Department of Finance, Goethe University Frankfurt, Germany

1. Optimal Dynamic Consumption and Portfolio Choice forPooled Annuity FundsbyMichael Z. StamosARIA, Quebec City, 2007 Department of Finance, Goethe University Frankfurt, Germany

2. Introduction (I) Individual Self-Annuitization (ISA)*:Ruin risk and utility implications well understood Optimal annuitization strategies**:Optimal asset allocation and timing of (variable/fixed) life-annuity purchases Current consensus: mortality risk should be hedgede.g. Mitchell et al. (1999) and Brown et al. (2001) report utility gains of around 40% ! *e.g. Merton (1971), Albrecht and Maurer (2002), Ameriks et al. (2001), Bengen (1994, 1997), Dus et al. (2005), Ho et al. (1994), Hughen et al. (2002), Milevsky (1998, 2001), Milevsky and Robinson (2000), Milevsky et al. (1997), and Pye (2000, 2001). **e.g.Yaari (1965), Richard (1975), Babbel and Merrill (2006), Cairns et al. (2006), Koijen et al. (2006), Milevsky and Young (2007), Milevsky et al. (2006), andHorneff et al. (2006a,2006b,2006c,2007) Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds

3. Introduction (II) Alternative mortality hedge: Group Self-Annuitization (GSA) Construction of a Pooled Annuity Fund: Individuals pool their retirement wealth into an annuity fund in case one participant dies, survivors share the released funds (mortality credit) In fact: GSA very common since families self-insure(Kotlikoff and Spivak, 1981) Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 3/22

4. Introduction (III) Trade Offs between mutual fund, pooled annuity fund and life-annuity: Common characteristics: assets underlying the different wrappers can be broadly diversified GSA / life-annuity versus mutual fund earn the mortality credit but lose bequest potential lost flexibility since purchase is irrevocable (due to severe adverse selection) GSA versus life-annuity: group bears some mortality risk, but has to pay no risk premium to owners of an insurance provider Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 4/22

5. Prior Literature on Pooled Annuity Funds Piggott, Valdez, and Detzel (2005): The Simple Analytics of Pooled Annuity Funds,” The Journal of Risk and Insurance, 72 (3), 497–520. Mechanics of pooled annuity funds: recursive evolution of payments over time given that investment returns and mortality deviate from expectation But, no explicit modeling of risks Valdez, Piggott, and Wang (2006): “Demand and adverse selection in a pooled annuity fund,” Insurance: Mathematics and Economics, 39, 251–266. Two period model Result: adverse selection problem inherent in life annuities markets is alleviated in pooled annuity funds Rational: investors of pooled annuity funds cannot exploit perfectly the gains of adverse selection since mortality credit is stochastic Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 5/22

6. Contributions Derivation of the optimal consumption and portfolio choice for pooled annuity funds Iflis the number of homogenous participants:l = 1:Individual Self Annuitization1< l < ∞: Group Self Annuitizationl → ∞: Ideal Life-Annuity Integration of Individual / Group-Self-Annuitization and Life Annuity in a Merton continuous time framework with stochastic investment horizon Prior literature on life-annuities sets the payout pattern exogenously (via so-called “assumed interest rate”, AIR) Evaluation of self-insurance effectiveness for various pool sizes Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 6/22

7. Population Model I • Stochastic time of death of investor i determined by inhomogeneous Poisson Process Ni with jump intensity l(t) • Probability of no-jump (surviving) between t and s > t: with l(t)according to Gompertz Law (fits empirical data, easy to estimate). Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 7/22

8. Population Model II Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 8/22

9. Financial Markets • Risky asset e.g. globally diversified stock portfolio: • Riskless asset e.g. local money market: • Parsimonious asset model to focus on effects of mortality risk Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 9/22

10. Wealth Dynamics: Total Annuity Fund Wealth • L0 homogenous participants pool their wealth Wi,0 in annuity fund (AF) • Dynamics of the total fund value • ct: continuous withdrawal-rate from pooled annuity fund • pt: portfolio weights Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 10/22

11. Wealth Dynamics: Individual Wealth (I) • Fraction of AF assets belonging to individual i • Reallocation of individual wealth in case j dies: • Dynamics of individual’s wealth (Ito’s Lemma): Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 11/22

12. Wealth Dynamics: Individual Wealth (II) • If all investors have equal share hi : • Expected instantaneous mortality credit: • Instantaneous variance of mortality credit: Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 12/22

13. Wealth Dynamics: Mortality Credit (I) • Long-run expected mortality credit for finite pools with size l: expected growth-rate between t and s due to reallocation of wealth MC(t,s): deterministic mortality credit if l → ∞ Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 13/22

14. Wealth Dynamics: Mortality Credit Annualized (II) Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 14/22

15. Optimization Problem • Investors have CRRA preferences • Optimize expected utility by choosing ct and pt subject to: • 1<Lt< ∞ • Lt=1 • Lt → ∞ Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 15/22

16. Analytical Results • Optimal stock fraction for all cases: • Optimal Consumption fraction • Lt = 1: • Lt → ∞: • 1<Lt< ∞: Solve ODEs numerically for f(l,t) and plug into c(l,t) Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 16/22

17. Numerical Results: Optimal Withdrawal Rate/Payout Profile Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 17/22

18. Numerical Results: Optimal Withdrawal Rate/Payout Profile Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 18/22

19. Numerical Results: Expected Consumption Path Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 19/22

20. Numerical Results: Welfare Increase Relative to l=1 Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 20/22

21. Conclusion Integration of Individual / Group-Self-Annuitization and Life Annuity in a Merton continuous time framework with stochastic investment horizon Derivation of the optimal consumption and portfolio choice for l = 1:Individual Self Annuitization 1<l< ∞ : Group-Self-Annuitization l → ∞ : Life-Annuity with optimal payout profile GSA is an effective mortality hedge Utility gains almost as high as those of optimal and actuarially fair life-annuities Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 21/22

22. Thank You for Your Attention!Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity FundsMichael Z. StamosDepartment of Finance, Goethe University (Frankfurt) Goethe University Frankfurt