Optimal investment with a value-at-risk constraint

Abstract

We consider constrained investment problem with the objective of minimizing the ruin probability. In thispaper, we formulate the cash reserve and investment model for theinsurance company and analyze the value-at-risk ($VaR$) in a shorttime horizon. For risk regulation, we impose it as a risk constraintdynamically. Then the problem becomes minimizingthe probability of ruin together with the imposed risk constraint.By solving the corresponding Hamilton-Jacobi-Bellman equations, wederive analytic expressions for the optimal value function andthe corresponding optimal strategies. Looking at the value-at-risk alone, we showthat it is possible to reduce the overall risk by an increased exposure to therisky asset. This is aligned with the risk diversification effectfor negative correlated oruncorrelated risky asset with the stochastic of the fundamental insurance business.Moreover, studying theoptimal strategies, we find that a different investment strategy will be in placedepending on the Sharpe ratio of the risky asset.