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Optimal retirement consumption with a stochastic force of mortality. (English) Zbl 1284.91162

Summary: We extend the lifecycle model (LCM) of consumption over a random horizon (also known as the Yaari model) to a world in which (i) the force of mortality obeys a diffusion process as opposed to being deterministic, and (ii) consumers can adapt their consumption strategy to new information about their mortality rate (also known as health status) as it becomes available. In particular, we derive the optimal consumption rate and focus on the impact of mortality rate uncertainty versus simple lifetime uncertainty – assuming that the actuarial survival curves are initially identical – in the retirement phase where this risk plays a greater role.{
}In addition to deriving and numerically solving the partial differential equation (PDE) for the optimal consumption rate, our main general result is that when the utility preferences are logarithmic the initial consumption rates are identical. But, in a constant relative risk aversion (CRRA) framework in which the coefficient of relative risk aversion is greater (smaller) than one, the consumption rate is higher (lower) and a stochastic force of mortality does make a difference.{
}That said, numerical experiments indicate that, even for non-logarithmic preferences, the stochastic mortality effect is relatively minor from the individual’s perspective. Our results should be relevant to researchers interested in calibrating the lifecycle model as well as those who provide normative guidance (also known as financial advice) to retirees.

MSC:

91B25 Asset pricing models (MSC2010)
91B30 Risk theory, insurance (MSC2010)
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[1] Attanasio, O.; Weber, G., Consumption and saving: models of intertemporal allocation and their implications for public policy, Journal of economic literature, 48, 693-751, (2010)
[2] Babbel, D., Merrill, C., 2006. Rational decumulation. Working paper, Wharton Financial Institutions Centre.
[3] Bauer, D.; Borger, J.; Zwiesler, H., The volatility of mortality, Asia-Pacific journal of risk and insurance, 3, 184-211, (2008)
[4] Bodie, Z.; Detemple, J.; Ortuba, S.; Walter, S., Optimal consumption-portfolio choice and retirement planning, Journal of economic dynamics and control, 28, 1115-1148, (2004) · Zbl 1179.91230
[5] ()
[6] Bommier, A.; Villeneuve, B., Risk aversion and the value of risk to life, Journal of risk and insurance, 79, 1, 77-103, (2012)
[7] Brillinger, D., A justification of some common laws of mortality, Transactions of the society of actuaries, 13, 116-119, (1961)
[8] Butler, M., Neoclassical life-cycle consumption: a textbook example, Economic theory, 17, 209-221, (2001) · Zbl 0970.91038
[9] Cairns, A.; Blake, D.; Dowd, K., A two-factor model for stochastic mortality with parameter uncertainty, Journal of risk and insurance, 73, 4, 687-718, (2006)
[10] Carrière, J., An investigation of the Gompertz law of mortality, Actuarial research clearing house, 2, 1-34, (1994)
[11] Chiang, A., Elements of dynamic optimization, (1992), Waveland Press Long Grove, IL
[12] Cocco, J., Gomes, F., 2009. Longevity risk and retirement savings. Working paper, London Business School.
[13] Dahl, M., Stochastic mortality in life insurance: market reserves and mortality-linked insurance contracts, Insurance: mathematics and economics, 35, 113-136, (2004) · Zbl 1075.62095
[14] Davies, J., Uncertain lifetime, consumption and dissaving in retirement, Journal of political economy, 89, 561-577, (1981)
[15] Deaton, A., Saving and liquidity constraints, Econometrica, 59, 5, 1221-1249, (1991)
[16] Dothan, L., On the term structure of interest rates, The journal of financial economics, 6, 59-69, (1978)
[17] Dybvig, P., Liu, H., 2005. Lifetime consumption and investment: retirement and constrained borrowing. Working paper, John M. Olin School of Business. · Zbl 1245.91044
[18] Feigenbaum, J., Can mortality risk explain the consumption hump?, Journal of macroeconomics, 30, 844-872, (2008)
[19] Fisher, I., The theory of interest: As determined by impatience to spend income and opportunity to invest it, (1930), Macmillan New York · JFM 56.1108.04
[20] Gutterman, S.; Vanderhoof, I., Forecasting changes in mortality: a search for a law of causes and effects, North American actuarial journal, 2, 4, 135-138, (1998) · Zbl 1081.91602
[21] Kingston, G.; Thorp, S., Annuitization and asset allocation with HARA utility, Journal of pension economics and finance, 4, 3, 225-248, (2005)
[22] Kotlikoff, L., Economics’ approach to financial planning, Journal of financial planning, 21, 3, 42-52, (2008)
[23] Lachance, M., Optimal onset and exhaustion of retirement savings in a life-cycle model, Journal of pension economics and finance, 11, 1, 21-52, (2012)
[24] Lee, R.; Carter, L., Modeling and forecasting the time series of U.S. mortality, Journal of the American statistical association, 87, 659-671, (1992)
[25] Leung, S., Uncertain lifetime, the theory of the consumer and the life cycle hypothesis, Econometrica, 62, 5, 1233-1239, (1994) · Zbl 0812.90026
[26] Levhari, D.; Mirman, L., Savings and consumption with an uncertain horizon, Journal of political economy, 85, 2, 265-281, (1977)
[27] Lucas, R., Asset prices in an exchange economy, Econometrica, 46, 1429-1445, (1978) · Zbl 0398.90016
[28] Menoncin, F., The role of longevity bonds in optimal portfolios, Insurance: mathematics and economics, 42, 348-358, (2008) · Zbl 1141.91537
[29] Merton, R., Optimal consumption and portfolio rules in a continuous time model, Journal of economic theory, 3, 4, 373-413, (1971) · Zbl 1011.91502
[30] Milevsky, M.; Huang, H., Spending retirement on planet vulcan: the impact of longevity risk aversion on optimal withdrawal rates, Financial analysts journal, 67, 2, 45-58, (2010)
[31] Milevsky, M.; Promislow, S., Mortality derivatives and the option to annuitize, Insurance: mathematics and economics, 29, 299-318, (2001) · Zbl 1074.62530
[32] Modigliani, F., Lifecycle, individual thrift and the wealth of nations, American economic review, 76, 3, 297-313, (1986)
[33] Modigliani, F.; Brumberg, R., Utility analysis and the consumption function: an interpretation of cross-section data, (), 388-436
[34] Norberg, R., Forward mortality and other vital signs – are they the way forward?, Insurance: mathematics and economics, 47, 2, 105-112, (2010) · Zbl 1231.91459
[35] Park, M., An analytical solution to the inverse consumption function with liquidity constraints, Economics letters, 92, 389-394, (2006) · Zbl 1254.91433
[36] Phelps, E., The accumulation of risk capital: a sequential utility analysis, Econometrica, 30, 4, 729-743, (1962) · Zbl 0126.36402
[37] Pitacco, E.; Denuit, M.; Haberman, S.; Olivieri, A., Modelling longevity dynamics for pensions and annuity business, (2008), Oxford University Press UK · Zbl 1163.91005
[38] Post, T., 2010. Individual welfare gains from deferred life annuities under stochastic mortality. Working paper, Netspar.
[39] Ramsey, F., A mathematical theory of saving, The economic journal, 38, 152, 543-559, (1928)
[40] Richard, S., Optimal consumption, portfolio and life insurance rules for an uncertain lived individual in a continuous time model, Journal of financial economics, 2, 187-203, (1975)
[41] Stevens, R., 2009. Annuity decisions with systematic longevity risk. Working paper, Netspar.
[42] Wallmeier, M.; Zainhofer, F., How to invest over the lifecycle: insights from theory, Journal für betriebswirtschaft, 56, 4, 219-244, (2007)
[43] Wills, S.; Sherris, M., Securitization, structuring and pricing of longevity risk, Insurance: mathematics and economics, 46, 1, 173-185, (2010) · Zbl 1231.91251
[44] Yaari, M., On the consumer’s lifetime allocation process, International economic review, 5, 3, 304-317, (1964)
[45] Yaari, M., Uncertain lifetime, life insurance and the theory of the consumer, The review of economic studies, 32, 2, 137-150, (1965)
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