Optimal Investment, Consumption and Life Insurance Problem with Stochastic Environments

Stanley Jere, Elias R. Offen and Othusitse Basmanebothe

Journal of Mathematics Research, 2022, vol. 14, issue 4, 33

Abstract: Optimal investment, consumption and life insurance problem with stochastic environments for a CRRA wage-earner is solved in this study. The wage-earner invests in the financial market with one risk-free security, one risky security, receives labor income and has a life insurance policy in the insurance market. A life insurance policy is purchased to hedge the financial wealth for the beneficiary in case of wage earner premature death. The interest rate and the volatility are stochastic. The stochastic interest rate dynamics of risk-free security follow a Ho-Lee model and the risky security follow Heston’s model with stochastic volatility parameter dynamics following a Cox-Ingersoll-Ross (CIR) model. The objective of the wage-earner is to allocate wealth between risky security and risk-free security but also buy a life insurance policy during the investment period to maximize the expected discounted utilities derived from consumption, legacy and terminal wealth over an uncertain lifetime. By applying Bellman's optimality principle, the associated HJB PDE for the value function is established. The power utility function is employed for our analysis to obtain the value function and optimal policies. Finally, numerical examples and simulations are provided.

Date: 2022
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