Asset allocation and annuity-purchase strategies to minimize the probability of financial ruin. (English) Zbl 1130.91031

Summary: In this paper, we derive the optimal investment and annuitization strategies for a retiree whose objective is to minimize the probability of lifetime ruin, namely the probability that a fixed consumption strategy will lead to zero wealth while the individual is still alive. Recent papers in the insurance economics literature have examined utility-maximizing annuitization strategies. Others in the probability, finance, and risk management literature have derived shortfall-minimizing investment and hedging strategies given a limited amount of initial capital. This paper brings the two strands of research together. Our model pre-supposes a retiree who does not currently have sufficient wealth to purchase a life annuity that will yield her exogenously desired fixed consumption level. She seeks the asset allocation and annuitization strategy that will minimize the probability of lifetime ruin. We demonstrate that because of the binary nature of the investor’s goal, she will not annuitize any of her wealth until she can fully cover her desired consumption with a life annuity. We derive a variational inequality that governs the ruin probability and the optimal strategies, and we demonstrate that the problem can be recast as a related optimal stopping problem which yields a free-boundary problem that is more tractable. We numerically calculate the ruin probability and optimal strategies and examine how they change as we vary the mortality assumption and parameters of the financial model. Moreover, for the special case of exponential future lifetime, we solve the (dual) problem explicitly. As a byproduct of our calculations, we are able to quantify the reduction in lifetime ruin probability that comes from being able to manage the investment portfolio dynamically and purchase annuities.


91B30 Risk theory, insurance (MSC2010)
91B28 Finance etc. (MSC2000)
49L20 Dynamic programming in optimal control and differential games
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
60H30 Applications of stochastic analysis (to PDEs, etc.)
35R35 Free boundary problems for PDEs
35R60 PDEs with randomness, stochastic partial differential equations
Full Text: DOI Link


[1] Bowers, Actuarial Mathematics (1997)
[2] Brown, Private Pensions, Mortality Risk, and the Decision to Annuitize, J. Public Econ. 82 (1) pp 29– (2001)
[3] Browne, Optimal Investment Policies for a Firm with a Random Risk Process: Exponential Utility and Minimizing the Probability of Ruin, Math. Oper. Res. 20 (4) pp 937– (1995) · Zbl 0846.90012
[4] Browne, Beating a Moving Target: Optimal Portfolio Strategies for Outperforming a Stochastic Benchmark, Finance Stoch. 3 pp 275– (1999a) · Zbl 1047.91025
[5] Browne, The Risk and Rewards of Minimizing Shortfall Probability, J. Portfolio Manag. 25 (4) pp 76– (1999b)
[6] Browne, Reaching Goals by a Deadline: Digital Options and Continuous-Time Active Portfolio Management, Adv. Appl. Probab 31 pp 551– (1999c) · Zbl 0963.91053
[7] Brugiavini, Uncertainty Resolution and the Timing of Annuity Purchases, J. Public Econ. 50 pp 31– (1993)
[8] Copeland , C. 2002 An Analysis of the Retirement and Pension Plan Coverage Topical Module of SIPP http://www.ebri.org
[9] Cummins, Bill Rewards Buyers of Life Annuities; Plan Would Give Retirees Half of the Income Tax-Free, Star Tribune (2004)
[10] Davidoff , T. J. Brown P. Diamond 2003 Annuities and Individual Welfare
[11] Davis, Portfolio Selection with Transaction Costs, Math. Oper. Res. 15 pp 676– (1990) · Zbl 0717.90007
[12] Dixit, Investment under Uncertainty (1994)
[13] Duffie, Hedging in Incomplete Markets with HARA Utility, J. Econ. Dyn. Contr. 21 pp 753– (1997) · Zbl 0899.90026
[14] Feldstein, Individual Risk in an Investment-Based Social Security System, Am. Econ. Rev 91 (4) pp 1116– (2001)
[15] Friedman, A Variational Inequality Approach to Financial Valuation of Retirement Benefits Based on Salary, Finance Stoch 6 (3) pp 273– (2002)
[16] Gerber, An Introduction to Mathematical Risk Theory (1979) · Zbl 0431.62066
[17] Harrison, Instantaneous Control of Brownian Motion, Math. Oper. Res 8 (3) pp 439– (1983) · Zbl 0523.93068
[18] Huang, Ruined Moments in Your Life: How Good Are the Approximations?, Insurance: Math. Econ 34 (3) pp 421– (2004) · Zbl 1188.91233
[19] Kapur , S. M. Orszag 1999 A Portfolio Approach to Investment and Annuitization during Retirement
[20] Karatzas, Methods of Mathematical Finance (1998)
[21] Koo, Consumption and Portfolio Selection with Labor Income: A Continuous Time Approach, Math. Fin 8 pp 49– (1998) · Zbl 0911.90030
[22] Milevsky , M. A. V. R. Young 2003 Annuitization and Asset Allocation · Zbl 1163.91440
[23] Milevsky, Self-Annuitization and Ruin in Retirement, N. Am. Actuarial J 4 (4) pp 112– (2000) · Zbl 1083.60515
[24] Moore , K. S. V. R. Young 2006 Optimal and Simple, Nearly Optimal Rules for Minimizing the Probability of Ruin in Retirement
[25] Neuberger , A. 2002 Optimal Annuitization Strategies
[26] Øksendal, Stochastic Differential Equations: An Introduction with Applications (1998) · Zbl 0897.60056
[27] Parikh, The Evolving U.S. Retirement System, The Actuary pp 2– (2003)
[28] Poterba , J. M. 1997 The History of Annuities in the United States
[29] Richard, Optimal Consumption, Portfolio and Life Insurance Rules for an Uncertain Lived Individual in a Continuous Time Model, J. Financ. Econ 2 pp 187– (1975)
[30] Roy, Safety First and the Holding of Assets, Econometrica 20 pp 431– (1952) · Zbl 0047.38805
[31] Shreve, Optimal Investment and Consumption with Transaction Costs, Ann. Appl. Probab 4 (3) pp 206– (1994) · Zbl 0813.60051
[32] Society of Actuaries, Risks of Retirement-Key Findings and Issues (2004)
[33] VanDerhei , J. C. Copeland 2003 Can America Afford Tomorrow’s Retirees: Results from the EBRI-ERF Retirement Security Projection Model http://www.ebri.org
[34] Wilmott, Option Pricing: Mathematical Models and Computation (2000)
[35] Yaari, Uncertain Lifetime, Life Insurance and the Theory of the Consumer, Rev. Econ. Stud 32 pp 137– (1965)
[36] Young, Optimal Investment Strategy to Minimize the Probability of Lifetime Ruin, N. Am. Actuarial J 8 (4) pp 106– (2004) · Zbl 1085.60514
[37] Zariphopoulou, Investment/Consumption Models with Transaction Costs and Markov-Chain Parameters, SIAM J. Control Optim 30 pp 613– (1992)
[38] Zariphopoulou, Introduction to Mathematical Finance 57 pp 101– (1999)
[39] Zariphopoulou, Handbook of Stochastic Analysis and Applications (2001)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.