An approximate solution for the power utility optimization under predictable returns

TL;DR

This paper develops an approximate analytical solution for power utility portfolio optimization assuming predictable returns, using normal distribution assumptions and gradient descent, validated through simulation comparisons.

Contribution

It introduces a novel approximate solution method combining analytical derivation and machine learning techniques for portfolio choice under predictable returns.

Findings

The approximate solution closely matches traditional Taylor expansion results.

Gradient descent effectively solves the portfolio optimization problem.

Simulation shows the method's robustness and accuracy.

Abstract

This work derives an approximate analytical single period solution of the portfolio choice problem for the power utility function. It is possible to do so if we consider that the asset returns follow a multivariate normal distribution. It is shown in the literature that the log-normal distribution seems to be a good proxy of the normal distribution in case if the standard deviation of the last one is way smaller than its mean. So we can use this property because this happens to be true for gross portfolio returns. In addition, we present a different solution method that relies on the machine learning algorithm called Gradient Descent. It is a powerful tool to solve a wide range of problems, and it was possible to implement this approach to portfolio selection. Besides, the paper provides a simulation study, where we compare the derived results with the well-known solution, which uses a Taylor series expansion of the utility function.