Kelly Allocation for Normal Returns and Risk-Averse Investors
Summary
The document asks how the Kelly fraction depends on the mean and volatility of strategy returns under a normal-return assumption, while recognizing that this assumption idealizes real trading outcomes. The main answer reframes the problem as continuous-time portfolio choice: an investor allocates wealth between a risky asset and a risk-free asset and chooses an allocation to maximize expected utility at a future horizon.
With constant relative risk aversion and Black–Scholes asset dynamics, the response gives a constant risky-asset allocation proportional to the excess expected return and inversely proportional to risk aversion times variance. This shows that the allocation depends on investor preferences, not just the return distribution. A second answer supplies a different Kelly-style expression and a small-mean approximation, but the excerpt does not reconcile its assumptions with the utility-based solution. The formulas therefore should not be treated as interchangeable without specifying the investment model, utility criterion, and return convention.
Key ideas
- The main response formulates allocation as expected-utility maximization over risky and risk-free assets.
- Under CRRA utility and Black–Scholes dynamics, the optimal risky allocation depends on excess return, risk aversion, and variance.
- Higher risk aversion reduces the optimal exposure for a given excess return and variance.
- A separate answer provides another Kelly expression and approximation without reconciling the assumptions.
- The normal-return premise is an idealization, and the excerpt does not analyze fat tails or implementation constraints.
Tags
Full text
# Kelly criterion for normally distributed returns
# Kelly criterion for normally distributed returns
If the returns of my strategy are distributed like 𝒩[μ,σ], what is the optimal fraction of capital to invest in each single trade, as a function μ and σ? Help!
PS. I know that normally distributed returns are an abstraction. But I'd like to grasp the concept in an ideal world, before exploring the implications of fat tails on the formula...
## Answer by starovoitovs (score 10, accepted)
https://quant.stackexchange.com/a/43350
This problem can be expressed as the original Merton's portfolio problem.
Consider wealth process defined by SDE
$$ d X _ { t } = \frac { X _ { t } \alpha _ { t } } { S _ { t } } d S _ { t } + \frac { X _ { t } \left( 1 - \alpha _ { t } \right) } { S _ { t } ^ { 0 } } d S _ { t } ^ { 0 } $$
where $\alpha_t$ is proportion of the investment in the risky asset $S_t$, and $S_t^0$ is the risk-free asset.
Optimality criterion may depend on the risk aversion of the investor, and the problem is to maximize expected utility of the investor for appropriate utility function $U$:
$$ E \left[ U \left( X _ { T } \right) \right] \rightarrow \max $$
Classical choice of the utility function is CRRA:
$$ u ( x ) = \frac { x ^ { 1 - \gamma } } { 1 - \gamma } $$
where $\gamma$ is constant and corresponds to the risk-aversion of the investor.
If the asset $S_t$ follows Black-Scholes dynamics (in conformance with your assumption of log-normal returns)
$$ \begin{aligned} d S _ { t } ^ { 0 } & = r S _ { t } ^ { 0 } d t \\ d S _ { t } & = \mu S _ { t } d t + \sigma S _ { t } d W _ { t } \end{aligned} $$
remarkably there is a closed-form solution which it is to invest a constant proportion of wealth in the risky asset
$$ \alpha_t = \frac { \mu - r } { \gamma \sigma ^ { 2 } } $$
Notice that the solution can be interpreted as the mean-variance trade-off.
## Answer by elemolotiv (score 2)
https://quant.stackexchange.com/a/43345
ok I found it 🙂and this works for any distribution, not just the normal distribution
$f^*=\frac μ {σ^2 + μ^2} \approx \frac μ {σ^2} \space if μ\llσ$
here the steps: https://www.dropbox.com/s/4nqd5yfk2xcuag5/kelly.pdfShown 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.