)</span></span></span></span></span></span></span>
<p>Or, to put it more succinctly,</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mi>y</mi></mfrac><mo>⋅</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>eq. 3</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\tag*{eq. 3} h(t) = - \frac {1} {y} \cdot \frac {dy} {dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.20188em;vertical-align:-0.8804400000000001em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.0574399999999997em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="tag"><span class="strut" style="height:2.25188em;vertical-align:-0.8804400000000001em;"></span><span class="mord text"><span class="mord">eq. 3</span></span></span></span></span></span>
<p>And call it <em>continuous</em> hazard rate <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>.</p>
</div>
</details>
<details>
<summary>
Exploring if our hazard rate is the same as Gompertz’s intensity of mortality
</summary>
<div>
<p>I don’t know about you, but I am quite happy with the tidy results that we have just achieved.</p>
<p>There is still a remaining question, though. How can we tell if the hazard rate <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> that we have found is the same concept as Gompertz’s “intensity of mortality”?</p>
<p>A way to investigate this is to calculate the hazard rate from the data points available to Gompertz, and then plot them to check if they resemble the exponential <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">Be^{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span> of his 1825 paper. If they do, that would be good evidence that he had our hazard rate in mind, and not something else.</p>
<p>So, from <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mi>q</mi><mi mathvariant="normal">.</mi><mtext> </mtext><mn>2</mn></mrow><annotation encoding="application/x-tex">eq. \medspace 2</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mord">.</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">2</span></span></span></span> above, the discrete hazard rate <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(k)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span> is</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></mfrac><mo>⋅</mo><mfrac><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo lspace="0em" rspace="0em">+</mo><mi mathvariant="normal">Δ</mi><mi>k</mi><mo stretchy="false">)</mo><mo>−</mo><mi>y</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><mrow><mi mathvariant="normal">Δ</mi><mi>k</mi></mrow></mfrac></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>eq. 2</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\tag*{eq. 2} h(k) = - \frac {1} {y(k)} \cdot \frac {y(k \mathclose + \Delta k) - y(k)} {\Delta k}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25744em;vertical-align:-0.936em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.936em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.113em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose"><span class="mord">+</span></span><span class="mord">Δ</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal" style="margin-right:0.03148em;">k</span><span class="mclose">)</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="tag"><span class="strut" style="height:2.363em;vertical-align:-0.936em;"></span><span class="mord text"><span class="mord">eq. 2</span></span></span></span></span></span>
<p>Using it to calculate and plot the hazard rate found in early mortality tables:</p>
<div class="warning-narrow-screen">
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<div class="charts">
<canvas id="canvas-18th-century-hazard"></canvas>
</div>
<p>So? What do you think? Does it resemble an exponential?</p>
<p>Personally, I cannot fit a exponential with my eyes only.</p>
<p>To make our eyeballing a bit easier, let’s take advantage of the fact that any exponential curve becomes a straight line on logarithmic scale. Let’s switch the <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi></mrow><annotation encoding="application/x-tex">y</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span>-axis to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>l</mi><mi>o</mi><mi>g</mi></mrow><annotation encoding="application/x-tex">log</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.01968em;">l</span><span class="mord mathnormal">o</span><span class="mord mathnormal" style="margin-right:0.03588em;">g</span></span></span></span> scale. And now that we there, let’s also overlay an exponential function on top of the data points:</p>
<div class="warning-narrow-screen">
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<div class="charts">
<canvas id="canvas-18th-century-log-hazard"></canvas>
</div>
<p class="right"><small>— Download <a href="/assets/data/18th-century-log-hazard.csv">chart data in .csv</a></small></p>
<p>Now we can see it! From <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn></mrow><annotation encoding="application/x-tex">25</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">25</span></span></span></span> to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>75</mn></mrow><annotation encoding="application/x-tex">75</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">75</span></span></span></span> years it is visually plausible that the hazard rate from early mortality tables resembles an exponential, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">Be^{x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span>.</p>
<p>Of course this is still eyeballing (feel free to download the data and work out the details yourself), but it is now more believable that we have indeed recreated what Gompertz had in mind with intensity of mortality.</p>
<p>Incidentally, the name intensity of mortality did not stick. In fact, today that same function has multiple names depending on the field you come from. In Statistics, it is <a href="https://en.wikipedia.org/wiki/Survival_analysis#Hazard_function_and_cumulative_hazard_function">hazard rate</a> (or hazard function). In Actuarial science, it is known as <a href="https://en.wikipedia.org/wiki/Force_of_mortality">force of mortality</a>. In Engineering, it is called <a href="https://en.wikipedia.org/wiki/Failure_rate#Failure_Rate_in_the_Discrete_Sense">failture rate</a>. If anything, so many synonyms attest to the wide applicability of the concept.</p>
</div>
</details>
<details>
<summary>
Pioneers wrestled with data issues for decades throughout the 19th century
</summary>
<div>
<p>The pioneers continued to wrestle with mortality data for several decades over the 19th century. The inadequacy and poor quality of datasets led not only to dead ends in analyses (<a href="https://en.wikipedia.org/wiki/Garbage_in,_garbage_out">garbage in, garbage out</a>), but also to significant financial losses to insurers.</p>
<p>Case in point, the losses the British government took after the <a href="https://www.legislation.gov.uk/ukpga/Geo3/48/142/contents/enacted">Life Annuity Act of 1808</a>. The Act, passed in the midst of the <a href="https://en.wikipedia.org/wiki/Napoleonic_Wars">Napoleonic Wars</a>, aimed to help funding Britain’s war efforts. but it ended up being a big mistake.</p>
<p>Casey Rothschild describes the colorful sequence of events in a <a href="https://doi.org/10.1016/j.jpubeco.2009.01.002">2009 paper</a>:</p>
<blockquote>
<p>Prior to the Life Annuity Act of 1808, British government debt consisted almost exclusively of <a href="https://en.wikipedia.org/wiki/Consol_(bond)">Consols</a> — coupon bonds with infinite maturity. The explicit goal of the Act was to replace them with finite-lived debt by allowing individuals to exchange Consols for life annuities. Since Consols were tradable assets, the act effectively opened a life-annuity market.</p>
<p>Annuities sold under this act made twice-yearly tax-exempt payments. The size of these payments depended on the interest rate (the market Consol price) and the age of the annuitant (the nominees). Prices were designed to be actuarially fair; to that end, they were priced to be 2% more expensive than the actuarially fair price implied by the Northampton life table.</p>
<p>Shortly after passage of the Act, there appears to have been a recognition that the use of the Northampton tables was leading to large government losses. Ray D. Murphy writes <em>[in <a href="https://books.google.com/books/about/Sale_of_Annuities_by_Governments.html?id=vP3YAAAAMAAJ">Sale of Annuities by Governments</a>]</em> that it “was wholly unsuitable as a measure of the lower rates of mortality experienced by a self-selected group of annuitants. It was not long before this shortcoming was brought to the attention of the <a href="https://en.wikipedia.org/wiki/Exchequer">Exchequer</a>.”</p>
</blockquote>
<p>You, astute reader, will remember that we have already seen data from the Northampton life table before. (In any case, we will revisit it in a moment.)</p>
<p>Back to Rothschild to learn how the story ended:</p>
<blockquote>
<p>In 1823, Parliament finally took active steps to address this perceived mispricing by commissioning <a href="https://en.wikipedia.org/wiki/John_Finlaison">John Finlaison</a> to study the mortality experience of the early annuitants. His 1829 report developed a new set of gender-specific life tables based on the observed mortality experience of these nominees. After some debate and a brief suspension of the life annuity program, Parliament determined to resume it with gender-specific pricing based on these new tables.<sup id="fnref:4" role="doc-noteref"><a href="#fn:4" class="footnote" rel="footnote">4</a></sup></p>
</blockquote>
<p>In effect, here is a visual comparison between Finlaison’s data on Government Annuitants and <a href="https://en.wikipedia.org/wiki/Richard_Price">Richard Price</a>’s Northampton table:</p>
<div class="warning-narrow-screen">
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<div class="charts">
<canvas id="canvas-gov-annuitants-comparison-log-hazard"></canvas>
</div>
<p>As Craig Turnbull writes in his <a href="https://www.amazon.com/History-British-Actuarial-Thought/dp/3319331825">A History of British Actuarial Thought</a>:</p>
<blockquote>
<p>Finlaison’s analysis evidenced the profound differences between the government annuity mortality experience and the mortality rates assumed in the Northampton table: for example, at age 60, the Northampton table was implying a mortality rate that was fully double that experienced by the government annuitants!</p>
<p>Beyond the impact it had on government policy, Finlaison’s report was an important milestone for actuarial thought. [… T]he scale and rigour of Finlaison’s analysis was of a different order to these earlier works.</p>
</blockquote>
</div>
</details>
<p>Fast forward to the 2010s. Although there are still limitations in mortality data for ages above <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn></mrow><annotation encoding="application/x-tex">100</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">100</span></span></span></span>, we have access to data of much higher quality than the pioneers ever had.</p>
<p>The Gompertzian exponential does fit present-day mortality remarkably well. The fit encompasses an age period of approximately <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>70</mn></mrow><annotation encoding="application/x-tex">70</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">70</span></span></span></span> years and a range of hazard rates covering four orders of magnitude — from <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.01</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">0.01\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.80556em;vertical-align:-0.05556em;"></span><span class="mord">0.01%</span></span></span></span> to <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">10\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.80556em;vertical-align:-0.05556em;"></span><span class="mord">10%</span></span></span></span>.</p>
<p>Recall that the Gompertz hazard function is</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>B</mi><msup><mi>e</mi><mrow><mi>C</mi><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">h(x) = B e^{C x}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">x</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8913309999999999em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913309999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mathnormal mtight" style="margin-right:0.07153em;">C</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span></span>
<p>Here are the constants <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span></span></span></span> and <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span></span></span></span>, and age periods (<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>x</mi></mrow><annotation encoding="application/x-tex">x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">x</span></span></span></span> values) that fit the Gompertzian function to recent mortality data from the U.S., England & Wales, and Brazil:</p>
<table>
<thead>
<tr>
<th style="text-align: left">Country</th>
<th style="text-align: left">Sex</th>
<th style="text-align: left">Age period</th>
<th style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">\mathit{B}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathit">B</span></span></span></span></th>
<th style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathit{C}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathit">C</span></span></span></span></th>
</tr>
</thead>
<tbody>
<tr>
<td style="text-align: left">USA</td>
<td style="text-align: left">M</td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn></mrow><annotation encoding="application/x-tex">25</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">25</span></span></span></span>-<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn></mrow><annotation encoding="application/x-tex">100</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">100</span></span></span></span> y.o.</td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.0001240</mn></mrow><annotation encoding="application/x-tex">0.0001240</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.0001240</span></span></span></span></td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.07759</mn></mrow><annotation encoding="application/x-tex">0.07759</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.07759</span></span></span></span></td>
</tr>
<tr>
<td style="text-align: left">USA</td>
<td style="text-align: left">F</td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn></mrow><annotation encoding="application/x-tex">25</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">25</span></span></span></span>-<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn></mrow><annotation encoding="application/x-tex">100</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">100</span></span></span></span> y.o.</td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.0000471</mn></mrow><annotation encoding="application/x-tex">0.0000471</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.0000471</span></span></span></span></td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.08588</mn></mrow><annotation encoding="application/x-tex">0.08588</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.08588</span></span></span></span></td>
</tr>
<tr>
<td style="text-align: left">E&W</td>
<td style="text-align: left">M</td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn></mrow><annotation encoding="application/x-tex">25</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">25</span></span></span></span>-<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn></mrow><annotation encoding="application/x-tex">100</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">100</span></span></span></span> y.o.</td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.0000340</mn></mrow><annotation encoding="application/x-tex">0.0000340</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.0000340</span></span></span></span></td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.09348</mn></mrow><annotation encoding="application/x-tex">0.09348</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.09348</span></span></span></span></td>
</tr>
<tr>
<td style="text-align: left">E&W</td>
<td style="text-align: left">F</td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn></mrow><annotation encoding="application/x-tex">25</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">25</span></span></span></span>-<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn></mrow><annotation encoding="application/x-tex">100</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">100</span></span></span></span> y.o.</td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.0000154</mn></mrow><annotation encoding="application/x-tex">0.0000154</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.0000154</span></span></span></span></td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.09947</mn></mrow><annotation encoding="application/x-tex">0.09947</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.09947</span></span></span></span></td>
</tr>
<tr>
<td style="text-align: left">Brazil</td>
<td style="text-align: left">M</td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn></mrow><annotation encoding="application/x-tex">25</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">25</span></span></span></span>-<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>80</mn></mrow><annotation encoding="application/x-tex">80</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">80</span></span></span></span> y.o.</td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.0002798</mn></mrow><annotation encoding="application/x-tex">0.0002798</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.0002798</span></span></span></span></td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.06562</mn></mrow><annotation encoding="application/x-tex">0.06562</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.06562</span></span></span></span></td>
</tr>
<tr>
<td style="text-align: left">Brazil</td>
<td style="text-align: left">F</td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn></mrow><annotation encoding="application/x-tex">25</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">25</span></span></span></span>-<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>80</mn></mrow><annotation encoding="application/x-tex">80</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">80</span></span></span></span> y.o.</td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.0000621</mn></mrow><annotation encoding="application/x-tex">0.0000621</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.0000621</span></span></span></span></td>
<td style="text-align: left"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.08105</mn></mrow><annotation encoding="application/x-tex">0.08105</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.08105</span></span></span></span></td>
</tr>
</tbody>
</table>
<p>You may ask: Where did this table come from? Or, more specifically, how did I find the exponentials that fit the data?</p>
<p>I ran a simple <a href="https://en.wikipedia.org/wiki/Linear_regression">linear regression</a> over recent data provided by the national bureaus of statistics (or equivalent) of these three countries.<sup id="fnref:5" role="doc-noteref"><a href="#fn:5" class="footnote" rel="footnote">5</a></sup></p>
<p>To convince you that these Gompertzian curves do fit quite well the best data that we have available, let’s plot the official data points and overlay the fitted lines. For males we have the following:</p>
<div class="warning-narrow-screen">
<p class="muted small">⚠️ Your screen is almost too narrow for the interactive chart below. Please rotate your
device for a better experience.</p>
</div>
<div class="charts">
<canvas id="canvas-recent-mortality-males"></canvas>
</div>
<p class="right"><small>— Download <a href="/assets/data/recent-mortality-males.csv">chart data in .csv</a></small></p>
<p>So, suppose you are <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>40</mn></mrow><annotation encoding="application/x-tex">40</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">40</span></span></span></span>-year old English man.</p>
<p>The “official” data indicates that your chances of dying at this age are <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.1474</mn><mi mathvariant="normal">%</mi><mo>≈</mo><mn>0.15</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">0.1474\% \approx 0.15\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.80556em;vertical-align:-0.05556em;"></span><span class="mord">0.1474%</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.80556em;vertical-align:-0.05556em;"></span><span class="mord">0.15%</span></span></span></span>. That is, according to death statistics in 2010-2012 in England and Wales, roughly <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>15</mn></mrow><annotation encoding="application/x-tex">15</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">15</span></span></span></span> in every <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>10</mn><mtext>,</mtext><mn>000</mn></mrow><annotation encoding="application/x-tex">10\text{,}000</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord">10</span><span class="mord text"><span class="mord">,</span></span><span class="mord">000</span></span></span></span> English and Welsh men aged <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>40</mn></mrow><annotation encoding="application/x-tex">40</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">40</span></span></span></span> died at that age.</p>
<p>The Gompertz law that we fitted to the data predicts that</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>0.0000340</mn><msup><mi>e</mi><mrow><mn>0.09348</mn><mo>∗</mo><mn>40</mn></mrow></msup><mo>=</mo><mn>0.001430</mn></mrow><annotation encoding="application/x-tex">0.0000340e^{0.09348 * 40} = 0.001430</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em;"></span><span class="mord">0.0000340</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0.09348</span><span class="mbin mtight">∗</span><span class="mord mtight">40</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.001430</span></span></span></span></span>
<p>which is off to the “actual” number by</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mn>0.1474</mn><mi mathvariant="normal">%</mi><mo>−</mo><mn>0.1430</mn><mi mathvariant="normal">%</mi></mrow><mrow><mn>0.1474</mn><mi mathvariant="normal">%</mi></mrow></mfrac><mo>≈</mo><mn>3</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">\frac {0.1474\%-0.1430\%} {0.1474\%} \approx 3\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1685600000000003em;vertical-align:-0.74156em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.1474%</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.1474%</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">0.1430%</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.74156em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.80556em;vertical-align:-0.05556em;"></span><span class="mord">3%</span></span></span></span></span>
<p>I don’t know about you, but getting a <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>3</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">3\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.80556em;vertical-align:-0.05556em;"></span><span class="mord">3%</span></span></span></span> error from a quick and simple 2-parameter model seems great to me!</p>
<p>Here is the equivalent chart for females:</p>
<div class="warning-narrow-screen">
<p class="muted small">⚠️ Your screen is almost too narrow for the interactive chart below. Please rotate your
device for a better experience.</p>
</div>
<div class="charts">
<canvas id="canvas-recent-mortality-females"></canvas>
</div>
<p class="right"><small>— Download <a href="/assets/data/recent-mortality-females.csv">chart data in .csv</a></small></p>
<p>If you are a <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>26</mn></mrow><annotation encoding="application/x-tex">26</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">26</span></span></span></span>-year old American woman, your probability of dying before your <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>27</mn></mrow><annotation encoding="application/x-tex">27</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">27</span></span></span></span>th birthday is <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>0.0006456</mn><mo>≈</mo><mn>0.065</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">0.0006456 \approx 0.065\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.0006456</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.80556em;vertical-align:-0.05556em;"></span><span class="mord">0.065%</span></span></span></span>. In other words, according to U.S. death statistics in 2017, approximately <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>6</mn></mrow><annotation encoding="application/x-tex">6</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">6</span></span></span></span> in every <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>100</mn><mtext>,</mtext><mn>000</mn></mrow><annotation encoding="application/x-tex">100\text{,}000</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord">100</span><span class="mord text"><span class="mord">,</span></span><span class="mord">000</span></span></span></span> American women aged <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>26</mn></mrow><annotation encoding="application/x-tex">26</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">26</span></span></span></span> died.</p>
<p>The prediction of our Gompertzian curve is</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mn>0.0000471</mn><msup><mi>e</mi><mrow><mn>0.08588</mn><mo>∗</mo><mn>26</mn></mrow></msup><mo>=</mo><mn>0.0004392</mn></mrow><annotation encoding="application/x-tex">0.0000471e^{0.08588 * 26} = 0.0004392</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8641079999999999em;vertical-align:0em;"></span><span class="mord">0.0000471</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8641079999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">0.08588</span><span class="mbin mtight">∗</span><span class="mord mtight">26</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0.0004392</span></span></span></span></span>
<p>And the error is</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mrow><mn>0.06456</mn><mi mathvariant="normal">%</mi><mo>−</mo><mn>0.04392</mn><mi mathvariant="normal">%</mi></mrow><mrow><mn>0.06456</mn><mi mathvariant="normal">%</mi></mrow></mfrac><mo>≈</mo><mn>32</mn><mi mathvariant="normal">%</mi></mrow><annotation encoding="application/x-tex">\frac {0.06456\%-0.04392\%} {0.06456\%} \approx 32\%</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.1685600000000003em;vertical-align:-0.74156em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.427em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.06456%</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">0.06456%</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">−</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">0.04392%</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.74156em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">≈</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.80556em;vertical-align:-0.05556em;"></span><span class="mord">32%</span></span></span></span></span>
<p>Why is the error much higher for this second example?</p>
<p>The explanation is clear. You can spot in both charts that the fit is much worse near the start and the end of the age periods — i.e., around <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>25</mn></mrow><annotation encoding="application/x-tex">25</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">25</span></span></span></span>-<span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>35</mn></mrow><annotation encoding="application/x-tex">35</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">35</span></span></span></span> y.o. and above the age of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mn>95</mn></mrow><annotation encoding="application/x-tex">95</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">95</span></span></span></span>.</p>
<p>I will explore these two edge cases in future essays. There is a lot of very intringuing stuff going on both edges!</p>
<p>For instance, youngsters are particularly affected by what is known as “extrinsic” causes of mortality. These are infections, accidents and crime — things that come from “outside”. In fact, if we remove the deaths from extrinsic causes and leave just the ones due to “intrinsic” issues, it seems that the fit to the Gompertz law expands to younger ages very nicely!</p>
<p>As for the mortality of the eldest, due to sparse data,<sup id="fnref:6" role="doc-noteref"><a href="#fn:6" class="footnote" rel="footnote">6</a></sup> it is still not entirely clear to this day if the hazard rate flattens out — or perhaps it even decreases. (I don’t have a strong opinion on the matter yet. I still have several papers on the topic to read and think about.)</p>
<p>I hope you will join me in future essays on these and related topics.</p>
<p>Finally, if you haven’t yet, feel free to scroll up, find you age and eyeball your chances of dying this year. You may not be from any of these three countries. In this case, of course the estimate will be rougher but probably still in the correct order of magnitude.</p>
<h2 id="further-reading">Further reading</h2>
<details>
<summary>
Appendix A — The famous modification proposed by Makeham in the 1860s
</summary>
<div>
<p>In the decades following Gompertz’s original papers, it was clear for everyone involved in Actuarial science that the Gompertzian exponential did not fit the data over the entire human lifespan. Despite that, the lure of finding the <em>actual</em> law of mortality continued to motivate the proposal of several other expressions.</p>
<p>For example, M. E. Ogborn wrote the following in a <a href="https://doi.org/10.1017/S0020268100054044">1953 paper</a>:</p>
<blockquote>
<p>Gompertz, himself, <a href="https://doi.org/10.1017/S204616740004369X">suggested in 1860</a> a formula based on an amalgamation of several of his curves with different constants, and various combinations have been suggested from time to time,<sup id="fnref:7" role="doc-noteref"><a href="#fn:7" class="footnote" rel="footnote">7</a></sup> together with others of a different type.</p>
<p>Perhaps the best illustration of the point of view is given by the mathematical formula <a href="https://doi.org/10.1017/S2046167400043688">proposed in 1871</a> by the Danish mathematician, <a href="https://en.wikipedia.org/wiki/Thorvald_N._Thiele">Thorvald Thiele</a>, to express the rate of mortality throughout the whole of life.</p>
<p>In Thiele’s <em>[7-parameter]</em> formula,</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mi>A</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>B</mi><mi>x</mi></mrow></msup><mo>+</mo><mi>C</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>E</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msup><mo>+</mo><mi>F</mi><msup><mi>G</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">h(t) = A e^{-Bx} + C e^{-D(x-E)^2} + F G^x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.974661em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.891331em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:1.1202499999999997em;vertical-align:-0.08333em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:1.0369199999999998em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">x</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913142857142857em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.7143919999999999em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.7143919999999999em;"><span style="top:-3.113em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span>
<p>the last term, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>F</mi><msup><mi>G</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">F G^x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.13889em;">F</span><span class="mord"><span class="mord mathnormal">G</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span>, is a Gompertz curve to represent old-age mortality and the first, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>B</mi><mi>x</mi></mrow></msup></mrow><annotation encoding="application/x-tex">A e^{-Bx}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8413309999999999em;vertical-align:0em;"></span><span class="mord mathnormal">A</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8413309999999999em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.05017em;">B</span><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span></span>, a decreasing Gompertz curve to represent the mortality of infancy. The middle term, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>C</mi><msup><mi>e</mi><mrow><mo>−</mo><mi>D</mi><mo stretchy="false">(</mo><mi>x</mi><mo>−</mo><mi>E</mi><msup><mo stretchy="false">)</mo><mn>2</mn></msup></mrow></msup></mrow><annotation encoding="application/x-tex">C e^{-D(x-E)^2}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.9869199999999998em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.07153em;">C</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9869199999999998em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.02778em;">D</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">x</span><span class="mbin mtight">−</span><span class="mord mathnormal mtight" style="margin-right:0.05764em;">E</span><span class="mclose mtight"><span class="mclose mtight">)</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.8913142857142857em;"><span style="top:-2.931em;margin-right:0.07142857142857144em;"><span class="pstrut" style="height:2.5em;"></span><span class="sizing reset-size3 size1 mtight"><span class="mord mtight">2</span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span></span>, is a form of the normal curve of error <em>[likely to account for the hump in mortality experienced by young adults]</em>.</p>
</blockquote>
<p>The contribution that ended up being the most famous was proposed by <a href="https://en.wikipedia.org/wiki/William_Makeham">William Makeham</a> in the 1860s. He wrote in a <a href="https://doi.org/10.1017/S2046165800002823">1866 paper</a>:</p>
<blockquote>
<p>Mr. Gompertz’s theory of the law of mortality is, that the vital power, or the “power to oppose destruction,” loses equal proportions in equal times; and consequently that the intensity of mortality, which is inversely proportional to this power, is represented by a series in geometrical progression.</p>
<p>It has, however, invariably been found that the ratio of progression, instead of remaining constant throughout the whole period of life, as the theory supposes, is, on the contrary, subject to a slow but continued increase with age; in consequence of which it has been found necessary to change the constants at least once, but generally twice, in the construction of a complete table of mortality.</p>
<p>I venture to think, in a more scientific manner, by supposing the intensity of mortality to be represented by a series not purely geometrical, but consisting of the sum of two terms, the one a constant quantity and the other geometrical. That is, instead of representing the intensity of mortality by an expression of the form <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>B</mi><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">Be^x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span> we represent it by one of the form <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>A</mi><mo>+</mo><mi>B</mi><msup><mi>e</mi><mi>x</mi></msup></mrow><annotation encoding="application/x-tex">A + Be^x</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.76666em;vertical-align:-0.08333em;"></span><span class="mord mathnormal">A</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.68333em;vertical-align:0em;"></span><span class="mord mathnormal" style="margin-right:0.05017em;">B</span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.664392em;"><span style="top:-3.063em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mathnormal mtight">x</span></span></span></span></span></span></span></span></span></span></span>.</p>
</blockquote>
<p>He then offered the folllowing rationale for his proposal:</p>
<blockquote>
<p>From the age of 15, or thereabouts, the normal law of mortality, of which we are in search, is characterized by an increasing progression throughout; the rate of increase, however, being at first very slow, and gradually gaining in rapidity with increased age.</p>
<p>It is this characteristic which renders the formula before described (consisting of a constant combined with an increasing geometrical series) singularly well adapted to represent the law in question from adolescence to extreme old age — a satisfactory proof of which assertion I hope to give on a future occasion, when I propose also to examine the results of an extension of the formula to all periods of life.</p>
</blockquote>
<p>Makeham’s “satisfactory proof” on “a future occasion” came out the next year. In 1867 he <a href="https://doi.org/10.1017/S2046166600003238">published another paper</a> expanding on his hypothesis.</p>
<p>Makeham used data from five U.K. mortality series from the 1800s to support his argument. To check it out visually, let’s plot the data points from all five data series as presented by him:</p>
<div class="warning-narrow-screen">
<p class="muted small">⚠️ Your screen is almost too narrow for the interactive chart below. Please rotate your
device for a better experience.</p>
</div>
<div class="charts">
<canvas id="canvas-makeham_1867-log-hazard"></canvas>
</div>
<p>Now, let’s overlay two curves on top of his data points: a Gompertz’s exponential (in <span style="color:#c8d0d9;">light gray</span>) and the Makeham’s proposed modification (<span style="color:#57606c;">dark gray</span>).</p>
<p>From the chart below, it is pretty clear that Makeham’s modification did improve the fit to the data available at his time:</p>
<div class="warning-narrow-screen">
<p class="muted small">⚠️ Your screen is almost too narrow for the interactive chart below. Please rotate your
device for a better experience.</p>
</div>
<div class="charts">
<canvas id="canvas-makeham_1867-log-hazard-fit"></canvas>
</div>
<p class="right"><small>— Download <a href="/assets/data/makeham_1867-hazard-fit.csv">chart data in .csv</a></small></p>
<p>Part of the appeal of Makeham’s modification is that it adds just one additional parameter to the expression (for a total of three). From a modeler standpoint, too many parameters isn’t good — given enough parameters, any data set can be fitted.<sup id="fnref:8" role="doc-noteref"><a href="#fn:8" class="footnote" rel="footnote">8</a></sup></p>
<p>As much as it was an improvement, the Gompertz-Makeham law is <em>not</em> the <em>ultimate</em> law of mortality. Makeham avoided infancy and childhood altogether in his 1867 paper. In fact, he wrote at the time:</p>
<blockquote>
<p>I must postpone to a future opportunity the examination of so interesting and important a subject as the mortality of infancy and childhood.</p>
</blockquote>
</div>
</details>
<details>
<summary>
Appendix B — Survival function <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> in terms of hazard rate <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>
</summary>
<div>
<p>We have already have <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> in terms of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> from <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>e</mi><mi>q</mi><mi mathvariant="normal">.</mi><mtext> </mtext><mn>3</mn></mrow><annotation encoding="application/x-tex">{eq. \medspace 3}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.8388800000000001em;vertical-align:-0.19444em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="mord mathnormal" style="margin-right:0.03588em;">q</span><span class="mord">.</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord">3</span></span></span></span></span> (see below). Let’s work out a few more steps to express <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> in terms of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>, and come full circle.</p>
<p>Recall that</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><mo>−</mo><mfrac><mn>1</mn><mi>y</mi></mfrac><mo>⋅</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>eq. 3</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\tag*{eq. 3} h(t) = - \frac {1} {y} \cdot \frac {dy} {dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.20188em;vertical-align:-0.8804400000000001em;"></span><span class="mord">−</span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.0574399999999997em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span><span class="tag"><span class="strut" style="height:2.25188em;vertical-align:-0.8804400000000001em;"></span><span class="mord text"><span class="mord">eq. 3</span></span></span></span></span></span>
<p>By the <a href="https://en.wikipedia.org/wiki/Fundamental_theorem_of_calculus">Fundamental theorem of Calculus</a>, we can write</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∫</mo><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>d</mi><mi>t</mi></mrow><mo>=</mo><mo>−</mo><mo>∫</mo><mrow><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">(</mo><mrow><mfrac><mn>1</mn><mi>y</mi></mfrac><mo>⋅</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>⋅</mo><mi>d</mi><mi>t</mi></mrow><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">)</mo></mrow><mo>+</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\int {h(t) \cdot dt} = - \int {\Bigg( {\frac {1} {y} \cdot \frac {dy} {dt} \cdot dt} \Bigg)} + c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.22225em;vertical-align:-0.86225em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="delimsizing size4">(</span></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span><span class="mord"><span class="delimsizing size4">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span></span></span></span>
<p>When thinking about how to solve the integral at right-hand side, we recollect that the <a href="https://en.wikipedia.org/wiki/Natural_logarithm">natural logarithm</a> somehow resembles it.</p>
<p>Moreover, because we defined <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">y > 0</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.7335400000000001em;vertical-align:-0.19444em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.64444em;vertical-align:0em;"></span><span class="mord">0</span></span></span></span>, we can summon the <a href="https://en.wikipedia.org/wiki/Chain_rule">Chain rule</a> as:</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mrow><mi>ln</mi><mo></mo><mi>y</mi></mrow><mo>=</mo><mfrac><mi>d</mi><mrow><mi>d</mi><mi>y</mi></mrow></mfrac><mrow><mi>ln</mi><mo></mo><mi>y</mi></mrow><mo>⋅</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>=</mo><mfrac><mn>1</mn><mi>y</mi></mfrac><mo>⋅</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac></mrow><annotation encoding="application/x-tex">\frac {d} {dt} {\ln {y}} = \frac {d} {dy} {\ln {y}} \cdot \frac {dy} {dt} = \frac {1} {y} \cdot \frac {dy} {dt}</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.05744em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mop">ln</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.25188em;vertical-align:-0.8804400000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mop">ln</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.0574399999999997em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:2.20188em;vertical-align:-0.8804400000000001em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:2.0574399999999997em;vertical-align:-0.686em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span></span></span></span></span>
<p>And insert this result into the previous expression</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∫</mo><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>d</mi><mi>t</mi></mrow><mo>=</mo><mo>−</mo><mo>∫</mo><mrow><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">(</mo><mrow><mfrac><mn>1</mn><mi>y</mi></mfrac><mo>⋅</mo><mfrac><mrow><mi>d</mi><mi>y</mi></mrow><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mo>⋅</mo><mi>d</mi><mi>t</mi></mrow><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">)</mo></mrow><mo>+</mo><mi>c</mi><mo>=</mo><mo>−</mo><mo>∫</mo><mrow><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">(</mo><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mrow><mi>ln</mi><mo></mo><mi>y</mi></mrow><mo>⋅</mo><mi>d</mi><mi>t</mi></mrow><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">)</mo></mrow><mo>+</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\int {h(t) \cdot dt} = - \int {\Bigg( {\frac {1} {y} \cdot \frac {dy} {dt} \cdot dt} \Bigg)} + c = - \int {\Bigg( { \frac {d} {dt} {\ln {y}} \cdot dt} \Bigg)} + c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.22225em;vertical-align:-0.86225em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="delimsizing size4">(</span></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.32144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord">1</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.8804400000000001em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.3714399999999998em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span><span class="mord"><span class="delimsizing size4">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="delimsizing size4">(</span></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mop">ln</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span><span class="mord"><span class="delimsizing size4">)</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span></span></span></span>
<p>By the Fundamental theorem of Calculus again, we have</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∫</mo><mrow><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">(</mo><mrow><mfrac><mi>d</mi><mrow><mi>d</mi><mi>t</mi></mrow></mfrac><mrow><mi>ln</mi><mo></mo><mi>y</mi></mrow><mo>⋅</mo><mi>d</mi><mi>t</mi></mrow><mo fence="false" stretchy="true" minsize="3em" maxsize="3em">)</mo></mrow><mo>=</mo><mi>ln</mi><mo></mo><mi>y</mi><mo>+</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\int {\Bigg( { \frac {d} {dt} {\ln {y}} \cdot dt} \Bigg)} = \ln {y} + c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:3.0000299999999998em;vertical-align:-1.25003em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord"><span class="delimsizing size4">(</span></span><span class="mord"><span class="mord"><span class="mopen nulldelimiter"></span><span class="mfrac"><span class="vlist-t vlist-t2"><span class="vlist-r"><span class="vlist" style="height:1.37144em;"><span style="top:-2.314em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span></span><span style="top:-3.23em;"><span class="pstrut" style="height:3em;"></span><span class="frac-line" style="border-bottom-width:0.04em;"></span></span><span style="top:-3.677em;"><span class="pstrut" style="height:3em;"></span><span class="mord"><span class="mord mathnormal">d</span></span></span></span><span class="vlist-s"></span></span><span class="vlist-r"><span class="vlist" style="height:0.686em;"><span></span></span></span></span></span><span class="mclose nulldelimiter"></span></span><span class="mord"><span class="mop">ln</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span><span class="mord"><span class="delimsizing size4">)</span></span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span></span></span></span>
<p>Thus,</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mo>∫</mo><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>d</mi><mi>t</mi></mrow><mo>=</mo><mo>−</mo><mi>ln</mi><mo></mo><mi>y</mi><mo>+</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\int {h(t) \cdot dt} = - \ln {y} + c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:2.22225em;vertical-align:-0.86225em;"></span><span class="mop op-symbol large-op" style="margin-right:0.44445em;position:relative;top:-0.0011249999999999316em;">∫</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">⋅</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mord mathnormal">d</span><span class="mord mathnormal">t</span></span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:0.8888799999999999em;vertical-align:-0.19444em;"></span><span class="mord">−</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mop">ln</span><span class="mspace" style="margin-right:0.16666666666666666em;"></span><span class="mord"><span class="mord mathnormal" style="margin-right:0.03588em;">y</span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span></span></span></span>
<p>Finally, by the definition of logarithms</p>
<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mtable width="100%"><mtr><mtd width="50%"></mtd><mtd><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>e</mi><mrow><mo>−</mo><mo>∫</mo><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>d</mi><mi>t</mi></mrow></mrow></msup><mo>+</mo><mi>c</mi></mrow></mtd><mtd width="50%"></mtd><mtd><mtext>eq. 4</mtext></mtd></mtr></mtable><annotation encoding="application/x-tex">\tag*{eq. 4} y(t) = e^{- \int {h(t) \cdot dt}} + c</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2777777777777778em;"></span></span><span class="base"><span class="strut" style="height:1.05983em;vertical-align:-0.08333em;"></span><span class="mord"><span class="mord mathnormal">e</span><span class="msupsub"><span class="vlist-t"><span class="vlist-r"><span class="vlist" style="height:0.9765em;"><span style="top:-3.1130000000000004em;margin-right:0.05em;"><span class="pstrut" style="height:2.7em;"></span><span class="sizing reset-size6 size3 mtight"><span class="mord mtight"><span class="mord mtight">−</span><span class="mspace mtight" style="margin-right:0.19516666666666668em;"></span><span class="mop op-symbol small-op mtight" style="margin-right:0.19445em;position:relative;top:-0.0005599999999999772em;">∫</span><span class="mspace mtight" style="margin-right:0.19516666666666668em;"></span><span class="mord mtight"><span class="mord mathnormal mtight">h</span><span class="mopen mtight">(</span><span class="mord mathnormal mtight">t</span><span class="mclose mtight">)</span><span class="mbin mtight">⋅</span><span class="mord mathnormal mtight">d</span><span class="mord mathnormal mtight">t</span></span></span></span></span></span></span></span></span></span><span class="mspace" style="margin-right:0.2222222222222222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222222222222222em;"></span></span><span class="base"><span class="strut" style="height:0.43056em;vertical-align:0em;"></span><span class="mord mathnormal">c</span></span><span class="tag"><span class="strut" style="height:1.2265000000000001em;vertical-align:-0.25em;"></span><span class="mord text"><span class="mord">eq. 4</span></span></span></span></span></span>
<p>There it is, <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">y(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span> expressed in terms of <span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML"><semantics><mrow><mi>h</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">h(t)</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:1em;vertical-align:-0.25em;"></span><span class="mord mathnormal">h</span><span class="mopen">(</span><span class="mord mathnormal">t</span><span class="mclose">)</span></span></span></span>.</p>
<p>This is especially useful for when the hazard rate is known and we intend to find its corresponding survival curve. That is exactly what we will do in a future interactive essay (soon!).</p>
</div>
</details>
<p class="small center muted">· · ·</p>
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scaleLabel: {
labelString: axes.y.label,
display: true,
}
}],
xAxes: [{
type: axes.x.scale,
position: 'bottom',
gridLines: {
color: applyA(Chart.defaults.global.defaultColor, 0.1),
zeroLineColor: applyA(Chart.defaults.global.defaultColor, 0.3),
zeroLineWidth: 1.5,
drawBorder: false,
},
ticks: {
min: 0,
// max: 120,
},
scaleLabel: {
labelString: axes.x.label,
display: true
}
}]
}
}
};
// Callbacks
// https://github.com/chartjs/Chart.js/blob/171a7e3a7a23b1d4a9d37ed2b2495698969b79d5/src/core/core.tooltip.js#L626
// sort tooltip labels in descending order (by value)
config.options.tooltips.itemSort = function(a, b){
return (b.value - a.value)
}
if (ttLabelFn) {
config.options.tooltips.callbacks.label = ttLabelFn;
}
if (config.options.tooltips.mode == 'x') {
config.options.tooltips.callbacks.title = tooltipTitleFn;
}
if (axes.x.ticksCallback) {
config.options.scales.xAxes[0].ticks.callback = axes.x.ticksCallback;
}
if (axes.y.ticksCallback) {
config.options.scales.yAxes[0].ticks.callback = axes.y.ticksCallback;
}
overwriteChartGlobalStyle();
const chart = new Chart(context, config);
charts.push(chart);
}
</script>
<script>
function buildCharts() {
const linearAge = {
label: 'Age (y.o.)',
scale: 'linear',
}
let survAxes = {
y: {
label: 'Fraction of persons alive',
scale: 'linear',
},
x: linearAge
}
let linearAxes = {
y: {
label: 'Hazard rate',
scale: 'linear',
},
x: linearAge
}
let logAxes = {
y: {
label: 'Hazard rate · log scale',
scale: 'logarithmic',
ticksCallback: logTicksFn,
},
x: linearAge
}
let tooltipsFn = tooltipLabelFn // = null
// Survival curves – 18th century, Europe
const earlySurvData = [
{'label': 'France (Déparcieux, 1760)', 'data': deparcieuxSurvData, 'color': c01PrimBlue},
{'label': 'Sweden – Males (Wargentin, 1766)', 'data': swedenMalesSurvData, 'color': c02PrimOrange},
{'label': 'Northampton, UK (Price, 1771)', 'data': northamptonSurvData, 'color': c03PrimGray},
{'label': 'Carlisle, UK (Milne, 1815)', 'data': carlisleSurvData, 'color': c04SecGray},
]
const earlySurvDataset = createDatasets(earlySurvData, null)
const earlySurvDatasetWithExp = createDatasets([].concat(earlySurvData, [{'label': 'Exponential Decay (exp(-x/30))', 'data': expDecay30SurvData, 'color': c09SecOrange, 'type': 'line'}, {'label': 'Exponential Decay (exp(-x/50))', 'data': expDecay50SurvData, 'color': c10ForGray, 'type': 'line'}]), null)
createChart('canvas-18th-century-survival', earlySurvDataset, 'scatter', survAxes, 'Survival curves · 18th-century Europe', tooltipsFn)
createChart('canvas-18th-century-survival-exp', earlySurvDatasetWithExp, 'scatter', survAxes, 'Exponential decays vs. Survival curves', tooltipsFn)
// Hazard rates – 18th century, Europe
const earlyHazardData = [
{'label': 'France (Déparcieux, 1760)', 'data': deparcieuxHzdData, 'color': c01PrimBlue},
{'label': 'Sweden – Males (Wargentin, 1766)', 'data': swedenMalesHzdData, 'color': c02PrimOrange},
{'label': 'Northampton, UK (Price, 1771)', 'data': northamptonHzdData, 'color': c03PrimGray},
{'label': 'Carlisle, UK (Milne, 1815)', 'data': carlisleHzdData, 'color': c04SecGray},
]
const earlyHazardDataset = createDatasets(earlyHazardData, null)
const earlyHazardDatasetWithExp = createDatasets([].concat(earlyHazardData, [{'label': 'Exponential (0.002 * exp(x/20))', 'data': expGrowth20Hazard, 'color': c10ForGray, 'type': 'line'}]), null)
createChart('canvas-18th-century-hazard', earlyHazardDataset, 'scatter', linearAxes, 'Hazard rates · 18th-century Europe', tooltipLabelPercentFn)
createChart('canvas-18th-century-log-hazard', earlyHazardDatasetWithExp, 'scatter', logAxes, 'Hazard rates · 18th-century Europe', tooltipLabelPercentFn)
// Northampton vs Government Annuitants
const govAnnuitantsCompData = [
{'label': 'Northampton, UK (Price, 1771)', 'data': northamptonHzdData, 'color': c03PrimGray},
{'label': 'Government Annuitants (Finlaison, 1829)', 'data': governmentAnnuitantsHzdData, 'color': c05SecBlue},
]
const govAnnuitantsCompDataset = createDatasets(govAnnuitantsCompData, null)
createChart('canvas-gov-annuitants-comparison-log-hazard', govAnnuitantsCompDataset, 'scatter', logAxes, 'Hazard rates · Northampton vs. Government Annuitants', tooltipLabelPercentFn)
// Makeham 1867
const makeham1867HazardData = [
{'label': 'Government Annuitants (Finlaison, 1829)', 'data': governmentAnnuitantsHzdData, 'color': c05SecBlue},
{'label': 'Combined Experience of Seventeen Life Assurance Offices (1843)', 'data': combinedExperienceHzdData, 'color': c07TerGray},
{'label': 'Friendly Societies – Males (Finlaison, 1853)', 'data': friendlySocMalesHzdData, 'color': c06PrimRed},
{'label': 'Peerage Families (Bailey & Day, 1861)', 'data': peerageFamiliesHzdData, 'color': c08TerBlue},
{'label': 'Clergy of England and Wales (Hodgson & Brown, 1865)', 'data': clergyHzdData, 'color': c09SecOrange},
]
const makeham1867HazardDataset = createDatasets(makeham1867HazardData, "mediumPoints")
createChart('canvas-makeham_1867-log-hazard', makeham1867HazardDataset, 'scatter', logAxes, 'Hazard rates · 19th-century United Kingdom', tooltipLabelPercentFn)
const makeham1867HazardFitData = [
{'label': 'Exponential (0.0007 * exp(x/16))', 'data': expGrowthHazard, 'color': c10ForGray, 'type': 'line'},
{'label': 'Linear + Exponential (0.0045 + 0.0005 * exp(x/15.5))', 'data': linearExpGrowthHazard, 'color': c04SecGray, 'type': 'line'},
{'label': 'Government Annuitants (Finlaison, 1829)', 'data': governmentAnnuitantsHzdData, 'color': c05SecBlue, 'alpha': 0.5},
{'label': 'Combined Experience of Seventeen Life Assurance Offices (1843)', 'data': combinedExperienceHzdData, 'color': c07TerGray, 'alpha': 0.5},
{'label': 'Friendly Societies – Males (Finlaison, 1853)', 'data': friendlySocMalesHzdData, 'color': c06PrimRed, 'alpha': 0.5},
{'label': 'Peerage Families (Bailey & Day, 1861)', 'data': peerageFamiliesHzdData, 'color': c08TerBlue, 'alpha': 0.5},
{'label': 'Clergy of England and Wales (Hodgson & Brown, 1865)', 'data': clergyHzdData, 'color': c09SecOrange, 'alpha': 0.5},
]
const makeham1867HazardFitDataset = createDatasets(makeham1867HazardFitData, "mediumPoints")
createChart('canvas-makeham_1867-log-hazard-fit', makeham1867HazardFitDataset, 'scatter', logAxes, 'Hazard rates vs. Gompertzian exponential vs. Gompertz-Makeham law', tooltipLabelPercentFn)
// Recent data - ELT 17, USA, Brasil
const recentMalesData = [
{'label': 'England & Wales (ONS ELT 17, 2015)', 'data': elt17Males2015RawData, 'color': c01PrimBlue, 'alpha': 0.5},
{'label': 'E&W Fitted', 'data': fitELT17MalesHazard, 'color': c01PrimBlue, 'type': 'line'},
{'label': 'Brazil (IBGE, 2019)', 'data': braMales2018RawData, 'color': c03PrimGray, 'alpha': 0.5},
{'label': 'BRA Fitted', 'data': fitBraMalesHazard, 'color': c03PrimGray, 'type': 'line'},
{'label': 'USA (CDC NCHS, 2019)', 'data': usaMales2017RawData, 'color': c05SecBlue, 'alpha': 0.5},
{'label': 'USA Fitted', 'data': fitUSAMalesHazard, 'color': c05SecBlue, 'type': 'line'},
]
const recentMalesDataset = createDatasets(recentMalesData, "smallPoints")
createChart('canvas-recent-mortality-males', recentMalesDataset, 'scatter', logAxes, 'Hazard rates – Males · 2010s E&W, USA, Brazil', tooltipLabelPercentFn);
const recentFemalesData = [
{'label': 'England & Wales (ONS ELT 17, 2015)', 'data': elt17Females2015RawData, 'color': c02PrimOrange, 'alpha': 0.5},
{'label': 'E&W Fitted', 'data': fitELT17FemalesHazard, 'color': c02PrimOrange, 'type': 'line'},
{'label': 'Brazil (IBGE, 2019)', 'data': braFemales2018RawData, 'color': c04SecGray, 'alpha': 0.5},
{'label': 'BRA Fitted', 'data': fitBraFemalesHazard, 'color': c04SecGray, 'type': 'line'},
{'label': 'USA (CDC NCHS, 2019)', 'data': usaFemales2017RawData, 'color': c06PrimRed, 'alpha': 0.5},
{'label': 'USA Fitted', 'data': fitUSAFemalesHazard, 'color': c06PrimRed, 'type': 'line'},
]
const recentFemalesDataset = createDatasets(recentFemalesData, "smallPoints")
createChart('canvas-recent-mortality-females', recentFemalesDataset, 'scatter', logAxes, 'Hazard rates – Females · 2010s E&W, USA, Brazil', tooltipLabelPercentFn);
}
</script>
<script>
// colors taken from tableau.ColorBlind10 theme
// https://nagix.github.io/chartjs-plugin-colorschemes/colorchart.html
const c01PrimBlue = '#1170aa'
const c02PrimOrange = '#fc7d0b'
const c03PrimGray = '#a3acb9'
const c04SecGray = '#57606c'
const c05SecBlue = '#5fa2ce'
const c06PrimRed = '#c85200'
const c07TerGray = '#7b848f'
const c08TerBlue = '#a3cce9'
const c09SecOrange = '#ffbc79'
const c10ForGray = '#c8d0d9'
// David Forfar 2006 - Early Mortality Tables
const deparcieuxSurvData = [{x: 0, y: 1}, {x: 5, y: 0.56}, {x: 15, y: 0.50}, {x: 25, y: 0.45}, {x: 35, y: 0.41}, {x: 45, y: 0.36}, {x: 55, y: 0.31}, {x: 65, y: 0.23}, {x: 75, y: 0.12}, {x: 85, y: 0.03}, {x: 95, y: 0}, {x: 100, y: 0}]
const swedenMalesSurvData = [{x: 0, y: 1}, {x: 5, y: 0.65}, {x: 15, y: 0.58}, {x: 25, y: 0.53}, {x: 35, y: 0.47}, {x: 45, y: 0.41}, {x: 55, y: 0.32}, {x: 65, y: 0.21}, {x: 75, y: 0.09}, {x: 85, y: 0.02}, {x: 95, y: 0}, {x: 100, y: 0}]
const northamptonSurvData = [{x: 0, y: 1}, {x: 5, y: 0.54}, {x: 15, y: 0.47}, {x: 25, y: 0.41}, {x: 35, y: 0.34}, {x: 45, y: 0.28}, {x: 55, y: 0.20}, {x: 65, y: 0.13}, {x: 75, y: 0.06}, {x: 85, y: 0.01}, {x: 95, y: 0}, {x: 100, y: 0}]
const carlisleSurvData = [{x: 0, y: 1}, {x: 5, y: 0.67}, {x: 15, y: 0.63}, {x: 25, y: 0.59}, {x: 35, y: 0.54}, {x: 45, y: 0.47}, {x: 55, y: 0.41}, {x: 65, y: 0.30}, {x: 75, y: 0.17}, {x: 85, y: 0.04}, {x: 95, y: 0}, {x: 100, y: 0}]
// Makeham 1867
const governmentAnnuitantsSurvData = [{x: 15, y: 9732.2}, {x: 29, y: 8463.1}, {x: 43, y: 7253}, {x: 57, y: 5845.3}, {x: 71, y: 3676.1}, {x: 85, y: 820.7}]
const peerageFamiliesSurvData = [{x: 14, y: 9780}, {x: 28, y: 8702.8}, {x: 42, y: 7617.6}, {x: 56, y: 6286.2}, {x: 70, y: 3990.3}, {x: 84, y: 979.3}]
const combinedExperienceSurvData = [{x: 20, y: 9337.5}, {x: 32, y: 8478.5}, {x: 44, y: 7532.3}, {x: 56, y: 6197.5}, {x: 68, y: 4053.3}, {x: 80, y: 1303.7}]
const friendlySocMalesSurvData = [{x:18, y:6333}, {x:19, y:6289.7}, {x:20, y:6253.5}, {x:21, y:6201.8}, {x:22, y:6156}, {x:23, y:6108.4}, {x:24, y:6063}, {x:25, y:6020.5}, {x:26, y:5978.1}, {x:27, y:5937.9}, {x:28, y:5893.9}, {x:29, y:5849.7}, {x:30, y:5804.3}, {x:31, y:5754.9}, {x:32, y:5710.6}, {x:33, y:5670.6}, {x:34, y:5623.2}, {x:35, y:5576.9}, {x:36, y:5529.6}, {x:37, y:5486.4}, {x:38, y:5438}, {x:39, y:5385.2}, {x:40, y:5331.4}, {x:41, y:5270.4}, {x:42, y:5217.2}, {x:43, y:5162.9}, {x:44, y:5104.6}, {x:45, y:5044.3}, {x:46, y:4983.6}, {x:47, y:4922.9}, {x:48, y:4857}, {x:49, y:4792.1}, {x:50, y:4721.7}, {x:51, y:4651.6}, {x:52, y:4581.3}, {x:53, y:4501.1}, {x:54, y:4427.2}, {x:55, y:4345.2}, {x:56, y:4259.3}, {x:57, y:4158.1}, {x:58, y:4058.3}, {x:59, y:3947.4}, {x:60, y:3853.1}, {x:61, y:3746.8}, {x:62, y:3659}, {x:63, y:3555.3}, {x:64, y:3439.2}, {x:65, y:3326.7}, {x:66, y:3212.1}, {x:67, y:3069.2}, {x:68, y:2936}, {x:69, y:2801.6}, {x:70, y:2642.5}, {x:71, y:2497}, {x:72, y:2341.1}, {x:73, y:2192.9}, {x:74, y:2055.9}, {x:75, y:1905.2}, {x:76, y:1756.3}, {x:77, y:1593.1}, {x:78, y:1459.6}, {x:79, y:1285.6}, {x:80, y:1156.6}, {x:81, y:1000}, {x:82, y:881.8}]
const clergySurvData = [{x: 24, y: 10049.2}, {x: 37, y: 9366.1}, {x: 50, y: 8325.3}, {x: 63, y: 6262.1}, {x: 76, y: 2777.5}, {x: 89, y: 223.9}]
// Data points from Mathematical functions
const expDecay30SurvData = []
for (let i = 0; i < 100; i += 2) {
expDecay30SurvData.push({x: i, y: Math.exp(-i / 30).toPrecision(3)})
}
const expDecay50SurvData = []
for (let i = 0; i < 100; i += 2) {
expDecay50SurvData.push({x: i, y: Math.exp(-i / 50).toPrecision(3)})
}
const expGrowth20Hazard = []
for (let i = 25; i < 80; i += 2) {
expGrowth20Hazard.push({x: i, y: 0.002 * Math.exp(i / 20).toPrecision(3)})
}
const expGrowthHazard = []
for (let i = 35; i < 85; i += 2) {
expGrowthHazard.push({
x: i,
y: 0.0007 * Math.exp(i / 16).toPrecision(4)
})
}
const linearExpGrowthHazard = []
for (let i = 15; i < 85; i += 2) {
linearExpGrowthHazard.push({
x: i,
y: 0.0045 + 0.0005 * Math.exp(i / 15.5).toPrecision(4)
})
}
// Hazard helper function
let hazardFn = function (el, idx, arr) {
if (idx == 0) return
if (el.y != 0) {
return {
x: el.x,
y: (-1) * (parseFloat(el.y - arr[idx - 1].y) / parseFloat(el.x - arr[idx - 1].x) / parseFloat(el.y)).toPrecision(2)
}
}
}
const deparcieuxHzdData = deparcieuxSurvData.map(hazardFn)
const northamptonHzdData = northamptonSurvData.map(hazardFn)
const swedenMalesHzdData = swedenMalesSurvData.map(hazardFn)
const carlisleHzdData = carlisleSurvData.map(hazardFn)
const governmentAnnuitantsHzdData = governmentAnnuitantsSurvData.map(hazardFn)
const peerageFamiliesHzdData = peerageFamiliesSurvData.map(hazardFn)
const combinedExperienceHzdData = combinedExperienceSurvData.map(hazardFn)
const friendlySocMalesHzdData = friendlySocMalesSurvData.map(hazardFn)
const clergyHzdData = clergySurvData.map(hazardFn)
// Recent mortality data
const elt17Males2015RawData = [{x: 0, y: 0.004766}, {x: 1, y: 0.000306}, {x: 2, y: 0.000207}, {x: 3, y: 0.000147}, {x: 4, y: 0.000115}, {x: 5, y: 0.000099}, {x: 6, y: 0.000091}, {x: 7, y: 0.000088}, {x: 8, y: 0.000088}, {x: 9, y: 0.000089}, {x: 10, y: 0.000091}, {x: 11, y: 0.000094}, {x: 12, y: 0.0001}, {x: 13, y: 0.000112}, {x: 14, y: 0.000134}, {x: 15, y: 0.000172}, {x: 16, y: 0.000233}, {x: 17, y: 0.000311}, {x: 18, y: 0.000391}, {x: 19, y: 0.000457}, {x: 20, y: 0.000496}, {x: 21, y: 0.000516}, {x: 22, y: 0.000529}, {x: 23, y: 0.000537}, {x: 24, y: 0.000543}, {x: 25, y: 0.00055}, {x: 26, y: 0.000566}, {x: 27, y: 0.000594}, {x: 28, y: 0.000634}, {x: 29, y: 0.000682}, {x: 30, y: 0.000728}, {x: 31, y: 0.000763}, {x: 32, y: 0.000796}, {x: 33, y: 0.000838}, {x: 34, y: 0.000895}, {x: 35, y: 0.000969}, {x: 36, y: 0.00106}, {x: 37, y: 0.001163}, {x: 38, y: 0.001274}, {x: 39, y: 0.001379}, {x: 40, y: 0.001474}, {x: 41, y: 0.001576}, {x: 42, y: 0.001706}, {x: 43, y: 0.001858}, {x: 44, y: 0.002017}, {x: 45, y: 0.00217}, {x: 46, y: 0.002314}, {x: 47, y: 0.002455}, {x: 48, y: 0.002623}, {x: 49, y: 0.002839}, {x: 50, y: 0.00311}, {x: 51, y: 0.003435}, {x: 52, y: 0.003798}, {x: 53, y: 0.004178}, {x: 54, y: 0.004587}, {x: 55, y: 0.005061}, {x: 56, y: 0.005599}, {x: 57, y: 0.006153}, {x: 58, y: 0.006709}, {x: 59, y: 0.007326}, {x: 60, y: 0.008049}, {x: 61, y: 0.008812}, {x: 62, y: 0.009538}, {x: 63, y: 0.010312}, {x: 64, y: 0.011291}, {x: 65, y: 0.012482}, {x: 66, y: 0.013763}, {x: 67, y: 0.015122}, {x: 68, y: 0.016734}, {x: 69, y: 0.018712}, {x: 70, y: 0.020908}, {x: 71, y: 0.023125}, {x: 72, y: 0.0254}, {x: 73, y: 0.027864}, {x: 74, y: 0.030658}, {x: 75, y: 0.033931}, {x: 76, y: 0.037755}, {x: 77, y: 0.042067}, {x: 78, y: 0.046868}, {x: 79, y: 0.052464}, {x: 80, y: 0.059114}, {x: 81, y: 0.066728}, {x: 82, y: 0.075115}, {x: 83, y: 0.084348}, {x: 84, y: 0.094629}, {x: 85, y: 0.10611}, {x: 86, y: 0.118909}, {x: 87, y: 0.133183}, {x: 88, y: 0.149149}, {x: 89, y: 0.166598}, {x: 90, y: 0.184091}, {x: 91, y: 0.200675}, {x: 92, y: 0.218727}, {x: 93, y: 0.241585}, {x: 94, y: 0.2686}, {x: 95, y: 0.297347}, {x: 96, y: 0.326313}, {x: 97, y: 0.356129}, {x: 98, y: 0.38647}, {x: 99, y: 0.41785}, {x: 100, y: 0.449217}, {x: 101, y: 0.48015}, {x: 102, y: 0.505257}, {x: 103, y: 0.529348}, {x: 104, y: 0.555803}, {x: 105, y: 0.58012}, {x: 106, y: 0.600178}, {x: 107, y: 0.625666}, {x: 108, y: 0.651023}, {x: 109, y: 0.676172}, {x: 110, y: 0.701065}, {x: 111, y: 0.725677}, {x: 112, y: 0.75104}]
const elt17Females2015RawData = [{x:0, y:0.003824}, {x:1, y:0.000238}, {x:2, y:0.000176}, {x:3, y:0.000133}, {x:4, y:0.000107}, {x:5, y:0.00009}, {x:6, y:0.000079}, {x:7, y:0.000073}, {x:8, y:0.00007}, {x:9, y:0.000069}, {x:10, y:0.000071}, {x:11, y:0.000076}, {x:12, y:0.000084}, {x:13, y:0.000094}, {x:14, y:0.000108}, {x:15, y:0.000124}, {x:16, y:0.000143}, {x:17, y:0.000162}, {x:18, y:0.000178}, {x:19, y:0.000191}, {x:20, y:0.0002}, {x:21, y:0.000208}, {x:22, y:0.000216}, {x:23, y:0.000226}, {x:24, y:0.000239}, {x:25, y:0.000254}, {x:26, y:0.000271}, {x:27, y:0.000289}, {x:28, y:0.000309}, {x:29, y:0.000332}, {x:30, y:0.000359}, {x:31, y:0.00039}, {x:32, y:0.000425}, {x:33, y:0.000465}, {x:34, y:0.000508}, {x:35, y:0.000555}, {x:36, y:0.000607}, {x:37, y:0.000662}, {x:38, y:0.000721}, {x:39, y:0.000786}, {x:40, y:0.00086}, {x:41, y:0.000947}, {x:42, y:0.001042}, {x:43, y:0.001142}, {x:44, y:0.001246}, {x:45, y:0.001355}, {x:46, y:0.001472}, {x:47, y:0.001603}, {x:48, y:0.001754}, {x:49, y:0.001931}, {x:50, y:0.002139}, {x:51, y:0.002369}, {x:52, y:0.002606}, {x:53, y:0.002847}, {x:54, y:0.003101}, {x:55, y:0.003379}, {x:56, y:0.003693}, {x:57, y:0.004054}, {x:58, y:0.004458}, {x:59, y:0.004895}, {x:60, y:0.005342}, {x:61, y:0.00577}, {x:62, y:0.006197}, {x:63, y:0.006696}, {x:64, y:0.00732}, {x:65, y:0.008048}, {x:66, y:0.008854}, {x:67, y:0.009764}, {x:68, y:0.01083}, {x:69, y:0.012055}, {x:70, y:0.013398}, {x:71, y:0.014837}, {x:72, y:0.016397}, {x:73, y:0.018125}, {x:74, y:0.020131}, {x:75, y:0.022536}, {x:76, y:0.02539}, {x:77, y:0.028692}, {x:78, y:0.032455}, {x:79, y:0.036714}, {x:80, y:0.041548}, {x:81, y:0.04716}, {x:82, y:0.053771}, {x:83, y:0.061347}, {x:84, y:0.069722}, {x:85, y:0.078882}, {x:86, y:0.088986}, {x:87, y:0.10049}, {x:88, y:0.113837}, {x:89, y:0.129012}, {x:90, y:0.145126}, {x:91, y:0.161889}, {x:92, y:0.180129}, {x:93, y:0.200354}, {x:94, y:0.222724}, {x:95, y:0.246836}, {x:96, y:0.272288}, {x:97, y:0.299053}, {x:98, y:0.327165}, {x:99, y:0.356678}, {x:100, y:0.38763}, {x:101, y:0.419763}, {x:102, y:0.452164}, {x:103, y:0.484643}, {x:104, y:0.516616}, {x:105, y:0.548223}, {x:106, y:0.578589}, {x:107, y:0.609029}, {x:108, y:0.639523}, {x:109, y:0.66876}, {x:110, y:0.697334}, {x:111, y:0.724789}, {x:112, y:0.75104}, {x:113, y:0.776042}, {x:114, y:0.799785}]
const braMales2018RawData = [{x:0, y:0.013305251}, {x:1, y:0.000912361}, {x:2, y:0.000600557}, {x:3, y:0.000462823}, {x:4, y:0.000382937}, {x:5, y:0.000330729}, {x:6, y:0.000294797}, {x:7, y:0.000270192}, {x:8, y:0.000254978}, {x:9, y:0.000249142}, {x:10, y:0.000254396}, {x:11, y:0.00027451}, {x:12, y:0.000316127}, {x:13, y:0.000390211}, {x:14, y:0.000514428}, {x:15, y:0.001024113}, {x:16, y:0.00131045}, {x:17, y:0.001570685}, {x:18, y:0.001783584}, {x:19, y:0.00195508}, {x:20, y:0.002126731}, {x:21, y:0.002292854}, {x:22, y:0.002402379}, {x:23, y:0.002440102}, {x:24, y:0.002424355}, {x:25, y:0.002384455}, {x:26, y:0.002350482}, {x:27, y:0.002331569}, {x:28, y:0.002342369}, {x:29, y:0.002377347}, {x:30, y:0.002417145}, {x:31, y:0.002454886}, {x:32, y:0.002504029}, {x:33, y:0.00256626}, {x:34, y:0.002641968}, {x:35, y:0.00273272}, {x:36, y:0.002837271}, {x:37, y:0.002953552}, {x:38, y:0.003081158}, {x:39, y:0.003223464}, {x:40, y:0.003383443}, {x:41, y:0.003567167}, {x:42, y:0.003780202}, {x:43, y:0.004026866}, {x:44, y:0.004306197}, {x:45, y:0.004612683}, {x:46, y:0.004945825}, {x:47, y:0.005311632}, {x:48, y:0.005712027}, {x:49, y:0.006146991}, {x:50, y:0.00661607}, {x:51, y:0.007118684}, {x:52, y:0.007655583}, {x:53, y:0.008227478}, {x:54, y:0.008836773}, {x:55, y:0.009495883}, {x:56, y:0.010201377}, {x:57, y:0.010939109}, {x:58, y:0.011705937}, {x:59, y:0.01251561}, {x:60, y:0.013386462}, {x:61, y:0.014341832}, {x:62, y:0.015398021}, {x:63, y:0.016573681}, {x:64, y:0.017875219}, {x:65, y:0.019271166}, {x:66, y:0.020790484}, {x:67, y:0.022512871}, {x:68, y:0.024481977}, {x:69, y:0.026688499}, {x:70, y:0.029072112}, {x:71, y:0.031624578}, {x:72, y:0.03441465}, {x:73, y:0.037470656}, {x:74, y:0.040801086}, {x:75, y:0.044391345}, {x:76, y:0.048255282}, {x:77, y:0.052447815}, {x:78, y:0.057008029}, {x:79, y:0.061964767}]
const braFemales2018RawData = [{x:0, y:0.01135075}, {x:1, y:0.000762192}, {x:2, y:0.000475833}, {x:3, y:0.000354694}, {x:4, y:0.000286331}, {x:5, y:0.000242503}, {x:6, y:0.000212689}, {x:7, y:0.000192274}, {x:8, y:0.000179258}, {x:9, y:0.00017316}, {x:10, y:0.000174698}, {x:11, y:0.00018585}, {x:12, y:0.000220354}, {x:13, y:0.0002638}, {x:14, y:0.000305164}, {x:15, y:0.000345144}, {x:16, y:0.000393491}, {x:17, y:0.000432514}, {x:18, y:0.000457133}, {x:19, y:0.000471272}, {x:20, y:0.000484349}, {x:21, y:0.000501258}, {x:22, y:0.000518537}, {x:23, y:0.00053725}, {x:24, y:0.00055793}, {x:25, y:0.000579402}, {x:26, y:0.000603218}, {x:27, y:0.000632872}, {x:28, y:0.000669955}, {x:29, y:0.000713526}, {x:30, y:0.000763392}, {x:31, y:0.000816566}, {x:32, y:0.000870033}, {x:33, y:0.000922208}, {x:34, y:0.000975901}, {x:35, y:0.001036198}, {x:36, y:0.001106734}, {x:37, y:0.00118751}, {x:38, y:0.001280255}, {x:39, y:0.001385684}, {x:40, y:0.001500579}, {x:41, y:0.001628138}, {x:42, y:0.001776007}, {x:43, y:0.001947448}, {x:44, y:0.002139665}, {x:45, y:0.002350509}, {x:46, y:0.002573159}, {x:47, y:0.002801908}, {x:48, y:0.0030331}, {x:49, y:0.003271567}, {x:50, y:0.003528935}, {x:51, y:0.003810047}, {x:52, y:0.004110319}, {x:53, y:0.004431284}, {x:54, y:0.004776873}, {x:55, y:0.00515692}, {x:56, y:0.005572668}, {x:57, y:0.006018791}, {x:58, y:0.00649634}, {x:59, y:0.007014587}, {x:60, y:0.007583564}, {x:61, y:0.008218058}, {x:62, y:0.008930942}, {x:63, y:0.009734577}, {x:64, y:0.010633007}, {x:65, y:0.011615678}, {x:66, y:0.012694255}, {x:67, y:0.013901034}, {x:68, y:0.01525454}, {x:69, y:0.016757646}, {x:70, y:0.018383764}, {x:71, y:0.020150681}, {x:72, y:0.022118388}, {x:73, y:0.024320167}, {x:74, y:0.026756896}, {x:75, y:0.029376389}, {x:76, y:0.032199577}, {x:77, y:0.035326334}, {x:78, y:0.038812682}, {x:79, y:0.042664106}]
const usaFemales2017RawData = [{x:0, y:0.005226429}, {x:1, y:0.000339063}, {x:2, y:0.000207538}, {x:3, y:0.000159215}, {x:4, y:0.000138854}, {x:5, y:0.000125572}, {x:6, y:0.000112958}, {x:7, y:0.000103622}, {x:8, y:9.66915E-05}, {x:9, y:9.23788E-05}, {x:10, y:9.20186E-05}, {x:11, y:9.78959E-05}, {x:12, y:0.000112694}, {x:13, y:0.000137937}, {x:14, y:0.000171663}, {x:15, y:0.000209989}, {x:16, y:0.00025021}, {x:17, y:0.00029255}, {x:18, y:0.000335662}, {x:19, y:0.000379035}, {x:20, y:0.000424451}, {x:21, y:0.000470575}, {x:22, y:0.000513099}, {x:23, y:0.000549985}, {x:24, y:0.00058268}, {x:25, y:0.000613459}, {x:26, y:0.000645689}, {x:27, y:0.000681731}, {x:28, y:0.000724494}, {x:29, y:0.000773766}, {x:30, y:0.000828459}, {x:31, y:0.000884733}, {x:32, y:0.000939507}, {x:33, y:0.000989236}, {x:34, y:0.001035546}, {x:35, y:0.001086647}, {x:36, y:0.001144198}, {x:37, y:0.001202918}, {x:38, y:0.001264072}, {x:39, y:0.001332405}, {x:40, y:0.001414246}, {x:41, y:0.001513007}, {x:42, y:0.001626327}, {x:43, y:0.001750333}, {x:44, y:0.001883346}, {x:45, y:0.002024771}, {x:46, y:0.002182611}, {x:47, y:0.002365926}, {x:48, y:0.002583846}, {x:49, y:0.00283627}, {x:50, y:0.003105002}, {x:51, y:0.003390508}, {x:52, y:0.003711457}, {x:53, y:0.004066173}, {x:54, y:0.004441261}, {x:55, y:0.004829319}, {x:56, y:0.005221006}, {x:57, y:0.00561329}, {x:58, y:0.006011385}, {x:59, y:0.006429093}, {x:60, y:0.006880441}, {x:61, y:0.007371199}, {x:62, y:0.00790339}, {x:63, y:0.008480929}, {x:64, y:0.009110869}, {x:65, y:0.009792981}, {x:66, y:0.010567729}, {x:67, y:0.011436491}, {x:68, y:0.012473745}, {x:69, y:0.013659073}, {x:70, y:0.014881229}, {x:71, y:0.016529232}, {x:72, y:0.018210264}, {x:73, y:0.020010984}, {x:74, y:0.021903079}, {x:75, y:0.02432197}, {x:76, y:0.02689858}, {x:77, y:0.029885624}, {x:78, y:0.033412755}, {x:79, y:0.037064902}, {x:80, y:0.041477967}, {x:81, y:0.046149768}, {x:82, y:0.051680829}, {x:83, y:0.05858719}, {x:84, y:0.065586209}, {x:85, y:0.072854951}, {x:86, y:0.081114702}, {x:87, y:0.091617569}, {x:88, y:0.103240602}, {x:89, y:0.116040625}, {x:90, y:0.130061284}, {x:91, y:0.145328909}, {x:92, y:0.161848158}, {x:93, y:0.179598287}, {x:94, y:0.198529512}, {x:95, y:0.21856083}, {x:96, y:0.239578903}, {x:97, y:0.261438966}, {x:98, y:0.283967406}, {x:99, y:0.306966633}]
const usaMales2017RawData = [{x:0, y:0.006302347}, {x:1, y:0.000422686}, {x:2, y:0.000287435}, {x:3, y:0.000224945}, {x:4, y:0.000158165}, {x:5, y:0.000155541}, {x:6, y:0.000138456}, {x:7, y:0.000124364}, {x:8, y:0.000110411}, {x:9, y:9.82037E-05}, {x:10, y:9.39768E-05}, {x:11, y:0.000107861}, {x:12, y:0.000151557}, {x:13, y:0.000231895}, {x:14, y:0.00034139}, {x:15, y:0.000461328}, {x:16, y:0.000584161}, {x:17, y:0.000717579}, {x:18, y:0.000858598}, {x:19, y:0.001001487}, {x:20, y:0.001147071}, {x:21, y:0.001285942}, {x:22, y:0.001402833}, {x:23, y:0.001489993}, {x:24, y:0.001553819}, {x:25, y:0.001609192}, {x:26, y:0.00166366}, {x:27, y:0.0017133}, {x:28, y:0.001761547}, {x:29, y:0.001810307}, {x:30, y:0.001858758}, {x:31, y:0.001907458}, {x:32, y:0.001959197}, {x:33, y:0.002014129}, {x:34, y:0.002071627}, {x:35, y:0.002139015}, {x:36, y:0.002211334}, {x:37, y:0.002276815}, {x:38, y:0.002332931}, {x:39, y:0.002389836}, {x:40, y:0.002462646}, {x:41, y:0.002565524}, {x:42, y:0.002700979}, {x:43, y:0.002869693}, {x:44, y:0.00306605}, {x:45, y:0.003280112}, {x:46, y:0.003520103}, {x:47, y:0.00380392}, {x:48, y:0.004145976}, {x:49, y:0.004546886}, {x:50, y:0.004978296}, {x:51, y:0.005440931}, {x:52, y:0.005965096}, {x:53, y:0.006548528}, {x:54, y:0.00716992}, {x:55, y:0.007803182}, {x:56, y:0.008444663}, {x:57, y:0.009116184}, {x:58, y:0.009838087}, {x:59, y:0.010619323}, {x:60, y:0.01146991}, {x:61, y:0.012361295}, {x:62, y:0.013260341}, {x:63, y:0.014139737}, {x:64, y:0.015018866}, {x:65, y:0.015941706}, {x:66, y:0.017026432}, {x:67, y:0.018188728}, {x:68, y:0.019483164}, {x:69, y:0.02099042}, {x:70, y:0.022447998}, {x:71, y:0.024630912}, {x:72, y:0.026569555}, {x:73, y:0.029040491}, {x:74, y:0.031539336}, {x:75, y:0.034644466}, {x:76, y:0.038148299}, {x:77, y:0.042249702}, {x:78, y:0.04652179}, {x:79, y:0.051400777}, {x:80, y:0.056782775}, {x:81, y:0.062514141}, {x:82, y:0.069451943}, {x:83, y:0.077621564}, {x:84, y:0.08615537}, {x:85, y:0.095450498}, {x:86, y:0.105788276}, {x:87, y:0.118526995}, {x:88, y:0.132437468}, {x:89, y:0.147541076}, {x:90, y:0.163838893}, {x:91, y:0.181308255}, {x:92, y:0.199899748}, {x:93, y:0.219535381}, {x:94, y:0.240107596}, {x:95, y:0.261480212}, {x:96, y:0.283490926}, {x:97, y:0.30595538}, {x:98, y:0.328673065}, {x:99, y:0.351434112}]
// Fitted exponential curves
const fitELT17MalesHazard = []
for (let i = 26; i < 100; i += 1) {
fitELT17MalesHazard.push({
x: i,
y: (Math.pow(10, -4.4673) * Math.pow(10, i * 0.0406)).toPrecision(4)
})
}
const fitELT17FemalesHazard = []
for (let i = 26; i < 100; i += 1) {
fitELT17FemalesHazard.push({
x: i,
y: (Math.pow(10, -4.8111) * Math.pow(10, i * 0.0432)).toPrecision(4)
})
}
const fitBraMalesHazard = []
for (let i = 26; i < 80; i += 1) {
fitBraMalesHazard.push({
x: i,
y: (Math.pow(10, -3.5531) * Math.pow(10, i * 0.0285)).toPrecision(4)
})
}
const fitBraFemalesHazard = []
for (let i = 26; i < 80; i += 1) {
fitBraFemalesHazard.push({
x: i,
y: (Math.pow(10, -4.2067) * Math.pow(10, i * 0.0352)).toPrecision(4)
})
}
const fitUSAMalesHazard = []
for (let i = 26; i < 100; i += 1) {
fitUSAMalesHazard.push({
x: i,
y: (Math.pow(10, -3.9064) * Math.pow(10, i * 0.0337)).toPrecision(4)
})
}
const fitUSAFemalesHazard = []
for (let i = 26; i < 100; i += 1) {
fitUSAFemalesHazard.push({
x: i,
y: (Math.pow(10, -4.3268) * Math.pow(10, i * 0.0373)).toPrecision(4)
})
}
buildCharts()
</script>
<div class="footnotes" role="doc-endnotes">
<ol>
<li id="fn:1" role="doc-endnote">
<p>One can read commentary about issues with datas in <a href="https://doi.org/10.1017/S204616580000126X">Makeham 1860</a>, <a href="https://doi.org/10.1017/S2046165800002823">Makeham 1866</a>, <a href="https://doi.org/10.1017/S2046166600003238">Makeham 1867</a>, and <a href="https://doi.org/10.1017/S2046167400045912">Sutton 1884</a>. <a href="#fnref:1" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:2" role="doc-endnote">
<p>de Moivre based his calculations on a table assembled by <a href="https://en.wikipedia.org/wiki/Edmond_Halley">Edmond Halley</a> — the astronomer of comet’s fame. Halley used mortality data from Wrocław, Poland, and published his table in a 1693 paper, <a href="http://www.pierre-marteau.com/editions/1693-mortality.html">An Estimate of the Degrees of the Mortality of Mankind</a>. <a href="#fnref:2" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:3" role="doc-endnote">
<p>A good, short, readable paper with the historical context of early <a href="https://en.wikipedia.org/wiki/Actuarial_science">Actuarial science</a> is <a href="https://doi.org/10.1017/S0020268100054044">Ogborn 1953</a>. <a href="#fnref:3" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:4" role="doc-endnote">
<p>I couldn’t get my hands at the full Finlaison tables. They were originally published in 1829 as a <a href="https://books.google.com/books/about/Report_of_John_Finlaison_Actuary_of_the.html?id=tbXrGwAACAAJ">Report of John Finlaison, Actuary of the National Debt, on the Evidence and Elementary Facts on which the Tables of Life Annuities are Founded: Ordered by the House of Commons, to be Printed 31 March 1829</a>. The 10-volume set <a href="https://www.amazon.com/History-Actuarial-Science-10-Set/dp/1851961437/">History of Actuarial Science</a>, edited by Steven Haberman and Trevor A. Sibbett and published in 1995, also has the tables (and much more). The set is the historical <em><a href="https://en.wiktionary.org/wiki/vade_mecum">vade mecum</a></em> of the field, your local library might have it. Look for the 1829 Finlaison tables in <a href="https://books.google.com/books?id=L8XuAAAAMAAJ">Volume 2</a>. <a href="#fnref:4" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:5" role="doc-endnote">
<p>Our data sources are:</p>
<ul>
<li><a href="https://stacks.cdc.gov/view/cdc/79487">United States Life Tables, 2017</a> (<a href="https://www.cdc.gov/nchs/data/nvsr/nvsr68/nvsr68_07-508.pdf">PDF</a>) by Elizabeth Arias and Jiaquan Xu (<a href="https://www.cdc.gov/nchs/products/life_tables.htm">CDC/NCHS Life Tables</a>), published in 2019</li>
<li><a href="https://www.ons.gov.uk/peoplepopulationandcommunity/birthsdeathsandmarriages/lifeexpectancies/bulletins/englishlifetablesno17/2015-09-01">English Life Tables No. 17: 2010 to 2012</a> (<a href="https://webarchive.nationalarchives.gov.uk/20160106044025/https://ons.gov.uk/ons/guide-method/method-quality/specific/population-and-migration/demography/english-life-tables--no--17---2010-2012.pdf">PDF</a>) by Jakub Bijak, Erengul Dodd, Jonathan J. Forster, and Peter W. F. Smith, published in 2015</li>
<li><a href="https://biblioteca.ibge.gov.br/index.php/biblioteca-catalogo?view=detalhes&id=73097">Tábua completa de mortalidade para o Brasil – 2018</a> (<a href="https://biblioteca.ibge.gov.br/visualizacao/periodicos/3097/tcmb_2018.pdf">PDF</a>) by Instituto Brasileiro de Geografia e Estatística (<a href="https://www.ibge.gov.br/en/home-eng.html">IBGE</a>), published in 2019</li>
</ul>
<p><a href="#fnref:5" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:6" role="doc-endnote">
<p>For instance, in the paper about ELT 17 Methodology (<a href="https://webarchive.nationalarchives.gov.uk/20160106044025/https://ons.gov.uk/ons/guide-method/method-quality/specific/population-and-migration/demography/english-life-tables--no--17---2010-2012.pdf">PDF</a>) the authors write: “Graduation (smoothing) is particularly important at the oldest ages, where exposure numbers are small and data are sparse.” <a href="#fnref:6" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:7" role="doc-endnote">
<p>Another example is <a href="https://en.wikipedia.org/wiki/Thomas_Rowe_Edmonds">Thomas Rowe Edmonds</a>’s 1832 book <a href="http://hdl.handle.net/2027/coo1.ark:/13960/t7np2nd8c">Life Tables, Founded Upon the Discovery of a Numerical Law Regulating the Existence of Every Human Being, Ilustrated by a New Theory of the Causes Producing Health and Longevity</a>. Edmonds proposed the use of three exponential curves to account for the hazard rate throughout all ages. Each exponential curve corresponded to one of the “three grand divisions of life”: Infancy (from birth to 8 years), Manhood (from 12 to 55 years), and Old Age (from 55 to end of life). Incidentally, it was Edmonds who first coined the name “force of mortality”, which is used in Actuarial science to this day and replaced Gompertz’s “intensity of mortality”. <a href="#fnref:7" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
<li id="fn:8" role="doc-endnote">
<p><a href="https://en.wikipedia.org/wiki/John_von_Neumann">John von Neumann</a> is often quoted having said: “With four parameters I can fit an elephant, and with five I can make him wiggle his trunk.” For an actual elephant-fitting function, check out this cool 2010 paper, <a href="https://doi.org/10.1119/1.3254017">Drawing an elephant with four complex parameters</a>. <a href="#fnref:8" class="reversefootnote" role="doc-backlink">↩</a></p>
</li>
</ol>
</div>