The Kelly criterion · quants.md

Having an edge is half the problem. Bet too little and you leave growth on the table; bet too much and you go broke even with positive expectancy. Kelly gives the fraction that maximises long-run growth.

The discrete bet

You win $b$ times your stake with probability $p$ and lose your stake with probability $q = 1 - p$ . Betting a fraction $f$ of wealth each time, the expected log growth per bet is

G(f) = p \ln(1 + bf) + q \ln(1 - f)

Maximising gives the Kelly fraction:

f^* = \frac{bp - q}{b} = \frac{\text{edge}}{\text{odds}}

Worked example

An even-money bet ( $b = 1$ ) that wins 55% of the time: $f^* = (0.55 - 0.45)/1 = 10\%$ of wealth per bet. The growth rate is $G(0.10) \approx 0.5\%$ per bet.

The continuous version

For a strategy with expected excess return $\mu$ and volatility $\sigma$ (per unit of capital), the growth rate of leveraged wealth at leverage $f$ is approximately

g(f) = f\mu - \frac{f^2\sigma^2}{2}

maximised at

f^* = \frac{\mu}{\sigma^2}, \qquad g(f^*) = \frac{\mu^2}{2\sigma^2} = \frac{SR^2}{2}

So the maximum achievable growth rate is set by the Sharpe ratio. A Sharpe-1 strategy can compound at most at about 50% a year; a Sharpe-0.5 strategy at about 12.5%.

The shape of the growth curve

$g(f)$ is a parabola. Two facts follow:

Betting twice Kelly gives a growth rate of exactly zero. Beyond that, growth is negative: you will almost surely lose money while having an edge.
Betting half Kelly gives 75% of the maximum growth with half the volatility and far shallower drawdowns.

Why professionals bet half Kelly or less

Kelly assumes you know $\mu$ and $\sigma$ exactly. You don’t. If your estimate of the edge is 2× too high, “full Kelly” is really twice Kelly, which grows at zero. Half Kelly is insurance against your own overconfidence, and its drawdowns are tolerable to a human.

Kelly and drawdowns

Under full Kelly, the probability of ever seeing your wealth fall to a fraction $x$ of its peak is roughly $x$ itself. There is a 50% chance of halving at some point and a 10% chance of losing 90%. Most people cannot hold through that, which is one more argument for fractional Kelly.

Kelly is not martingale

Kelly sizes as a fixed fraction of current wealth, so it bets less after losses. Martingale doubles after losses, betting more as wealth shrinks. Kelly can never go to zero in theory; martingale eventually always meets the losing streak that does. Many specs in the vault use martingale averaging — read them with this in mind.

The discrete bet

The continuous version

The shape of the growth curve

Kelly and drawdowns

Put it to work