Optimal Gradual Annuitization:Optimal Gradual Annuitization:Quantifying the Cost of Switching to Quantifying the Cost of Switching to
AnnuitiesAnnuities
by
Wolfram Horneff*, Raimond Maurer*, and Michael Stamos*
*Department of Finance, Goethe University Frankfurt, Germany
2007 IME Conference, Piraeus, Greece
Goethe University Frankfurt, Germany
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Introduction
Increasing public awareness of longevity insurance PAYGO vs. privately funded pension system DB vs. DC
Developing a strategy for retirement is key (Retirement assets in the US: 15 Trillions)
Individuals have the role of risk managers
Main risks: (1) capital market risks(2) mortality
Building portfolios of mutual funds and life-annuities Intertemporaneous mix Contemporaneously mix Advantage: access both equity premium and longevity insurance
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Policy/Regulatory Relevance
UK: accumulated occupational pension assets has to be annuitized by age 75
Germany’s “Riester” plans provide a tax inducement if life annuity payments begin to pay out at age 85
In the US, annuitization not compulsory for 401(k) plans Low annuity demand tax laws require minimum distributions to begin at age 70 ½
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Annuity Mechanics I: Mortality Credit
Simple 1-period example: Alternative 1: direct bond investmentAlternative 2: invest in bonds through annuity
Real interest rate: r = 2%, survival prob.: p = 90%
Age dependent return profile
End-of-year payoff
Initial Investment Alive Dead
(1) 100 (in bond) 100(1+r)=102
100(1+r)=102
(2)
End-of-year payoff
Initial Investment Alive Dead
(1) 100 (in bond) 100(1+r)=102
100(1+r)=102
(2) 100 (in annuity) 100(1+r)/p=113.3 0
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Annuity Mechanics II
Immediate Constant Payout Life Annuity: like a fixed coupon corporate bond (default: time of death)
Pricing:
Mortality credit is compensation for illiquidity: Once purchased annuities have to be held until deathOpportunity costs: no equity premium, lack of bequest
potential, inflexibility
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Most Related Insurance Literature
Blake, Cairns, and Dowd, (2003), Pensionmetrics II: Stochastic pension plan design during the distribution phase, Insurance: Mathematics and Economics. Numerical derivation of complete switching time (with varying
bequest motive and annuity costs)
Milevsky and Young, 2007, Annuitization and Asset Allocation, Journal of Economic Dynamics and Control. Derived analytically optimal asset allocation and complete switching
time Also, gradual annuitization derived for restrictive case (up to solution
of ODEs) (constant force of mortality, no annuity cost) But, lack of pre-existing pension income, bequest motives
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Contributions
Comparison of different annuitization strategies: Complete stochastic switching Partial stochastic switching Gradual annuitization
Dynamic optimization of asset allocation (stocks, bonds and annuities) and consumption
Robustness: Pre-existing pensions (e.g. public and/or occupational) Non additive utility (Epstein/Zin) Bequest motives Annuity costs
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The Model: Capital Markets
Riskless bonds: Rf: riskless growth rate in real terms (= 1.02)
Risky stocks: Rt ~ LN(,)
: expected growthrate (= 1.06) : standard deviation (= 18 percent)
Bonds and stocks can be traded each year
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The Model: Wealth Evolution
Contemporary budget constraint Gradual Annuitization Partial Switching Complete Switching
Wt: cash on handMt: amount invested in bonds St: amount invested in stocksPRt: amount invested in annuity Ct: consumption
Cash on hand in t + 1 conditional on survivalGradual Annuitization Partial Switching Complete Switching
Wt+1: next period cash on handLt+1: sum of annuity payments
Pt+1: annuity payoutsYt+1: public pension income.
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The Model: Preferences
Preferences as in Epstein and Zin (1989) are described by
level of relative risk aversion (= 5)
elasticity of intertemporal substitution (= 0.2)
personal discount factor (= 0.96)
k: strength of the bequest motive ( =0)
ps: subjective survival probabilities (population average [male])
C: consumption
B: bequest
Choose Ct, Mt, St and PRt in the way that Vt is maximized
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The Model: Optimization Problems
Complete Switching:
Partial Switching:
Gradual Annuitization:
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The Model: Numerical Solution
Normalize all variables with public pension income
Dynamic stochastic optimization of the Bellman equation in a 3-dimensional state space: Discretize continuous state variables:
- Normalized wealth- Normalized sum of annuity payouts
Discrete state variables:- Age- Switched or not
Calculations of expectations (multiple integral): gaussian cubature integration
One period optimization: numerical constrained maximization Value function derived by piecewise bi-cubic-splines Policy function by bi-cubic splines and neighboring interpolation
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Optimal Policies: Demand for Annuities (No Bequest)
Wealth Age
Annuity Payouts = 0
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Value of Postponing Annuitization
VSW(PR,.) < VSW(PR=0,.)No purchase region
PR0
VSW(PR,.) ≥ VSW(PR=0,.)Purchase region
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OpportunityCosts
VSW(PR=
VSW(PR=0,.)
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Annuitization Frontier and Expected Value of Purchases
Initial Multiple of Pension Income = 6
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Optimal Expected Annuity Stock Fraction for Various CRRA
Initial Multiple of Pension Income = 6
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The Impact of IES on
Initial Multiple of Pension Income = 6
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Equivalent Increase in Financial Wealth (Percentage Points)
Usually Switching restrictions show moderate utility losses High utility losses for high IES
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Conclusions
Hedging longevity risk is valuable for the retiree
Trade offs: Age effect: (1) increasing mortality credit (mortality risk),
(2) decreasing human capital/pension wealth Wealth effect: the higher wealth on hand compared to bond-like human
capital, the lower is the stock demand
Switching restrictions: annuitization postponed: wait until mortality credit high enough to
compensate higher opportunity costs High welfare losses for (1) Switching at 65 and (2) High IES
Future work: Interest rate risk and inflation risk Alternative longevity insurance Implications of Taxes and Housing (Reverse Mortgage)