Refund Clause of Contributions with Predetermined Interest under CEV Model

Abstract

This work studies the optimal control strategy for a pension plan with refund clause of contri-
butions with predetermined interest under constant elasticity of variance (CEV) in a defined
contribution (DC) pension plan. A model which mandates fund managers to refund dead mem-
bers’ accumulations with predetermined interest to their next of kin during the accumulation
phase is considered. Also considered herein are investments in a bank security and stock where
the stock market price is driven by the CEV model and the remaining accumulations are equally
distributed between the remaining members. Furthermore, the game theoretic approach is use
in establishing an optimization problem from the extended Hamilton Jacobi Bellman (HJB)
equation which is a non-linear partial differential equation (PDE). Using mean variance utility
function and method of variable separation, explicit solutions of the optimal control strategy
and the efficient frontier are obtained. Finally, Numerical simulations and theoretical analysis
are used to study the effect of the elasticity parameter β and some other parameters on the
optimal control strategy with observations that the elasticity parameter affects the investment
strategy of the fund manager significantly. Also, we observed that the optimal control strategy
employed by the fund manager is inversely proportional to the risk aversion coefficient, initial
fund size, instantaneous volatility and predetermined interest rate but directly proportional to
time.