Kelly Bet Sizing for Symmetric Odds and Unequal Payoffs

# Answer by olaker (score 16, accepted)


# What is the optimal strategy when there is an equal chance for gain or loss but the size of the potential gain is larger?












I'm investigating a situation where the chance for gain or loss is the same, but the amount gained is greater than the amount that is lost. For example, the gain would be about 30% of the trade amount, and the loss would be 23% of the trade amount. While there is slightly more to it than that, that is the core of it -- random/even chance for hitting the gain or the loss the way the trade is structured, and approximately the percentages indicated. Please note either the gain or loss will be reached.

If one has amount A to invest, what considerations need to be taken into account to make a situation like this profitable, or is it not possible for it to be profitable (e.g. due to many successive losses)?

## Answer by olaker (score 16, accepted)

https://quant.stackexchange.com/a/2766

This is practically a textbook case begging for the Kelly criterion.

In your specific example, the optimal trade size is $f^*A$, where $f^*$ maximizes the average rate of return $$\mathbb{E}[\log (X)]=0.5\log(1+0.3f)+0.5\log(1-0.23f).$$ Here $f$ is the fraction of the current capital to trade. A straightforward calculation yields that $$f^*=\frac{0.3-0.23}{2\times 0.3\times 0.23}\approx 0.5072$$

In general, if you expect to gain $gX$ with probability $p$ or lose $lX$ with probability $q$ on a trade of the size $X$, then the optimal (Kelly) bet is $$f^*=\frac{pg-ql}{gl}.$$

Some caveats might be worth noting.

- The Kelly framework assumes that sequential trades are (sufficiently) independent.

- Since the exact payoffs $g$, $l$ and probabilities $p$, $q$ are typically not known, it is safer to bet less than Kelly. Betting double the optimal Kelly bet reduces the growth rate of capital to zero (see e.g. "Good and Bad Properties of the Kelly Criterion" by Bill Ziemba).