Optimal mean-variance investment strategy under value-at-risk constraints

TL;DR

This paper derives explicit optimal mean-variance investment strategies in a continuous-time setting with a VaR constraint, analyzing how the constraint influences investment decisions through stochastic control methods.

Contribution

It provides closed-form solutions for mean-variance strategies with VaR constraints using stochastic LQ control and HJB equations, extending classical models.

Findings

VaR constraints alter the optimal investment strategy.

Explicit formulas for strategies with and without VaR constraints.

Numerical examples illustrate the impact of VaR constraints.

Abstract

This paper is devoted to study the effects arising from imposing a value-at-risk (VaR) constraint in mean-variance portfolio selection problem for an investor who receives a stochastic cash flow which he/she must then invest in a continuous-time financial market. For simplicity, we assume that there is only one investment opportunity available for the investor, a risky stock. Using techniques of stochastic linear-quadratic (LQ) control, the optimal mean-variance investment strategy with and without VaR constraint are derived explicitly in closed forms, based on solution of corresponding Hamilton-Jacobi-Bellman (HJB) equation. Furthermore, some numerical examples are proposed to show how the addition of the VaR constraint affects the optimal strategy.