Kelly criterion

In probability theory, the Kelly criterion, or Kelly strategy or Kelly formula, or Kelly bet, is a formula used to determine to optimal size of a series of bets. Under some simplifying assumptions, the Kelly strategy will do better than any essentially different strategy in the long run, with probability 1. It was described by J. L. Kelly, Jr, in a 1956 issue of the "Bell System Technical Journal"J. L. Kelly, Jr, A New Interpretation of Information Rate, Bell System Technical Journal, 35, (1956), 917–926] . Edward O. Thorp demonstrated the practical use of the formula in a 1961 address to the "American Mathematical Society"E. O. Thorp, Fortune's Formula: The Game of Blackjack, American Mathematical Society, January 1961] and later in his books "Beat the Dealer"E. O. Thorp, Beat the dealer: a winning strategy for the game of twenty-one. A scientific analysis of the world-wide game known variously as blackjack, twenty-one, vingt-et-un, pontoon or Van John, Blaisdell Pub. Co (1962), ASIN: B0006AY2QW] (for gambling) and "Beat the Market"Edward O. Thorp and Sheen T. Kassouf, Beat the Market: A Scientific Stock Market System, Random House (1967), ISBN: 978-0394424392] (with Sheen Kassouf, for investing).

Although the Kelly strategy's promise of doing better than any other strategy seems compelling, some economists have argued strenuously against it.William Poundstone, Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street, Hill and Wang, New York, 2005. ISBN 0809046377] The conventional alternative is utility theory which says bets should be sized to maximize the expected utility of the outcome (to an individual with logarithmic utility, the Kelly bet maximizes utility, so there is no conflict). Even Kelly supporters usually argue for fractional Kelly (betting a fixed fraction of the amount recommended by Kelly) for a variety of practical reasons, including some unrealistic assumptions of the Kelly proof.E. O. Thorp, The Kelly Criterion: Part I, Wilmott Magazine, May 2008]

In recent years, Kelly has become a part of mainstream investment theoryS.A. Zenios and W.T. Ziemba, Handbook of Asset and Liability Management, North Holland (2006), ISBN: 978-0444508751] and the claim has been made that well-known successful investors including Warren BuffettMohnish Pabrai, The Dhandho Investor: The Low - Risk Value Method to High Returns, Wiley (2007), ISBN: 978-0470043899] and Bill GrossE. O. Thorp, The Kelly Criterion: Part II, Wilmott Magazine, September 2008] use Kelly methods. William Poundstone wrote a extensive popular account of the history of Kelly betting in "Fortune's Formula"William Poundstone, Fortune's Formula: The Untold Story of the Scientific Betting System That Beat the Casinos and Wall Street, Hill and Wang, New York, 2005. ISBN 0809046377] .

tatement

For simple bets with two outcomes, one involving losing the entire amount bet, and the other involving winning the bet amount multiplied by the payoff odds, the Kelly bet is:: $f^\{*\}\; =\; frac\{bp\; -\; q\}\{b\}\; ,\; !$where
* "f"* is the fraction of the current bankroll to wager;
* "b" is the odds received on the wager;
* "p" is the probability of winning;
* "q" is the probability of losing, which is 1 − "p".

As an example, if a gamble has a 40% chance of winning ("p" = 0.40, "q" = 0.60), but the gambler receives 2-to-1 odds on a winning bet ("b" = 2), then the gambler should bet 10% of the bankroll at each opportunity ("f"* = 0.10), in order to maximize the long-run growth rate of the bankroll. If the gambler has zero edge, i.e. if "b" = "q"/"p", then the gambler should bet nothing. If the edge is negative ("b" < "q"/"p") the formula gives a negative result, indicating that the gambler should take the other side of the bet. For example, in standard American roulette, the bettor is offered an even money payoff (b = 1) on red, when there are 18 red numbers and 20 non-red numbers on the wheel (p = 18/38). The Kelly bet is -0.0526, meaning the gambler should be 5.26% of bankroll that red will not come up. Unfortunately, the casino reserves this bet for itself, so a Kelly gambler will bet zero.

For even-money bets (i.e. when "b" = 1), the formula can be simplified to:: $f^\{*\}\; =\; p\; -\; q\; .\; !$Since q = 1-p, this simplifies further to: $f^\{*\}\; =\; 2p\; -\; 1\; .\; !$

Proof

The proof that maximizing the expected value of the logarithm of wealth will do better than any essentially different strategy in the long run, with probability 1, is not difficult, but it is too long to present here. It can be found in Kelly's original paperJ. L. Kelly, Jr, A New Interpretation of Information Rate, Bell System Technical Journal, 35, (1956), 917–926] and many other sources.

The two-payoff form of the formula given above can be easily derived using differential calculus. The expected value of the logarithm of wealth is:: $p\; mbox\{ln\}(1\; +\; bf^\{*\})\; +\; q\; mbox\{ln\}(1\; -\; f^\{*\})\; !$because there is probability of p of winning b"f"* and probability q of losing b. Taking the derivative with respect to "f"* and setting it to zero gives:: $pb(1\; -\; f^\{*\})\; =\; q(1\; +\; bf^\{*\})\; !$which can be rewritten:: $pb-q\; =\; (p+q)bf^\{*\}\; !$Since p+q=1, the result follows immediately.

Reasons to Bet Less than Kelly

A natural assumption is that taking more risk increases the probability of both very good and very bad outcomes. One of the most important ideas in Kelly is that betting more than the Kelly amount "decreases" the probability of very good results, while still increasing the probability of very bad results. Since in reality we seldom know the precise probabilities and payoffs, and since overbetting is worse than underbetting, it makes sense to err on the side of caution and bet less than the Kelly amount.

Kelly assumes sequential bets that are independent (later work generalizes to bets that have sufficient independence). That may be a good model for some gambling games, but generally does not apply in investing and other forms of risk-taking. Suppose an investor is offered 10 different bets with 40% chance of winning and 2 to 1 payoffs (this is the example used above). Considering the bets one at a time, Kelly says to bet 10% of wealth on each, which means the investor's entire wealth is at risk. That risks ruin, especially if the payoffs of the bets are correlated.

The Kelly property appears "in the long run" (that is, it is an asymptotic property). To a real person, it matters whether the properties emerge over dozens of bets, or trillions of bets, or more. It makes sense to consider not just the long run, but where losing a bet might leave you in the short and medium term as well. A related point is that Kelly assumes the only important thing is long-term wealth. Most people also care about about the path to get there. Two people dying with the same amount of money need not have had equally happy lives. Kelly betting leads to highly volatile short-term outcomes which many people find unpleasant, even if they believe they will do well in the end.

One of the most unrealistic assumptions in the Kelly derivation is that wealth is both the goal and the limit to what you can bet. Most people cannot bet their entire wealth, for example it is illegal to bet your future human capital (you cannot sell yourself into slavery). On the other hand, people can bet money they do not have by borrowing. A person who is allowed to bet more than his wealth might choose to bet more than Kelly (if you know you can always borrow a new stake, it makes sense to take more risk) while someone who is constrained to bet much less than his wealth (say a young college graduate with high lifetime potential earnings but no cash or credit) is forced to bet less.

Bernoulli

In a 1738 article, Daniel Bernoulli suggested that when you have a choice of bets or investments you should choose that with the highest geometric mean of outcomes. This is mathematically equivalent to the Kelly criterion, although the motivation is entirely different (Bernoulli wanted to resolve the St. Peterburg Paradox). The Bernoulli article was not translated into EnglishDaniel Bernoulli, Exposition of a New Theory on the Measurement of Risk, Econometrica, 22(1), (english translation: 1956, original article:1738), 23–36] until 1956 but the work was well-known among mathematicians and economists.

Cited References

See also

*Gambling and information theory

External links

* [http://www.racing.saratoga.ny.us/kelly.pdf Original Kelly paper]
* [http://www.mathandpoker.com/Browne-Whitt-1996.pdf Bayesian Kelly Criterion]
* [http://www.sbrforum.com/Betting+Tools/Kelly+Calculator.aspx Multi-variable Kelly Calculator for Sports Bettors]
* [http://groups.google.com/group/rec.gambling.poker/msg/7bb09884cfac7678 Kelly Criterion by Tom Weideman]
* [http://www.cisiova.com/betsizing.asp Generalized Kelly Criterion For Multiple Outcomes and Financial Investors]
* [http://www.goldengem.co.uk/portfolio.html portfolio analyzer which maximizes expected log return (Kelly criterion) for one risk free bond and a group of risky alpha Levy stable assets]