Portfolio selection based on a benchmark process with dynamic value-at-risk constraints

Portfolio selection is an essential issue in finance. It studies how to allocate one’s wealth in a basket of securities to maximize the return and minimize the risk. And dynamic portfolio selection based on a benchmark process is one of the most important types. Different from the existing literature, we impose a dynamic risk control on it. As a matter of fact, performing an optimal portfolio strategy in the light of a dynamic portfolio formulation does not eliminate the possibility of an investor going to bankruptcy or even more serious situations in a volatile financial market before the terminal time, so it is reasonable and necessary to impose a dynamic risk control on the instantaneous wealth throughout the investment horizon to ensure that the investment behavior can proceed and we intend to address this interesting issue in this paper. More specifically, we investigate the dynamic portfolio selection problem based on a benchmark process coupled with a dynamic value-at-risk constraint. By stochastic dynamic programming techniques, we derive the corresponding Hamilton–Jacobi–Bellman equation. Moreover, the optimal portfolio strategies are obtained by Lagrange multiplier method. To verify the model, two numerical examples are illustrated. The results show the difference of optimal portfolio strategies with and without the dynamic VaR constraint: the composition of the risky assets is constant but the investment proportion is reduced as the VaR constraint becomes binding. This research can provide a good decision-making reference for risk-averse investors.