Hostname: page-component-586b7cd67f-l7hp2 Total loading time: 0 Render date: 2024-11-22T16:29:32.288Z Has data issue: false hasContentIssue false
  • English
  • Français

Some Notes on the Dynamics and Optimal Control of Stochastic Pension Fund Models in Continuous Time

Published online by Cambridge University Press:  29 August 2014

Andrew Cairns
Andrew Cairns*
Affiliation:
Heriot-Watt University, United Kingdom
*
Department of Actuarial Mathematics and Statistics, Heriot-Watt UniversityEdinburgh EH 14 4AS, Scotland
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

This paper discusses the modelling and control of pension funds.

A continuous-time stochastic pension fund model is proposed in which there are n risky assets plus the risk-free asset as well as randomness in the level of benefit outgo. We consider Markov control strategies which optimise over the contribution rate and over the range of possible asset-allocation strategies.

For a general (not necessarily quadratic) loss function it is shown that the optimal proportions of the fund invested in each of the risky assets remain constant relative to one another. Furthermore, the asset allocation strategy always lies on the capital market line familiar from modern portfolio theory.

A general quadratic loss function is proposed which provides an explicit solution for the optimal contribution and asset-allocation strategies. It is noted that these solutions are not dependent on the level of uncertainty in the level of benefit outgo, suggesting that small schemes should operate in the same way as large ones. The optimal asset-allocation strategy, however, is found to be counterintuitive leading to some discussion of the form of the loss function. Power and exponential loss functions are then investigated and related problems discussed.

The stationary distribution of the process is considered and optimal strategies compared with dynamic control strategies.

Finally there is some discussion of the effects of constraints on contribution and asset-allocation strategies.

Keywords

Continuous timestochastic differential equationasset-allocationcontribution strategyBellman equationoptimal controlconstraints
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 30 , Issue 1 , May 2000 , pp. 19 - 55
Copyright
Copyright © International Actuarial Association 2000

References

Black, F. and Jones, R. (1988) Simplifying portfolio insurance for corporate pension plans. Journal of Portfolio Management 14(4), 33–37.CrossRefGoogle Scholar
Black, F. and Perold, A. (1992) Theory of constant proportion portfolio insurance. Journal of Economic Dynamics and Control 16, 403–426.CrossRefGoogle Scholar
Boulier, J-F., Trussant, E. and Florens, D. (1995) A dynamic model for pension funds management. Proceedings of the 5th AFIR International Colloquium 1, 361–384.Google Scholar
Boulier, J-F., Michel, S. and Wisnia, V. (1996) Optimizing investment and contribution policies of a defined benefit pension fund. Proceedings of the 6th AFIR International Colloquium 1, 593–607.Google Scholar
Cairns, A.J.G. (1995) Pension funding in a stochastic environment: the role of objectives in selecting an asset-allocation strategy. Proceedings of the 5th AFIR International Colloquium 1, 429–453.Google Scholar
Cairns, A.J.G. (1996) Continuous-time stochastic pension fund modelling. Proceedings of the 6th AFIR International Colloquium 1, 609–624.Google Scholar
Cairns, A.J.G., Blake, D. and Dowd, K. (2000). Optimal dynamic asset allocation for defined contribution pensionplans. To appear in Proceedings of the 10th AFIR International Colloquium.Google Scholar
Cairns, A.J.G. and Parker, G. (1997) Stochastic pension fund modelling. Insurance Mathematics and Economics 21, 43–79.CrossRefGoogle Scholar
Davis, M.H.A. and Norman, A.R. (1990) Portfolio selection with transaction costs. Mathematics of Operations Research 15, 676–713.CrossRefGoogle Scholar
Deelstra, G., Grasselli, M. and Koehl, P-F. (1999) Optimal investment strategies in a CIR framework. Preprint.CrossRefGoogle Scholar
De Schepper, A., Teunen, M. and Goovaerts, M. (1994) An analytical inversion of a Laplace transform related to annuities certain. Insurance: Mathematics and Economics 14, 33–37.Google Scholar
Dufresne, D. (1988) Moments of pension contributions and fund levels when rates of return are random. Journal of the Institute of Actuaries 115, 535–544.CrossRefGoogle Scholar
Dufresne, D. (1989) Stability of pension systems when rates of return are random. Insurance: Mathematics and Economics 8, 71–76.Google Scholar
Dufresne, D. (1990) The distribution of a perpetuity, with applications to risk theory and pension funding. Scandinavian Actuarial Journal 1990, 39–79.CrossRefGoogle Scholar
Fleming, W.H. and Rishel, R.W. (1975) Deterministic and stochastic optimal control. Springer-Verlag, New York.CrossRefGoogle Scholar
Föllmer, H. and Schweizer, M. (1993) A microeconomic approach to diffusion models for stock prices. Mathematical Finance 3, 1–23. (Erratum: 4:285.)CrossRefGoogle Scholar
Gerber, H.U. and Shiu, E.S. (2000) Investing for retirement: optimal capital growth and dynamic asset allocation. To appear in North American Actuarial Journal 4 (2).Google Scholar
Haberman, S. and Sung, J-H. (1994) Dynamic approaches to pension funding. Insurance: Mathematics and Economics 15, 151–162.Google Scholar
Johnson, N.L., Kotz, S. and Balakrishnan, N. (1994) Continuous Univariate Distributions: Volume I, 2nd Edition. Wiley, New York.Google Scholar
Keel, A. and Müller, H.H. (1995) Efficient portfolios in the asset liability context. ASTIN Bulletin 25, 33–48.CrossRefGoogle Scholar
Merton, R.C. (1971) Optimal consumption and portfolio rules in a continuous-time model. Journal of Economic Theory 3, 373–413.CrossRefGoogle Scholar
Merton, R.C. (1990) Continuous-Time Finance. Blackwell, Cambridge, Mass.Google Scholar
O'Brien, T. (1986) A stochastic-dynamic approach to pension funding. Insurance: Mathematics and Economics 5, 141–146.Google Scholar
O'Brien, T. (1987) A two-parameter family of pension contribution functions and stochastie optimization. Insurance: Mathematics and Economics 6, 129–134.Google Scholar
øksendal, B. (1998) Stochastic Differential Equations, 5th Edition. Springer-Verlag, Berlin.CrossRefGoogle Scholar
Sharpe, W.F. and Tint, L.G. (1990) Liabilities – a new approach. Journal of Portfolio Management Winter 1990, 5–10.CrossRefGoogle Scholar
Siegmann, A.H. and Lucas, A. (1999) Continuous-time dynamic programming for ALM with risk-averse loss functions. Proceedings of the 9th AFIR International Colloquium 2, 183–193.Google Scholar
Shreve, S.E. and Soner, H.M. (1994) Optimal investment and consumption with transaction costs. Annals of Applied Probability 4, 609–692.CrossRefGoogle Scholar
Wilkie, A.D. (1985) Portfolio selection in the presence of fixed liabilities. Journal of the Institute of Actuaries 112, 229–277.CrossRefGoogle Scholar
Wise, A. (1984) The matching of assets to liabilities. Journal of the Institute of Actuaries 111, 445–501.CrossRefGoogle Scholar
Yor, M. (1992) Sur certaines foncionnelles exponentielles du mouvement brownien réel. Journal of Applied Probability 29, 202–208.CrossRefGoogle Scholar