Optimal portfolios under a value-at-risk constraint

This paper looks at the optimal portfolio problem when a value-at-risk constraint is imposed. This provides a way to control risks in the optimal portfolio and to fulfil the requirement of regulators on market risks. The value-at-risk constraint is derived for n risky assets plus a risk-free asset and is imposed continuously over time. The problem is formulated as a constrained utility maximization problem over a period of time. The dynamic programming technique is applied to derive the Hamilton-Jacobi-Bellman equation and the method of Lagrange multiplier is used to tackle the constraint. A numerical method is proposed to solve the HJB-equation and hence the optimal constrained portfolio allocation. Under this formulation, we find that investments in risky assets are optimally reduced by the imposed value-at-risk constraint.