Legg Mason Capital Management CEO Bill Miller, among the most consistently successful investors of recent decades, wrote two years ago that "the Kelly criterion is integral to the way we manage money." BusinessWeek September 26, 2005

Last year, Charles Munger recommended an excellent book, Fortunes Formula by William Poundstone. The book discusses John Kellys formula, the Kelly criterion. The Kelly criterion is a formula for wagering the optimal amount of ones bankroll to maximize the growth rate of that bankroll. Such wagers must be repeatable and possess positive expected values. John L. Kelly first described the Kelly criterion in a Bell System Technical Journal article in August 1956. The Kelly criterion maximizes long-term growth of ones bankroll while eliminating total ruin by not allowing any bet to reach 100% of ones bankroll. The Kelly criterion or formula will help one understand how to size their investment positions according to ones edge.

The Kelly criterion or formula is Edge/Odds = f. Edge is the expected value of the bet or in this case investment. Odds reflect the markets expectation for how much a person would win if they were successful, and f represents the percentage of ones bankroll they should wager.

In order to successfully apply this formula one must possess an edge. Back in the 1950s the edge on horse races was the private wire. This private wire was the equivalent of insider information in investing. The edge, the Kelly criterions numerator, is the expected value of a proposition. The expected value is composed of subjective probabilities, an extremely difficult task unless one has a strong understanding of the situation. This is the reason Buffett advocates investing in situations which reside within ones circle of competence.

"we try to exert a Ted Williams kind of discipline. In his book The Science of Hitting, Ted explains that he carved the strike zone into 77 cells, each the size of a baseball. Swinging only at balls in his "best" cell, he knew, would allow him to bat .400; reaching for balls in his "worst" spot, the low outside corner of the strike zone, would reduce him to .230. In other words, waiting for the fat pitch would mean a trip to the Hall of Fame; swinging indiscriminately would mean a ticket to the minors." Warren Buffett, 1997 Berkshire Hathaway Letter to Shareholders

Looking at Warren Buffetts investments, he seems to have structured his portfolio in a way similar to the Kelly criterion. When Mr. Buffett ran a private partnership in the 1950s he invested as much as 40% of the partnerships assets in American Express. He placed such a large portion of his partnership in this one investment because that is where he maintained an edge.

Looking at Berkshire Hathaways portfolio we can see multiple position sizes reflecting a similar portfolio allocation as the Kelly criterion would imply. In the 2006 Annual Report, Coca-Cola comprised 15.7% of Berkshire s $61.5 billion dollar equity portfolio, while the smallest position M&T Bank Corporation represents only 1.3% of the portfolio. You can see from both these examples why I believe he is intuitively applying the Kelly criterion.

2006 Wesco Annual Meeting:

Question (Shai Dardashti, Dardashti Capital Management): Would you use Kelly criteria to size positions?

Charles Munger: Oh, that is a very good question. The first time I read about that sizing system, my take was that it seemed plausible to me, but I havent run that formula through my head and I wont. You couldnt apply it to the investment operations Ive run [I think because of Berkshire s size], but the gist of it in terms of sizing your bet makes sense. Whoever developed that formula has an approach to life similar to mine.

The extreme volatility of the Kelly criterion can punish an investor with more than a 50% loss on his/her bankroll. Due to this extreme volatility most people apply a half or a quarter Kelly to the results of their equation. Most investors possess psychological biases that make an investor unable to properly deal with a 50% loss, even though in the long run they would be maximizing ones bankroll.

An example of this extreme volatility can be found in Mr. Mugners past investment partnership. When he ran his partnership in the 1960s he invested in as little as three equities. As noted in the table below, his partnership compounded money at 20% per annum, net of fees, from 1962-1972. If the years 1973-1975 are included, the returns are dragged down to 13.7% net of fees ( Poor Charlies Almanack and Martin Capital Management 2006 annual letter). This result would be unbearable for many investors; they would sell out before the incredible performance of 1975 lifted their returns back up. The returns are extraordinary but they display the volatility that one must be willing to stomach.

Anyone managing money should look into using the Kelly criterion. Whether one chooses to scale it down due to volatility is up to them. Not using the Kelly criterion or a concentrated portfolio like Mr. Buffett and Mr. Munger seems sub-optimal as the Kelly criterion maximizes the long-term growth of ones portfolio. To learn further about the Kelly criterion, reference William Poundstones book Fortunes Formula.

Further Reading on Kelly criterion:

http://www.businessweek.com/magazine/content/05_39/b3952138.htm

http://www.capatcolumbia.com/MM%20LMCM%20reports/Size%20Matters.pdf

Kelly criterion Calculator:

http://www.albionresearch.com/kelly/default.php

Last year, Charles Munger recommended an excellent book, Fortunes Formula by William Poundstone. The book discusses John Kellys formula, the Kelly criterion. The Kelly criterion is a formula for wagering the optimal amount of ones bankroll to maximize the growth rate of that bankroll. Such wagers must be repeatable and possess positive expected values. John L. Kelly first described the Kelly criterion in a Bell System Technical Journal article in August 1956. The Kelly criterion maximizes long-term growth of ones bankroll while eliminating total ruin by not allowing any bet to reach 100% of ones bankroll. The Kelly criterion or formula will help one understand how to size their investment positions according to ones edge.

The Kelly criterion or formula is Edge/Odds = f. Edge is the expected value of the bet or in this case investment. Odds reflect the markets expectation for how much a person would win if they were successful, and f represents the percentage of ones bankroll they should wager.

In order to successfully apply this formula one must possess an edge. Back in the 1950s the edge on horse races was the private wire. This private wire was the equivalent of insider information in investing. The edge, the Kelly criterions numerator, is the expected value of a proposition. The expected value is composed of subjective probabilities, an extremely difficult task unless one has a strong understanding of the situation. This is the reason Buffett advocates investing in situations which reside within ones circle of competence.

"we try to exert a Ted Williams kind of discipline. In his book The Science of Hitting, Ted explains that he carved the strike zone into 77 cells, each the size of a baseball. Swinging only at balls in his "best" cell, he knew, would allow him to bat .400; reaching for balls in his "worst" spot, the low outside corner of the strike zone, would reduce him to .230. In other words, waiting for the fat pitch would mean a trip to the Hall of Fame; swinging indiscriminately would mean a ticket to the minors." Warren Buffett, 1997 Berkshire Hathaway Letter to Shareholders

Looking at Warren Buffetts investments, he seems to have structured his portfolio in a way similar to the Kelly criterion. When Mr. Buffett ran a private partnership in the 1950s he invested as much as 40% of the partnerships assets in American Express. He placed such a large portion of his partnership in this one investment because that is where he maintained an edge.

Looking at Berkshire Hathaways portfolio we can see multiple position sizes reflecting a similar portfolio allocation as the Kelly criterion would imply. In the 2006 Annual Report, Coca-Cola comprised 15.7% of Berkshire s $61.5 billion dollar equity portfolio, while the smallest position M&T Bank Corporation represents only 1.3% of the portfolio. You can see from both these examples why I believe he is intuitively applying the Kelly criterion.

2006 Wesco Annual Meeting:

Question (Shai Dardashti, Dardashti Capital Management): Would you use Kelly criteria to size positions?

Charles Munger: Oh, that is a very good question. The first time I read about that sizing system, my take was that it seemed plausible to me, but I havent run that formula through my head and I wont. You couldnt apply it to the investment operations Ive run [I think because of Berkshire s size], but the gist of it in terms of sizing your bet makes sense. Whoever developed that formula has an approach to life similar to mine.

The extreme volatility of the Kelly criterion can punish an investor with more than a 50% loss on his/her bankroll. Due to this extreme volatility most people apply a half or a quarter Kelly to the results of their equation. Most investors possess psychological biases that make an investor unable to properly deal with a 50% loss, even though in the long run they would be maximizing ones bankroll.

An example of this extreme volatility can be found in Mr. Mugners past investment partnership. When he ran his partnership in the 1960s he invested in as little as three equities. As noted in the table below, his partnership compounded money at 20% per annum, net of fees, from 1962-1972. If the years 1973-1975 are included, the returns are dragged down to 13.7% net of fees ( Poor Charlies Almanack and Martin Capital Management 2006 annual letter). This result would be unbearable for many investors; they would sell out before the incredible performance of 1975 lifted their returns back up. The returns are extraordinary but they display the volatility that one must be willing to stomach.

Anyone managing money should look into using the Kelly criterion. Whether one chooses to scale it down due to volatility is up to them. Not using the Kelly criterion or a concentrated portfolio like Mr. Buffett and Mr. Munger seems sub-optimal as the Kelly criterion maximizes the long-term growth of ones portfolio. To learn further about the Kelly criterion, reference William Poundstones book Fortunes Formula.

Further Reading on Kelly criterion:

http://www.businessweek.com/magazine/content/05_39/b3952138.htm

http://www.capatcolumbia.com/MM%20LMCM%20reports/Size%20Matters.pdf

Kelly criterion Calculator:

http://www.albionresearch.com/kelly/default.php

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