Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds by Michael Z. Stamos ARIA, Quebec City, 2007 Department of Finance, Goethe University Frankfurt, Germany

2/22Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 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 hedged e.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), and Horneff et al. (2006a,2006b,2006c,2007)

3/22Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 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)

4/22Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 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

5/22Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 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

6/22Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds Contributions  Derivation of the optimal consumption and portfolio choice for pooled annuity funds  If l is the number of homogenous participants: l = 1: Individual Self Annuitization 1< l < ∞ : Group Self Annuitization l → ∞ : 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

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

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

 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 9/22Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds Financial Markets

10/22Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds Wealth Dynamics: Total Annuity Fund Wealth  L 0 homogenous participants pool their wealth W i,0 in annuity fund (AF)  Dynamics of the total fund value  c t : continuous withdrawal-rate from pooled annuity fund   t : portfolio weights

11/22Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 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):

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

13/22Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 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 → ∞

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

15/22Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds Optimization Problem  Investors have CRRA preferences  Optimize expected utility by choosing c t and  t subject to:  1<L t < ∞  L t =1  L t → ∞

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

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

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

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

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

21/22Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds 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

Thank You for Your Attention! Optimal Dynamic Consumption and Portfolio Choice for Pooled Annuity Funds Michael Z. Stamos Department of Finance, Goethe University (Frankfurt) Goethe University Frankfurt