1. Optimal Gradual Annuitization: Quantifying the Cost of Switching to Annuities Optimal Gradual Annuitization: Quantifying the Cost of Switching to Annuities by Wolfram Horneff*, Raimond Maurer*, and Michael Stamos* *Department of Finance, Goethe University Frankfurt, Germany 2007 IME Conference, Piraeus, Greece Goethe University Frankfurt, Germany

2. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities2/19 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

3. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities3/19 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

4. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities4/19 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 End-of-year payoff Initial InvestmentAliveDead (1) 100 (in bond) 100(1+r) =102 100(1+r) =102 (2) End-of-year payoff Initial InvestmentAliveDead (1) 100 (in bond) 100(1+r) =102 100(1+r) =102 (2) 100 (in annuity) 100(1+r)/p =113.30

5. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities5/19 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

6. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities6/19 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

7. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities7/19 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

8. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities8/19 The Model: Capital Markets  Riskless bonds:  R f : riskless growth rate in real terms (= 1.02)  Risky stocks:  R t ~ LN( ,  )   : expected growthrate (= 1.06)   : standard deviation (= 18 percent)  Bonds and stocks can be traded each year

9. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities9/19 The Model: Wealth Evolution  Contemporary budget constraint Gradual Annuitization Partial Switching Complete Switching W t : cash on hand M t : amount invested in bonds S t : amount invested in stocks PR t : amount invested in annuity C t : consumption  Cash on hand in t + 1 conditional on survival Gradual Annuitization Partial Switching Complete Switching W t+1 : next period cash on hand L t+1 : sum of annuity payments P t+1 : annuity payouts Y t+1 : public pension income.

10. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities10/19 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) p s :subjective survival probabilities (population average [male]) C :consumption B :bequest  Choose C t, M t, S t and PR t in the way that V t is maximized

11. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities11/19 The Model: Optimization Problems Complete Switching: Partial Switching: Gradual Annuitization:

12. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities12/19 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

13. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities13/19 Optimal Policies: Demand for Annuities (No Bequest) Wealth Age Annuity Payouts = 0

14. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities14/19 Value of Postponing Annuitization V SW (PR,.) < V SW (PR=0,.) No purchase region PR 0 V SW (PR,.) ≥ V SW (PR=0,.) Purchase region V SW (PR,.) PR ind Opportunity Costs V SW (PR=  V SW (PR=0,.)

15. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities15/19 Annuitization Frontier and Expected Value of Purchases Initial Multiple of Pension Income = 6

16. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities16/19 Optimal Expected Annuity Stock Fraction for Various CRRA Initial Multiple of Pension Income = 6

17. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities17/19 The Impact of IES on Initial Multiple of Pension Income = 6

18. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities18/19 Equivalent Increase in Financial Wealth (Percentage Points)  Usually Switching restrictions show moderate utility losses  High utility losses for high IES

19. 07/11/07Gradual Annuitization: Quantifying the Costs of Switching to Annuities19/19 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)