Kelly Criterion

The Kelly criterion is a formula for deciding what fraction of capital to commit to a bet or investment with a positive expected return, so as to maximise the long-run growth rate of wealth. Betting less than the Kelly fraction grows capital more slowly. Betting more increases risk and, beyond a point, lowers growth and eventually leads to ruin. It was published in 1956 by John L. Kelly Jr., a researcher at Bell Labs, in a paper that linked gambling to information theory.

The criterion is used by gamblers, by traders and by some fund managers as a guide to position size. Its best-known advocate is the mathematician Edward Thorp, who applied it first to blackjack and then to financial markets. It sits alongside mean-variance portfolio optimization as one of the main theoretical answers to the question of how much to stake.

For a bet with two outcomes, the Kelly fraction is:

• $f^{*}$ is the fraction of current capital to stake.
• $p$ is the probability of winning.
• $q$ is the probability of losing, equal to $1 - p$.
• $b$ is the net odds: the amount won per unit staked.

Suppose a bet pays even money, so $b = 1$, and the chance of winning is 55 percent. The Kelly fraction is $(1 \times 0.55 - 0.45) / 1 = 0.10$, or 10 percent of capital on each bet. For a bet that wins 40 percent of the time but pays 2 to 1, the fraction is $(2 \times 0.40 - 0.60) / 2 = 0.10$ as well. If the numerator $bp - q$ is zero or negative, the bet has no edge and the criterion says to stake nothing.

For an asset whose returns are roughly continuous, such as a share or a trading strategy, the corresponding rule is:

Here $\mu$ is the expected annual return, $r$ is the risk-free rate and $\sigma^{2}$ is the variance of annual returns. A strategy with an expected excess return of 6 percent and a volatility of 20 percent has a Kelly fraction of $0.06 / 0.04 = 1.5$, meaning 150 percent of capital, a position that requires leverage. The formula equals the Sharpe ratio divided by volatility, so strategies with better risk-adjusted returns receive larger allocations.

Wealth that is reinvested grows by multiplication, not addition, so what matters over many bets is the average of the logarithm of the growth factor, not the average payoff. The Kelly fraction is the stake that maximises this expected logarithm. A bettor who stakes everything on a favourable bet has the highest expected value on each round but will lose everything at the first loss. By the law of large numbers, a bettor who follows Kelly will, over a long enough series, almost certainly end up with more than one who follows any substantially different fixed strategy. This is the effect of compounding at the highest sustainable rate.

Full Kelly staking is aggressive. The path of wealth is very volatile, and large falls are normal: under the continuous model, the chance of capital falling to half its starting level at some point is one in two. In practice most users stake a fixed fraction of the Kelly amount. Half Kelly gives up about a quarter of the growth rate in exchange for half the volatility and a much smaller maximum drawdown.

The stronger reason for caution is that the inputs are never known. In a casino game the probabilities can be calculated. In markets, $p$, $b$, $\mu$ and $\sigma$ are estimates from limited data. Overestimating the edge leads to overbetting, and betting twice the true Kelly fraction reduces the long-run growth rate to zero. Estimates taken from backtesting tend to be too optimistic because of overfitting, which makes a reduced fraction a sensible margin for error.

The criterion assumes that the same kind of bet can be repeated many times, that winnings are reinvested and that stakes can be any size. It also assumes the bettor cares only about long-run growth. The economist Paul Samuelson argued that this is one preference among many: an investor with a different attitude to risk, described by a different utility function, would rationally choose a different stake. Market returns have fat tails and change over time, so a fraction based on a normal distribution understates the danger of extreme losses. With several simultaneous positions the simple formulas no longer apply, and the correlated version must be solved jointly, which brings it close to the methods used in volatility targeting and risk-based allocation.