How the Kelly Criterion Relates to the Sharpe Ratio
Summary
The document contrasts the discrete Kelly formula for sizing a binary wager with the continuous Kelly rule used for strategies whose returns vary continuously. For a strategy with mean excess return and return volatility, the continuous rule expresses optimal leverage in terms of those quantities, or equivalently the Sharpe ratio and volatility. It also gives the relationship between portfolio Sharpe ratio and maximum compounded growth under the stated assumptions, with the risk-free rate often omitted for simplicity.
The answers caution against treating the payout odds in the discrete formula as a Sharpe ratio. Kelly is a position-sizing framework, while the Sharpe ratio measures excess return relative to volatility; their connection arises through shared return inputs and particular modeling assumptions. The discrete formulation limits the bet to available capital because a loss can consume the stake, whereas continuous-return models can imply leverage above one. These relationships should not be generalized without checking the return distribution and assumptions behind the model.
Key ideas
- The discrete Kelly criterion sizes a wager using its success probability and payoff odds.
- Continuous Kelly sizing depends on expected excess return and return variance.
- The Sharpe ratio is mean excess return divided by return volatility, not the payout odds in a binary bet.
- Continuous Kelly leverage can exceed available capital under its model assumptions.
- The stated link between Sharpe ratio and compounded growth depends on the continuous framework.
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Full text
# Kelly criterion and Sharpe ratio
# Kelly criterion and Sharpe ratio
Whats the relationship between the Kelly criterion and the Sharpe ratio?
$$ f=\frac{p(b+1)-1}{b} $$
where $f$ is a percentage of how much capital to place on a bet, $p$ is the probability of success, and $b$ is the payout odds (eg. 3 dollars for ever 1 dollar bet).
Is $b$ (the payout ratio) also the Sharpe ratio? I am having a hard time understanding what Ernie is refering to when he is connecting the two concepts.
## Answer by madilyn (score 18, accepted)
https://quant.stackexchange.com/a/7199
The Sharpe ratio $S_i$ of a strategy indexed by $i$ is given by the ratio of the mean excess return $m_i$ to the standard deviation of returns $\sigma_i$,
The formula you have quoted is the discrete Kelly criterion. That's not so useful in trading, where the outcomes are continuous. The continuous Kelly criterion states that for every $i$th strategy with Sharpe ratio $S_i$ and standard deviation of returns $\sigma_i$, you should be leveraged $f_i = m_i/\sigma_i^2 = S_i/\sigma_i$.
Note of difference between the discrete and continuous criteria: The Kelly criterion is designed to protect your equity from "ruin", so it will never tell you to bet more than what you have in the discrete case - because when you "lose", you lose the complete bet you've placed. The leverage $f_i$ will always be $<1$ in the discrete case. On the other hand, in the continuous case, your leverage can be $>1$.
Let us assume we have a portfolio with an overall Sharpe ratio $S$. What Ernie is talking about is that the maximum compounded growth rate $g$ is given by $g = r + S^2/2$. We usually drop the risk-free rate (unless we post treasuries for margin), so we have $g = S^2/2$.
## Answer by Matt Wolf (score 6)
https://quant.stackexchange.com/a/7200
I would not put too much weight on any relationship between Sharpe ratio and Kelly criterion. The two are simply not logically related other than they both share common inputs. Kelly relates to sizing your position while Sharpe ratios relate your excess returns to the volatility of those.
As long as you find common inputs you can always setup a mathematical relationship between two equations.
Yes, both relate to risk but thats as far as I would go in relating one concept to the other.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.