How Return Variance Affects Optimal Kelly Bets

Summary

The document considers how variance affects optimal growth under Kelly betting. It defines the objective as expected log wealth after investing a fraction of the portfolio in a risky return, with the remainder earning a risk-free rate. The author presents a claimed result: when two return distributions have the same raw moments except for a larger second moment in one, and both have support bounded below by zero, the distribution with the smaller second moment yields at least as high an optimal expected log return.

The claim is framed as a theorem the author wants to know whether others have established, rather than as a fully documented proof or empirical test. The note also says that, in general, larger odd moments and smaller even moments can improve the objective. These conclusions depend on technical assumptions, including the existence of moments and the stated support condition; the document does not spell out a proof or discuss how the results extend to practical return distributions.

Key ideas

• Kelly betting maximizes expected logarithmic wealth by choosing a portfolio fraction.
• The author claims that greater variance can reduce optimal expected log returns when other raw moments match.
• The claim assumes distributions with lower support at zero and existing raw moments.
• The note suggests that higher odd moments and lower even moments may favor growth, but gives no proof details.

Tags

# Reference check: Increasing variance is not optimal for Kelly betting


# Reference check: Increasing variance is not optimal for Kelly betting












Kelly betting considers optimal betting when the player has finite resources. It can be shown that one is optimizing the following integral: $$r(x)=\int \ln(r'(1-x)+xy)d\omega$$ For $r'$ the risk-free rate, $\omega$ the distribution of the returns, and $x$ is the fraction of their portfolio one bets. Let $x^*\in[0,\infty]$ denote the optimal fraction to bet, and $r^*=r(x^*)$.

I have proven that for any two distributions $\theta$ and $\omega$ such that: $$ \inf \text{supp}(\theta) = \inf \text{supp}(\omega) = 0 $$ If the raw moments all exist, and that they are all equal except $\theta$ has strictly larger second moment, then $\omega$ has a larger or equal $r^*$ than $\theta$. Equality only occurs iff $x^*=0$.

In English, this says that, under some reasonable technical conditions, if one is presented with two almost identical bets except one has larger variance, one should take the bet with lower variance.

In general, it can also be shown that larger odd moments and smaller even moments lead to higher returns.

I am wondering if this theorem is known, and if any other theorems of this sort are known.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.