Optimal pension fund management with stochastic additional contribution rate in a defined contribution pension plan

Abstract

This paper examines an optimal pension fund management in which a pension plan member (PPM) make a flow of contributions that comprises of two parts: mandatory contribution with fixed contribution rate, and additional voluntary contribution (AVC) with stochastic contribution rate. The mandatory part is a fixed fraction of a PPM's stochastic salary while the latter part is assume to be stochastic with contribution growth rate (CGR) and volatility depending on the contribution rate, the salary and the wealth processes of a PPM over time. The sum of the two contribution rates make up the total stochastic contribution rate of a PPM. Two asset classes are consider: the risky and the riskless assets. The latter pays a stochastic interest rate that follows uncontrolled Vasisek model. Also, it is assume that the PPM consume some portion of his or her wealth continuously over time. This paper aims at determining the optimal investment, optimal consumption rate, the optimal contribution rate and effect of AVC rate on the investment portfolio. The optimal investment portfolio, optimal contribution rate, optimal consumption rate and explicit expression of the CGR of a PPM's contribution rate are obtained. We found that the investment portfolio depend inversely on the contribution rate and the interest rate, and directly on the contribution of a PPM. The intertemporary hedging portfolio strategies with respect to contribution rate, contribution process and interest rate are obtained. Furthermore, we found that the CGR depend on the optimal wealth, optimal contribution rate, interest rate, salary, investment risk, salary risk, contribution risk, interest rate risk, and coefficient of constant relative risk averse of a PPM. We also found that the CGR depend directly on the salary and the contribution processes and inversely on the contribution rate and interest rate of a PPM over time. Also, we found that the CGR react to changes in the salary process and its risk, wealth process, contribution rate and contribution risk, interest rate and interest rate risk, and coeffecient of risk averse.