# On Investment Model for a CARA Pension Scheme Member with Return of Contributions Clause for Mortgage Housing Scheme

## Keywords:

Optimal investment distribution, pension scheme member, Ito’s lemma, Mortgage Housing Scheme, Return Clause## Abstract

**Communication in Physical Sciences, 2024, 11(3): 548-558**

**Authors: Ase M. Esabai ^{*} Edikan E. Akpanibah and Sylvanus K. Samaila**

**Received: 12 March 2024/Accepted: 20 June 2024**

This paper investigates a pension scheme member’s (PSM) portfolio in a defined contributory (DC) pension plan with a return of premium for mortgage scheme and charge on balance(CB) for time-consistent utility. A portfolio with one risk-free asset and two risky assets which are modelled using the geometric Brownian motion (GBM) process is considered, such that the instantaneous volatilities of the risky assets form a 2×2 positive definite matrix with . The PSMs interested in the mortgage housing scheme are modelled using Abraham De Moivre’s force function, and the optimization problem is obtained using a dynamic programming approach. Using Legendre transformation and dual theory with variable change technique, the optimal value function (OVF), and optimal investment distribution (OID) are obtained for a PSM with utility function exhibiting constant absolute risk averse (CARA). Furthermore, some numerical analyses were carried out and was observed that factors such as risk-free interest rate, risk-averse coefficient, entry age of PSM, initial wealth and CB were critical in developing an OID with this kind of return clause.

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