Optimal Decision on Dynamic Insurance Price and Investment Portfolio of an Insurer in a Competitive Market

Abstract

In this article, we establish a model of insurance pricing with the assumptions that the insurance price, investment returns and insured losses are correlated stochastic processes, while also considering the affect of the demand on the price. The objective of the pricing model is to maximize the expected utility of the terminal wealth of an insurer. We construct a Hamilton–Jacobi–Bellman (HJB) equation and determine the optimal price of an insurance product and optimal investment portfolio of an insurer simultaneously by solving that HJB equation. We also carry out sensitivity analysis. The results of our analysis show that elasticity of insurance demand greatly affects the optimal solutions. Notably, quantity of insurance demanded affects the optimal allocation to risky assets in the insurer’s investment portfolio. Therefore, the demand function for insurance must be considered in management of insurance firm. Our results also show that the drift and volatility of the process of insurance price will affect the investment strategy, in addition to the effect of the drift and volatility of investment process itself. Finally, the drift and volatility of investment stochastic process will affect the insurance price in the cases when the elasticity of demand is large, but that influence is negligible with small elasticity of demand.