Department of Finance, Goethe University
Frankfurt, Germany
Optimal Dynamic Consumption and Portfolio Choice for
Pooled Annuity Funds
by
Michael Z. Stamos
ARIA, Quebec City, 2007
Introduction (I)
 Individual Self-Annuitization (ISA)*:
Ruin risk and utility implications well understood
 Optimal annuitization strategies**:
Optimal asset allocation and timing of (variable/fixed) lifeannuity 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)
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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
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)
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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 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
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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 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
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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).
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Population Model II
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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
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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 fund
 pt: 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 pt 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
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Goethe University
Frankfurt
Thank You for Your Attention!
Optimal Dynamic Consumption and Portfolio Choice for
Pooled Annuity Funds
Michael Z. Stamos
Department of Finance, Goethe University (Frankfurt)