department of finance, goethe university frankfurt, germany

Optimal Dynamic Consumption and Portfolio Choice forPooled 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 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)


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)


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


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


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


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).


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


Financial Markets


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


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):


Wealth Dynamics: Individual Wealth (II)

If all investors have equal share hi :

Expected instantaneous mortality credit:

Instantaneous variance of mortality credit:


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 → ∞


Wealth Dynamics: Mortality Credit Annualized (II)


Optimization Problem

Investors have CRRA preferences

Optimize expected utility by choosing ct and t subject to:

1<Lt< ∞

Lt=1

Lt → ∞


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)


Numerical Results: Optimal Withdrawal Rate/Payout Profile


Numerical Results: Expected Consumption Path


Numerical Results: Welfare Increase Relative to l=1


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. StamosDepartment of Finance, Goethe University (Frankfurt)

Goethe University Frankfurt

department of finance, goethe university frankfurt, germany

Documents