Health and (other) Asset Holdings

Transcription

1 Health and (other) Asset Holdings Julien Hugonnier 1,3 Florian Pelgrin 2 Pascal St-Amour 2,3 1 École Polytechnique Fédérale de Lausanne (EPFL) 2 HEC, University of Lausanne 3 Swiss Finance Institute October 14, 2009 P. St-Amour Health and (other) Asset Holdings 1 / 43

2 Strong links health and financial status/decisions Health, wealth on portfolio, health expenditures: Dependent Variable Impact of Risky port. Health expend. Labor income Variable share of wealth share of wealth Wealth (+) ( ) Health (+) ( ) pre-retire. (++) post-rerire. (+) Should treat portfolio/health expend. as joint decision process, (H t, W t ) (π t, I t ) (H t+s, W t+s ),... (almost) Never done. P. St-Amour Health and (other) Asset Holdings 2 / 43

3 Theoret. explan.: Two segmented strands of research Health Econ. Fin. Econ. This paper Health investment health expend. mortality risk health dynamics health effects: -utility -income -mortality health insur. Portfolio/savings consumption asset allocation life cycle P. St-Amour Health and (other) Asset Holdings 3 / 43

4 Main elements of model Standard financial asset allocation [Merton, 1971] IID returns, constant investment set intermediate consumption utility, Health investment model [Grossman, 1972] health as human capital locally deterministic process P. St-Amour Health and (other) Asset Holdings 4 / 43

5 Main elements of model Additional features: Preferences: Generalized recursive: VNM as special case. Non-homothetic: Min. subsistence cons. Health effects: (partially) Endogenous mortality Positive effects on labor income Technology: Convex health adjustment costs Decreasing returns in mortality control Life cycle: Different pre- post-retirement health elasticities of income Life cycle properties for all variables P. St-Amour Health and (other) Asset Holdings 5 / 43

6 Solution concepts Dual effects of health on income, mortality: Proceed in two steps 1 Abstract from endogenous mortality risk: Closed forms, 2 Allow endogenous mortality risk: No closed-form solutions. Perturbation method around first-step benchmark, Characterize solutions in (Wt, H t ) space. Advantages: 1 Analytical tractability: No calibration exercise for comparative statics. 2 Econometric tractability: Conditionally linear estimated optimal rules. P. St-Amour Health and (other) Asset Holdings 6 / 43

7 Main findings Data Exo. mortality Endo. mortality Portfolios H t (+) (+) (+) W t (+) (+) (+) Health invest. H t ( ) (+) ( ) W t ( ) ( ) ( ) *: In certain areas of (W t, H t ) space. P. St-Amour Health and (other) Asset Holdings 7 / 43

8 Empirical analysis Fully structural econometrics: Dynamic theoretical model with predictions in closed-forms optimal portfolio, health investment shares. Cross-sectional estimation using HRS data. Econometric tractability: Conditional linear optimal rules: SRF estimation. Can recover structural parameters from SRF estimated parameters. P. St-Amour Health and (other) Asset Holdings 8 / 43

9 Empirical analysis Main estimation results confirm theoretical model relevance: Health technology parameters: Rapid depreciation of health in absence of invest. Adjustments feasible, but strongly diminishing returns Mortality distribution parameters: Important incompressible mortality, but mortality is partially controllable Predicted longevity in accord with data Dual effects of H t are relevant. Preference parameters Subsistence consumption important Realistic risk aversion, EIS VNM preferences rejected P. St-Amour Health and (other) Asset Holdings 9 / 43

10 Related literature authors similarities differences [Edwards, 2008] asset selection no mortality risks health non-storable perm. health expend. if sick no preventive expend. no health-dep. income [Hall and Jones, 2007] endo. mortality no asset selection health investment aggreg. health spend. convex adjust. non structural econometrics spec. of prefs. [Yogo, 2008] health investment no health-dep. income asset selection no life cycle health can be sold calibration optimal annuities mkt. exogenous mortality only P. St-Amour Health and (other) Asset Holdings 10 / 43

11 Outline of the talk 1 Introduction Motivation and outline Related literature 2 Data Description of data set Relevant co-movements 3 Model Health dynamics, survival and income dynamics Preferences and budget constraint The decision problem 4 Optimal rules Exogenous mortality Endogenous mortality 5 Econometric analysis Econometric model 6 Estimation results Unrestricted reduced-form parameters Structural parameters 7 Conclusion P. St-Amour Health and (other) Asset Holdings 11 / 43

12 Data description Health and Retire. Survey (HRS) resp. aged 51+, 5 th wave (2000), Financial wealth: W j = Safe j + Bonds j + Risky j safe assets (check. and saving accounts, money mkt. funds, CD s, gov. savings bonds and T-bills) bonds (corp., muni. and frgn. bonds, and bond funds) risky assets (stock and equity mutual funds) Self-reported health level (poor, fair, good, v. good, excel.) Health investment Medical expenditures (doctor visits, outpatient surg., home, hosp. and nurs. home care, prescr. drugs,... ) OOP (unins. cost over prev. 2 yrs.) P. St-Amour Health and (other) Asset Holdings 12 / 43

13 HRS data: Effects of health, wealth Table: Summary stats. by net fin. wealth and health for retired agents Net financial wealth quintile Health Fair Wealth $6,114 $596 $12,683 $59,366 $514,602 P(risky > 0) 2% 1% 14% 42% 74% risky assets 2% 1% 7% 24% 42% Health inv. share 245% 710% 46% 12% 2% Good Wealth $10,911 $718 $13,094 $64,108 $436,456 P(risky > 0) 5% 2% 19% 45% 77% risky assets 5% 3% 12% 24% 45% Health inv. share 79% 476% 31% 7% 1% Very Good Wealth $7,108 $960 $13,578 $64,905 $467,585 P(risky > 0) 7% 4% 24% 52% 82% risky assets 61% 7% 12% 27% 50% Health inv. share 86% 188% 21% 5% 1% P. St-Amour Health and (other) Asset Holdings 13 / 43

14 HRS data: Effects of health on income Table: Income and health regression All Non-retired Retired A. Individual income Constant ** *** (0.0021) (0.0051) (0.0012) Health *** *** *** (0.0006) (0.0014) (0.0004) Observations 19,571 8,836 10,735 B. Household income Constant ** *** *** (0.0022) (0.0053) (0.0013) Health *** *** *** (0.0007) (0.0014) (0.0004) Observations 19,571 8,836 10,735 P. St-Amour Health and (other) Asset Holdings 14 / 43

15 Health dynamics and survival dh t = ( It α Ht 1 α ) δh t dt, H0 > 0, (1) [ ] t < τ t + s = λ(ht ) = λ 0 + λ 1 (2) 1 lim s 0 s P t P 0 [τ > t] = E 0 [ e R t 0 λ(hs)ds] (3) H ξ t Health as human capital, locally deterministic [Grossman, 1972] Convex adj. costs [Ehrlich, 2000, Ehrlich and Chuma, 1990] Poisson mortality [Ehrlich and Yin, 2005, Hall and Jones, 2007] Incompressible mortality λ0, Path dependence of health decisions. P. St-Amour Health and (other) Asset Holdings 15 / 43

16 Income dynamics Y t Y (t, H t ) = 1 {t T } Y e t + 1 {t>t } Y r t (4) Y i t Y i (H t ) = y i + β i H t, (5) Two employment phases i = e (employed) or i = r (retired) Health-dependent labor income, Higher wages to agents in better health, less absent from work, better access to promotions. Differences in pre- post- retirement fixed income (e.g. pensions) and health sensitivity. P. St-Amour Health and (other) Asset Holdings 16 / 43

17 Standard approaches under endo. mortality Standard approach: U t = 1 {τ>t} E t [ τ t u(x; γ) = x1 γ (1 γ) e ρ(s t) u(c s )ds] life always preferable u(x; γ < 1) x 0 u(x; γ > 1) death always preferable P. St-Amour Health and (other) Asset Holdings 17 / 43

18 Preferences Our approach: Abandon VNM U t = 1 {τ>t} E t [ τ f (c, v) = t vρ 1 1/ε γ ) ] σ s (U) 2 ds (6) 2U s [ (c ) a 1 1/ε 1]. (7) v ( f (c s, U s ) Generalized recursive [Duffie and Epstein, 1992, Schroder and Skiadas, 1999]. f ( ) h.d. 1 U( ) h.d. 1 Ut, c t a in same metric ct a 0 U t 0 life always preferable by monotonicity. Non-homothetic for a 0, Health-, time-indep., no bequest. P. St-Amour Health and (other) Asset Holdings 18 / 43

19 Financial market and budget constraint S 0 t = e rt (8) ds t = µs t dt + σs t dz t, S 0 > 0, (9) dw t = (rw t + Y t I t c t )dt + W t π t σ(dz t + θdt), (10) Riskless and risky assets, Constant investment set, Incomplete markets. P. St-Amour Health and (other) Asset Holdings 19 / 43

20 Iso-morphism 1- With health-dependent preferences: V (t, W t, H t ) = sup U t (c) (π,c,i ) s.t. dw t = (rw t + y + βh t I t c t )dt + W t π t σ(dz t + θdt) V (t, W t, H t ) = sup U t (x + βh), (π,x,i ) s.t. dw t = (rw t + y I t x t )dt + W t π t σ(dz t + θdt). P. St-Amour Health and (other) Asset Holdings 20 / 43

21 Iso-morphism 2- With complete market + infinite horiz. + endo. discount. [ τ ( U t = 1 {τ>t} E t f (c s, U s ) γ ) ] σ s (U) 2 ds 2U s = 1 {τ>t} U o t, t where, U o t = E t [ t e R s t λ(hu)du ( f (c s, U o s ) γ 2U o s σ s (U o ) 2 ) ds ]. P. St-Amour Health and (other) Asset Holdings 21 / 43

22 Optimal rules Two channels for health effects: 1 Income y t = y 0 + βh t 2 Mortality λ(h t ) = λ 0 + λ 1 H ξ t Abstract first from mortality (λ 1 = 0) to highlight income effects, then re-introduce mortality. P. St-Amour Health and (other) Asset Holdings 22 / 43

23 Optimal rules: Exo. mortality (λ 1 = 0) Solution concept exogenous mortality: 1 Choose I to max. disposable wealth: P(t, H t ) = sup I 0 [ ] ( ) E t ξ t,s Y (s, Hs ) a I s ds s.t. (1) t 2 Replace N t N(t, W t, H t ) = W t + P(t, H t ). Choose c, π to max. util. s.t. process for dn. (11) P. St-Amour Health and (other) Asset Holdings 23 / 43

24 Optimal rules: Exo. mortality (λ 1 = 0) variable λ 0 H W N(t, W, H) = W + B(t)H + C(t) (+) (+) I0 s = W 1 H ( αb(t) ) 1 1 α (+) ( ) data ( ) ( ) V 0 = ρ(a/ρ) 1 1 ε N(t, W, H) ( ) (+) (+) π 0 = θ γσw N(t, W, H) (+) (+) data (+) (+) c 0 = a + A N(t, W, H) ( ) (+) (+) C(t) e r(s t)( Y (s, 0) a ) ds t ( ) A ερ + (1 ε) r λ γ θ2 *: if B(t)H + C(t) < 0, : if ε < 1 ( ) P. St-Amour Health and (other) Asset Holdings 24 / 43

25 Optimal rules: Endo. mortality (λ 1 > 0) HJB equation: λ(h)v = max (π,c,i ) {L π,c V + f (c, V ) γ(πσwv W ) 2 } 2V (12) where, L π,c = t + (H 1 α I α δh) H + ((r + πσθ)w + Y c I ) W (πσw ) WW (13) P. St-Amour Health and (other) Asset Holdings 25 / 43

26 Optimal rules: Endo. mortality (λ 1 > 0) Main problem: No closed-form solutions when λ 1 0 Solution: n th order expansion around exo. mortality benchmark solution λ 1 = 0 V V 0 + λ 1 V m! λm 1 V m + 1 n! λn 1V n, V m V m (t, W, H) = m V (t, W, H) λ m 1 λ1 =0 Once V solved, substitute in FOC, expand again X = X 0 + λ 1 X 1,... to get closed-form solutions. P. St-Amour Health and (other) Asset Holdings 26 / 43

27 Optimal rules: Endo. mortality (λ 1 > 0) variable effect of λ 1 λ(t, H) = λ 0 + λ 1 H ξ > λ 0 N(t, W, H) = W + B(t)H + C(t) V = V 0 λ1 (t)v H ξ 0 < V 0 π 1 = π 0 c 1 = c 0 λ1 (t)(1 ε)an(t, W, H) < c H ξ 0 I 1 = I 0 (t, H) + λ1 (t)(αb(t)) α H ξ 1 α ηn(t, W, H) > I0 C(t) e r(s t)( Y (s, 0) a ) ds t ( ) A ερ + (1 ε) r λ γ θ2 : if ε < 1 P. St-Amour Health and (other) Asset Holdings 27 / 43

28 Optimal rules: Endo. mortality (λ 1 > 0) Two effects of λ 1 > 0: 1 Higher mortality risk λ(h t ) = λ 0 + λ 1 H ξ t, but,..., 2... can partially offset through higher I t Impact on optimal rules: V < V 0, (because λ(h t ) ) I 1 > I 0, (because λ(h t ) ) π unchanged, c 1 < c 0 if ε > 1. P. St-Amour Health and (other) Asset Holdings 28 / 43

29 Optimal rules: Endo. mortality (λ 1 > 0) Figure: Comparative statics of the first order rules IH s < 0 IH s > 0 Financial wealth C(t) I W s > 0 I W s < 0 W = G(t, H) πw > 0 πw < 0 0 W = P(t, H) 0 H(t) H(t) Health status π H > 0 O P. St-Amour Health and (other) Asset Holdings 29 / 43

30 Econometric model Structural estimation of optimal rules: π j W j = θ π,0 (t j )W j + θ π,1 (t j )H j + θ π,2 (t j ) + ɛ π,j, (14) I j = θ I,1 (t j )H j + θ I,2 (t j )H ξ j W j + θ I,3 (t j )H 1 ξ j + θ I,4 (t j )H ξ j + ɛ I,j, (15) Problems: [ɛ π,j, ɛ I,j ] N (0, Σ) Pre-retire. is heavy computation. use post-retire. only. Censored dependent variable use Tobit. Very non-linear in parameters + s.t. non-linear constraints 2-step proc. P. St-Amour Health and (other) Asset Holdings 30 / 43

31 Econometric model Back-out structural parameters: θ π,0 = θ γσ, θ I,1 = (δ + J) 1 α, θ π,1 = θ π,0 B, θ I,2 = αλ 1ξ(J + δ) (1 α)(ξj + A), θ π,2 = θ π,0 C, θ I,3 = θ I,2 B, θ I,4 = θ I,2 C, (16) where C C(T ) = (y r a)/r, and (J, B) solves J = (αb) α 1 α δ, (17) β r B = r + αδ (1 α)j. (18) P. St-Amour Health and (other) Asset Holdings 31 / 43

32 Econometric model Table: Summary of calibrated and estimated parameters Item Symbol Calibrated Estimated Income dynamics (Eq.(5)) Constant y r Health sensitivity β r Financial markets (Eqs.(8),(9)) Interest rate r Std. error risky return σ Mean risky return µ Health dynamics (Eq.(1)) Convexity α Depreciation δ Death intensity (Eq.(2)) Convexity ξ [3.8, 4.7] Exogenous λ 0 Health sensitivity λ 1 Preferences (Eqs.(6),(7)) Discount rate ρ Risk aversion γ Subsistence cons. a EIS ε P. St-Amour Health and (other) Asset Holdings 32 / 43

33 SRF Results Table: SRF parameter estimates expected sign ξ = 3.8 ξ = 4.2 ξ = 4.7 A. Labor income (Eq.(5)) y r (+) *** *** *** (0.0013) (0.0013) (0.0013) β r (+) *** *** (0.0004) (0.0004) (0.0004) B. Risky asset levels (Eq.(14)) θ π,0 (+) *** *** *** (0.0060) (0.0060) (0.0060) θ π,1 (+) *** *** *** (0.0022) (0.0022) (0.0022) θ π,2 ( ) *** *** *** (0.0081) (0.0081) (0.0081) C. Health expenditure levels (Eq.(15)) θ I,1 (+) *** *** *** (0.0002) (0.0002) (0.0002) θ I,2 (+) (0.0011) (0.0010) (0.0009) θ I,3 (+) *** *** *** (0.0039) (0.0046) (0.0057) θ I,4 ( ) *** *** *** (0.0039) (0.0046) (0.0057) P. St-Amour Health and (other) Asset Holdings 33 / 43

34 SRF Results: Health investments Figure: Actual vs predicted health investment levels Health investment ($1, 000) Actual Predicted Age P. St-Amour Health and (other) Asset Holdings 34 / 43

35 SRF Results: Portfolios Figure: Actual vs predicted portfolio levels 20 Actual Predicted Portfolio ($10, 000) Age P. St-Amour Health and (other) Asset Holdings 35 / 43

36 Structural parameters Table: Structural parameter estimates ξ = 3.8 ξ = 4.2 ξ = 4.7 A. Preferences (Eqs.(6),(7)) γ *** *** *** (0.0125) (0.0125) (0.0125) a *** *** *** (0.0005) (0.0005) (0.0005) ε *** *** *** (0.0000) (0.0000) (0.0000) B. Health dynamics (Eq.(1)) α *** *** *** (0.0580) (0.0620) (0.0565) δ *** *** *** (0.0128) (0.0191) (0.0149) C. Death intensity (Eq.(2)) λ *** *** *** (0.0000) (0.0002) (0.0001) λ *** *** *** (0.0009) (0.0023) (0.0010) P. St-Amour Health and (other) Asset Holdings 36 / 43

37 Structural parameters Interpretation structural parameters: Robustness: All parameters reasonably robust to calibrated ξ. γ 1.76: Realistic [0, 10]. a > 0: significant (quasi-homothetic prefs.), large ($24,800/yr.) C < 0 π w > 0 ε 0.28 < 1: Agents substitute poorly across time; λ 0 > 0 c 0 ; λ 1 > 0 c 1 < c 0. ε 1/γ: Reject separable VNM preferences in absence of mortality risk. P. St-Amour Health and (other) Asset Holdings 37 / 43

38 Structural parameters Interpretation structural parameters (cont d): α 0.23: Significant convexities health adjustment costs. δ 0.28: Very rapid depreciation health stock absent I λ : Significant (exo. mortality), large. λ : Small (perturbation correct), significant (endo. mortality). P. St-Amour Health and (other) Asset Holdings 38 / 43

39 Implied variables Can compute expected lifetime at optimum: l(h) = 1 ( 1 λ ) 1 λ 0 H ξ Φ, (19) where, ) Φ 1 = λ 0 + ξ ((αb(t )) α 1 α δ > 0 (20) Independent of wealth, age can compare with estimates from US life tables [Lubitz et al., 2003] P. St-Amour Health and (other) Asset Holdings 39 / 43

40 Implied variables Table: Implied variables Data ξ = 3.8 ξ = 4.2 ξ = 4.7 A. Life expectancy l(1) l(2) l(3) l(4) l(5) B. Thresholds in Figure 1 H H P(H) G(H) G(H) P. St-Amour Health and (other) Asset Holdings 40 / 43

41 Conclusion Close feedbacks H t, W t π t, I t H t+s, W t+s : Need to be studied jointly. No joint model. This model: at interface between Health and Fin. Econ. Consumption/portfolio/health investment in presence of mortality + financial risks. Health has 2 effects: Endogenous mortality; Human capital (labor income). Convex health and mortality adjust. costs Preferences: Non-expected utility life always valuable Quasi-homothetic P. St-Amour Health and (other) Asset Holdings 41 / 43

42 Conclusion Main theoretical findings: Exogenous mortality: Incomplete success Can reproduce co-movements of portfolios if subsistence consumption sufficiently high Cannot reproduce co-movements of health investments. Endogenous mortality: Potential success Exists regions of state space where can reproduce all co-movements. Potential for poverty traps. P. St-Amour Health and (other) Asset Holdings 42 / 43

43 Conclusion Empirical evaluation: Structural estimation of dynamic model in cross-section. Realistic parameters. Main empirical findings in line with theoretical results. P. St-Amour Health and (other) Asset Holdings 43 / 43

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46 Optimum consumption and portfolio rules in a continuous-time model. Journal of Economic Theory, 3(4): Schroder, M. and Skiadas, C. (1999). Optimal consumption and portfolio selection with stochastic differential utility. Journal of Economic Theory, 89(1): Yogo, M. (2008). Portfolio choices in retirement: Health risk and the demand for annuities, housing and risky assets. manuscript, Finance Department, The Wharton School, University of Pennsylvania. P. St-Amour Health and (other) Asset Holdings 43 / 43