A Closed-Form Solution of the Multi-Period Portfolio Choice Problem for a Quadratic Utility Function

TL;DR

This paper derives a comprehensive closed-form solution for multi-period portfolio optimization under quadratic utility, applicable with or without a riskless asset, and demonstrates its effectiveness through real data comparisons.

Contribution

It provides a novel, general closed-form solution for multi-period portfolio choice that requires minimal assumptions on asset return distributions and correlations.

Findings

Solution is expressed in terms of conditional mean vectors and covariance matrices.

When asset returns are independent, the solution simplifies to single-period Markowitz optimization.

The multi-period solution with a riskless asset involves proportional weights scaled by time-varying factors.

Abstract

In the present paper, we derive a closed-form solution of the multi-period portfolio choice problem for a quadratic utility function with and without a riskless asset. All results are derived under weak conditions on the asset returns. No assumption on the correlation structure between different time points is needed and no assumption on the distribution is imposed. All expressions are presented in terms of the conditional mean vectors and the conditional covariance matrices. If the multivariate process of the asset returns is independent it is shown that in the case without a riskless asset the solution is presented as a sequence of optimal portfolio weights obtained by solving the single-period Markowitz optimization problem. The process dynamics are included only in the shape parameter of the utility function. If a riskless asset is present then the multi-period optimal portfolio weights are proportional to the single-period solutions multiplied by time-varying constants which are depending on the process dynamics. Remarkably, in the case of a portfolio selection with the tangency portfolio the multi-period solution coincides with the sequence of the simple-period solutions. Finally, we compare the suggested strategies with existing multi-period portfolio allocation methods for real data.