risk management - 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

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.