Hostname: page-component-cd9895bd7-8ctnn Total loading time: 0 Render date: 2024-12-27T11:08:41.309Z Has data issue: false hasContentIssue false
  • English
  • Français

On Merton’s Problem for Life Insurers

Published online by Cambridge University Press:  17 April 2015

Mogens Steffensen
Mogens Steffensen*
Affiliation:
Laboratory of Actuarial Mathematics, Institute of Mathematical Sciences, University of Copenhagen, Universitetsparken 5, DK-2100 Copenhagen Ø, Denmark, e-mail: mogens@math.ku.dk
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 deals with optimal investment and redistribution of the free reserves connected to life and pension insurance contracts in form of dividends and bonus. Formulated appropriately this problem can be viewed as a modification of Merton’s problem of optimal consumption and investment with a very particular form of consumption and utility hereof. Both are linked to a finite state Markov chain. We distinguish between utility optimization of dividends, where a semi-explicit result is obtained, and utility optimization of bonus payments. The latter connects to the financial notion of durable goods and allows for an explicit solution only in very special cases.

Keywords

Markov chainDividends and bonusPower utilityBellman equationDurable goods
Type
Articles
Information
ASTIN Bulletin: The Journal of the IAA , Volume 34 , Issue 1 , May 2004 , pp. 5 - 25
Copyright
Copyright © ASTIN Bulletin 2004

References

Björk, T. (1994) Stokastisk kalkyl och kapitalmarknadsteori. Lecture notes, Royal Institute of Technology, Stockholm.Google 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(1), 19–55.10.2143/AST.30.1.504625CrossRefGoogle Scholar
Fleming, W.H. and Rishel, R.W. (1975) Deterministic and Stochastic Optimal Control, Springer-Verlag.10.1007/978-1-4612-6380-7CrossRefGoogle Scholar
Hansen, M.S. (2001) Optimal portfolio policies and the bonus option in a pension fund. Technical report, Department of Finance, University of Odense.Google Scholar
Hindi, A. and Huangc, C. (1993) Optimal consumption and portfolio rules with durability and local substitution. Econometrica 61(1), 85–121.CrossRefGoogle Scholar
Hoem, J.M. (1969) Markov chain models in life insurance. Blätter der Deutschen Gesellschaft für Versicherungsmathematik 9, 91–107.Google Scholar
Jensen, B.A. and Sørensen, C. (2001) Paying for minimum interest rate guarantees: Who should compensate who? European Financial Management 7, 183–211.CrossRefGoogle Scholar
Korn, R. and Kraft, H. (2001) A stochastic control approach to portfolio problems with stochastic interest rates. SIAM Journal of Control and Optimization 40(4), 250–1269.Google Scholar
Korn, R. and Krekel, M. (2002) Optimal portfolios with fixed consumption or income streams. Technical report, Fraunhofer ITWM, Germany.Google Scholar
Merton, R.C. (1969) Lifetime portfolio selection under uncertainty: The continuous time case. Review of Economics and Statistics 51, 247–257.CrossRefGoogle Scholar
Merton, R.C. (1971) Optimum consumption and portfolio rules in a continuous time model. Journal of Economic Theory 3, 373–413; Erratum 6 (1973); 213–214.10.1016/0022-0531(71)90038-XCrossRefGoogle Scholar
Norberg, R. (1999) A theory of bonus in life insurance. Finance and Stochastics 3(4), 373–390.10.1007/s007800050067CrossRefGoogle Scholar
Richard, S.F. (1975) Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model. Journal of Financial Economics 2, 187–203.CrossRefGoogle Scholar
Steffensen, M. (2000a) Contingent claims analysis in life and pension insurance. Proceedings AFIR 2000, 587–603.Google Scholar
Steffensen, M. (2000b) A no arbitrage approach to Thiele’s differential equation. Insurance: Mathematics and Economics 27, 201–214.Google Scholar
Steffensen, M. (2001) On valuation and control in life and pension insurance. Ph.D.-thesis, Laboratory of Actuarial Mathematics, University of Copenhagen.Google Scholar