Goethe University Frankfurt

1. Life-Cycle Asset Allocation with Annuity Markets:Is Longevity Insurance a Good Deal?byWolfram J. Horneff, Raimond H. Maurer, and Michael Z. StamosDepartment of Finance, Goethe University (Frankfurt)(ARIA, Quebec City, 2007) Goethe University Frankfurt

2. Motivation Rising life expectancies, low birth rates -> worldwide shift to privately funded pension systems Household risk management: Uncertain capital market returns Uncertain labor income Uncertain time of death (mortality risk) Questions: What is the optimal dynamic portfolio choice with constant life annuities, stocks, and bonds? What are the welfare effects of purchasing a life-annuity in a realistically calibrated life-cycle model? 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

3. Outline Motivation Prior Literature The Model Key Results Conclusion 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 3/16

4. Prior Literature Long Horizon Asset Allocation with Stochastic Investment Opportunity Set: Brandt (1999 JF), Brennan and Xia (2000 EurFR,2002 JF), Campbell and Viceira (1999 QJE,2001 AER), Campbell, Chan, and Viceira (2003 JFE), Wachter (2002 JFQA,2003 JET), … Labor Income Implications on Portfolio Coice: Bodie, Merton, and Samuelson (1992 JEDC), Cocco, Gomes, and Maenhout (2005 RFS), Heaton and Lucas (1997 MD), Viceira (2001 JF). Also literature on housing, entrepreneurial risk, and taxes… Mortality Risk and Annuity Markets: Koijen, Nijman, and Werker (2006 WP), Cairns, Blake, and Dowd (2006, JEDC) 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 4/16

5. Contributions Implications of annuity markets on household portfolio choice Optimization of the annuitization strategy over entire life-cycle: gradual annuitization possible Consideration of labor income risk and bequest motives Sensitivity analysis including common explanations for limited annuity participation 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 5/16

6. The Model: Life-Annuity Market Immediate Constant Payout Life Annuity: like a fixed coupon corporate bond (default: time of death) Pricing: Mortality credit is compensation for: Lack of bequest potential Lost flexibility Mortality credit 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 6/16

7. The Model: Labor Income Process The process of labor income follows during (see Cocco et al. (2005) ) Working life t≤K f(t): deterministic function of age Pt: permanent component with innovation Nt Ut: transitory income shock Logarithms of Nt and Ut: multivariate normal distributed with means zero, with volatilities sN, sU and correlation zero. Retirement: t>K 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 7/16

8. The Model: Wealth Accumulation The budget constraint is Wt: wealth on hand Mt: amount invested in riskless bonds St: amount invested in risky stocks PRt: amount invested in annuities Ct: consumption. The individual’s cash on hand in t + 1 is given by Lt+1: sum of annuity payments Yt+1: labor income Rf: riskless growth rate of bonds Rt+1: risky growth rate of stocks 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 8/16

9. The Model: Preferences Preferences as in Epstein and Zin (1989) are described by r: level of relative risk aversion y:elasticity of intertemporal substitution b: personal discount factor k: the strength of the bequest motive ps: personal survival probabilities Optimization problem: 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 9/16

10. Results: Optimal Asset Allocation with Annuities Optimal policies: cash on hand w allocated in Stocks s(w,l,t) Bonds m(w,l,t) New annuities pr(w,l,t) Consumption c(w,l,t) Policies depend on normalized cash on hand w, normalized annuity income l, and age t(Normalization with permanent income) 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 10/16

11. Stylized Case without Loads and Bequest (Figure 1) Stocks s(w,l=0,t) Annuities pr(w,l=0,t) Bonds m(w,l=0,t) w w w age age age Motives to hold liquid wealth: (1) stock demand, (2) buffer stock savings Age effect: (1) increasing mortality credit (mortality risk), (2) decreasing human capital, and (3) labor income uncertainty Wealth effect: the higher wealth on hand compared to bond-like human capital, the lower is the relative stock demand 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 11/16

12. Asset allocation: Gradual shift from liquid savings to illiquid annuities First crowding out of bonds then of stocks Expected Life-Cycle Profile (Figure 3-4) 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 12/16

13. Cost effect: annuitization postponed to age 59 Bequest effect: additional liquid wealth motive, but still substantial annuity demand Expected Life-Cycle Profile (Figure 3-4) With loads With loads and bequest motives 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 13/16

14. Sensitivity of annuity demand regarding to factors deemed to explain the annuity puzzle: costs, bequest, bad health, high pension income Robustness Analysis of Annuity Demand: Table II 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 14/16

15. Welfare Analysis: Table III Equivalent Increase in Financial Wealth: additional financial wealth needed to compensate for the utility loss if no annuities available. 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 15/16

16. Conclusions: Longevity Insurance a Good Deal? Endogenizing the annuitization strategy shows Gradual purchase optimal Timing of annuity purchase crucial (Age effect, Wealth effect) Model predicts empirically found timing of annuity purchase Mortality credit high enough to compensate for forfeit bequest potential, illiquidity and lack of equity premium Welfare increase equivalent to 10-30% more cash on hand Outlook: Allow for variable payout annuities Model could be used to add behavioral explanations: e.g. informational costs 08/06/2007 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal? 16/16

17. Thank You for Your Attention!Life-Cycle Asset Allocation with Annuity Markets:Is Longevity Insurance a Good Deal?Wolfram J. Horneff, Raimond H. Maurer, and Michael Z. StamosDepartment of Finance, Goethe University (Frankfurt) Goethe University Frankfurt

18. AppendixLife-Cycle Asset Allocation with Annuity Markets:Is Longevity Insurance a Good Deal?Wolfram J. Horneff, Raimond H. Maurer, and Michael Z. StamosDepartment of Finance, Goethe University (Frankfurt) Goethe University Frankfurt

19. Technical Appendix: Numerical Solution Dynamic optimization problem in a three-dimensional state space Continuous state variables: Normalized wealth Normalized annuity payouts Discrete state variable: Age Calculations of expectations (multiple integral): quadrature integration One period optimization: numerical constrained minimization Policy functions derived by cubic-splines interpolation 12/13/2006 Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?