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Optimal consumption and portfolio choice of retirees with longevity risk*

Published online by Cambridge University Press:  10 October 2013

ALEXANDER KREMER ,
FRIEDRICH LIESE ,
SUSANNE HOMÖLLE  and
JOHANN CLAUSEN
ALEXANDER KREMER
Affiliation:
Chair for Statistics, University of Rostock
FRIEDRICH LIESE
Affiliation:
Department of Mathematics, University of Rostock
SUSANNE HOMÖLLE*
Affiliation:
Chair of Banking and Finance, University of Rostock
JOHANN CLAUSEN
Affiliation:
Hamburg
*
(e-mail: susanne.homoelle@uni-rostock.de)
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Abstract

The question how to optimize consumption and portfolio choice over the life cycle has been widely discussed in the literature so far. In this paper we concentrate on a retiree's optimal consumption and portfolio selection over his remaining years of life. We apply the logistic model of mortality thus modeling the empirically observed increase of mortality during the retirement period. The optimal consumption strategy and portfolio choice are established by reducing the Hamilton-Jacobi-Bellmann equation to the explicit solution of an ordinary differential function (ODF) that includes the mortality rate. A general finding is that the Merton-Samuelson result of constant portfolio choice for a constant mortality is confirmed for arbitrary mortality. The portfolio choice is only influenced by risk and return of assets and the retirees’ risk aversion. To get the specific optimal consumption strategy in a realistic situation the logistic model of mortality has been fitted to the data of the Statistical Yearbook for the Federal Republic of Germany 2006/2008. The optimal initial value for the ODF is obtained by numerical methods. The solution provides a large increase in the ratio of optimal consumption to wealth up to about 92 years followed by a sharp decrease. A bequest motive dampens the magnitudes of the ups and downs of the consumption ratio but does not change the basic shape.

Keywords

G11 (JEL)retirement periodmortalitylife time distributionasset allocation
Type
Articles
Information
Journal of Pension Economics & Finance , Volume 13 , Issue 3 , July 2014 , pp. 227 - 249
Copyright
Copyright © Cambridge University Press 2013 

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Footnotes

*

An earlier version of this paper was presented at the Annual Conference of the Swiss Society for Financial Market Research in Zurich. We thank the participants of these meeting, especially Alexandra Dias, and Roland Rau for many helpful comments. We are deeply indebted to an anonymous referee for providing valuable comments that led to a considerable improvement of the paper.

References

Babbel, D. F. and Merril, C. B. (2007) Rational Decumulation. Wharton Financial Institutions Center, Working Paper.CrossRefGoogle Scholar
Bodie, Z., Merton, R. and Samuelson, W. F. (1992) Labor supply flexibility and portfolio choice in a life-cycle model. Journal of Economic Dynamics and Control, 16: 427–449.CrossRefGoogle Scholar
Bodie, Z., Detemple, J. and Rindisbacher, M. (2009) Life-cycle finance and the design of pension plans. Annual Review of Financial Economics, 1: 249–286.CrossRefGoogle Scholar
Cagetti, M. (2003) Wealth accumulation over the life cycle and precautionary savings. Journal of Business and Economic Statistics, 21: 339–353.CrossRefGoogle Scholar
Cocco, J. F., Gomes, F. J. and Maenhout, P. J. (2005) Consumption and portfolio choice over the life-cycle. Review of Financial Studies, 18: 491–533.CrossRefGoogle Scholar
Davidoff, T., Brown, J. and Diamond, P. (2005) Annuities and individual welfare. American Economic Review, 95: 1573–1590.CrossRefGoogle Scholar
DAV-Unterarbeitsgruppe Rentnersterblichkeit (2005) Herleitung der DAV-Sterbetafel 2004 R für Rentenversicherungen. Blätter der DGVFM, Band XXVII, Heft 2.Google Scholar
de Moivre, A. (1725) Annuities upon Lives: Or the Valuation of Annuities upon any Number of Lives; as also of Reversions. To which is Added, An Appendix concerning the Expectations of Life, and Probabilities of Survivorship. London.Google Scholar
Dummann, K. (2008) Retirement saving and attitude towards financial intermediaries – Evidence for Germany. Working Paper No. 99, Thünen-Series of Applied Economic Theory, University of Rostock.Google Scholar
Edwards, E. (2008) Health risk and portfolio choice. Journal of Business and Economic Statistics, 26: 472–485.CrossRefGoogle Scholar
Feinstein, J. S. and Lin, C.-Y. (2006) Elderly Asset Management. Working Paper.CrossRefGoogle Scholar
Fleming, W. H. and Soner, M. H. (1993) Controlled Markov Processes and Viscosity Solutions. Berlin: Springer.Google Scholar
Gampe, J. (2010) Human Mortality Beyond Age 110. In Supercentenarians, Demographic Research Monographs, 7. Heidelberg: Springer.Google Scholar
Gerrad, R., Haberman, S. and Vigna, E. (2003) Optimal investment choices post retirement in a defined contribution pension scheme. Insurance: Mathematics and Economics, 35: 321–345.Google Scholar
Gerrad, R., Haberman, S. and Vigna, E. (2012) Choosing the optimal annuitization time post-retirement. Quantitative Finance, 12:7: 1143–1159.CrossRefGoogle Scholar
Gompertz, B. (1825) On the nature of the function expressive of the law of human mortality, and on a new mode of determining the value of life contingencies. Philosophical Transactions of the Royal Society of London, 115: 513–585.Google Scholar
Hakansson, N. (1969) Optimal investment and consumption strategies under risk, an uncertain lifetime, and insurance. International Economic Review, 10: 443–446.CrossRefGoogle Scholar
Horneff, W. J., Maurer, R. H., Mitchell, O. and Dusy, I. (2008 a) Following the rules: integrating asset allocation and annuitization in retirement portfolios. Insurance: Mathematics and Economics, 42: 396–408.Google Scholar
Horneff, W. J., Maurer, R. H. and Stamos, M. Z. (2008 b) Life-cycle asset allocation with annuity markets. Journal of Economic Dynamics and Control, 32: 3590–3612.CrossRefGoogle Scholar
Hubbard, R. G., Skinner, J. and Zeedes, S. P. (1994) The importance of precautionary motives in explaining individual and aggregate saving. Carnegie-Rochester Conference Series on Public Policy, 40: 59–125.CrossRefGoogle Scholar
Inkmann, J., Lopes, P. and Michaelides, A. (2011) How deep is the annuity market participation puzzle? Review of Financial Studies, 24: 279–319.CrossRefGoogle Scholar
Kannisto, V. (1992) Paper presented at the Workshop on old age mortality. Odense University, Odense, Denmark, June 1992.Google Scholar
Karatzas, I. and Shreve, S. E. (1998) Methods of Mathematical Finance. New York: Springer.Google Scholar
Keynes, J. M. (1936) The General Theory of Employment, Interest and Money. London, New York: Macmillan.Google Scholar
Klos, A., Langer, T. and Weber, M. (2003) Über kurz oder lang – welche Rolle spielt der Anlagehorizont bei Investments? Zeitschrift für Betriebswirtschaft, 73: 733–765.Google Scholar
Korn, R. (1997) Optimal Portfolios. Singapore: World Scientific.CrossRefGoogle Scholar
Lindbergson, M. (2001) Mortality among the elderly in Sweden 1988–1997. Scandinavian Actuarial Journal, 1: 79–94.CrossRefGoogle Scholar
Makeham, W. (1860) On the law of mortality and the construction of annuity tables. Journal of the Institute of Actuaries, 8: 301–310.Google Scholar
Maurer, R. and Somova, B. (2009) Better Outcomes for Future Retirees: How Can We Secure Better Outcomes for Future Retirees? Brussels.Google Scholar
Merton, R. (1969) Lifetime portfolio selection under uncertainty: the continuous-time case. Review of Economics and Statistics, 51: 247–257.CrossRefGoogle Scholar
Merton, R. (1971) Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3: 373–413.CrossRefGoogle Scholar
Milevsky, M. A. and Young, V. R. (2007) The timing of annuitization: investment dominance and mortality risk. Insurance, Mathematics and Economics, 40: 134–144.CrossRefGoogle Scholar
Milevsky, M. A., Moore, K. and Young, V. R. (2006) Asset allocation and annuity-purchase strategies to minimize the probability of financial ruin. Mathematical Finance, 16: 647–671.CrossRefGoogle Scholar
Modigliani, F. (1986) Life cycle, individual thrift, and the wealth of nations. American Economic Review, 76: 297–313.Google Scholar
Modigliani, F. and Brumberg, R. (1954) Utility analysis and the consumption function: an interpretation of cross-section data. In Kurihara, K. (ed.), Post-Keynesian Economics. New Brunswick: NJ. Rutgers University Press, 388–436.Google Scholar
Pham, H. (2009) Continuous-time stochastic control and optimization with financial applications. Berlin: Springer.CrossRefGoogle Scholar
Phelps, E. (1962) The accumulation of risky capital: a sequential utility analysis. Econometrica, 30: 729–743.CrossRefGoogle Scholar
Polyak, I. (2005) New advice to retirees: spend more at first, cut back later. New York Times, September 25, 2005.Google Scholar
Richard, S. (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
Samuelson, P. (1969) Lifetime portfolio selection by dynamic stochastic programming and long-horizon effect. Review of Economics and Statistics, 51: 239–246.CrossRefGoogle Scholar
Samuelson, P. (1989) The judgement of economic science on rational portfolio management: indexing, timing, and long-horizon effect. Journal of Portfolio Management, 16: 4–12.CrossRefGoogle Scholar
Stabile, G. (2006) Optimal timing of the annuity purchase: a combined control and optimal stopping problem. International Journal of Theoretical and Applied Finance, 9: 151–170.CrossRefGoogle Scholar
Thatcher, A. R. (1999) The long-term pattern of adult mortality and the highest attained age. Journal of the Royal Statistical Society, Series A, 162(1): 5–43.CrossRefGoogle ScholarPubMed
Thatcher, A. R., Kannisto, V. and Vaupel, J. W. (1999) The force of mortality at age 80 to 120. Monographs on Population Aging, 5. Odense: University Press of Southern Denmark.Google Scholar
Thorley, S. (1995) The time-diversification controversy. Financial Analysts Journal, 51: 68–76.CrossRefGoogle Scholar
Wang, T. and Young, V. R. (2012) Maximimizing the utility of consumption with commutable life annuities. Insurance: Mathematics and Economics, 51: 352–369.Google Scholar
Weibull, E. (1939) A statistical theory of the strength of materials. Ingeniors Vetenskaps Akademien, 151–3: 45–55.Google Scholar
Whitaker, B. (2005) Managing retirement, after you really retire. New York Times, October 16, 2005.Google Scholar
Wilmoth, J. R., Andreev, K., Jdanov, D. and Glei, D. A. (2007) Methods protocol for the human mortality database. Available online at http://www.mortality.org, 08/14/2012.Google Scholar
Yaari, M. (1965) Uncertain lifetime, life insurance, and the theory of the consumer. Review of Economic Studies, 32: 137–150.CrossRefGoogle Scholar
Yogo, M. (2009) Portfolio choice in retirement: health risk and the demand for annuities, housing, and risky assets. NBER Working Paper 15307.CrossRefGoogle Scholar
Zeng, L. (2008) Optimal Consumption and Portfolio Choice for Retirees. Working Paper.Google Scholar