Health Insurance, Portfolio Choices and Retirement Incentives

Transcription

1 Health Insurance, Portfolio Choices and Retirement Incentives Daniele Marazzina Joint work with Emilio Barucci and Enrico Biffis Emilio Barucci and Daniele Marazzina Dipartimento di Matematica F. Brioschi Politecnico di Milano - Enrico Biffis Department of Finance Imperial College, London XVI WORKSHOP ON QUANTITATIVE FINANCE Parma, January 29-30, 2015

2 Introduction The agent is exposed to the following risks: mortality risk (death) market risk (investment choice) aging risk (increase in the mortality rate and in future medical expenses) health shock risk (lower wage and leisure stock, medical expenses, increase in the mortality rate) We study the optimal retirement problem assuming that the agent chooses the leisure rate during her working period together with the portfolio, consumption, insurance, bequest she chooses the (irreversible) retirement date after the retirement the agent fully fully enjoys leisure, makes investment-consumption-bequest decisions Life and Health insurances are available on the market Risk Hedging and Retirement Incentives: Portfolio choices, consumption, work, insurance provide a way to hedge risks: retirement incentives, portfolio-consumption choices. Different health insurance coverages provide different incentives: policy implications. We calibrate the model on real data for elderlies (60 years old people).

3 Literature Since the seminal contributions of Merton (1969), (1971) several papers have investigated optimal consumption and portfolio choices in an intertemporal setting. The aim is to reconcile the normative insights with empirical evidence. Problems: the Merton problem becomes much complicated. Our goal: to have a parsimonious continuous time model allowing a semi-closed form solution. Literature: 1. analytical solutions for Endogenous Retirement problem 2. the calibration of simple models to Real Data and policy implications for insurance-health.

4 Literature 1- Endogenous Retirement The literature has addressed the problem assuming a constant wage with a fixed labor rate [1], or an endogenous choice of the labor rate [2,3], or with a stochastic wage but with no flexibility on the labor market [4] or with flexibility (optimal leisure rate) [5]. 1. Farhi & Panageas (2007) Saving and Investing for Early Retirement: a Theoretical Analysis 2. Choi & Shim (2006) Disutility, Optimal Retirement, and Portfolio Selection 3. Choi, Shim & Shin (2008) Optimal Portfolio, Consumption-Leisure and Retirement Choice Problem with CES Utility 4. Dybvig & Liu (2005) Lifetime Consumption and Investment: Retirement and Constrained Borrowing 5. Barucci & Marazzina (2012) Optimal investment, stochastic labor income and retirement

5 Literature 2- Policy implications Some contributions Life insurance over the life-cyle: Life insurance, Social Security, and household consumption (Hong and Rios-Rull, 2012) Labour supply and retirement behavior: Role of medical expenses and insurance (Rust and Phelan, 1997; Blau and Gilleskie, 2006, 2008) Role of health, wealth, and wages (French, 2005) Role of health insurance and Medicare (French and Jones, 2011) Why do the elderly save? Role of (uncertainty in) medical expenses, life expectancies (De Nardi et al., 2010). Common features 1. Health/mortality risk and out-of-pocket medical expenses are important drivers of retirement decisions. 2. No (semi-)explicit solutions: e.g., estimation based on Method of Simulated Moments. 3. Comparative statics difficult: effects of health, wealth, and wages difficult to disentangle.

6 Our Model The model aims at clarifying the main trade-offs driving labour supply and retirement decisions under health/mortality risk, while retaining analytical tractability. We capture the stylized features of the random environment with a regime-switching framework. We derive analytical solutions that can easily be used for comparative statics and policy experiments. Casual calibration of the model provides results in line with the empirical evidence on job exit rates and asset allocation in the presence of heterogeneity in life expectancies and medical expenses (e.g., French and Jones, 2011).

7 The Market The agent can invest in both the money market and the stock market For simplicity, we assume that the stock market consists of a single risky asset Thus we have a risk-free instantaneous interest rate r a risky asset whose price evolves as ds(t) = bsdt + σsdb(t), S(0) = S 0

8 Consumption, Portfolio & Leisure Processes We denote by c(t), θ S (t) and l(t) the Consumption, Risky Asset Portfolio and Leisure processes, respectively c(t) is a nonnegative process [ l(t) 0, l ] {L} is the leisure indicator (rate of leisure at time t) l 1 is the maximum rate of leisure the agent can choose before retirement utility function (from consumption and leisure rate) u cl (t, c, l) = (l(t)1 α c α (t)) 1 γ 1 γ θ S (t) represents the money invested in the risky asset at time t Let us denote by τ r 0 the retirement date chosen by the agent; the retirement decision is assumed to be irreversible: during her working life the agent chooses the leisure rate, after retirement the agent fully enjoys leisure time Therefore 0 l(t) l 1 if 0 t < τ r, l(t) = L > 1 if t τ r

9 Exogenous Health and Aging Shocks Mortality is described by a Cox process with intensity λ d (t), that is driven by two types of shocks, a health shock and an aging shock The agent is exposed to an exogenous health shock that occurs at the random time τ h distributed according to an independent Poisson process of parameter λ h The health shock leads to Instantaneously medical expenses of size M (stochastic) Jump in mortality intensity of size h Wage drop from w to w w(t) := w + (w w)1 τh t > 0 Maximum leisure drop from l to l l(t) := l + (l l)1τh t > 0 Before τ h, the agent s mortality intensity could increase by a jump of size a at the random time τ a (aging shock), distributed according to an independent Poisson process of parameter λ a. The aging shock also increase the size of medical expenses M M(t) = M1 τa t + M1 τa>t, with E[M] E[M].

10 Exogenous Health and Aging Shock The health deterioration experienced at times τ a and τ h is assumed to be irreversible Denoting by N i (t) := 1 τi t the indicator of the jump at time τ i, i {a, h}, we can label the resulting four possible health states as state best (on the event {N a = N h = 0}) state good (on the event {N a = 1, N h = 0}) state poor (on the event {N a = 0, N h = 1}) state worst (on the event {N a = N h = 1}) The mortality intensity is therefore λ d (t) := λ + an a(t )1 τa<τ h + h N h (t ) On {N h = 0} On {N h = 1} On {N a = 0} λ d = λ λ d = λ + h On {N a = 1} λ d = λ + a λ d = λ + a + h

11 Insurance The agent can insure against medical expenses and death risk By paying a health insurance premium θ h λ h E t [M] the agent will receive a benefit of amount θ h M(t) in case the health shock (τ h ) occurs Life Insurance Bequest By paying a life insurance premium λ d (θ d W ) the agent will receive a benefit θ d W in case death occurs When θ d (t) < W (t), we interpret the life insurance contract as an annuity utility function (from bequest) with k d > 0 u d (t, θ d ) = (k dθ d (t)) α(1 γ) 1 γ

12 The Problem The agent maximizes the expected utility Given the initial wealth W (0), we look for an admissible strategy that maximizes the objective function Value Function V(W (0)) = sup {c,l,θs,θ d,θ h,τ r } E [ + 0 ] e t β(s)ds 0 u(t, c(t), l(t), θ d (t))dt u(t, c(t), l(t), θ d (t)) = u cl (t, c(t), l(t)) + λ d (t)u d (t, θ d (t)) β(t) = δ + λ d (t), δ being the subjective discount rate subject to the budget constraint

13 Budget Constraint The wealth process W (t) (no retirement incentives are considered here) satisfies the dynamic budget constraint { dw (t) = (1 N d (t )) [Y (l(t), w(t)) c(t)] dt + θ S (t) (bdt + σdb(t)) + (W (t) θ S (t))rdt (1 N h (t ))λ h θ h (t)e t [M(t)]dt M(t)(1 θ h (t ))dn h (t) } λ d (t)(θ d (t) W (t))dt + (θ d (t ) W (t ))dn d (t) Wage process Y (l(t), w(t)) = (1 l(t) p )w(t)

14 The Duality Approach Literature He & Pagés (1993) solve via the duality approach the optimal consumption-portfolio problem with a stochastic tradable labor income Karatzas & Wang (2000) address a mixed optimal stopping/control problem: the agent chooses the portfolio and the consumption rate and the horizon of the problem deciding the final date as a stopping time Choi, Shim & Shin (2008) and Farhi & Panageas (2007) adapt these techniques to address the optimal investment-portfolio problem allowing the agent to choose the retirement date

15 Post Retirement Problem The objective function can be written as [ τr ] E e t β(s)ds 0 u(t, c(t), l(t), θ d (t))dt + e τr 0 β(s)ds U(τ r, W (τ r )) 0 where U(τ r, W (τ r )) is the optimal expected utility attainable at time τ r with wealth W (τ r ) In τ r the agent solves the optimal consumption-portfolio-bequest problem with initial wealth W (τ r ) subject to the budget constraint U(τ r, W (τ r )) is the indirect utility function associated with this maximization problem

16 Dual Utility Functions The Convex Conjugate of the utility function u(t, c, l, θ d ) is ũ(t, z) = max c 0, 0 l l(t) :=ũ cl (t, z) + λ d (t)ũ d (z) ( ) u cl (t, c, l) (c + w(t)l p )z + λ d (t) max u d (θ d ) θ d z θ d The Convex Conjugate of the utility function U(t, w) is Ũ(τ, z) = sup U(τ, w) wz w 0 Both ũ(t, z), and Ũ(τ, z) can be computed analytically.

17 The Dual Problem Dual objective function [ τr E e { } t 0 β(s)ds ũ (t, z(t)) + z(t) (w(t) (1 N h (t ))λ h E t [M(t)]) dt 0 + e ] τr 0 β(s)ds Ũ (τ r, z(τ r )) where z(t) = λe t 0 β(s)ds H(t), z(0) = λ λ being a Lagrange Multiplier H(t) := e (r+ 1 2 Θ2 )t ΘZ(t) e t 0 λ d (s)ds is the state-price-density process Θ := b r σ is the market price of financial risk

18 Solve the Dual Problem 1. We define [ τ r φ(t, z) := sup E e s β(u)du 0 {ũ(s, z(s))+(w(s) (1 N h (s ))λ h E s [M(s)])z(s)} ds τ r >t t +e ] τr 0 β(s)ds Ũ(τ r, z(τ r )) z(t) = z 2. The value function is equal to V(W (0)) = inf [φ(0, λ) + λw (0)] λ>0 = φ(0, λ ) + λ W (0) 3. Once the dual problem is solved (i.e., the value of φ is computed) find the optimal λ and thus optimal process z = λ t0 e β(s)ds H(t) find optimal strategies τ r = inf{t 0 : z (t) z(t)}, z(t) being a critical threshold for example, for 0 t < min{τ r, τ h} (before retirement and the health shock) and z z c = αp 1 α w(l ) p ( ( ) ) 1 α αp α(1 γ) 1 l = z 1 α w γ+(1 p)(α(1 γ) 1)

19 Compute φ and z - 1/3 We consider the above problem in a Regime-Switching framework Being in state i {b, g, p, w} (best, good, poor, worst) at time t < τ r or τ r φ i (t, z) = φ(t, z), z i = z(t), ũ i (z) = ũ(t, z), Ũ i (z) = Ũ(τr, z) Defining Φ i (z) = e t 0 β(s)ds φ i (t, z) and LΦ i = β i Φ i + (β i r)z Φ i z Θ2 z 2 2 Φ i z 2

20 Compute φ and z - 2/3 Compute Φ j and the free boundary z j, j = p, w, such that LΦ j (z) + ũ j (z) + zw = 0 if z > z j Φ j (z) = Ũj(z) and Φ j(z) = Ũ j (z) if z z j Once Φ w is known, compute Φ g and the free boundary z g such that LΦ g (z) + ũ g(z) + z(w λ h E[M]) λ h Φ g + λ h Φ w = 0 if z > z g Φ g (z) = Ũg (z) and Φ g (z) = Ũ g (z) if z z g Once Φ p and Φ g are known, compute Φ b and the free boundary z b such that LΦ b (z) + ũ b (z) + z(w λ h E[M]) (λ h + λ a)φ b + λ h Φ p + λ aφ g = 0 if z > z b Φ b (z) = Ũb(z) and Φ b(z) = Ũ b(z) if z z b If we assume that at t = 0 the agent is in the best state, it holds φ(0, λ) = Φ b (λ)

21 Compute φ and z - 3/3 The solution of the above free-boundary problem for the health state Φ w is given by if z z w if z w < z < z := ( αp 1 α w ) α(1 γ) 1 if z max{z w, z} Φ w (z) = Ũw (z) Φ w (z) = a 1z n+ + a 2z n + Az 1+ 1 α(1 γ) 1 + Bz Φ w (z) = a 3z n + Cz + Dz (1 γ)(α(1 p) 1) γ+(1 p)(α(1 γ) 1) + Ez 1+ 1 α(1 γ) 1 where z w is the unique zero of a given function and all the coefficients are known analytically Similar results hold for Φ p, Φ g and Φ b

22 Casual Calibration French E., Jones J.B., The effects of health insurance and self-insurance on retirement behaviour, Econometrica, 79-3, 2011 The authors estimate a life-cycle model for 60-year old agents with different risk preferences, wealth/wages, and health characteristics. We present numerical results based on casual calibration of the model to Health and Retirement Study (HRS) data and preference estimates provided in French&Jones (2011) The HRS is a sample of individuals aged in 1992, together with their spouses, who have been surveyed every two years since then We focus on male households and look at the optimal decisions made by a 60 years old from today onwards we use preference parameters based on the Preference Index developed by French&Jones (2011).

23 Numerical Results French&Jones (2011) use three questions from the HRS to estimate a person s willingness to work. The index is discretized into three values: high, low, and out. In this talk we provide results for the Average Agent Bequest k d = 2.5 Aging Process λ = 0.033, λ a = 0.200, a = Health Shock λ h = 0.125, h = Financial Market r = 0.030, b = 0.060, σ = Preferences δ = , α = 0.550, γ = 7.49 Labor income p = 2, w G = 3, w G = 2 Initial Wealth W0 G = 27.9 Medical Expenses M = %N a(t ) W0 G and w G, w G are given in USD, and are affected by a taxation of wealth and wage of 33% and 40%, respectively (Gross values)

24 Moreover, we set market parameters: r = 0.03; b = 0.06; σ = 0.2 and L = 1.2, l = 1, l = 0.8 leisure increases by 20% at retirement w.r.t. the best and good state p = 2 Figure: Net wage as a function of leisure for the average agent in line with Aaronson & French (2004) switching from full time to half time (i.e., increasing leisure from zero to 0.5) would generate a 25% drop in gross income

25 Retirement Decision The agent s optimal retirement τr coincides with the first time t 0 such that the optimal wealth exceeds the health state dependent threshold. The health shock determines a switch to a lower threshold target. Figure: Retirement thresholds for the average agent (net wealth)

26 Optimal Strategies We consider the behavior of the agent over all the life span including post retirement. Note that the agent s path is eventually characterized by a change of health state, therefore depending on the simulation path an agent starting in state b may go to state p or to g and then to w. We also characterize the average agent by different starting health status, assuming eventually an instantaneous aging and/or health shock, e.g., if at time 0 we assume that the agent s health status is w we assume instantaneous aging and health shock. Our choice to concentrate our attention on choices of a 60-year old agent is due to the fact that the two state wage dynamics is not reach enough to model the evolution and the variability of the wage observed for a young agent.

27 Optimal Strategies Figure: Optimal strategies for the average agent. Average values computed on the basis of 5000 simulations for the state variables and a time horizon of 30 years. Expected retirement date (assuming initial state best ): τ r = 8.33.

28 Optimal Strategies Figure: Optimal strategies for the average agent w.r.t the wealth.

29 Retirement Incentives In addition to the baseline model, where only private health insurance is available and is decided endogenously, we consider three forms of health insurance: Health coverage is provided by the employer while the agent is actively working (tied coverage). Health coverage is provided by the employer also while the agent is retired (tied-retiree coverage). Health coverage is provided by the employer when the agent is retired (retiree coverage). Health coverage pays the medical expenses M when the health shock occurs.

30 Retirement Decision in presence of Health Coverage Health insurance b g p w No insurance tied retiree tied-retiree Table: Retirement thresholds for an average agent aged 60. The availability of retiree coverage induces an immediate retire As expected, the retirement thresholds for the after shock states (p and w) do not change The presence of tied coverage induce the agent to postpone her retirement decision, while the presence of a retiree coverage, even if coupled with a tied one, accelerates retirement with respect to the baseline model without insurance

31 Figure: Optimal strategies for the average agent assuming a tied coverage.

32 We only consider the b and g health states since they are the unique states in which the insurance plays a role to hedge the health shock risk. With respect to the baseline model, the availability of tied coverage reduces working hours before retirement (at time 0, leisure is 0.68 in the baseline model, 0.75 if the tied coverage is considered). The delta amounts to a 10%. The rationale being that the agent has no more to work hard to insure himself against health shocks. On the other hand, employer provided health insurance leads to greater stock market participation (θ S (0) = 3.07 in the baseline model, 3.78 in the tied coverage case). This effect can be explained according to the fact that employer s health insurance reduces wealth variance, there is no need to acquire health insurance, these two effects lead the agent to invest more in the risky asset. The negative optimal allocation to health insurance in this case suggests that the agent is overinsured and is trying to undo the employer provided coverage.

33 Conclusions We obtain distributions of job exit times that are in line with the empirical evidence (and expected retirement between ages 67-68, depending on individual characteristics and insurance availability). If we consider optimal strategies beyond retirement, the model yields a jump in consumption at the retirement date, whose size is consistent with the empirical evidence. The patterns and levels of the optimal investment/insurance/consumption strategies are overall in line with the extant empirical literature (e.g., French, 2005). Work in progress includes extended numerical analysis (comparative statics), estimation of the model on PSID data, and the design of relevant policy experiments (e.g., age-dependent health insurance constraints/subsidies, like Medicare).

34 Complete market We extend the initial market to a complete market, introducing an aging insurance The agent can insure against the aging shock τ a, by paying an insurance premium θ aλ a to receive a benefit of amount θ a in case the aging shock occurs Therefore wealth process W (t) satisfies the dynamic budget constraint { dw (t) = (1 N d (t )) [Y (l(t), w(t)) c(t)] dt + θ S (t) (bdt + σdb(t)) + (W (t) θ S (t))rdt (1 N h (t ))λ h θ h (t)e t [M(t)]dt M(t)(1 θ h (t ))dn h (t) + (1 N h (t )) ( (1 N a(t ))λ aθ a(t)dt + θ a(t ))dn a(t)) } λ d (t)(θ d (t) W (t))dt + (θ d (t ) W (t ))dn d (t) We prove that in the new market the agent optimal strategy can be realized setting θ a 0, i.e., her optimal strategy can be realized in the initial incomplete market.

35 Dual Utility Function and Optimal Strategies ( ) α(1 γ) 1 Let t < τ h (before the health shock) and z z := α αp w 1 α The dual utility from consumption and leisure rate is ũ cl (t, z) = ( l 1 α ĉ α ) 1 γ 1 γ (ĉ + w l p )z where ĉ = ( αp ( ) ) 1 α(1 γ) 1 1 α w l p and l γ+(1 p)(α(1 γ) 1) 1 αp = α z 1 α w The optimal consumption and leisure strategies are thus given by c = ( αp 1 α w(l ) p and l = α 1 ( ) ) 1 α(1 γ) 1 γ+(1 p)(α(1 γ) 1) αp z 1 α w

36 Figure: One simulation: aging shock (b g) occurs after 4.11 years and retirement occurs after 5.89 years, due to the health shock (g w).

37 Figure: One simulation: no aging shock occurs, retirement occurs after years, due to the health shock (b p).

38 Figure: Optimal strategies for the average agent assuming a retiree coverage.

39 Figure: Optimal strategies for the average agent assuming a tied-retiree coverage.