The optimal mean–variance investment strategy under value-at-risk constraints

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