# Stochastic Funding of a Defined Contribution Pension Plan with Proportional Administrative Costs and Taxation under Mean-Variance Optimization Approach

### Abstract

This paper aim at studying a mean-variance portfolio selection problem with stochastic salary, proportional administrative costs and taxation in the accumulation phase of a defined contribution (DC) pension scheme. The fund process is subjected to taxation while the contribution of the pension plan member (PPM) is tax exempt. It is assumed that the flow of contributions of a PPM are invested into a market that is characterized by a cash account and a stock. The optimal portfolio processes and expected wealth for the PPM are established. The efficient and parabolic frontiers of a PPM portfolios in mean-variance are obtained. It was found that capital market line can be attained when initial fund and the contribution rate are zero. It was also found that the optimal portfolio process involved an inter-temporal hedging term that will offset any shocks to the stochastic salary of the PPM.### References

I. Bajeux-Besnainou and R. Portait, “Dynamic asset allocation in a mean-variance framework”, Management Science, 44(1998), S79-S95.

P. Battocchio and F. Menoncin, “Optimal pension management in a stochastic framework”, Insurance: Mathematics and Economics, 34(2004), pp. 79-95.

T. Bielecky, H. Jim, S. Pliska, and X. Zhou, “Continuous-time mean-variance portfolio selection with bankruptcy prohibition”, Mathematical Finance, 15(2005), 213-244.

D. Blake, D. Wright and Y. Zhang, Optimal funding and investment strategies in defined contribution pension plans under Epstein-Zin utility, Discussion paper, The pensions Institute, Cass Business School, City University, UK, 2008.

J. F. Boulier, S. J. Huang and G. Taillard, “Optimal management under stochastic interest rates: The case of a protected defined contribution pension fund”, Insurance: Mathematics and Economics, 28(2001), 173-189.

A. J. G. Cairns, D. Blake, and K. Dowd, “Stochastic lifestyling: Optimal dynamic asset allocation for defined contribution pension plans”, Journal of Economic Dynamic and Control, 30(2006), 843-877.

M. Chiu and D. Li, “Asset and liability management under a continuous-time mean-variance optimization framework”, Insurance: Mathematics and Economics, 39(2006), 330-355.

G. Deelstra, M. Grasselli and P. Koehl, “Optimal investment strategies in a CIR framework”, Journal of Applied Probability, 37(2000), 936-946.

P. Devolder, M. Bosch Princep and I. D. Fabian, “Stochastic optimal control of annuity contracts”, Insurance: Mathematics and Economics, 33(2003), pp. 227-238.

M. Di Giacinto, S. Federico and F. Gozzi, “Pension funds with a minimum guarantee: a stochastic control approach”, Finance and Stochastic, (2010).

J. Gao, Stochastic optimal control of DC pension funds, Insurance: Mathematics and Economics, 42(2008), 1159-1164.

R. Gerrard, S. Haberman and E. Vigna, “Optimal investment choices post retirement in a defined contribution pension scheme”, Insurance: Mathematics and Economics, 35(2004), 321-342.

S. Haberman and E. Vigna, “Optimal investment strategies and risk measures in defined contribution pension schemes”, Insurance: Mathematics and Economics, 31(2002), 35-69.

B. H jgaard and E. Vigna, Mean-variance portfolio selection and efficient frontier for defined contribution pension schemes, Technical report R-2007-13, Department of Mathematical Sciences, Aalborg University, 2007.

R. Josa-Fombellida and J. Rinc n-Zapatero, “Mean-variance portfolio and contribution selection in stochastic pension funding”, European Journal of Operational Research, 187(2008), 120-137.

R. Korn and M. Krekel, Optimal portfolios with fixed consumption or income streams. Working paper, University of Kaiserslautern, 2001.

D. Li and W.-L. Ng, “Optimal dynamic portfolio selection: multiperiod mean-variance formulation”, Mathematical Finance, 10(2000), pp. 387-406.

C. I. Nkeki,. On optimal portfolio management of the accumulation phase of a defined contributory pension scheme. Ph.D thesis, Department of Mathematics, University of Ibadan, Ibadan, Nigeria, 2011.

C. I. Nkeki and C. R. Nwozo , “Variational Form of Classical Portfolio Strategy and Expected Wealth for a Defined Contributory Pension Scheme”. Journal of Mathematical Finance 2(2012), 132-139.

H. Richardson, “A minimum variance result in continuous trading portfolio optimization”, Management Science, 35(1989), 1045-1055.

E. Vigna, On efficiency of mean-variance based portfolio selection in DC pension schemes, Collegio Carlo Alberto Notebook 154, 2010.

X. Zhou and D. Li, “Continuous-time mean-variance portfolio selection: A stochastic LQ framework,” Applied Mathematics and Optimization, 42(2000), 19-33.

C. I. Nkeki, “Mean-Variance Portfolio Selection Problem with Stochastic Salary for a Defined Contribution Pension Scheme: A Stochastic Linear-Quadratic Framework”, Statistics,optimization and information computing, 1(2013), 62-81.

S. Mannor and J. N. Tsitsiklis, “Mean-variance optimization in Markov decision processes”, Proceedings of the 28th international conference on machine learning Bellevue, WA, USA, (2011).

*Statistics, Optimization & Information Computing*,

*2*(4), 323-338. https://doi.org/10.19139/soic.v2i4.82

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