Kelly Criterion

If we bet a fraction $f$ of our capital on each round, and a win resulted in a gain of $f\times b$, and a loss resulted in a loss of $f \times a$, then after $S$ winning rounds and $F$ losing rounds, $n=F+S$, we would have a portfolio worth

We want to know what fraction $f$ of our capital would maximize that. Since the $log$ function is a monotonically increasing function (fancy speak for: it always increases as $x$ increases), it is equivalent to maximize $\log{W_n}$ and this will prove an easier task.

The thing we want to maximize is our expected wealth after $n$ rounds

On the right-hand side, by linearity of expectation of the fact that $\log{(1+fb)}$ and $\log{(1-fb)}$ are not random variables, just deterministic scalars, we can pull them out and simplify to:

If we have chance $p$ of a bet winning, and $(1-p)$ of it losing, then each bet is a Bernoulli trial and after $n$ rounds the results will be binomially distributed. The mean of the number of winning bets will therefore be $\mathbb{E}(S)=np$, and the mean of the losing bets will be $\mathbb{E}(F)=n(1-p)$

Jensen asidePermalink

For a strictly concave function like the log, Jensen’s Inequality tells us that

so

hence we are also maximizing the expected wealth $\mathbb{E}(W_n)$

Maximising with the usual calculus wayPermalink

Differentiating

Solving for $f$ with a bit of algebra gives

This is the optimal fraction to bet given chances of winning and the payout scheme

ExamplesPermalink

Coming soon…