Category Archives: Math

How Fast is Exponential Growth? (Or, Yao Ming Confronts the Vastness of the Universe)


via How Fast is Exponential Growth? (Or, Yao Ming Confronts the Vastness of the Universe).

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Mental Model: Continuous Compounding

A mental model is an explanation of a thought process about how something works in the real world. Mental models help shape our behaviour and define our approach to solving problems (akin to a personal algorithm) and carrying out tasks. Mental models have been studied by cognitive scientists as part of efforts to understand how humans know, perceive, make decisions, and construct behavior in a variety of environments. Charles Munger provides a concept of “Elementary, Worldly Wisdom” which consists of a set of mental models framed as a solving problems of business, finance and investing. According to Munger, only 80 or 90 important models will carry about 90% of the freight in making you a worldly-wise person.

Articles on Continuous Compounding as a Mental Model for Investing

In Three consecutive Letters to Partners, Warren Buffett wrote of the “Joys of Compounding”. You can find the excerpts below, but also recommended is the coverage of the letters in an East Coast Asset Management Letter to Investors from 3Q2010.

Buffett on the Joys of Compounding

1963 LtS

“I have it from unreliable sources that the cost of the voyage Isabella originally underwrote for Columbus was approximately $30,000. This has been considered at least a moderately successful utilization of venture capital. Without attempting to evaluate the psychic income derived from finding a new hemisphere, it must be pointed out that even had squatter’s rights prevailed, the whole deal was not exactly another IBM. Figured very roughly, the $30,000 invested at 4% compounded annually would have amounted to something like $2,000,000,000,000 (that’s $2 trillion for those of you who are not government statisticians) by 1962. Historical apologists for the Indians of Manhattan may find refuge in similar calculations. Such fanciful geometric progressions illustrate the value of either living a long time, or compounding your money at a decent rate. I have nothing particularly helpful to say on the former point.

The above table8 indicates the compounded value of $1,000,000 at 4%, 8% 12, and 16% for 10, 20 and 30 years. It is always startling to see how relatively small differences in rates add up to very significant sums over a period of years. That is why, even though we are shooting for more, we feel that a few percentage points advantage over the Dow is a very worthwhile achievement. It can mean a lot of dollars over a decade or two.”

1964 LtS

“Now to the pulse-quickening portion of our essay. Last year, in order to drive home the point on compounding, I took a pot shot at Queen Isabella and her financial advisors. You will remember they were euchred into such an obviously low-compound situation as the discovery of a new hemisphere.

Since the whole subject of compounding has such a crass ring to it, I will attempt to introduce a little class into this discussion by turning to the art world. Francis I of France paid 4,000 ecus in 1540 for Leonardo da Vinci’s Mona Lisa. On the off chance that a few of you have not kept track of the fluctuations of the ecu 4,000 converted out to about $20,000. If Francis had kept his feet on the ground and he (and his trustees) had been able to find a 6% after-tax investment, the estate now would be worth something over $1,000,000,000,000,000. That’s $1 quadrillion or over 3,000 times the present national debt, all from 6%. I trust this will end all discussion in our household about any purchase of paintings qualifying as an investment. However, as I pointed out last year, there are other morals to be drawn here. One is the wisdom of living a long time. The other impressive factor is the swing produced by relatively small changes in the rate of compound. Above are shown the gains from $1,000,000 compounded at various rates.

It is obvious that a variation of merely a few percentage points has an enormous effect on the success of a compounding (investment) program. It is also obvious that this effect mushrooms as the period lengthens. If, over a meaningful period of time, Buffett Partnership can achieve an edge of even a modest number of percentage points over the major investment media, its function will be fulfilled.”

1965 LtS

“Our last two excursions into the mythology of financial expertise have revealed that purportedly shrewd investments by Isabella (backing the voyage of Columbus) and Francis I (original purchase of Mona Lisa) bordered on fiscal lunacy. Apologists for these parties have presented an array of sentimental trivia. Through it all, our compounding tables have not been dented by attack. Nevertheless, one criticism has stung a bit. The charge has been made that this column has acquired a negative tone with only the financial incompetents of history receiving comment. We have been challenged to record on these pages a story of financial perspicacity which will be a bench mark of brilliance down through the ages.

One story stands out. This, of course, is the saga of trading acumen etched into history by the Manhattan Indians when they unloaded their island to that notorious spendthrift, Peter Minuit in 1626. My understanding is that they received $24 net. For this, Minuit received 22.3 square miles which works out to about 621,688,320 square feet. While on the basis of comparable sales, it is difficult to arrive at a precise appraisal, a $20 per square foot estimate seems reasonable giving a current land value for the island of $12,433,766,400 ($12 1/2 billion). To the novice, perhaps this sounds like a decent deal. However, the Indians have only had to achieve a 6 1/2% return (The tribal mutual fund representative would have promised them this.) to obtain the last laugh on Minuit. At 6 1/2%, $24 becomes $42,105,772,800 ($42 billion) in 338 years, and if they just managed to squeeze out an extra half point to get to 7%, the present value becomes $205 billion. So much for that. Some of you may view your investment policies on a shorter term basis. For your convenience, we include our usual table indicating the gains from compounding $1,000,000 at various rates.

This table indicates the financial advantages of:
(1) A long life (in the erudite vocabulary of the financial sophisticate this is referred to as the Methusalah Technique)
(2) A high compound rate
(3) A combination of both (especially recommended by this author)
To be observed are the enormous benefits produced by relatively small gains in the annual earnings rate. This explains our attitude which while hopeful of achieving a striking margin of superiority over average investment results, nevertheless, regards every percentage point of investment return above average as having real meaning.”


How many times do you need to fold a piece of paper to make it reach the moon? on
(The answer is 42…)

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