# On the optimal asset allocation strategy for a defined contribution pension system with refund clause of premium with predetermined interest under Heston's volatility model

Volume 13, Issue 1, pp 53--64
Publication Date: September 11, 2019 Submission Date: December 17, 2018 Revision Date: May 19, 2019 Accteptance Date: June 03, 2019
• 272 Views

### Authors

Edikan E. Akpanibah - Department of Mathematics and Statistics, Federal University Otuoke, P.M.B 126, Yenagoa, Bayelsa State, Nigeria. Bright O. Osu - Department of Mathematics, Michael Okpara University of Agriculture, Umudike, Abia State, Nigeria. Silas A. Ihedioha - Department of Mathematics, Plateau State University Bokkos, P.M.B 2012 Jos, Plateau state, Nigeria.

### Abstract

In this paper, we study optimal asset allocation strategy for a defined contribution (DC) pension fund with return of premium clause under Heston's volatility model in mean-variance utility frame work. In this model, members' next of kin are allowed to withdraw their family members' accumulated premium with predetermined interest. Also, investments in one risk free asset and one risky asset are considered to help increase the accumulated funds of the remaining members in order to meet their retirement needs. Using the actuarial symbol, we formulize the problem as a continuous time mean-variance stochastic optimal control problem. We establish an optimization problem from the extended Hamilton Jacobi Bellman equations using the game theoretic approach and solve the optimization problem to obtain the optimal allocation strategy for the two assets, the optimal fund size and also the efficient frontier of the pension members. We analyze numerically the effect of some parameters on the optimal allocation strategy and deduce that as the initial wealth, predetermined interest rate and risk averse level increases, the optimal allocation policy for the risky asset (equity) decreases. Furthermore, we give a theoretical comparison of our result with an existing result and observed that the optimal allocation policy whose return is with predetermined interest is higher compared to that without predetermined interest.

### Keywords

• DC pension fund
• extended HJB equation
• optimal allocation policy
• refund of contribution clause
• interest rate

•  91G10
•  91B30

### References

• [1] E. E. Akpanibah, S. K. Samaila, Stochastic strategies for optimal investment in a defined contribution (DC) pension fund, Int. J. Appl. Sci. Math. Theory, 3 (2017), 48--55

• [2] P. Battocchio, F. Menoncin, Optimal pension management in a stochastic framework, Insurance Math. Econom., 34 (2004), 79--95

• [3] T. Bjork, A. Murgoci, A general theory of Markovian time inconsistent stochastic control problems, Stockholm School of Economics (Working Paper), 2010 (2010), 55 pages

• [4] J.-F. Boulier, S. J. Huang, G. Taillard, Optimal management under stochastic interest rates: the case of a protected defined contribution pension fund, Insurance Math. Econom., 28 (2001), 173--189

• [5] A. J. G. Cairns, D. Blake, K. Dowd, Stochastic lifestyling: optimal dynamic asset allocation for defined contribution pension plans, J. Econom. Dynam. Control, 30 (2006), 843--877

• [6] M. Di Giacinto, S. Federico, F. Gozzi, Pension funds with a minimum guarantee: a stochastic control approach, Finance Stoch., 15 (2011), 297--342

• [7] J. W. Gao, Stochastic optimal control of DC pension funds, Insurance Math. Econom., 42 (2008), 1159--1164

• [8] J. W. Gao, Optimal portfolios for DC pension plans under a CEV model, Insurance Math. Econom., 44 (2009), 479--490

• [9] L. He, Z. X. Liang, Optimal financing and dividend control of the insurance company with fixed and proportional transaction costs, Insurance Math. Econom., 44 (2009), 88--94

• [10] L. He, Z. X. Liang, The optimal investment strategy for the DC plan with the return of premiums clauses in a mean-variance framework, Insurance Math. Econom., 53 (2013), 643--649

• [11] D. P. Li, X. M. Rong, H. Zhao, B. Yi, Equilibrium investment strategy for DC pension plan with default risk and return of premiums clauses under CEV model, Insurance Math. Econom., 72 (2017), 6--20

• [12] Z. X. Liang, J. P. Huang, Optimal dividend and investing control of an insurance company with higher solvency constraints, Insurance Math. Econom., 49 (2011), 501--511

• [13] B. O. Osu, E. E. Akpanibah, C. Olunkwa, Mean-Variance Optimization of portfolios with return of premium clauses in a DC pension plan with multiple contributors under constant elasticity of variance model, Int. J. Math. Comput. Sci., 12 (2018), 85--90

• [14] B. O. Osu, E. E. Akpanibah, B. I. Oruh, Optimal investment strategies for defined contribution (DC) pension fund with multiple contributors via Legendre transform and dual theory, Int. J. Pure Appl. Res., 2 (2017), 97--105

• [15] D.-L. Sheng, X. M. Rong, Optimal time consistent investment strategy for a DC pension with the return of premiums clauses and annuity contracts, Discrete Dyn. Nat. Soc., 2014 (2014), 13 pages

• [16] J. W. Xiao, Z. Hong, C. L. Qin, The constant elasticity of variance (CEV) model and the Legendre transform-dual solution for annuity contracts, Insurance Math. Econom., 40 (2007), 302--310

• [17] Y. Zeng, Z. F. Li, Optimal time consistent investment and reinsurance policies for mean-variance insurers, Insurance Math. Econom., 49 (2011), 145--454

• [18] C. B. Zhang, X. M. Rong, Optimal investment strategies for DC pension with astochastic salary under affine interest rate model, Discrete Dyn. Nat. Soc., 2013 (2013), 11 pages