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Comonotonic approximations for the probability of lifetime ruin*

Published online by Cambridge University Press:  18 May 2011

KOEN VAN WEERT ,
JAN DHAENE  and
MARC GOOVAERTS
KOEN VAN WEERT
Affiliation:
Department of Accountancy, Finance and Insurance, K.U.Leuven, Naamsestraat 69, B-3000 Leuven, Belgium (e-mail: [email protected])
JAN DHAENE
Affiliation:
Department of Accountancy, Finance and Insurance, K.U.Leuven, Naamsestraat 69, B-3000 Leuven, Belgium Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands
MARC GOOVAERTS
Affiliation:
Department of Accountancy, Finance and Insurance, K.U.Leuven, Naamsestraat 69, B-3000 Leuven, Belgium Department of Quantitative Economics, University of Amsterdam, Roetersstraat 11, 1018 WB Amsterdam, The Netherlands
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Abstract

This paper addresses the issue of lifetime ruin, which is defined as running out of money before death. Taking into account the random nature of the remaining lifetime, we discuss how a retiree should invest in order to avoid lifetime ruin. We also discuss the conditional time of lifetime ruin and the notion of bequest or wealth at death.

Using analytical approximations based on comonotonicity, we provide a new approach which is easy to understand and leads to very accurate results without computationally complex calculations. Our analytical approach avoids simulation, which allows to solve very general optimal portfolio selection problems.

Type
Articles
Information
Journal of Pension Economics & Finance , Volume 11 , Issue 2 , April 2012 , pp. 285 - 309
Copyright
Copyright © Cambridge University Press 2011

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References

Albrecher, H., Dhaene, J., Goovaerts, M. and Schoutens, W. (2005) Static hedging of Asian options under Lévy models: the comonotonicity approach. Journal of Derivatives, 12(3): 63–72.CrossRefGoogle Scholar
Albrecht, P. and Maurer, R. (2001) Self-annuitization, ruin risk in retirement and asset allocation: the annuity benchmark. Proceedings of the 11th International AFIR Colloquium, Toronto, vol 1, 19–37.Google Scholar
Asmussen, S. and Royas-Nandayapa, (2005) Sums of dependent lognormal random variables: asymptotics and simulation. Stochastic Series at Department of Mathematical Sciences, University of Aarhus, Research Report number 469.Google Scholar
Bayraktar, E. and Young, V. (2009) Minimizing the lifetime shortfall or shortfall at death. Insurance: Mathematics and Economics, 44: 447–458.Google Scholar
Björk, T. (1998) Arbitrage Theory in Continuous Time. Oxford: Oxford University Press, p. 311.CrossRefGoogle Scholar
Black, F. and Scholes, M. (1973) The pricing of options and corporate liabilities. Journal of Political Economy, 81(May–June): 637–659.CrossRefGoogle Scholar
Cesari, R. and Cremonini, D. (2003) Benchmarking, portfolio insurance and technical analysis: a Monte Carlo comparison of dynamic strategies of asset allocation. Journal of Economic Dynamics and Control, 27: 987–1011.CrossRefGoogle Scholar
Dhaene, J., Denuit, M., Goovaerts, M. J., Kaas, R. and Vyncke, D. (2002 a) The concept of comonotonicity in actuarial science and finance: theory. Insurance: Mathematics and Economics, 31(1): 3–33.Google Scholar
Dhaene, J., Denuit, M., Goovaerts, M. J., Kaas, R. and Vyncke, D. (2002 b) The concept of comonotonicity in actuarial science and finance: applications. Insurance: Mathematics and Economics, 31(2): 133–161.Google Scholar
Dhaene, J., Vanduffel, S., Goovaerts, M. J., Kaas, R. and Vyncke, D. (2005) Comonotonic approximations for optimal portfolio selection problems. Journal of Risk and Insurance, 72(2): 253–301.CrossRefGoogle Scholar
Dhaene, J., Vanduffel, S., Tang, Q., Goovaerts, M. J., Kaas, R. and Vyncke, D. (2006) Risk measures and comonotonicity: a review. Stochastic Models, 22: 573–606.CrossRefGoogle Scholar
Dickson, D., Hardy, M. and Waters, H. (2009) Actuarial Mathematics for Life Contingent Risks. Cambridge: Cambridge University Press, p. 493.CrossRefGoogle Scholar
Dufresne, D. (2004 a) Stochastic life annuities. North American Actuarial Journal, 11: 136–157.CrossRefGoogle Scholar
Dufresne, D. (2004 b) The log-normal approximation in financial and other computations. Advances in Applied Probability, 36: 747–773.CrossRefGoogle Scholar
Goovaerts, M. and Shang, Z. (2010) On the pricing of path-dependent options with stochastic time. Working paper.Google Scholar
Hoedemakers, T., Darkiewicz, G. and Goovaerts, M. (2005) Approximations for life annuity contracts in a stochastic financial environment. Insurance: Mathematics and Economics, 37: 239–269.Google Scholar
Huang, H., Milevsky, M. A. and Wang, J. (2004) Ruined moments in your life: how good are the approximations? Insurance: Mathematics and Economics, 34(3): 421–447.Google Scholar
Kaas, R., Dhaene, J. and Goovaerts, M. J. (2000) Upper and lower bounds for sums of random variables. Insurance: Mathematics and Economics, 27: 151–168.Google Scholar
Lévy, H. (2004) Asset return distributions and the investment horizon. The Journal of Portfolio Management, Spring 2004: 47–62.CrossRefGoogle Scholar
McDonald, A., Cairns, A., Gwilt, P. and Miller, K. (1998) An international comparison of recent trends in mortality. British Actuarial Journal, 4: 3–141.CrossRefGoogle Scholar
McNeil, A., Frey, R. and Embrechts, P. (2005) Quantitative Risk Management, Concepts, Techniques and Tools. Princeton, NJ: Princeton University Press, p. 538.Google Scholar
Merton, R. (1971) Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3: 373–413.CrossRefGoogle Scholar
Merton, R. (1990) Continuous Time Finance. Cambridge: Blackwell, p. 752.Google Scholar
Milevsky, M., Ho, K. and Robinson, C. (1997) Asset allocation via the conditional first exit time or how to avoid outliving your money. Review of Quantitative Finance and Accounting, 9(1): 53–70.CrossRefGoogle Scholar
Milevsky, M., Moore, K. and Young, V. (2005). Asset allocation and annuity-purchase strategies to minimize the probability of financial ruin. Mathematical Finance, 16(4): 647–671.CrossRefGoogle Scholar
Milevsky, M. and Posner, S. (1998 a) Asian options, the sum of lognormals, and the reciprocal gamma distribution. Journal of Financial and Quantitative Analysis, 33(3): 409–422.CrossRefGoogle Scholar
Milevsky, M. and Posner, S. (1998 b) A theoretical investigation of randomized asset allocation strategies. Applied Mathematical Finance, 5(2): 117–130.CrossRefGoogle Scholar
Milevsky, M. and Robinson, C. (2000) Self-annuitization and ruin in retirement. North American Actuarial Journal, 4(4): 112–124.CrossRefGoogle Scholar
Milevsky, M. and Robinson, C. (2005). A sustainable spending rate without simulation. Financial Analysts Journal, 61: 89–100.CrossRefGoogle Scholar
Moore, K. and Young, V. (2006). Optimal and simple, nearly optimal rules for minimizing the probability of financial ruin in retirement. North American Actuarial Journal, 10(4): 145–161.CrossRefGoogle Scholar
Rubinstein, M. (1991) Continuously rebalanced investment strategies. The Journal of Portfolio Management, Fall: 78–81.CrossRefGoogle Scholar
Stout, R. G. and Mitchell, J. B. (2006) Dynamic retirement withdrawal planning. Financial Services Review, 15: 117–131.Google Scholar
Valdez, E., Dhaene, J., Maj, M. and Vanduffel, S. (2009) Bounds and approximations for sums of dependent log-elliptical random variables. Insurance: Mathematics and Economics, 44: 385–397.Google Scholar
Vanduffel, S., Hoedemakers, T. and Dhaene, J. (2005) Comparing approximations for risk measures of sums of non-independent lognormal random variables. North American Actuarial Journal, 9(4): 71–82.CrossRefGoogle Scholar
Young, V. (2004) Optimal investment strategy to minimize the probability of lifetime ruin. North American Actuarial Journal, 8(4): 106–126.CrossRefGoogle Scholar