Folie 1 - American Risk and Insurance Association
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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 Ct11/y bEt ptsVt11r 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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11/y
1 r
1
11/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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