Hostname: page-component-848d4c4894-x5gtn Total loading time: 0 Render date: 2024-05-26T14:37:59.781Z Has data issue: false hasContentIssue false

A defined benefit pension plan model with stochastic salary and heterogeneous discounting

Published online by Cambridge University Press:  01 December 2022

Ricardo Josa-Fombellida*
Affiliation:
Dpto. de Estadística e Investigación Operativa and IMUVa, Universidad de Valladolid, Valladolid, Spain
Paula López-Casado
Affiliation:
Universidad de Valladolid, Valladolid, Spain
Jorge Navas
Affiliation:
Dpt. de Matemàtica Econòmica, Financera i Actuarial and BEAT, Universitat de Barcelona, Barcelona, Spain
*
*Corresponding author. E-mail: ricardo.josa@uva.es

Abstract

We study the time-consistent investment and contribution policies in a defined benefit stochastic pension fund where the manager discounts the instantaneous utility over a finite planning horizon and the final function at constant but different instantaneous rates of time preference. This difference, which can be motivated for some uncertainties affecting payoffs at the end of the planning horizon, will induce a variable bias between the relative valuation of the final function and the previous payoffs and will lead the manager to show time-inconsistent preferences. Both the benefits and the contribution rate are proportional to the total wage of the workers that we suppose is stochastic. The aim is to maximize a CRRA utility function of the net benefit relative to salary in a bounded horizon and to maximize a CRRA final utility of the fund level relative to the salary. The problem is solved by means of dynamic programming techniques, and main results are illustrated numerically.

Type
Research Article
Copyright
© The Author(s), 2022. Published by Cambridge University Press on behalf of The International Actuarial Association

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Ahmad, Z., Ibrahim, H. and Tuyon, J. (2017) Institutional investor behavioral biases: Syntheses of theory and evidence. Management Research Review, 40, 578603.CrossRefGoogle Scholar
Arnold, L. (1974) Stochastic Differential Equations. Theory and Applications. New York: John Wiley and Sons.Google Scholar
Ayton, P., Clacher, I., Strong, C., Thoma, V. and Weiss-Cohen, L. (2021) Pension Trustee Decision Making. London, UK: Actuarial Research Centre, Institute and Faculty of Actuaries.Google Scholar
Battocchio, P. and Menoncin, F. (2004) Optimal pension management in a stochastic framework. Insurance: Mathematics and Economics, 34,7995.Google Scholar
Battocchio, P., Menoncin, F. and Scaillet, O. (2007) Optimal asset allocation for pension funds under mortality risk during the accumulation and decumulation phases. Annals of Operations Research, 152, 141165.CrossRefGoogle Scholar
Berkelaar, A. and Kouwenberg, R. (2003) Retirement saving with contribution payments and labor income as a benchmark for investments. Journal of Economic Dynamics and Control, 27, 10691097.CrossRefGoogle Scholar
Cairns, A.J.G. (2000) Some notes on the dynamics and optimal control of stochastic pension fund models in continuous time. ASTIN Bulletin, 30, 1955.CrossRefGoogle Scholar
Cairns, A.J.G., Blake, D. and Dowd, K. (2006) Stochastic lifestyling: Optimal dynamic asset allocation for defined-contribution pension plans. Journal of Economic Dynamics and Control, 30, 843877.CrossRefGoogle Scholar
Chang, S.C, Tzeng, L.Y. and Miao, J.C.Y. (2003) Pension funding incorporating downside risks. Insurance: Mathematics and Economics, 32, 217228.Google Scholar
Chen, A. and Delong, Ł. (2015) Optimal investment for a defined-contribution pension scheme under a regime-switching model. ASTIN Bulletin, 45, 397419.CrossRefGoogle Scholar
Chen, S. and Hao, Z. (2013) Funding and investment decisions in a stochastic defined benefit pension with regime switching. Lithuanian Mathematical Journal, 53, 161180.CrossRefGoogle Scholar
Delong, L., Gerrard, R. and Haberman, S. (2008) Mean-variance optimization problems for an accumulation phase in a defined benefit plan. Insurance: Mathematics and Economics, 42, 107118.Google Scholar
de-Paz, A., Marín-Solano, J. and Navas, J. (2013) A consumption-investment problem with heterogeneous discounting. Mathematical Social Sciences, 66, 221232.CrossRefGoogle Scholar
de-Paz, A., Marín-Solano, J., Navas, J. and Roch, O. (2014) Consumption, investment and life insurance strategies with heterogeneous discounting. Insurance: Mathematics and Economics, 54, 6675.Google Scholar
Devolder, P., Bosch, P.M. and Domínguez, F.I. (2003) Stochastic optimal control of annuity contracts. Insurance: Mathematics and Economics, 33, 227238.Google Scholar
Ekeland, I. and Pirvu, T. (2008) Investment and consumption without commitment. Mathematics and Financial Economics, 2(1), 5786.CrossRefGoogle Scholar
Frederick, S., Loewenstein, G. and O’Donoghue, T. (2002) Time discounting and time preference: A critical review. Journal of Economic Literature, 40, 351401.CrossRefGoogle Scholar
Gao, J. (2009) Optimal portfolios for DC pension plans under a CEV model. Insurance: Mathematics and Economics, 44, 479490.Google Scholar
Gerrard, R., Haberman, S. and Vigna, E. (2006) The management of decumulation risks in a defined contribution pension scheme. North American Actuarial Journal, 10, 84110.CrossRefGoogle Scholar
Haberman, S. and Sung, J.H. (1994) Dynamics approaches to pension funding. Insurance: Mathematics and Economics, 15, 151162.Google Scholar
Hodgson, T., Hall, M., Urwin, R., Gao, J. and Khanna, S. (2021) Global top 300 pension funds. A Thinking Ahead Institute and Pensions & Investments joint study. Retrieved from https://www.thinkingaheadinstitute.org/content/uploads/2021/09/PI-300-2021.pdf.Google Scholar
Jackson, M.O. and Yariv, L. (2015) Collective dynamic choice: The necessity of time inconsistency. American Economic Journal: Microeconomics, 7, 150178.Google Scholar
Josa-Fombellida, R., López-Casado, P. and Rincón-Zapatero, J.P. (2018) Portfolio optimization in a defined benefit pension plan where the risky assets are processes with constant elasticity of variance. Insurance: Mathematics and Economics, 82, 7386.Google Scholar
Josa-Fombellida, R. and Navas, J. (2020) Time consistent pension funding in a defined benefit pension plan with non-constant discounting. Insurance: Mathematics and Economics, 94, 142153.Google Scholar
Josa-Fombellida, R. and Rincón-Zapatero, J.P. (2004) Optimal risk management in defined benefit stochastic pension funds. Insurance: Mathematics and Economics, 34, 489503.Google Scholar
Josa-Fombellida, R. and Rincón-Zapatero, J.P. (2008a) Funding and investment decisions in a stochastic defined benefit pension plan with several levels of labor-income earnings. Computers and Operations Research, 35, 4763.CrossRefGoogle Scholar
Josa-Fombellida, R. and Rincón-Zapatero, J.P. (2008b) Mean-variance portfolio and contribution selection in stochastic pension funding. European Journal of Operational Research, 187, 120137.CrossRefGoogle Scholar
Josa-Fombellida, R. and Rincón-Zapatero, J.P. (2019) Equilibrium strategies in a defined benefit pension plan game. European Journal of Operational Research, 275, 374386.CrossRefGoogle Scholar
Karp, L. (2007) Non-constant discounting in continuous time. Journal of Economic Theory, 132, 557568.CrossRefGoogle Scholar
Le Cortois, O. and Menoncin, F. (2015) Portfolio optimisation with jumps: Illustration with a pension accumulation scheme. Journal of Banking and Finance, 60, 127137.CrossRefGoogle Scholar
Marín-Solano, J. and Patxot, C. (2012) Heterogeneous discounting in economic problems. Optimal Control Applications and Methods, 33, 3250.CrossRefGoogle Scholar
Merton, R.C. (1971) Optimal consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3, 373413.CrossRefGoogle Scholar
Teschl, G. (2012) Ordinary Differential Equations and Dynamical Systems. Rhode Island, USA: American Mathematical Society.CrossRefGoogle Scholar
Urbano, B., Jurado, A. and Rosado-Cebrián, B. (2021) What will retirement pensions be like? Analysis of Spanish future pensioner households in terms of poverty. Sustainability, 13, 1760. https://doi.org/10.3390/su13041760.CrossRefGoogle Scholar
Walter, W. (1998) Ordinary Differential Equations. New York: Springer.CrossRefGoogle Scholar
Weiss-Cohen, L., Ayton, P., Clacher, I. and Thoma, V. (2019) Behavioral biases in pension fund trustees’ decision making. Review of Behavioral Finance, 11, 128143. doi: 10.1108/RBF-05-2018-0049.CrossRefGoogle Scholar
Zou, B. and Cadenillas, A. (2017) Optimal investment and liability ratio policies in a multidimensional regime switching model. Risks, 5, 6. doi: 10.3390/risks5010006.CrossRefGoogle Scholar
Zou, Z., Chen, S. and Wedge, L. (2014) Finite horizon consumption and portfolio decisions with stochastic hyperbolic discounting. Journal of Mathematical Economics, 52, 7080.CrossRefGoogle Scholar