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OPTIMAL TIME-CONSISTENT PORTFOLIO AND CONTRIBUTION SELECTION FOR DEFINED BENEFIT PENSION SCHEMES UNDER MEAN–VARIANCE CRITERION

Part of: Mathematical finance Stochastic systems and control Mathematical economics

Published online by Cambridge University Press:  09 October 2014

XIAOQING LIANG ,
LIHUA BAI  and
JUNYI GUO
XIAOQING LIANG
Affiliation:
School of Mathematical Sciences, Nankai University, Tianjin 30007, SC, PR China email [email protected], [email protected], [email protected]
LIHUA BAI
Affiliation:
School of Mathematical Sciences, Nankai University, Tianjin 30007, SC, PR China email [email protected], [email protected], [email protected]
JUNYI GUO*
Affiliation:
School of Mathematical Sciences, Nankai University, Tianjin 30007, SC, PR China email [email protected], [email protected], [email protected]
*
✉[email protected]
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Abstract

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We investigate two mean–variance optimization problems for a single cohort of workers in an accumulation phase of a defined benefit pension scheme. Since the mortality intensity evolves as a general Markov diffusion process, the liability is random. The fund manager aims to cover this uncertain liability via controlling the asset allocation strategy and the contribution rate. In order to have a more realistic model, we study the case when the risk aversion depends dynamically on current wealth. By solving an extended Hamilton–Jacobi–Bellman system, we obtain analytical solutions for the equilibrium strategies and value function which depend on both current wealth and mortality intensity. Moreover, results for the constant risk aversion are presented as special cases of our models.

Keywords

time-consistent strategystochastic mortality intensity processmean–variance criterionHamilton–Jacobi–Bellman equation

MSC classification

Secondary: 93E20: Optimal stochastic control 91B30: Risk theory, insurance 91G10: Portfolio theory 91B70: Stochastic models
Type
Research Article
Information
The ANZIAM Journal , Volume 56 , Issue 1 , July 2014 , pp. 66 - 90
Copyright
Copyright © 2014 Australian Mathematical Society 

References

Björk, T. and Murgoci, A., “A general theory of Markovian time inconsistent stochastic control problems”, SSRN eLibrary, 2010; doi:10.2139/ssrn.1694759.Google Scholar
Björk, T., Murgoci, A. and Zhou, X. Y., “Mean-variance portfolio optimization with state-dependent risk aversion”, Math. Finance 24 (2012) 1–24; doi:10.1111/j.1467-9965.2011.00515.x.Google Scholar
Cairns, A. J., “Some notes on the dynamics and optimal control of stochastic pension fund models in continuous time”, Astin Bull. 30 (2000) 19–56; doi:10.2143/AST.30.1.504625.Google Scholar
Delong, Ł., Gerrard, R. and Haberman, S., “Mean–variance optimization problems for an accumulation phase in a defined benefit plan”, Insurance Math. Econom. 42 (2008) 107–118 ; doi:10.1016/j.insmatheco.2007.01.005.Google Scholar
He, L. and Liang, Z., “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 ; doi:10.1016/j.insmatheco.2013.09.002.Google Scholar
Heath, D. and Schweizer, M., “Martingales versus PDEs in finance: an equivalence result with examples”, J. Appl. Probab. 37 (2000) 947–957; doi:10.1239/jap/1014843075.Google Scholar
Hu, Y., Jin, H. and Zhou, X. Y., “Time-inconsistent stochastic linear–quadratic control”, SIAM J. Control Optim. 50 (2012) 1548–1572; doi:10.1137/110853960.Google Scholar
Jalen, L. and Mamon., R., “Valuation of contingent claims with mortality and interest rate risks”, Math. Comput. Modelling 49 (2009) 1893–1904; doi:10.1016/j.mcm.2008.10.014.Google Scholar
Josa-Fombellida, R. and Rincón-Zapatero, J. P., “Optimal risk management in defined benefit stochastic pension funds”, Insurance Math. Econom. 34 (2004) 489–503 ; doi:10.1016/j.insmatheco.2004.03.002.Google Scholar
Josa-Fombellida, R. and Rincón-Zapatero, J. P., “Mean–variance portfolio and contribution selection in stochastic pension funding”, European J. Oper. Res. 187 (2008) 120–137 ; doi:10.1016/j.ejor.2007.03.002.Google Scholar
Josa-Fombellida, R. and Rincón-Zapatero, J. P., “Stochastic pension funding when the benefit and the risky asset follow jump diffusion processes”, European J. Oper. Res. 220 (2012) 404–413 ; doi:10.1016/j.ejor.2012.01.033.Google Scholar
Li, D. and Ng, W.-L., “Optimal dynamic portfolio selection: multiperiod mean–variance formulation”, Math. Finance 10 (2000) 387–406; doi:10.1111/1467-9965.00100.Google Scholar
Li, Y. and Li, Z., “Optimal time-consistent investment and reinsurance strategies for mean–variance insurers with state dependent risk aversion”, Insurance Math. Econom. 53 (2013) 86–97 ; doi:10.1016/j.insmatheco.2013.03.008.Google Scholar
Markowitz, H., “Portfolio selection”, J. Finance 7 (1952) 77–91 ; doi:10.1111/j.1540-6261.1952.tb01525.x.Google Scholar
Ngwira, B. and Gerrard, R., “Stochastic pension fund control in the presence of Poisson jumps”, Insurance Math. Econom. 40 (2007) 283–292; doi:10.1016/j.insmatheco.2006.05.002.Google Scholar
Qian, L., Wang, W., Wang, R. and Tang, Y., “Valuation of equity-indexed annuity under stochastic mortality and interest rate”, Insurance Math. Econom. 47 (2010) 123–129 ; doi:10.1016/j.insmatheco.2010.06.005.Google Scholar
Strotz, R. H., “Myopia and inconsistency in dynamic utility maximization”, Rev. Econ. Stud. 23 (1955) 165–180; doi:10.2307/2295722.Google Scholar
Zeng, Y. and Li, Z., “Optimal time-consistent investment and reinsurance policies for mean-variance insurers”, Insurance Math. Econom. 49 (2011) 145–154; doi:10.1016/j.insmatheco.2011.01.001.Google Scholar
Zeng, Y., Li, Z. and Lai, Y., “Time-consistent investment and reinsurance strategies for mean–variance insurers with jumps”, Insurance Math. Econom. 52 (2013) 498–507 ; doi:10.1016/j.insmatheco.2013.02.007.Google Scholar
Zhou, X. Y. and Li, D., “Continuous-time mean-variance portfolio selection: a stochastic LQ framework”, Appl. Math. Optim. 42 (2000) 19–33; doi:10.1007/s002450010003.CrossRefGoogle Scholar