Transcript Folie 1 - American Risk and Insurance Association

Goethe University
Frankfurt
Life-Cycle Asset Allocation with Annuity Markets:
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)
Motivation
 Rising life expectancies, low birth rates -> worldwide shift to
privately funded pension systems
 Household risk management:
Uncertain capital market returns
Uncertain labor income
Uncertain 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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Outline
1.
2.
3.
4.
5.
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Motivation
Prior Literature
The Model
Key Results
Conclusion
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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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Contributions
Implications of annuity markets on household portfolio
choice
Optimization of the annuitization strategy over entire lifecycle: gradual annuitization possible
Consideration of labor income risk and bequest motives
Sensitivity analysis including common explanations for
limited annuity participation
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The Model: Life-Annuity Market
 Immediate Constant Payout Life Annuity: like a fixed
coupon corporate bond (default: time of death)
T
CFt  pta ( s)
CFt

 Pricing: PRt  
s t
Rf
1 
s t 1
s t 1 s t
 R f

a
pt ( s) 

 

T
 R sf t
Mortality credit
 Mortality credit is compensation for:
Lack of bequest potential
Lost flexibility
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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
Yt  exp f t  Pt U t ,
Pt  Pt 1 N t .
f(t):
Pt:
U t:
deterministic function of age
permanent component with innovation Nt
transitory income shock
Logarithms of Nt and Ut: multivariate normal distributed with means
zero, with volatilities sN, sU and correlation zero.
 Retirement: t>K
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Yt   exp f K PK
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The Model: Wealth Accumulation
 The budget constraint is
Wt  M t  St  PRt  Ct
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
Wt 1  M t R f  St Rt 1  Lt 1  Yt 1
Lt+1: sum of annuity payments
Yt+1: labor income
Rf: riskless growth rate of bonds
Rt+1: risky growth rate of stocks
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The Model: Preferences
 Preferences as in Epstein and Zin (1989) are described by


1 r
Vt   1  bpts Ct11/y  bEt ptsVt11r  1  pts k Bt 1


r:
y:
b:
k:
ps:






level of relative risk aversion
elasticity of intertemporal substitution
personal discount factor
the strength of the bequest motive
personal survival probabilities
 Optimization problem:

max

Ct , M t , St , PRt T0
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
11/y
1 r
1
11/y
V0
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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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Stylized Case without Loads and Bequest (Figure 1)
Stocks s(w,l=0,t)
w
age
Annuities pr(w,l=0,t)
w
age
Bonds m(w,l=0,t)
w
age
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
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Expected Life-Cycle Profile (Figure 3-4)
 Asset allocation:
 Gradual shift from
liquid savings to
illiquid annuities
 First crowding out
of bonds then of
stocks
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Expected Life-Cycle Profile (Figure 3-4)
With loads
With loads and bequest motives
Cost effect: annuitization postponed to age 59
Bequest effect: additional liquid wealth motive, but still substantial annuity demand
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Robustness Analysis of Annuity Demand: Table II
Sensitivity of annuity demand regarding to factors deemed to explain the
annuity puzzle: costs, bequest, bad health, high pension income
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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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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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Goethe University
Frankfurt
Thank You for Your Attention!
Life-Cycle Asset Allocation with Annuity Markets:
Is Longevity Insurance a Good Deal?
Wolfram J. Horneff, Raimond H. Maurer, and Michael Z. Stamos
Department of Finance, Goethe University (Frankfurt)
Goethe University
Frankfurt
Appendix
Life-Cycle Asset Allocation with Annuity Markets:
Is Longevity Insurance a Good Deal?
Wolfram J. Horneff, Raimond H. Maurer, and Michael Z. Stamos
Department of Finance, Goethe University (Frankfurt)
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
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