Portfolio optimization with a prescribed terminal wealth distribution

Abstract

This paper studies a portfolio allocation problem, where the goal is to reach a prescribed wealth distribution at a final time. We study this problem with the tools of optimal mass transport. We provide a dual formulation which is solved with a gradient descent algorithm. This involves solving an associated Hamilton–Jacobi–Bellman and Fokker–Planck equations with a finite difference method. Numerical examples for various prescribed terminal distributions are given, showing that we can successfully reach attainable targets. We then consider adding consumption during the investment process, to take into account distributions that are either not attainable, or sub-optimal.

Keywords:

• Portfolio allocation
• Wealth distribution target
• Optimal mass transport
• HJB
• Fokker–Planck

JEL Classifications:

• C60
• C61
• G11

Disclosure statement

No potential conflict of interest was reported by the author(s).

Notes

1 We denote $\mathcal{N}(\mu ,\sigma )$ a Normal distribution with mean μ and standard deviation σ.

Additional information

Funding

The Centre for Quantitative Finance and Investment Strategies has been supported by BNP Paribas. Ivan Guo has been partially supported by the Australian Research Council Discovery Project DP170101227.