# 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.

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## 2 Answers

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 $g = S^2/2$.

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Kristine, thank you very much for your response. Though I am still confused as to why there are two versions of the Kelly criterion, continuous and discrete.  I have a tendency to look at Sharpe ratios from a trading perspective. Are you saying the Sharpe ratio of your trading model is constantly changing along with the volatility of returns?? I guess my question is where does the continuity aspect come from? Not to mention, why doesn't the continuous Kelly not include probability of the outcome? – jessica Feb 3 at 7:12
Yes, your Sharpe ratio changes with the volatility of returns. That's why a corollary of the Kelly criterion requires you to rebalance your portfolio constantly. The continuity comes from the continuity of prices - a "loss" can be perceived as a continuous range of outcomes where your exit price is {0 ticks + commissions, 1 tick + commissions, 2 ticks + commissions, ...} less than your entry price. The probability of the outcome is implicit in $S_i$ and $\sigma_i$, which specify the probability distribution. – kristine Feb 3 at 7:43
So the "g" reflects the growth rate in the asset? – jessica Feb 3 at 8:12
Maximum compounded growth rate of your portfolio. – – kristine Feb 3 at 14:53

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.

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Few comments. (1) Excess returns, volatility and position sizing should not be looked at separately. (2) There is logical intuition that drives the relationship between the Sharpe ratio and the Kelly leverage; you want to increase your allocation to a strategy that you believe to have better risk-adjusted returns (Sharpe ratio). (3) In mathematics, when you derive a theorem or equation, then it holds. You cannot ignore the relationship between its variables on the basis of emotion, opinion or religion as suggested. You can, however, challenge the assumptions and steps of your derivation. – kristine Feb 3 at 15:23
Nobody argues with emotions here. Nowhere in any literature is a bet size (Kelly is not even widely accepted as a a standard financial asset position sizing tool) related to Sharpe ratio. Sharpe ratio represents risk adjusted return regardless of position size. The position size and your capital base, two incredibly important aspects when deriving optimal bet size, are non-existent in deriving risk adjusted return a-la Sharpe ratios. – Freddy Feb 3 at 15:45
(1) I agree that the Kelly formula has no place in standard practice, but this is off-topic. (2) However, with regards to the jessica's question, there is an intuitive relationship and it is well-defined in the original literature (edwardothorp.com/sitebuildercontent/sitebuilderfiles/…, equation 7.3) though glossed over because it is akin to redundant substitution. I should point out to you that the Sharpe ratio is defined as the excess return over the standard deviation of return. – kristine Feb 3 at 17:34

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@kristine, you continue to miss the point here. – Freddy Feb 3 at 17:51

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Thanks, Freddy. – kristine Feb 3 at 18:22
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