Adverse Selection in the Annuity Market and the Role for Social Security

Adverse Selection in the Annuity Market and the Role for Social Security Roozbeh Hosseini Arizona State University Quantitative Society for Pensions and Saving 2011 Summer Workshop

Social Security The largest government program in the U.S. Many debates over reform/privatization Central question to this debate - What useful aspects are lost (that market can t replicate)? This paper talks about one - Mandatory annuity insurance Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 2 / 61

Mandatory annuity insurance Is a key feature in almost all social security systems Can be desirable when there is adverse selection Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 3 / 61

Why is it desirable? If there is private information about mortality High mortality types - Annuitize smaller portion of their wealth Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 4 / 61

Why is it desirable? If there is private information about mortality High mortality types - Annuitize smaller portion of their wealth Insurers - Recognize this self-selection - Offer high prices that reflect mortality rate of long-lived A mandatory annuity insurance Forces everyone (including high mortality) to join Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 4 / 61

Why is it desirable? If there is private information about mortality High mortality types - Annuitize smaller portion of their wealth Insurers - Recognize this self-selection - Offer high prices that reflect mortality rate of long-lived A mandatory annuity insurance Forces everyone (including high mortality) to join Thereby Provides insurance at higher (implicit) rate of return Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 4 / 61

Question We know Social security has mandatory annuitization It can be a desirable feature Private markets cannot replicate it Question How important is it quantitatively? Feldstein s quote Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 5 / 61

This paper Develops model of annuity market with adverse selection Heterogeneous mortality Private information Market structure: linear contracts Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 6 / 61

This paper Develops model of annuity market with adverse selection Heterogeneous mortality Private information Market structure: linear contracts - Annuities: financial contracts, difficult to observe/monitor - Lack of observability Contracts are non-exclusive - Little evidence on screening in the market Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 6 / 61

This paper Develops model of annuity market with adverse selection Heterogeneous mortality Private information Market structure: linear contracts Calibrates the model to match US facts Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 6 / 61

This paper Develops model of annuity market with adverse selection Heterogeneous mortality Private information Market structure: linear contracts Calibrates the model to match US facts Compares welfare between three benchmarks Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 6 / 61

Three benchmarks Private annuity markets No social security Annuity is available only through private markets Current U.S. system Stylized features of U.S. social security Private markets Ex ante efficient allocations Solution to utilitarian planner s problem Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 7 / 61

Overall ex ante gains If welfare is evaluated ex ante i.e., before mortality type is realized, then... Welfare gains between - Private annuity markets and current US system - Current US system and ex ante efficient Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 8 / 61

Overall ex ante gains If welfare is evaluated ex ante i.e., before mortality type is realized, then... Welfare gains between - Private annuity markets and current US system 0.27% - Current US system and ex ante efficient Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 8 / 61

Overall ex ante gains If welfare is evaluated ex ante i.e., before mortality type is realized, then... Welfare gains between - Private annuity markets and current US system 0.27% - Current US system and ex ante efficient 0.64% Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 8 / 61

Overall ex ante gains If welfare is evaluated ex ante i.e., before mortality type is realized, then... Welfare gains between - Private annuity markets and current US system 0.27% - Current US system and ex ante efficient 0.64% 0.91% Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 8 / 61

Overall ex ante gains If welfare is evaluated ex ante i.e., before mortality type is realized, then... Welfare gains between - Private annuity markets and current US system 0.27% - Current US system and ex ante efficient 0.64% 0.91% Who loses and who gains ex post, i.e., after mortality type is realized? autarky Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 8 / 61

Social Security has two effects 1. Transfers from high mortality types to low mortality types About 9% suffer losses: high mortality - low survival About 91% percent gain: low mortality - high survival 2. Crowds out high mortality type in the annuity markert These are good risk types Market pool is populated by bad risk types high prices This price effect has negative welfare impact of 0.29 percent Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 9 / 61

Social Security has two effects 1. Transfers from high mortality types to low mortality types About 9% suffer losses: high mortality - low survival About 91% percent gain: low mortality - high survival 2. Crowds out high mortality type in the annuity markert These are good risk types Market pool is populated by bad risk types high prices This price effect has negative welfare impact of 0.29 percent Can we use laternative policy to minimize this effect? Yes! Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 9 / 61

Related literature Theoretical models: Abel(1986); Eichenbaum and Peled(1987); Eckstein, Eichenbaum and Peled(1985) Welfare enhancing role for mandatory annuitization Detecting AS: Finkelstein and Poterba(2002,2004,2006); Mitchell, Poterba,Warshawsky and Brown(1999); Friedman and Warshawsky (1990) Evidence for adverse selection in the annuity market Measure the value of access to actuarially fair annuity Estimate welfare cost of asymmetric information: Einav, Finkelstein and Shrimpf(2010) preference heterogeneity as well as risk heterogeneity Benefits of annuitization in social security: Hubbard and Judd (1987) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 10 / 61

Model Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 11 / 61

Environment: information Individuals have private type θ known at date zero θ indexes their mortality It determines their individual survival probabilities Distribution at date zero: G 0 (θ) The only heterogeneity is in θ The only risk is time of death Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 12 / 61

Environment: preferences Everyone lives between 0 and T and has preferences Where T β t P t (θ)[u(c t ) + β(1 x t+1 (θ))ξu(b t )] t=0 Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 13 / 61

Environment: preferences Everyone lives between 0 and T and has preferences Where T β t P t (θ)[u(c t ) + β(1 x t+1 (θ))ξu(b t )] t=0 P t (θ) : probability that type θ is alive at age t Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 13 / 61

Environment: preferences Everyone lives between 0 and T and has preferences Where T β t P t (θ)[u(c t ) + β(1 x t+1 (θ))ξu(b t )] t=0 P t (θ) : probability that type θ is alive at age t x t+1 (θ) : One period conditional survival x t+1 (θ) = P t+1(θ) P t(θ) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 13 / 61

Environment: preferences Everyone lives between 0 and T and has preferences Where T β t P t (θ)[u(c t ) + β(1 x t+1 (θ))ξu(b t )] t=0 P t (θ) : probability that type θ is alive at age t x t+1 (θ) : One period conditional survival x t+1 (θ) = P t+1(θ) P t(θ) ξ : weight on bequest, b t Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 13 / 61

Technology Inelastic labor supply up to age J < T n units of labor produces wn units of consumption good Saving technology R = 1 β Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 14 / 61

Annuity contracts Can be purchased at age J (last period before retirement) Makes survival contingent payment starting age J + 1 Unit cost of annuity is q Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 15 / 61

Individual s problem subject to max c t,k t+1,a 0 T β t P t (θ)[u(c t ) + β(1 x t+1 (θ))ξu(rk t+1 )] t=0 c t + k t+1 = Rk t + w(1 τ) for t < J c t + k t+1 + qa = Rk t + w(1 τ) for t = J c t + k t+1 = Rk t + a + z for t > J Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 16 / 61

Insurers Insurers do not observe individual demand for each type θ However, they know the demand function a(θ, q) They anticipate the fraction of total sales, purchased by θ df (θ) = a(θ; q)dg J(θ) a(θ; q)dgj (θ) Insurers use F (θ) to evaluate their profit Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 17 / 61

Annuity insurers problem max y 0 ( qy y T t=j+1 P t (θ) P J (θ) 1 R s t ) df (θ) F (θ) is anticipated distribution of pay-outs - Determines fraction of y sold to type θ - Taken as given by the insurer Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 18 / 61

Government Budget Constraint τw ( J t=0 ) P t (θ) dg R t 0 (θ) = z ( T t=j+1 ) P t (θ) dg R t 0 (θ) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 19 / 61

Equilibrium Households and firms optimize + markets clear F (θ) is consistent with individual decisions df (θ) = a(θ)dg J(θ) a(θ)dgj (θ) Government budget constraint Skip Example Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 20 / 61

Properties of Equilibrium: Two period case Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 21 / 61

Two lessons Use two period example to illustrate two properties 1 In this environment there is adverse selection - Equilibrium price is higher than aggregate risk 2 Increasing social security tax and benefit - Crowds out annuity market - Increases equilibrium price of annuity Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 22 / 61

A two period example subject to max u(c 1 ) + P u(c 2 ) c 1 + qa w(1 τ) c 2 a + z P is probability of survival (with distribution G(P )) Aggregate risk of survival is P dg(p ) The goal is to show in equilibrium q > P dg(p ) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 23 / 61

Adverse selection Consider the zero profit condition q a(p ; q)dg(p ) = P a(p ; q)dg(p ) }{{}}{{} Total sale Total expected payment Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 24 / 61

Adverse selection Consider the zero profit condition q = P a(p ; q)dg(p ) a(p ; q)dg(p ) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 24 / 61

Adverse selection Consider the zero profit condition a(p ; q)dg(p ) q = P a(p ; q)dg(p ) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 24 / 61

Adverse selection Consider the zero profit condition a(p ; q)dg(p ) q = P a(p ; q)dg(p ) }{{} df (P ) F(P) - Insurers use F (P ) to evaluate risk G(P) 0 P 0.5 1 Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 24 / 61

Adverse selection Consider the zero profit condition a(p ; q)dg(p ) q = P a(p ; q)dg(p ) }{{} df (P ) F(P) - Insurers use F (P ) to evaluate risk G(P) - a(p ; q) is increasing in P F (P ) is more skewed relative to G(P ) 0 P 0.5 1 Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 24 / 61

Adverse selection Consider the zero profit condition q = P df (P ) > P dg(p ) F(P) - Insurers use F (P ) to evaluate risk G(P) - a(p ; q) is increasing in P F (P ) is more skewed relative to G(P ) 0 P 0.5 1 Therefore, equilibrium price is higher than aggregate risk Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 24 / 61

Effect of increasing social security SS benefit is a substitute for annuity increasing SS reduces demand for annuity Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 25 / 61

Effect of increasing social security SS benefit is a substitute for annuity increasing SS reduces demand for annuity Annuity Demand by Type Low SS Tax - a(p ; q) is increasing in P 0 Probability of Survival 1 Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 25 / 61

Effect of increasing social security SS benefit is a substitute for annuity increasing SS reduces demand for annuity Annuity Demand by Type Low SS Tax High SS Tax - a(p ; q) is increasing in P - As SS tax goes up a(p ; q) shifts down And becomes steeper 0 Probability of Survival 1 Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 25 / 61

Effect of increasing social security SS benefit is a substitute for annuity increasing SS reduces demand for annuity Annuity Demand by Type Low SS Tax High SS Tax - a(p ; q) is increasing in P - As SS tax goes up a(p ; q) shifts down And becomes steeper price goes up 0 Probability of Survival 1 Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 25 / 61

Effect of increasing social security SS benefit is a substitute for annuity increasing SS reduces demand for annuity Annuity Demand by Type Low SS Tax High SS Tax - a(p ; q) is increasing in P - As SS tax goes up a(p ; q) shifts down And becomes steeper price goes up 0 Probability of Survival 1 What about wlefare? Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 25 / 61

Calibration Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 26 / 61

Calibration Mortality parameters - Survival probabilities, P t (θ), for each t and θ - Initial distribution of θ: G 0 (θ) Preference/technology parameters - Curvature of utility function - Weight on bequest - Return on saving and time preference Policy parameters - Social security tax and benefits Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 27 / 61

Calibrating mortality parameters Observe data on Average survival probabilities (from life tables) Individuals own assessment about longevity (from HRS) Use these observations to back out P t (θ) for each θ The distribution G 0 (θ) Need to impose restriction on P t (θ) Standard assumptions from demography Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 28 / 61

Assumptions on P t (θ) Let H t (θ) be cumulative mortality hazard for type θ, define P t (θ) = exp( H t (θ)) Assumption 1: θ shifts mortality hazard H t (θ) = θh t Assumption 2: Initial distribution of θ is gamma g 0 (θ) Gamma( 1 k, k) = kk θ k 1 exp( kθ) Γ(k) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 29 / 61

Assumptions on P t (θ) Let H t (θ) be cumulative mortality hazard for type θ, define P t (θ) = exp( H t (θ)) Assumption 1: θ shifts mortality hazard H t (θ) = θh t Assumption 2: Initial distribution of θ is gamma g 0 (θ) Gamma( 1 k, k) = kk k 1 exp( kθ) θ Γ(k) What are implications of these assumptions? Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 29 / 61

Implication of the assumption H t (θ) = θh t Suppose type θ has 50% chance of surviving to age t Then, type 2θ has 25% chance of surviving to the same age Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 30 / 61

Implication of the assumption H t (θ) = θh t Suppose type θ has 50% chance of surviving to age t Then, type 2θ has 25% chance of surviving to the same age Once P t (θ) (or H t (θ)) is known for one θ It is known for all θ Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 30 / 61

Identifying survival probabilities Unknowns are H t Parameter of distribution G 0 Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 31 / 61

Identifying survival probabilities Unknowns are H t Parameter of distribution G 0 T + 1 unknowns 1 unknown Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 31 / 61

Identifying survival probabilities Unknowns are H t Parameter of distribution G 0 T + 1 unknowns 1 unknown T + 2 unknowns Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 31 / 61

Identifying survival probabilities Unknowns are H t Parameter of distribution G 0 T + 1 unknowns 1 unknown T + 2 unknowns Life table gives population survival probabilities P t = P t (θ)dg 0 (θ) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 31 / 61

Identifying survival probabilities Unknowns are H t Parameter of distribution G 0 Life table Life table gives population survival probabilities P t = exp( θh t )dg 0 (θ) T + 1 unknowns 1 unknown (T + 1) unknowns 1 unknown Given G 0 (θ) the above identity can be solved to find H t Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 31 / 61

Identifying survival probabilities Unknowns are H t Parameter of distribution G 0 Life table Life table gives population survival probabilities P t = exp( θh t )dg 0 (θ) T + 1 unknowns 1 unknown (T + 1) unknowns 1 unknown Given G 0 (θ) the above identity can be solved to find H t How do we find G 0 (θ)? Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 31 / 61

Subjective survival prob. in HRS HRS asks individuals their subjective prob. of living to 75 Hurd & McGarry(1995,2002): responses are consistent with Life tables Ex post mortality experience Individuals health data Use Gan-Hurd-McFadden(2003) s method to estimate G 0 (θ) Details Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 32 / 61

Individual survival curves: P t (θ) 1 Lowest Mortality 0.8 0.6 P (!) t 0.4 Highest Mortality 0.2 0 30 40 50 60 70 80 90 100 Age Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 33 / 61

Individual survival curves: P t (θ) 1 0.8 0.6 P t (!) 90th percentile 0.4 0.2 10th percentile 0 30 40 50 60 70 80 90 100 Age Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 33 / 61

Individual survival curves: P t (θ) 1 0.8 0.6 P t (!) 90th percentile 0.4 0.2 10th percentile 0 30 40 50 60 70 80 90 100 Age Average Life Expectancy at 30: 44 yrs (74 years old) Standard deviation : 4 yrs Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 33 / 61

Profile of Life Expectancy by age 100 Life expectancy at each age 90 80 70 60 50 30 40 50 60 Age 70 80 90 100 Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 34 / 61

Profile of Life Expectancy by age 100 Life expectancy at each age 90 80 70 60 One Standard Deviation 50 30 40 50 60 Age 70 80 90 100 Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 34 / 61

Profile of Life Expectancy by age 100 Life expectancy at each age 90 80 70 60 90th %tile 10th %tile One Standard Deviation 50 30 40 50 60 Age 70 80 90 100 Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 34 / 61

Calibration: preferences + social security CRRA utility function u(c) = c1 γ 1 γ Preference parameters are chosen to match Fraction of pension wealth for 70 yrs old in HRS ξ = 0.8 Fraction of SS wealth for 70 yrs old in HRS γ = 1.47 Social security tax: chosen to match %45 replacement ratio Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 35 / 61

Calibration summary Parameter Value risk aversion, γ 1.47 weight on bequest, ξ 0.8 discount factor, β 0.97 5 return on savings, R 1.03 5 SS tax, τ 0.08 variance of g 0 (θ), σθ 2 = 1 k 0.12 Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 36 / 61

Fraction of wealth annuitized, average 80 70 Annuitized Wealth/Total Wealth 60 50 % 40 30 SS Wealth/Total Wealth 20 10 Pension Wealth/Total Wealth 0 65 70 75 80 85 90 95 100 Age Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 37 / 61

Fraction of wealth annuitized, average 80 70 Annuitized Wealth/Total Wealth 60 50 % 40 30 20 SS Wealth/Total Wealth! and " Chosen to Match These 10 Pension Wealth/Total Wealth 0 65 70 75 80 85 90 95 100 Age Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 37 / 61

Findings Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 38 / 61

Use the model to ask How does annuitization decision vary by mortality type? How do these decisions change by removing SS? Welfare comparison Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 39 / 61

Fraction of wealth annuitized at 70, by type 100 % of Wealth Annuitized 80 60 40 20 SS wealth / Total wealth 0 0 0.5 1 1.5 2 2.5 3 3.5 Low Mortality! High Mortality Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 40 / 61

Fraction of wealth annuitized at 70, by type 100 % of Wealth Annuitized 80 60 40 20 SS wealth / Total wealth Pension wealth / Total wealth 0 0 0.5 1 1.5 2 2.5 3 3.5 Low Mortality! High Mortality Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 40 / 61

Fraction of wealth annuitized at 70, by type 100 Annuitized wealth / Total wealth % of Wealth Annuitized 80 60 40 20 SS wealth / Total wealth Pension wealth / Total wealth 0 0 0.5 1 1.5 2 2.5 3 3.5 Low Mortality! High Mortality 60% hold annuity Consistent with evidence in HRS Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 40 / 61

Fraction of wealth annuitized at 70, by type 100 Annuitized wealth / Total wealth % of Wealth Annuitized 80 60 40 20 SS wealth / Total wealth Pension wealth / Total wealth 0 0 0.5 1 1.5 2 2.5 3 3.5 Low Mortality! High Mortality 60% hold annuity Consistent with evidence in HRS Johnson-Burman-Kobes(2004) evidence from HRS 43% of all adults (52% of males) hold pensions Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 40 / 61

Only market vs Current U.S. 100 % of Wealth Annuitized 80 60 40 20 Annuitized wealth / Total wealth (Current U.S. system) 0 0 0.5 1 1.5 2 2.5 3 3.5 Low Mortality! High Mortality Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 41 / 61

Only market vs Current U.S. % of Wealth Annuitized 100 80 60 40 20 Annuitized wealth / Total wealth (Only private market) Annuitized wealth / Total wealth (Current U.S. system) 0 0 0.5 1 1.5 2 2.5 3 3.5 Low Mortality! High Mortality Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 41 / 61

Ex post gain/loss 1.5 1 Welfare Gains/Losses from Introducing SS Across Mortality Types 91 percent of population gain on average 0.34% 0.5 0 0.5 % 1 1.5 2 9 percent of population lose on average 0.39% 2.5 3 0 0.5 1 1.5 2 2.5 3 Ex ante gain = 0.27% Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 42 / 61

Ex post gain/loss 1.5 Welfare Gains/Losses from Introducing SS Across Mortality Types 1 0.5 0 0.5 % 1 1.5 2 2.5 3 0 0.5 1 1.5 2 2.5 3 Counter-factual: fix price at the equilibiurm level without SS Without price increase the ex ante gain is 0.56% Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 42 / 61

Can we do better? Social security forces individuals to pool their mortality risk But keeps this pool separate from market pool This derives good risk types out of the market. Alternative policy: - Return contributions to people at retirement - Force them to buy annuity in the market Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 43 / 61

Can we do better? Social security forces individuals to pool their mortality risk But keeps this pool separate from market pool This derives good risk types out of the market. Alternative policy: - Return contributions to people at retirement - Force them to buy annuity in the market Ex ante welfare gain increases to 0.36% Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 43 / 61

Gains from implementing ex ante efficient What is the maximum ex ante welfare gain from policy? We need to find the solution to utilitarian planner s problem Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 44 / 61

Planner s problem [ T ] max β t P t (θ)[u(c t (θ)) + β(1 x t+1 (θ))ξu(b t (θ))] dg 0 (θ) t=0 subject to T t=0 P t (θ) R t [ c t (θ) + (1 x ] t+1(θ)) b t (θ) dg 0 (θ) = R w J t=0 P t (θ) R t dg 0 (θ) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 45 / 61

Planner s problem [ T ] max β t P t (θ)[u(c t (θ)) + β(1 x t+1 (θ))ξu(b t (θ))] dg 0 (θ) t=0 subject to T t=0 P t (θ) R t [ c t (θ) + (1 x ] t+1(θ)) b t (θ) dg 0 (θ) = R w J t=0 P t (θ) R t dg 0 (θ) Planner chooses consumption and bequest Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 45 / 61

Planner s problem [ T ] max β t P t (θ)[u(c t (θ)) + β(1 x t+1 (θ))ξu(b t (θ))] dg 0 (θ) t=0 subject to T t=0 P t (θ) R t [ c t (θ) + (1 x ] t+1(θ)) b t (θ) dg 0 (θ) = R w J t=0 P t (θ) R t dg 0 (θ) Notice : No I.C constraints! Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 45 / 61

Planner s problem [ T ] max β t P t (θ)[u(c t (θ)) + β(1 x t+1 (θ))ξu(b t (θ))] dg 0 (θ) t=0 subject to T t=0 P t (θ) R t [ c t (θ) + (1 x ] t+1(θ)) b t (θ) dg 0 (θ) = R w J t=0 P t (θ) R t dg 0 (θ) It turns out they don t bind Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 45 / 61

Planner s problem [ T ] max β t P t (θ)[u(c t (θ)) + β(1 x t+1 (θ))ξu(b t (θ))] dg 0 (θ) t=0 subject to T t=0 P t (θ) R t [ c t (θ) + (1 x ] t+1(θ)) b t (θ) dg 0 (θ) = R w J t=0 P t (θ) R t dg 0 (θ) Ex ante efficient allocations have very simple form Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 45 / 61

Ex ante efficient allocations Perfect insurance against risk type θ c t (θ) = c t (θ ) = c t b t (θ) = b t (θ ) = b t Perfect insurance against time of death, u (c t ) = βru (c t+1 ) = βrξu (b t ) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 46 / 61

Ex ante efficient allocations Perfect insurance against risk type θ c t (θ) = c t (θ ) = c t b t (θ) = b t (θ ) = b t Perfect insurance against time of death, assume Rβ = 1 u (c t ) = u (c t+1 ) = ξu (b t ) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 46 / 61

Ex ante efficient allocations Perfect insurance against risk type θ c t (θ) = c t (θ ) = c t b t (θ) = b t (θ ) = b t Perfect insurance against time of death, assume Rβ = 1 c t = c, b t = b and u (c) = ξu (b) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 46 / 61

Ex ante efficient allocations Perfect insurance against risk type θ c t (θ) = c t (θ ) = c t b t (θ) = b t (θ ) = b t Perfect insurance against time of death, assume Rβ = 1 Can be implemented by c t = c, b t = b and u (c) = ξu (b) - Type-independent social security tax and benefit - Type-independent survivors benefit Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 46 / 61

Implementation Ex ante efficient allocation can be implemented useing - Type-independent taxes: 0.14 (compare this to 0.08) - Replacement ratio: 0.71 (compare to 0.45) - Survival benefit before retirement (small) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 47 / 61

Comment There are two key assumptions 1 Only heterogeneity is in mortality 2 Individuals (and planner) are expected utility maximizers Type-independent policy is optimal Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 48 / 61

Ex post gain/loss 1.5 Welfare Gains/Losses from Introducing SS Across Mortality Types 1 0.5 0!0.5 %!1 Gains from implementing ex ante efficient allocation!1.5!2!2.5!3 0 0.5 1 1.5 2 2.5 3! Ex ante gain = 0.91% Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 49 / 61

Conclusion Goal of the paper - Measure the gains from mandatory annuitization in S.S Welfare gain from mandatory annuitization current U.S. system over private markets : 0.27% Large impact on price with negative welfare implications Simple policy change can aleviate this negative price effect Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 50 / 61

Extensions Introducing other heterogeneities - Heterogeneity in preference for bequest - The link between measures of income and mortality Detailed model of altruism and intergenerational link Alternative equilibrium notions Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 51 / 61

Backup slides Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 52 / 61

Sensitivity: Risk aversion 5 Overall Welfare Gains for Various Levels of Risk Aversion,! Ex ante gains in the benchmark model Ex ante gains under autarky 4 3 % 2 1 0!1!2 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Coefficient of Relative Risk Aversion,! Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 53 / 61

Sensitivity: Bequest Parameter 4 3.5 Overall Welfare Gains for Various Levels of Bequest Parameter,! Ex ante welfare gains in the benchmark model Ex ante welfare gains under autarky 3 2.5 % 2 1.5 1 0.5 0!0.5 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 Weight on Bequest,! Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 54 / 61

Consumption/Saving profiles (w/ SS) 1 Consumption Profile Across Mortality Types, with SS 2 1.8 1.6 Profile of Liquid Asset Holdings Across Mortality Types, with SS Lowest 5% of Mortality (!) Median Mortality (!) Highest 5% of Mortality (!) Mean 0.8 1.4 1.2 c (!) t 0.6 k t (!) 1 0.8 0.4 0.6 0.2 Lowest 5% of Mortality (!) Median Mortality (!) Highest 5% of Mortality (!) Mean 0 30 40 50 60 70 80 90 100 Age 0.4 0.2 0 30 40 50 60 70 80 90 100 Age Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 55 / 61

Consumption/Saving profiles (w/o SS) 1 Consumption Profile Across Mortality Types, without SS 2 1.8 1.6 Profile of Liquid Asset Holdings Across Mortality Types, without SS Lowest 5% of Mortality (!) Median Mortality (!) Highest 5% of Mortality (!) Mean 0.8 1.4 1.2 c t (!) 0.6 k t (!) 1 0.8 0.4 0.6 0.2 Lowest 5% of Mortality (!) Median Mortality (!) Highest 5% of Mortality (!) Mean 0 30 40 50 60 70 80 90 100 Age 0.4 0.2 0 30 40 50 60 70 80 90 100 Age Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 56 / 61

Backups: Calculations under autarky Welfare gains going from Private saving to current US system 2.85% Current US system to ex ante efficient 0.84% 3.71% When there is no annuity market, gains are large Go Back Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 57 / 61

Ex post gain/loss Welfare Gains/Losses from Introducing SS Across Mortality Types 8 6 % 4 Gains/Losses when Annuity Market Does not Exist 2 0!2 Ganins/Losses when Annuity Market Exists 0 0.5 1 1.5 2 2.5 3! Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 58 / 61

Estimation procedure What is observed in HRS Response to the question on subjective survival prob. Ex post mortality/survival Problem : there are many 0 s and 1 s in responses Solution: assume error in reports Type θ at age t makes report r with prob. f(r P 75(θ) P t(θ) ) Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 59 / 61

Estimation procedure (cont.) Observing report, r, we can estimate θ using Baye s rule Prior on θ is given by G t (θ) Report, r and f( ) can be used to form a posterior Use posterior mean as estimate for θ Use estimates to form likelihood functions for survival Estimate parameters of G t (θ) and f( ) using MLE Go Back Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 60 / 61

Feldstein s quote the existence of asymmetric information may justify a social insurance program (a government annuity in this case) but does not necessarily do so. The case for a mandatory annuity program depends on calculations that could be done but that have not yet been done. Martin Feldstein, presidential address (2005) Go Back Roozbeh Hosseini(ASU) AS in Annuity Market and the Role for SS 61 / 61