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Optimal Portfolio Strategies with Minimum Guarantee Maximizing the Expected Utility of Terminal Surplus for a Defined Contribution Pension Plan

Charles I. Nkeki, Chukwuma R. Nwozo


We consider the optimal portfolio strategies with minimum guarantee policies that maximize the expected utility of the terminal surplus of a pension plan member (PPM) in a defined contribution (DC) pension plan in a continuous-time framework. We assume a market structure characterized by a risk-free asset and n risky assets. The risk-free rate is assume to be deterministic. The risky assets and the salary of the PPM are stochastic and follow a geometric Brownian motion. We propose in this paper, an optimal investment policy, obtain by solving the Hamilton-Jacobi-Bellman (HJB) equation. We establish a unique method of solving explicitly, the non-linear partial differential equation using Constant Relative Risk Aversion (CRRA) utility function. We derive the present value of the future minimum guarantee at any time t as well as the terminal minimum guarantee. We find that the initial minimum guarantee of the PPM is equal to zero. We also find that the PPM’s optimal portfolio depend on the investment strategies, the flow of contributions and the flow of minimum guarantee over time. Therefore, the classical Merton rule needs modification. We also derive an explicit solution of the expected utility of the terminal surplus of a PPM using CRRA utility function. We find an interesting result that some proportion of wealth should be transfer into the bond market in order to secure the wealth of the PPM up to retirement period.


Optimal strategies, portfolio, minimum guarantee, pension fund, defined contribution, surplus.

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