Mathematical Model of a Degenerate-Pension Wealth Generation Strategies, with Constant Interest Rate: The Ito’ Product Law Approach

Abstract

This research seeks to develop completely new formulations for portfolio management strategies, for Degenerate-pension wealth, in a DC Pension scheme, with constant interest rate, during the wealth accumulation stage. The Pension plan member (PPM) invested money in savings account (a risk-free asset), in mutual benefit account; the Nigerian Stanbic IBTC Bank’s Money market (a risk-less asset), and in Stock (a risky asset), under the Geometric Brownian Motion (GBM) model. Using the Ito’ Product Law, an Ordinary Stochastic Differential Equation, representing the evolution of the Degenerate-Pension wealth optimization program was developed. Thereafter, a nonlinear Partial Differential Equation was obtained, using the associated Hamilton Jacobi Bellman (H.J.B) equation, for the optimality condition. The explicit solution of the constant relative risk aversion (CRRA) utility function was obtained, using Legendre transform, dual theory, and change of variable methods. It was established herein that the annuity term due to the chosen utility function vanishes, which depicts a sharp collapse in investment in risky assets. Theorem is constructed and proved on the Degenerate-pension wealth generation strategies.