Peter Lindmark

# Investing with the Kelly Criterion

March 08, 2007

Investing with the Kelly Criterion

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, Fortune’s Formula by William Poundstone. The book discusses John Kelly’s formula, the Kelly criterion. The Kelly criterion is a formula for wagering the optimal amount of one’s 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 one’s bankroll while eliminating total ruin by not allowing any bet to reach 100% of one’s bankroll. The Kelly criterion or formula will help one understand how to size their investment positions according to one’s 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 market’s expectation for how much a person would win if they were successful, and f represents the percentage of one’s bankroll they should wager.

In order to successfully apply this formula one must possess an edge. Back in the 1950’s the edge on horse races was the “private wire.” This “private wire” was the equivalent of insider information in investing. The edge, the Kelly criterion’s 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 one’s 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 Buffett’s 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 1950’s he invested as much as 40% of the partnership’s 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 Hathaway’s 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 haven’t run that formula through my head – and I won’t. You couldn’t apply it to the investment operations I’ve 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 investor’s 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 one’s bankroll.

An example of this extreme volatility can be found in Mr. Mugner’s past investment partnership. When he ran his partnership in the 1960’s 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 Charlie’s 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.

 Year Charlie Munger Inv. Partnership Dow 1962 20.1% -7.6% 1963 47.8% 20.6% 1964 33.1% 18.7% 1965 6.0% 14.2% 1966 8.3% -15.7% 1967 37.5% 19.0% 1968 27.0% 7.7% 1969 21.3% -11.6% 1970 -0.1% 8.7% 1971 20.6% 9.9% 1972 7.3% 18.2% 1973 -31.9% -13.1% 1974 -31.5% -23.1% 1975 73.2% 44.4%

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 one’s portfolio. To learn further about the Kelly criterion, reference William Poundstone’s book Fortune’s Formula.

Further Reading on Kelly criterion:

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

Kelly criterion Calculator:

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

### About the author:

Peter Lindmark
Charlie Tian, Ph.D. - Founder of GuruFocus. You can now pre-order his book Invest Like a Guru on Amazon.

 Currently 2.96/512345 Rating: 3.0/5 (26 votes)

Vooch - 9 years ago    Report SPAM
Peter,

Excellent article!

I've revised the Kelly Formula and it'll be in my upcoming book. My current research is on determining Expected Value properly.

Anyways, I like your comments and posts here at gurufocus.com and I'd like to get together with you sometime either in Omaha or Atlanta, but I won't be attending the Berkshire annual meeting this year - perhaps next year. I will be going to Atlanta sometime this summer.

If we get together, we'll talk about expected value, latticework, and other stuff.

- Vooch

Vooch - 9 years ago    Report SPAM
Here's something interesting, since we're talking about it, I just read from page 74 of Michael Mauboussin's "More Than You Know":

"The goal of an investor is to buy an asset below its expected value. Expected value is the weighted average for a distribution of possible outcomes. You calculate expected value by multiplying the payoff for a given outcome by the probability that outcome will occur."

Therein lies where I'm researching.

- Vooch

Eda Luo - 9 years ago    Report SPAM
I am also researching on the aspect of valuation these days and think a book "Valuation---measuring and managing the value of companies" by McKinsey Inc. is valid. It goes through different types of companies.
Plindmark - 9 years ago    Report SPAM
Drop me an email if you are in Omaha or Atlanta.

Looking forward to your book!
Vooch - 9 years ago    Report SPAM
plindmark,

Will do!

- Vooch

Teddy.rooney - 3 years ago    Report SPAM
I know the kelly criterion gets a bad rap a lot of times, but the truth is that it's the best money management system out there. When you put it up against any other system it wins hands down for the rate of return it produces relative to bankroll size and volatility. The reason people don't really understand it is because it is only half of the equation. Most people can't predict winners consistently or evaluate their edge accurately. Without those two things the kelly calculator won't help you much.