Optimal investment with a value-at-risk constraint

We consider constrained investment problem with the objective of minimizing the ruin probability. In this paper, we formulate the cash reserve and investment model for the insurance company and analyze the value-at-risk (VaR) in a short time horizon. For risk regulation, we impose it as a risk constraint dynamically. Then the problem becomes minimizing the probability of ruin together with the imposed risk constraint. By solving the corresponding Hamilton-Jacobi-Bellman equations, we derive analytic expressions for the optimal value function and the corresponding optimal strategies. Looking at the value-at-risk alone, we show that it is possible to reduce the overall risk by an increased exposure to the risky asset. This is aligned with the risk diversification effect for negative correlated or uncorrelated risky asset with the stochastic of the fundamental insurance business. Moreover, studying the optimal strategies, we find that a different investment strategy will be in place depending on the Sharpe ratio of the risky asset.