Life-Cycle Asset Allocation with Annuity Markets:Life-Cycle Asset Allocation with Annuity Markets:Is Longevity Insurance a Good Deal?Is Longevity Insurance a Good Deal?

by

Wolfram J. Horneff, Raimond H. Maurer, and Michael Z. Stamos

Department of Finance, Goethe University (Frankfurt)

(ARIA, Quebec City, 2007)

Goethe University Frankfurt

2/16Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

Motivation

Rising life expectancies, low birth rates -> worldwide shift to privately funded pension systems

Household risk management:Uncertain capital market returnsUncertain labor incomeUncertain time of death (mortality risk)

Questions:What is the optimal dynamic portfolio choice with

constant life annuities, stocks, and bonds?What are the welfare effects of purchasing a life-annuity

in a realistically calibrated life-cycle model?

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Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

Outline

1. Motivation

2. Prior Literature

3. The Model

4. Key Results

5. Conclusion

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Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

Prior Literature

Long Horizon Asset Allocation with Stochastic Investment Opportunity Set: Brandt (1999 JF), Brennan and Xia (2000 EurFR,2002 JF), Campbell and Viceira (1999 QJE,2001 AER), Campbell, Chan, and Viceira (2003 JFE), Wachter (2002 JFQA,2003 JET), …

Labor Income Implications on Portfolio Coice: Bodie, Merton, and Samuelson (1992 JEDC), Cocco, Gomes, and Maenhout (2005 RFS), Heaton and Lucas (1997 MD), Viceira (2001 JF).

Also literature on housing, entrepreneurial risk, and taxes…

Mortality Risk and Annuity Markets: Koijen, Nijman, and Werker (2006 WP), Cairns, Blake, and Dowd (2006, JEDC)

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Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

Contributions

Implications of annuity markets on household portfolio choice

Optimization of the annuitization strategy over entire life-cycle: gradual annuitization possible

Consideration of labor income risk and bequest motives

Sensitivity analysis including common explanations for limited annuity participation

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Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

The Model: Life-Annuity Market

Immediate Constant Payout Life Annuity: like a fixed coupon corporate bond (default: time of death)

Pricing:

Mortality credit is compensation for: Lack of bequest potentialLost flexibility

T

ts

R

at

tsf

tT

tsts

f

att

t

tsf

spR

CF

R

spCFPR

11

)(1

)(

Mortality credit

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The Model: Labor Income Process

The process of labor income follows during (see Cocco et al. (2005) )

Working life t≤K

f(t): deterministic function of age

Pt: permanent component with innovation Nt

Ut: transitory income shock

Logarithms of Nt and Ut: multivariate normal distributed with means zero, with volatilities N, U and correlation zero.

Retirement: t>K

.

,exp

1 ttt

ttt

NPP

UPtfY

Kt PKfY exp

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8/16Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

The Model: Wealth Accumulation

The budget constraint is

Wt: wealth on hand

Mt: amount invested in riskless bonds

St: amount invested in risky stocks

PRt: amount invested in annuities

Ct: consumption.

The individual’s cash on hand in t + 1 is given by

Lt+1: sum of annuity payments

Yt+1: labor income

Rf: riskless growth rate of bonds

Rt+1: risky growth rate of stocks

ttttt CPRSMW

1111 ttttftt YLRSRMW

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9/16Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

The Model: Preferences

Preferences as in Epstein and Zin (1989) are described by

level of relative risk aversion

elasticity of intertemporal substitution

personal discount factor

k: the strength of the bequest motive

ps: personal survival probabilities

Optimization problem:

/11

1

1

/1111

11

/11 11

tstt

sttt

stt BkpVpECpV

0,,, 0

max VT

tttt PRSMC

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Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

Results: Optimal Asset Allocation with Annuities

Optimal policies: cash on hand w allocated in

Stocks s(w,l,t)Bonds m(w,l,t)New annuities pr(w,l,t)Consumption c(w,l,t)

Policies depend on normalized cash on hand w, normalized annuity income l, and age t(Normalization with permanent income)

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Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

Stylized Case without Loads and Bequest (Figure 1)

Motives to hold liquid wealth: (1) stock demand, (2) buffer stock savings

Age effect: (1) increasing mortality credit (mortality risk), (2) decreasing human capital, and (3) labor income uncertainty

Wealth effect: the higher wealth on hand compared to bond-like human capital, the lower is the relative stock demand

agew agew

agew

Stocks s(w,l=0,t) Annuities pr(w,l=0,t) Bonds m(w,l=0,t)

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Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

Asset allocation:

Gradual shift from liquid savings to illiquid annuities

First crowding out of bonds then of stocks

Expected Life-Cycle Profile (Figure 3-4)

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Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

Cost effect: annuitization postponed to age 59

Bequest effect: additional liquid wealth motive, but still substantial annuity demand

Expected Life-Cycle Profile (Figure 3-4)

With loads With loads and bequest motives

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Sensitivity of annuity demand regarding to factors deemed to explain the annuity puzzle: costs, bequest, bad health, high pension income

Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

Robustness Analysis of Annuity Demand: Table II

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Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

Welfare Analysis: Table III

Equivalent Increase in Financial Wealth: additional financial wealth needed to compensate for the utility loss if no annuities available.

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Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

Conclusions: Longevity Insurance a Good Deal?

Endogenizing the annuitization strategy shows Gradual purchase optimal Timing of annuity purchase crucial (Age effect, Wealth effect) Model predicts empirically found timing of annuity purchase

Mortality credit high enough to compensate for forfeit bequest potential, illiquidity and lack of equity premium

Welfare increase equivalent to 10-30% more cash on hand

Outlook: Allow for variable payout annuities Model could be used to add behavioral explanations: e.g.

informational costs

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Thank You for Your Attention!

Life-Cycle Asset Allocation with Annuity Markets:Life-Cycle Asset Allocation with Annuity Markets:Is Longevity Insurance a Good Deal?Is Longevity Insurance a Good Deal?

Wolfram J. Horneff, Raimond H. Maurer, and Michael Z. StamosDepartment of Finance, Goethe University (Frankfurt)

Goethe University Frankfurt

Appendix

Life-Cycle Asset Allocation with Annuity Markets:Life-Cycle Asset Allocation with Annuity Markets:Is Longevity Insurance a Good Deal?Is Longevity Insurance a Good Deal?

Wolfram J. Horneff, Raimond H. Maurer, and Michael Z. StamosDepartment of Finance, Goethe University (Frankfurt)

Goethe University Frankfurt

19/15Life-Cycle Asset Allocation with Annuity Markets: Is Longevity Insurance a Good Deal?

Technical Appendix: Numerical Solution

Dynamic optimization problem in a three-dimensional state space

Continuous state variables: Normalized wealth Normalized annuity payouts

Discrete state variable: Age

Calculations of expectations (multiple integral): quadrature integration

One period optimization: numerical constrained minimization

Policy functions derived by cubic-splines interpolation

12/13/2006