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
Dec 15, 2015
Optimal Dynamic Consumption and Portfolio Choice forPooled Annuity Funds
by
Michael Z. Stamos
ARIA, Quebec City, 2007
Department of Finance, Goethe University Frankfurt, Germany
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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)
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Introduction (II)
Alternative mortality hedge: Group Self-Annuitization (GSA)
Construction of a Pooled Annuity Fund: Individuals pool their retirement wealth into an annuity
fundin case one participant dies, survivors share the released
funds (mortality credit)
In fact: GSA very common since families self-insure(Kotlikoff and Spivak, 1981)
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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
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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 modelResult: adverse selection problem inherent in life annuities
markets is alleviated in pooled annuity fundsRational: investors of pooled annuity funds cannot exploit
perfectly the gains of adverse selection since mortality credit is stochastic
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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 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
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Population Model I
Stochastic time of death of investor i determined by inhomogeneous Poisson Process Ni 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).
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
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Financial Markets
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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 fundt: portfolio weights
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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):
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Wealth Dynamics: Individual Wealth (II)
If all investors have equal share hi :
Expected instantaneous mortality credit:
Instantaneous variance of mortality credit:
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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 → ∞
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Wealth Dynamics: Mortality Credit Annualized (II)
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Optimization Problem
Investors have CRRA preferences
Optimize expected utility by choosing ct and t subject to:
1<Lt< ∞
Lt=1
Lt → ∞
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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)
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Numerical Results: Optimal Withdrawal Rate/Payout Profile
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Numerical Results: Optimal Withdrawal Rate/Payout Profile
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Numerical Results: Expected Consumption Path
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Numerical Results: Welfare Increase Relative to l=1
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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