Analysis of investment portfolio with 2 × 2 positive definite matrix and mortality risk under exponential utility

Abstrakt

This paper investigates a pension plan member’s (PPM) portfolio in a defined contributory (DC) pension plan with return clause and charge on balance exhibiting constant absolute risk averse (CARA). A portfolio comprising of a risk-free asset and two risky assets modelled by the geometric Brownian motion (GBM) process is considered, where the instantaneous volatilities of the two risky assets form a 2×2 matrix k = {k_a,b}_2×2 such that kk^T is positive definite. The Abraham De Moivre mortality force function is use to determine the mortality rate of PPMs during accumulation phase and an optimization problem is obtained from the Hamilton Jacobi Bellman (HJB) equation using dynamic programming approach. Using Legendre transformation and dual theory with variable change technique, the optimal value function (OVF), optimal control plan (OCP) and the optimal fund size (OFS) are obtained under exponential utility function. Furthermore, some numerical analysis of some sensitive parameters such as risk free interest rate, risk averse coefficient, entry age of PPM, charge on balance and OFS are presented to explain their impact on the OCP.