Kelly criterion

In probability theory, the Kelly criterion, or Kelly formula, is a formula used to maximize the long-term growth rate of repeated plays of a given gamble that has positive expected value. It was described by J. L. Kelly, Jr, in a 1956 issue of the Bell System Technical Journal. The formula specifies the percentage of the current bankroll to be bet at each iteration of the game. In addition to maximizing the growth rate in the long run, the formula has the added benefit of having zero risk of ruin; the formula will never allow a loss of 100% of the bankroll on any bet. An assumption of the formula is that currency and bets are infinitely divisible, which is not a concern for practical purposes if the bankroll is large enough.

Statement
The most general statement of the Kelly criterion is that long-term growth rate is maximized by finding the fraction f* of the bankroll that maximizes the expectation of the logarithm of the results. 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 following formula can be derived from the general statement:
 * $$ f^{*} = \frac{bp - q}{b}, \! $$

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 &minus; 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 or negative edge, i.e. if b &le; q/p, then the gambler should bet nothing.

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
 * $$ f^{*} = 2p - 1 . \! $$

The Kelly criterion was originally developed by AT&T Bell Laboratories physicist John Larry Kelly, Jr, based on the work of his colleague Claude Shannon, which applied to noise issues arising over long distance telephone lines. Kelly showed how Shannon's information theory could be applied to the problem of a gambler who has inside information about a horse race, trying to determine the optimum bet size. The gambler's inside information need not be perfect (noise-free) in order for him to exploit his edge. Kelly's formula was later applied by another colleague of Shannon's, Edward O. Thorp, both in blackjack and in the stock market.

Proof
The result is proved by maximising long term growth rate of the bankroll as a function of the fraction f*. In the limit that the same bet is repeated an infinite number of times, it can be shown that the bankroll will grow at a rate G(f*) given by
 * $$ G(f^{*}) = p log(1 + bf^{*}) + q log(1 - f^{*}) \! $$

Using calculus to calculate the maximum of this function proves the Kelly result.

Disadvantages
Using the Kelly system in practice does have drawbacks. While it guarantees that you will never lose all your bankroll, it does not guarantee that you will not lose money. When a series of serial bets are made the chance of dropping to 1/n of your bankroll is 1/n. Thus you have a 50% chance of at some point losing 50% of your bankroll, a 10% chance of dropping to 10%, and so on.

The optimum bet for the greatest growth of bankroll is making the full bet suggested by the Kelly criterion, but this produces a volatile result. There is a 1/3 chance of halving the bankroll before it is doubled. A popular alternative is to bet only half the amount suggested which gives three-quarters of the investment return with much less volatility. Where money would accumulate at 9.06% compound interest with full bets, it still accumulates at 7.5% for half-bets.

Over-betting beyond that suggested by Kelly is counter-productive as the long run growth rate will fall, dropping to zero when the Kelly bet is approximately doubled. Using half-Kelly bets also safeguards against being ruined by unknowingly overbetting, as it can be easy to over-estimate the true odds by a factor of two.

The above applies to a sequence of serial bets. It is better to diversify, as the gambler who for example bets on every horse at a race using the Kelly criterion makes on average a better long-run return than the gambler who only bets on one horse per race, and similarly for the diversified stock market investor.

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 Bernoulli article was not translated into English until 1954 (in an economics journal) and it is unlikely that Kelly was aware of it. For the investor who does not re-invest the profits, but only invests a set amount each time, this rule does not apply; instead the investor should choose the investment with the greatest arithmetic mean.