Goethe University Frankfurt, Germany

1. Goethe University Frankfurt, Germany Optimal Gradual Annuitization:Quantifying the Cost of Switching to AnnuitiesbyWolfram Horneff*, Raimond Maurer*, and Michael Stamos**Department of Finance, Goethe University Frankfurt, Germany2007 IME Conference, Piraeus, Greece

2. 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 Gradual Annuitization: Quantifying the Costs of Switching to Annuities

3. 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 Gradual Annuitization: Quantifying the Costs of Switching to Annuities

4. Annuity Mechanics I: Mortality Credit Simple 1-period example: Alternative 1: direct bond investment Alternative 2: invest in bonds through annuity Real interest rate: r = 2%, survival prob.: p = 90% Age dependent return profile Gradual Annuitization: Quantifying the Costs of Switching to Annuities

5. 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 death Opportunity costs: no equity premium, lack of bequest potential, inflexibility Mortality credit Gradual Annuitization: Quantifying the Costs of Switching to Annuities

6. 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 Gradual Annuitization: Quantifying the Costs of Switching to Annuities

7. 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 Gradual Annuitization: Quantifying the Costs of Switching to Annuities

8. The Model: Capital Markets • Riskless bonds: • Rf: riskless growth rate in real terms (= 1.02) • Risky stocks: • Rt~ LN(m,s) • m: expected growthrate (= 1.06) • s: standard deviation (= 18 percent) • Bonds and stocks can be traded each year Gradual Annuitization: Quantifying the Costs of Switching to Annuities

9. The Model: Wealth Evolution • Contemporary budget constraint Gradual Annuitization Partial Switching Complete Switching Wt: cash on hand Mt: amount invested in bonds St: amount invested in stocks PRt: amount invested in annuity Ct:consumption • Cash on hand in t + 1 conditional on survival Gradual Annuitization Partial Switching Complete Switching Wt+1:next period cash on hand Lt+1:sum of annuity payments Pt+1: annuity payouts Yt+1:public pension income. Gradual Annuitization: Quantifying the Costs of Switching to Annuities

10. The Model: Preferences Preferences as in Epstein and Zin (1989) are described by r: level of relative risk aversion (= 5) y:elasticity of intertemporal substitution (= 0.2) b: 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 Gradual Annuitization: Quantifying the Costs of Switching to Annuities

11. The Model: Optimization Problems Complete Switching: Partial Switching: Gradual Annuitization: Gradual Annuitization: Quantifying the Costs of Switching to Annuities

12. 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 Gradual Annuitization: Quantifying the Costs of Switching to Annuities

13. Optimal Policies: Demand for Annuities (No Bequest) Wealth Age Annuity Payouts = 0 Gradual Annuitization: Quantifying the Costs of Switching to Annuities

14. Value of Postponing Annuitization VSW(PR,.) VSW(PR=0,.) Opportunity Costs VSW(PR=e,.) PR PRind 0 VSW(PR,.) < VSW(PR=0,.) No purchase region VSW(PR,.) ≥ VSW(PR=0,.) Purchase region Gradual Annuitization: Quantifying the Costs of Switching to Annuities

15. Annuitization Frontier and Expected Value of Purchases Initial Multiple of Pension Income = 6 Gradual Annuitization: Quantifying the Costs of Switching to Annuities

16. Optimal Expected Annuity Stock Fraction for Various CRRA Initial Multiple of Pension Income = 6 Gradual Annuitization: Quantifying the Costs of Switching to Annuities

17. The Impact of IES on Initial Multiple of Pension Income = 6 Gradual Annuitization: Quantifying the Costs of Switching to Annuities

18. Equivalent Increase in Financial Wealth (Percentage Points) • Usually Switching restrictions show moderate utility losses • High utility losses for high IES Gradual Annuitization: Quantifying the Costs of Switching to Annuities

19. 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) Gradual Annuitization: Quantifying the Costs of Switching to Annuities