Life-cycle Portfolio Allocation When Disasters are Possible

Transcription

1 Life-cycle Portfolio Allocation When Disasters are Possible Daniela Kolusheva* November 2009 JOB MARKET PAPER Abstract In contrast to the predictions of life-cycle models with homothetic utility and risky labor income, the majority of US households do not hold any wealth in the stock market. Even among stockholders, the equity share out of net worth is modest. I develop a life-cycle model in which there are rare disasters, calibrated to match the joint empirical distribution of stock market crashes and macroeconomic contractions. The model provides an explanation for the low portfolio share in equity. About half of the decrease in stock holdings can be attributed to the correlation between stock market crashes and depressions. Furthermore, a 6% perceived probability of disaster is sufficient to deter the median household from investing in stocks during its working life. Small differences in expectations lead to dramatically different portfolios, consistent with the heterogeneity of asset allocation in data. * Department of Economics and International Business School, Brandeis University, 415 South Street, Waltham MA 02453, USA. danielak@brandeis.edu, Website: I would like to thank Blake LeBaron, George Hall, and Jens Hilscher for their valuable guidance, Catherine Mann, Dan Tortorice, Tanseli Savaser and seminar participants at Brandeis University for their insightful comments and discussion. 1

2 1. Introduction Consumer surveys reveal that the majority of US households do not hold any wealth in the stock market, either directly or indirectly through mutual funds and retirement-savings accounts. Even among stockholders, the average share of stocks out of net worth was only about 35% in 2004 and 29% in 2007 according to data from the Survey of Consumer Finances. Both facts are at odds with the portfolio-specialization predictions of life-cycle models with homothetic utility and non-tradable labor income. In these models households optimally invest all of their savings in stocks when young and middle-aged. Only towards the end of the life-cycle households start holding the risk-free asset. The high equity premium partially explains why stocks dominate the household s portfolio in these models. The other reason is the low correlation between labor income and stock returns in aggregate data. The present value of future labor income essentially acts as a risk-free asset in the household s total wealth (the sum of financial wealth and human capital), creating a strong incentive to hold equity. Both of these motives deepen the stockholding puzzle : if one can expect to earn more by holding stocks than by holding riskless financial assets and if stocks provide a hedge for uncertain labor income, what is it that keeps the majority of households out of the stock market and limits the stockholdings of the rest? This paper examines the role that tail events in the stock and labor markets play in explaining the stockholding puzzle. I develop a life-cycle model of consumption and asset allocation for a household with stochastic labor income who faces the risk of a rare economic disaster. When the disaster occurs, there is either a large negative shock to stock returns (a stock market crash), or a large negative shock to permanent income (a macroeconomic contraction), or both events occur simultaneously. The joint probability distribution of stock returns and labor income shocks during disasters is calibrated to match the empirical distribution of stock market crashes and depressions from Barro and Ursua (2009). I solve the model with disasters and use the derived policy functions to simulate an economy of ex-ante identical households who are subject to idiosyncratic realizations of income and stock return shocks. I also solve the model without disasters, to highlight what the novel implications are. The first contribution of this paper is to provide an explanation for the low portfolio share in stocks, conditional on stock market participation. There is a sizeable reduction in 2

3 stockholdings between ages 20 and 65 when the household is susceptible to (potentially coincident) disastrous shocks to permanent labor income and stock returns. Households invest only between 30% and 35% in stocks when middle-aged because the closer they get to retirement, the more vulnerable they are to disasters. Workers nearing retirement age run the risk of diminished economic well-being for the remainder of their life if they face a large negative shock to their permanent labor income. Similarly, a stock market crash close to retirement would destroy the household s nest egg at its zenith and leave the household no opportunity to re-build it. In contrast, the household is more flexible in the beginning of adult life. Holding relatively more stocks makes sense at this point, given the high equity premium, because the household would have time to catch up even if it experienced a large negative labor and/or portfolio shock. The older household is also less vulnerable to disasters than the middle-aged, but for a different reason. Since the pension it receives is risk-free, the household is insulated from shocks to permanent labor income after retirement. Faced with longevity risk (the risk of running out of resources), the retired investor increases stockholdings slightly relative to before retirement. Overall, the optimal response when disasters are possible is to shift the portfolio towards the riskless asset, foregoing the higher equity premium. The household thus accumulates less financial wealth, leading to a curtailed consumption profile. The utility cost of disasters to the consumer is 5.9%, which is non-negligible. A second contribution of this paper is to decompose the effect of disasters into the overall increase in the riskiness of labor income and stock returns, and the higher correlation between them during crises. Both crowd out stockholdings, but it is not clear which effect predominates. To examine their relative importance, I also solve and simulate the model with disasters under the assumption that stock market crashes and the negative shocks to permanent income are independent. In the model with independent disasters, the incentive of the household to invest in the risk-free asset falls by 50% during the working life. The utility cost of independent disasters is also substantially smaller. A third contribution of this paper is to establish that households who only anticipate disasters behave exactly the same way as those who actually experience disasters. I find that as long as the investor subjectively perceives a large enough probability of disaster, he might stay 3

4 away from the stock market completely, even if in reality disasters are not that common. In particular, a 6% perceived probability of disaster is sufficient to deter the median household from investing in stocks during its working life. Very small differences in the perceived probability of disasters lead to dramatically different portfolios, consistent with the heterogeneity of asset allocation in data. The rest of the paper is organized as follows: Section 2 summarizes results from the existing theoretical and empirical literature on life-cycle asset allocation. Section 3 contains the model and method of solution. Section 4 outlines the benchmark parameterization of preferences and the calibration of labor income and disasters. In Sections 5, I discuss the results from the model with disasters and contrast them with the traditional life-cycle model without disasters. Section 6 concludes. 2. Asset allocation over the lifecycle theory and evidence Sophisticated theoretical treatments of dynamic portfolio theory first appeared in the late 1960 s, with papers by Mossin (1968), Merton (1969) and Samuelson (1969). The classic result is that the optimal portfolio shares are constant over the lifecycle, i.e. independent of both age and wealth, and only vary with risk aversion, the mean equity premium and the variance of stock returns. This is driven by the following assumptions: (1) asset returns are independently and identically distributed, (2) agents have time-invariant and additively separable utility functions of the hyperbolic absolute risk aversion class (HARA), (3) there is no labor income or other nontradeable assets, and (4) markets are frictionless and complete. A well known problem of calibrating the classic model is the stockholding puzzle, which is the microeconomic equivalent of the equity premium puzzle: for plausible parameters of risk aversion and the equity premium and rate of return risk, the consumer should hold almost all of his portfolio in stocks. This prediction is at odds with empirical facts. It is well documented that a large fraction of the US population holds little or no stocks (Bertaut and Haliassos 1995, Blume and Zeldes 1993) and that participation varies systematically with factors such as wealth and age (Gentry and Hubbard 1998, King and Leape 1987). Stockholders are considerably wealthier in all asset categories, somewhat older and better educated. Households of low or moderate wealth hold almost no stocks (Heaton and Lucas, 2000b). Using US data from the PSID, Gakidis (1997) finds that households with a larger 4

5 probability of realizing extremely low wage income are less likely to participate in the stock market. For those who are participating, the probability of very low wages reduces the proportion invested in stocks. While wealth and stockholdings are positively correlated, there is no clear relationship between age and the fraction of wealth held in the stock market. On the one hand, age, cohort and time effects cannot be separately estimated without restrictions, because of the identification problem pointed out by Ameriks and Zeldes (2004). On the other hand, the relative importance of stockholding changes significantly depending on how broadly wealth is defined (Heaton and Lucas (2000b)). When liquid net worth is used as the metric, the share of stocks rises slightly with age for cohorts under 65, and then declines significantly for those over 65, producing a hump-shaped pattern over the lifecycle. When financial net worth is used, the age profile of stockholding is essentially flat. As a share of total wealth (financial wealth and human capital), stockholdings increase with age. Another puzzle is the significant heterogeneity of stockholdings which cannot be explained by heterogeneity of preferences (Curcuru, Heaton, Lucas and Moore (2006)). In response, recently there has been a resurgence of interest in the portfolio problem with a focus on incomplete markets in which background risks can affect portfolio rules. This greatly complicates finding analytical solutions to the portfolio problem and necessitates numerical simulations and calibration. A number of more recent papers have taken this approach, including Bertaut and Haliasson (1997), Cocco, Gomes and Maenhout (2005), Campbell, Cocco, Gomes and Maenhout (2001), Gakidis (1997), Heaton and Lucas (1997, 2000a), Koo (1995) and Viceira (1997). Curcuru, Heaton, Lucas and Moore (2006) identify several different sources of market incompleteness: 1) stochastic income, which cannot be insured due to moral hazard and adverse selection; 2) transaction costs; 3) limitations on the agent s ability to trade marketable securities (short sale restrictions or housing due to its bulkiness and illiquidity). In the context of life cycle portfolio allocation, the most important form of market incompleteness comes from human capital which generates a risky labor income stream not spanned by traded assets. Intuitively, one might expect that this additional risk would crowd out stockholdings. However, this is not what the standard life-cycle model with risky labor income implies. On the contrary, households in these models invest almost all of their portfolios in stocks. This is 5

6 at odds with the overall low participation in the stock market, as well as the modest level of stock investment by stock market participants. The reason life-cycle models with homothetic utility and uninsurable labor income imply such a strong preference for stocks is that labor income functions as a risk-free asset. In other words, although labor income is risky, it limits bad outcomes relative to investment income, which significantly reduces effective risk aversion. This finding is driven in part by the insignificant correlation between labor income and stock market returns from aggregate data. One possible venue to get the agents in risky labor models to hold less equity is if human capital had more stock-like qualities. Heaton and Lucas (2000b) examine proprietary business income which is considerably more variable and highly correlated with the stock market than wage income. They find that households with proprietary business income do not invest all wealth in the stock market. Benzoni, Collin-Dufresne and Goldstein (2007) assume a long-term cointegration between the stock market and aggregate labor income. This creates greater substitutability between stocks and human capital, despite the low contemporaneous correlation between labor income and the return on stocks, which reduces stockholdings. This leaves unexplained the fact that the majority of US households hold no stocks. With a positive equity premium and smooth preferences and return processes, all agents should hold at least a small amount of stocks. To get non-participation, Gomes and Michaelidis (2005) assume a fixed cost of stock market entry and find that it constrains young households from investing in the stock market. However, almost all households find it optimal to invest fully in the stock market when middle-aged. Thereafter, the age profile of stockholdings resembles closely the standard model with risky labor income and no fixed costs. Another venue to explain non-participation is a crowding-out effect from housing when the investor is borrowing-constrained. Flavin and Yamashita (2002) suggest that the demand for housing creates a highly levered position in real estate for younger households. This affects negatively their tolerance for stock market risk relative to older households who have paid down their mortgage. Cocco (2005) looks at the life-cycle optimization problem of homeowners and shows that house price risk crowds out stock holdings. Yao and Zhang (2005) examine the optimal dynamic portfolio decision for both owners and renters. Their model predicts that the percentage of liquid assets held as stock is roughly decreasing over time. In the Yao and Zhang (2005) model, homeowners hold a lower equity proportion in their financial net worth (bonds, 6

7 stocks and home equity) as a result of the substitution effect between housing and stocks. However, homeowners hold a higher equity proportion in their liquid portfolio (bonds and stocks) due to the diversification effect of housing as an investment. 3. A life-cycle model of consumption and portfolio choice 3.1 Preferences The model is set in discrete time and each period equals one year. The first period corresponds to the time when the investor starts earning labor income. The investor lives for a maximum of T periods. To account for an uncertain life-span, I use a survival function in the manner of Hubbard, Skinner and Zeldes (1995). Let p t denote the probability that the investor is alive in period t + 1, conditional on being alive in period t; then p j > 0 for all j and the initial and terminal probabilities are given by p 0 = 1 and p T = 0. Consider the decision problem of a household at age t, whose goal is to maximize the lifetime discounted utility from a composite non-durable consumption good C t : max u(c t ) + E t β s t p k T s=t+1 s 1 k=t u(c s ), where β is the subjective discount factor and E is the expectation operator. For tractability, u C t is chosen to be isoelastic with coefficient of relative risk aversion γ: 3.2 Asset returns u C t = C 1 γ t 1 γ for γ > 1. The investment opportunity set is composed of two financial assets: a risk-free bond with constant gross real return R f and a risky asset called stock. Stock returns are distributed as: R e,t = R e Dis R e ε t Mkt Cras h with probability π with probability 1 π MktCras h ln ε t ~ N μ e, ς e 2, (4) where N denotes the normal distribution. Stock market crashes occur with probability π MktCras h. If there is a crash, the gross real return on the risky asset R e Dis is drawn from the joint disaster (1) (2) (3) 7

8 distribution D, calibrated later to accord with the empirical distribution of stock market crashes and macroeconomic contractions 1. The parameter μ e is chosen so that expected value of the gross real stock returns is E[R e,t ] = R e. If the probability of stock market crashes is zero, μ e = ς 2 e /2, implying E ε t = 1. If the probability of stock market crashes is non-zero, μ e is rescaled so that the disaster is a mean-preserving event relative to the model without disasters. The investor is subject to borrowing and short-sale constraints, so that the portfolio share invested in the risky asset, denoted by ω t, must satisfy: 0 ω t 1. (5) 3.3 Labor income Following Zeldes (1989) and Carroll (1997), the household s stochastic labor income prior to retirement is the product of a permanent and a transitory component: Y t = Ρ t Θ t for t K, (6) Ρ t = Γ t Ρ t 1 Ψ t, (7) given an initial level P 1. The variable P t denotes permanent income in period t, defined as the labor income that would be earned in the absence of transitory shocks (i.e. Θ t = 1). As in Carroll (1997) and Gourinchas and Parker (2002), the logarithm of permanent income ln(ρ t ) evolves as a random walk with drift Γ t, calibrated to capture the typical hump-shaped pattern of earnings over the lifecycle. The random walk specification for permanent labor income is supported by the findings of Hubbard, Skinner and Zeldes (1995), who estimate a general firstorder autoregressive process and report the autocorrelation coefficient to be very close to unity. During the household s working life, permanent income is subject to an independently and identically distributed shock Ψ t : Ψ t = Dis Ψ t Ψ t with probability π MacroContr (8) with probability 1 π MacroContr 2 ln Ψ t ~ N μ Ψ, ς Ψ. (9) Macroeconomic contractions occur with probability π MacroContr. If there is a macroeconomic Dis contraction, the shock to permanent income Ψ t is drawn from the joint disaster distribution D, calibrated later to accord with the empirical distribution of stock market crashes and 1 See Section 4 Calibration for further details. 8

9 macroeconomic contractions. The parameter μ Ψ is chosen so that the shock to permanent income has mean E Ψ t = 1. If the probability of macroeconomic contraction is zero, μ Ψ = ς 2 Ψ /2. The model allows for contemporaneous dependence between the permanent component of labor income and stock returns. In the normal state, the correlation coefficient between stock returns and the shocks to permanent labor income is ρ Nor. During disasters, the dependence is characterized by the joint disaster distribution D. During the household s working life, labor income is also subject to an independently and identically distributed transitory shock Θ t ln Θ t ~N ς θ 2 2, ς (10) θ 2. The transitory shock has mean E Θ t = 1. Retirement is exogenous and deterministic, with all households retiring at age K. The retired household receives an earnings-based pension similar to a defined benefit plan. To facilitate the solution of the model by reducing dimensionality, the pension is modeled as a constant fraction λ of permanent labor income in the last working year: Y t = λρ K for t > K. (11) The assumption that labor income is exogenous is made for simplicity. In reality individuals decide how many hours to work and the amount of effort to exert, both of which influence labor income. The labor exogeneity assumption prevents the investor from compensating for bad portfolio return or labor income realizations by working more hours. Bodie, Merton and Samuelson (1992) examine the effect of the opportunity to smooth income shocks by adjusting labor supply and find that the optimal equity share is a decreasing function of age. Gomes, Kotlikoff and Viceira (2008) calibrate a life-cycle model with flexible labor supply, but variable labor supply in their model substantially increases the optimal equity holdings relative to the standard model, thus making it even harder to explain the stock market participation puzzle. 3.4 The investor s problem The timing of events is as follows: each period t, the investor starts with a certain amount of wealth, based on the saving and investment decisions from the previous periods, and then receives labor income Y t. Following Deaton (1991), denote the resources available for 9

10 consumption as cash-on-hand (M t ). The problem is to choose how much to consume (C t ) and how to allocate savings (A t ) between stocks and the risk-free asset. The transition equations are: A t = M t C t (12) M t+1 = Y t+1 + A t ω t R e,t ω t R f = Y t+1 + A t R p,t+1, (13) where R p,t+1 = ω t R e,t ω t R f = R f + R e,t+1 R f ω t is the return to the investor s portfolio. The investor s problem is to maximize equation (1) subject to (2) through (12), in addition to the usual non-negativity constraint on consumption. The control variables are T C t, ω t t=1 T. The state variables are t, M t, P t t=1. The solution to this problem consists of the policy rules for consumption and risky asset portfolio share as functions of the state variables: C t t, M t, P t and ω t t, M t, P t. The functional form of the labor income process post-retirement is chosen to ensure that the value function is homogenous with respect to current permanent labor income. Because of the scalability of the value function, it is possible to reduce the state space by normalizing with respect to permanent labor income. 2 The normalized variables are represented by lower case letters (m t = M t P t, c t = C t /P t, etc.). The Bellman equation for this problem prior to retirement is: v t m t = max c t 0,0 ω t 1 u c t + βp t E t Γ t+1 Ψ t+1 1 γ v t+1 m t+1 s.t. m t+1 = m t c t R p,t+1 /(Γ t+1 Ψ t+1 ) + Θ t+1. for t K The assumption about the labor income process after retirement is equivalent to Γ t>k = 1, P t>k = 1 and Θ t>k = λ. The Bellman equation after retirement thus simplifies to: v t m t = max c t 0,0 ω t 1 u c t + βp t E t v t+1 m t+1 s.t. m t+1 = m t c t R p,t+1 + λ. for t > K An analytical solution to this problem does not exist. Given the terminal condition (14) (15) (16) (17) v T+1 m T+1 = 0, (18) the policy functions can be derived numerically by backward induction. In the last period the investor optimally consumes all available resources and the value function corresponds to the indirect utility function. Substituting this value function in the Bellman equation gives the policy rules for the previous period, t = T 1. Using the policy rules, one can obtain the corresponding 2 Appendix A demonstrates that the solution to the normalized problem (14) yields the solution to the original problem via V t M t, P t = P t 1 γ v t (m t ). 10

11 value function in t = T 1 and then iterate this procedure backwards until t=1. See Appendix A for more details on how the model is solved. 4. Benchmark Parameterization and Calibration 4.1 Preference Parameters Table 1 contains the benchmark parameters. The model starts at age 20 for households without a college degree and 22 for households with a college degree. The investor retires at K=65, and dies with probability one at T+1=101. The conditional survival probabilities, p j for j=1,,t are calculated using the mortality tables from the National Center for Health Statistics (2003) 3. The benchmark parameter values for the subjective discount factor and the coefficient of relative risk aversion are set to β = 0.98 and γ = 5, respectively. 4.2 Labor Income Calibration I estimate the labor income equations (6), (7) and (11) using data from the Panel Study of Income Dynamics (PSID). In this section I provide a brief summary of the sample selection and the estimation methods. For more details, see Appendix B. As in Cocco, Gomes and Maenhout (2005) and Campbell, Cocco, Gomes and Maenhout (2001), labor income is defined broadly to control for possible mechanisms to self-insure against labor income risk. The labor income measure encompasses total reported labor income, social security and supplemental social security, unemployment and workers compensation, other welfare, total transfers (mainly help from relatives) and child support for the head of household and his spouse, if present. The predictable component of permanent labor income is modeled as a function of age and a vector of household characteristics Z t. I use two alternative specifications to capture the effect of age on income: age dummies and a third degree polynomial in age. The degree of the age polynomial is determined by the Bayes information criterion (BIC). The vector of individual characteristics Z t consists of family fixed effects and indicator variables for marital status and household size. To control for education, I split the sample into two groups: households without a college degree and college graduates. This procedure allows the age profiles to differ across 3 United States Life Tables, 2003 are available at ftp://ftp.cdc.gov/pub/health_statistics/nchs/publications/nvsr/54_14/ 11

12 education groups 4. Ideally, one should also control for occupation, but this is not possible with PSID data, because of the inconsistency in the categories for occupation and the lack of distinction in the data between the unemployed and people who are not in the labor force. This issue is mitigated by the presence of fixed effects, if the occupation of the head of household remains mostly unchanged over the time period covered by the sample. Table 2 presents the results from the labor income estimation by education group. All the coefficients in the cubic polynomial of age are significant and have the intuitive signs. The overall hump-shaped age profile matches the empirical findings in Attanasio (1995), Hubbard, Skinner and Zeldes (1995) and Gourinchas and Parker (2002). Figure 1 plots the age profiles by education group using two alternative specifications: a 3 rd degree polynomial and age dummies. The two methods produce very similar results. I use the more parsimonious third degree polynomial for the numerical solution of the model and I focus on the group with no college degree for the benchmark results. The replacement ratio λ which determines the amount of retirement income is calculated by taking the average labor income for retirees in a given education group and dividing it by the average labor income in the last working year prior to retirement 5. The replacement ratios for the two education groups are reported in Table 2. I use the variance decomposition method of Carroll and Samwick (1997) to compute the variance of the permanent and transitory shocks to income. The estimated variances for the two education groups are presented in Table 3. In line with previous findings 6, college graduates have higher variance of the permanent shock and lower variance of the temporary shock relative to households without a college degree. 4.3 Asset returns and correlations with labor income Using long-term data since 1870 for the United States, the real rate of return on stocks was 8.27% with a standard deviation of 18.66%, while the real rate of return on bills was 1.99% with a standard deviation of 4.82% (Barro and Ursua 2008). Based on these historical values, the constant real risk free rate for the model is set at 2%, the mean equity premium at 6% and the 4 See Attanasio (1995), Hubbard, Skinner and Zeldes (1995) for evidence on the different age profiles across education groups. 5 I use the median instead of the mean as the measure of central tendency, because the distribution of income is highly asymmetric. 6 See Cocco, Gomes and Maenhout (2005) 12

13 standard deviation of stock returns at 18%. 7. The correlation between stock returns and the permanent income shocks in normal times is calibrated using the labor income data from the PSID and the value-weighted NYSE index returns from CRSP. The details of the calibration are given in Appendix B. Table 4 reports the results for each of the two education groups. In both cases, the correlation between aggregate income growth and excess stock returns is not statistically different from zero during normal years. The calibration of market crashes and macroeconomic contractions is based on the empirical evidence in Barro and Ursua (2009). Because disasters are rare, pinning down the probability of a disaster and the distribution of market returns or the size of a macroeconomic contraction from historical data requires a long time series for many countries, along with the assumption of parameter stability over time and across countries. Table in Appendix C reproduces the Barro and Ursua (2009) list of market crashes and associated macroeconomic crises 8. Market crashes are defined as peak-to-trough cumulative real returns of -25% or worse; macroeconomic crises are defined as cumulative declines in GDP or consumption of 10% or more. The matching between market crashes and depressions follows a flexible scheme which considers both coincident and adjacent events. Using long-term data for 30 countries 9 up to 2006 comprising 3037 annual observations, Barro and Ursua (2009) identify 232 stock market crashes and 100 depressions. The average duration of disasters, measured by the interval between peak and trough for each event, is 4 years for macroeconomic crises and 3.2 years for stock market crashes. To express the probabilities of transitioning from the normal state into a market crash or a macroeconomic crisis, I estimate the ratio of the number of disasters to the number of normal years. The corresponding unconditional probabilities are 10.11% for stock market crashes and 3.79% for macro contractions. Table reveals that among the 232 stock-market crashes and 100 macro contractions, there were 71 matched events, occurring over overlapping or adjacent years. The percentage of stock market crashes that occur with depressions is 31%, and the percentage of depressions that occur 7 A common choice in the literature on lifecycle portfolio choice is to use a forward-looking equity premium of 4%, because a higher equity premium in those models leads to complete portfolio specialization. 8 For more detailed information on the construction of the data, please refer to Table 2 and the accompanying notes in Barro and Ursua (2009). 9 The sample comprises 20 OECD countries and 10 middle-income countries with long-term data on stock returns and macro aggregates (since at least the early 1930s). 13

14 with stock market crashes is 71%. Figure 2 shows the 3D histogram of matched stock market crashes and macro contractions. Based on 349 annual observations for matched events out of the sample of 3037 annual observations, the calibrated probability of transitioning from the normal state to a disaster that features both a market crash and a depression equals 71/( ) = 2.64%. The corresponding probability of a stock market crash and no contraction is 7.47% and the probability of a macro contraction and no stock market crash is 1.15%, as shown in Table 5. The joint distribution of stock returns and permanent income shocks during disasters D is calibrated to match the empirical distribution of observed stock market crashes and macroeconomic contractions from the Barro and Ursua (2009) dataset. In the matched market crash and macro contraction sample, the average stock return is -0.53, with an average duration of 3.8 years, and the average macroeconomic contraction size is 0.23, with an average duration of 4.1 years. In the 29 cases of depressions that were not associated with stock-market crashes, the average stock return is -0.09, with an average duration of 1.7 years, and the average macroeconomic contraction size is 0.17, with an average duration of 3.8 years. Among the 161 cases of stock-market crashes not associated with depressions, the average stock return is -0.43, with an average duration of 2.9 years, and the average contraction size is 0.01, with an average duration of 1.5 years. Matched events thus tend to last longer and feature larger negative shocks to stock returns and income. A salient feature of the disaster data that is not explicitly captured by the present model is the duration of the declines. This is equivalent to assuming that the important aspect of a disaster is the cumulative amount of the contraction, rather than whether the decline occurs in a single time period (as in the model) or is more realistically spread out over a number of years. Barro (2006) examines the sensitivity of the implied equity premium to varying the duration of crises between 0 and 5 years and finds that this does not impact significantly on the implied equity premium relative to the benchmark analysis which constrains the cumulative contractions to a single time period. 5. Benchmark results In order to highlight the novel implications of the life-cycle model with disasters, I also solve the household s problem when the probability of a market crash (π MktCras h ) and the 14

15 probability of a macroeconomic contraction (π MacroContr ) are both set to zero. I call this the model without disasters. 5.1 Policy functions Optimal portfolio rule In the complete-markets setting with no labor income, the optimal portfolio rule for an investor with CRRA utility who faces a constant investment opportunity set is constant, independent of wealth and age. For the benchmark parameters of risk aversion and the moments of the risky asset s excess return, the optimal equity share would be: ω complete markets = μ e γς e 2 = % (19) However, in a more realistic setting where risky labor income cannot be capitalized due to moral hazard, the portfolio rule is a function of financial wealth and age. Figure 3a plots the portfolio share invested in stocks in the model without disasters. The policy function is decreasing in both age and financial wealth. This is a general feature of life-cycle models with homothetic utility and non-tradable labor income (Bodie, Merton, and Samuelson (1992), Heaton and Lucas (2000b), Cocco, Gomes and Maenhout (2005)) 10, driven by the bond-like qualities of human capital. Because labor income is relatively stable and has zero correlation with stock returns in the model without disasters, human capital (the present discounted value of future labor income) acts as a substitute for the riskless asset. The portfolio share in stocks falls in age because younger households have greater incentive to diversify their large endowment of non-tradable human capital. At a fixed level of financial wealth, as the investor ages, the amount of future income (and of the risk-free asset holdings implicit in it) decreases, and therefore the agent holds a larger proportion of her financial portfolio in the riskless asset. Holding age constant, the portfolio rule without disasters is decreasing in financial wealth. An investor with little financial wealth tilts his portfolio relatively more towards stocks than the investor with a lot of financial wealth, because the poorer investor already has a relatively larger risk-free asset position out of total wealth (financial wealth and human capital) 10 See Wachter and Yogo (2009) for an interesting life-cycle model in which households have nonhomothetic utility over two types of consumption goods, basic and luxury. The nonhomethetic preference model predicts that the portfolio share rises in wealth, because households with higher permanent income are less risk averse, and allocate a higher share of their wealth to stocks. 15

16 due to his future income. In the limit, for very high end-of-period assets, the future income plays very little role and the investor s decision converges to the complete markets solution, given by equation (19). Figure 3b shows the optimal portfolio policy, as a function of normalized financial wealth, when disasters are possible. There is a pronounced decrease in the portfolio share in stocks relative to the model without disasters, especially prior to retirement. Both labor income and stock returns are more risky when there are disasters. Furthermore market crashes and macroeconomic contractions are positively correlated, meaning the risky labor income is now relatively more stock-like than in the case without disasters. As a result the household shifts its portfolio away from equity, both at a given age, and for a given level of wealth, to reduce its exposure. As can be seen in Figure 3b, there is an abrupt change in the optimal portfolio rule at retirement. At a given level of wealth, the portfolio share in stocks goes up and then gradually decreases with age. What changes at this point in the life-cycle is the assumption about labor income. Post-retirement labor income is constant and has no correlation with stock returns. The investor is still vulnerable to stock market crashes after retirement, but macroeconomic contractions no longer threaten labor income. Human capital maintains its bond-like qualities for the retired investor even in the model with disasters Optimal consumption rule Because the optimal portfolio share in stocks depends on wealth, the consumptionsavings decision determines where the portfolio rules are evaluated. Figure 4a shows the optimal consumption policy in the model without disasters as a function of normalized cash-on-hand and tracks how it evolves at different points in the lifecycle. In accord with the analytical results of Carroll and Kimball (1996), the consumption functions at any particular age are monotonically increasing and concave in cash-on-hand. The liquidity-constrained young household (age 20) consumes relatively more out of normalized resources than the middle-age household (age 40) which is actively saving for retirement. At retirement, the household faces a sharp drop in labor income (the replacement ratio λ is 0.75) which decreases resources available for consumption. Retirees draw down their savings instead of adding to them, which accounts for the rise in the consumption profile after retirement. At the end of life, in the absence of a bequest motive, the household optimally consumes all available resources. Figure 4b shows the optimal consumption 16

17 rule when disasters are possible. It is qualitatively very similar to the consumption rule without disasters. 5.2 Simulation Results Using the policy functions described in the previous section, I simulate the consumption and asset allocation profiles of 10,000 investors over the life-cycle. The households are ex-ante identical and have zero initial wealth, but are subject to idiosyncratic realizations of stock returns and labor shocks. I draw an initial level of permanent income for each household from a lognormal distribution, based on the estimates from the PSID 11. I repeat the same simulation exercise for the life-cycle model without disasters Implications for portfolio choice Table 6 reports the share of risky asset holdings implied by the two models. Panel A shows the mean, the 5 th, 50 th and 95 th percentiles of the cross-sectional distribution of stockholdings without disasters, tabulated by age groups. Panel B shows the analogous statistics for the model with disasters. For example, the mean share allocated to equity by investors in their 50s is 91% in the model without disasters and 29% in the model with disasters. Under the benchmark parameters without disasters, the average portfolio is aggressively invested in stocks while the household is young and middle-aged (see also Figure 5a which plots the mean, 5 th and 95 th percentiles of stockholdings for the model without disasters). Even after retirement, the share of the risky asset is very high (about 80%). The portfolio result is determined by the high historic equity premium and the zero correlation of labor income and stock returns. In contrast, when there are disasters, the average investor carries substantially less stocks in his portfolio. Figure 5b plots the mean, 5 th and 95 th percentiles of the cross-sectional distribution of stockholdings. There is a clear downward pattern of stockholding over the working life of the household. For example, the average investor holds 94% of financial wealth in equity in his 20s, 52% in his 30s, and 34% in his 40s. Just prior to retirement, the average portfolio is down to only 27% in stocks. 11 The mean of P 1 is normalized to one and the standard deviation of its logarithm is 0.09 for the No College benchmark group. 17

18 Intuitively, experiencing a disastrous labor shock in mid-life would sentence the household to diminished economic well-being for the rest of its life, because labor is persistent and the defined benefit pension depends on the level of permanent labor income prevailing just prior to retirement. The effect of a large income contraction would be even greater if it was coupled with a ruined financial portfolio as a result of a concurrent stock market crash. The damage to the household s nest egg becomes irreversible once the household is past its prime earning years. Financial wealth reaches its maximum when the household is in its early 60s, making it most vulnerable to a market crash at that point. Not surprisingly, the model with disasters predicts the average household should invest more cautiously in its 50s and early 60s. In contrast, the younger household is more flexible, in the sense that it would have time to catch up even if it experiences a negative labor shock early in life. The model with disasters predicts a relatively larger portfolio share in the beginning of adult life, which is not observed empirically. The literature has proposed other compelling explanations for why young households stay out of the stock market. For example, housing or fixed costs can crowd out stocks from the household s portfolio early in life (Cocco, 2005; Hu, 2005; Yao and Zhang, 2005). These explanations are beyond the scope of the present model, but are plausible determinants for the observed behavior of young households. After retirement, average stockholdings in the model with disasters rise, although they remain substantially lower than in the model without disasters. On the one hand, given the high historic equity premium, longevity risk creates an incentive to hold stocks. The household needs to ensure it has enough resources to last it until the end of its life. On the other hand, the increased riskiness of equity when there is a non-zero probability of a stock market crash crowds out stockholdings. Figure 5b demonstrates that the incentive to hold stocks prevails: the average risky share jumps up to 47% at retirement and increases gradually thereafter. The explanation for this puzzling result is that labor income in retirement is risk-free. The household effectively holds a buffer in the form of a steady defined-benefit pension, uncorrelated with stock returns. The higher the defined-benefit replacement ratio λ is, the higher the portfolio share after retirement. In addition, the more the household runs down its accumulated financial wealth, the larger portion of its total portfolio (financial wealth and human capital) becomes implicitly risk-free. As a result, the optimal portfolio choice, which is declining in wealth, dictates a shift toward equity. 18

19 In conclusion, allowing for rare disasters decreases the portfolio share in stocks substantially. The reduction is most pronounced for middle-aged households who are actively saving for retirement and are thus more vulnerable to disasters. The average middle-aged household allocates only 30% to 35% of its portfolio to risky assets, matching very closely the data for median stockholdings as a percentage of net worth from the 2007 Survey of Consumer Finances Implications for consumption and wealth accumulation Figure 6a shows the mean profiles of financial wealth, consumption and labor income without disasters. Figure 6b plots the mean profiles for the simulation with disasters. In both graphs, wealth and consumption are hump-shaped because of borrowing constraints in early adulthood, saving for retirement in midlife, and de-accumulation in the last phase of life. The young household is liquidity constrained and only saves a small portion of wealth as buffer stock to insulate consumption from negative shocks to labor income. In midlife, saving for retirement becomes a crucial determinant of behavior. Starting at about age 40-45, the household is able to boost saving due to the upward-sloping income profile which relaxes the borrowing constraint. Wealth accumulation increases significantly and peaks when the investor is in his early 60s. After retirement, the investor begins using up his accumulated financial resources to smooth consumption. Towards the very end of the life-cycle, consumption decreases slightly due to the higher effective discount rate from the rising mortality risk. Comparing the profiles on Figures 6a and 6b reveals that wealth accumulation with disasters is considerably lower than without disasters. This does not mean that the consumer lacks a precautionary motive to save more when there are disasters. Rather he invests considerably less in the stock market, foregoing the high average equity premium. How much wealth is accumulated depends more on the compound return to the financial portfolio than on the amount of savings that a borrowing-constrained household can set aside. In other words, the lower wealth profile is the price the consumer pays for being more conservative in his investment when there are disasters. To get a sense of the magnitude of this effect, I calculate an annual wealth loss metric, WealthLoss Dis, which equals the difference between the two profiles as a percentage of the profile without disasters, discounted to the beginning of adult life using the effective discount rate for the consumer: 19

20 t 1 WealthLoss Dis = β t 1 p k T t=1 k=0 E A t Dis E A t NoDis E A t NoDis /T=21% The possibility of rare disasters decreases wealth by 21% per year on average. To assess how disasters affect the consumer s welfare, I use standard consumptionequivalent variation. Specifically, I compute the constant consumption stream that would make the household as well-off in terms of expected utility as the average consumption stream with disasters: C Dis = (1 γ)v 1 Dis T t=1 β t 1 t 1 k=0 p k 1/(1 γ) where V 1 Dis is the expected value of the maximum discounted lifetime utility for an investor who faces the risk of disaster, given by: T V 1 Dis = E 1 β t 1 p k t=1 t 1 k=0 C t 1 γ 1 γ Then I calculate the constant consumption stream, C NoDis, that would make the household as well-off in terms of expected utility as the average consumption stream without disasters. The utility cost of disasters, UtilCost Dis, is the percentage loss in equivalent consumption from having to curtail consumption: UtilCost Dis = C Dis C NoDis CNoDis = 5.9% Put differently, the consumer would be willing to forego about 5.9% of consumption each year to eliminate the possibility of disasters. 5.3 What drives life-cycle portfolio choice with disasters: the overall increase in stock market and labor risk or the correlation between them? Disasters alter life-cycle portfolio allocation for two reasons: 1) the overall increase in the riskiness of stocks and permanent labor income and 2) the positive correlation between them during crises. Both crowd out stockholdings, but it is not clear what their relative magnitudes are. To answer this question, I shut down the correlation effect and simulate the model under the assumption that stock market returns and shocks to permanent income are independent. I call this the model with independent disasters.., (20) (21) (22) (23) 20

21 I set the probability of a disastrous equity shock equal to the unconditional probability of a stock market crash (10.11%) and the probability of a disastrous permanent income shock equal to the unconditional probability of contraction (3.79%). Under independence, these marginal probabilities imply that the probability of both disasters happening simultaneously is only 0.38%, compared to the 2.64% probability of matched events used previously. Panel C of Table 6 reports the mean and summary percentiles of the cross-sectional distribution of stockholdings for independent disasters. On average, households allocate 94% to stocks in their 30s, 72% in their 40s, and 55% in their 50s. Prior to retirement, the average portfolio is down to 47% in stocks. Shutting down the correlation between stock returns and labor risk lessens the incentive to invest in the risk-free asset. Figure 7 plots the mean portfolio share with independent disasters. During the working life, it lies roughly midway between the profile with dependent disasters and the profile without disasters. The correlation thus accounts for about half of the decrease in the optimal portfolio share before retirement. After retirement, the profiles with independent disasters and with dependent disasters are very close because labor income is no longer risky. The small disparity comes from the initial conditions: given different portfolio allocations over the working life, households will have different nest eggs at retirement. With independent disasters, the average household has more equity in its portfolio and is wealthier at retirement, so it allocates slightly more to the riskless asset thereafter. Figure 8c plots the mean consumption, wealth and labor profiles with independent disasters. The decrease in wealth relative to the model without disasters amounts to 15% per year when stock market crashes and macroeconomic contractions are not correlated. In contrast, when the disasters are dependent, wealth accumulation decreases by 21% (see equation (20)). The utility cost of independent disasters is also smaller. Evaluated at the mean, independent disasters cause a 1.7% decrease in the constant consumption stream that would make the household as well-off in terms of expected utility. When the disasters are correlated, the drop in the constant consumption stream is 5.9% (see equation (23)). In other words, about 71% of the utility cost of disasters could be attributed to the dependence between stock market returns and labor income during disasters. 21

22 5.4 The potential for disasters vs. realized disasters The final piece for understanding the stockholding puzzle in light of the possibility for rare disasters comes from the fact that the more substantial shift towards the risk-free asset happens not as a result of realized disasters, but in response to the potential for disasters. To quantify this effect, I calculate the average portfolio allocation over the life-cycle for the simulated households who actually experienced a disaster and compare it to the one for the remaining households, who only faced the threat of a disaster. Figure 9 plots the average life-cycle profiles with realized disasters and with potential disasters. The households who did not experience a disaster, but who solved their optimization problem anticipating one, allocate their portfolios very similarly to the ones who really experienced a disaster. The sum of the absolute differences in the two portfolio allocation profiles amounts to only 0.9% over the entire life-cycle. Thus, as long as an investor subjectively perceives a large enough probability of disaster, he might stay out from the stock market completely, even if in reality disasters are not that common. This naturally leads to the question How large should the perceived probability of disaster be for the median household to hold no stocks? Figure 10 plots the portfolio share profiles for different perceived probabilities of disaster, holding the expected equity premium constant. A 6% perceived probability of disaster is sufficient to induce the median household to stay completely out of the stock market for the majority of its working life. Table 7 reports the median portfolio share for different values of the perceived probability of disaster, tabulated by age groups. Small differences in the perceived probability of disaster lead to very different asset allocations prior to retirement. Focusing on the household between 50 and 59 when the amount of accumulated financial wealth is greatest, the median household allocates 94% of its portfolio to the risky asset when it assumes 0% probability of disasters. With 1% perceived probability, stockholdings fall to 47%. With the benchmark probability of coincident disasters at 2.64%, the portfolio share is only 29%. At 5% it goes down to 11%. Thus, differences in the perceived probability of disaster can explain heterogeneity as well as stock market non-participation. 22

23 6. Conclusion In this paper, I present a life-cycle model of consumption and portfolio choice with realistically calibrated uninsurable labor income and disaster risk that provides an explanation for a key feature of asset allocation data: moderate equity holdings for stock market participants over the working life. In the absence of disasters, even though labor income is risky, the optimal portfolio rules indicate that labor income serves as a substitute for the risk-free asset and the investor should be fully invested in equities. An empirically calibrated small probability of a disastrous stock market crash, likely to be accompanied by a bad labor income draw, substantially decreases the average allocation to equities, and therefore seems to be an important factor for explaining the data. The paper decomposes the disaster effect in two components: the overall increase in the riskiness of labor income and stock returns, and the higher correlation between them during crises. About half of the decrease in stockholdings can be attributed to the correlation between stock market crashes and depressions. Given this lower allocation to stocks, the household builds a smaller nest egg because it foregoes the high average equity premium. The increase in riskiness accounts for a 15% loss in wealth per annum relative to the model without disasters, and the correlation component is responsible for a 6% additional decrease. As a result of the lower wealth accumulation and the stronger precautionary saving motive, consumption is curtailed. Disasters generate high utility costs: 1.7% when stock market crashes and depressions are independent, and 5.9% when they are correlated. Because the change in optimal behavior is driven by the potential for disaster, rather than conditional on a disaster actually taking place, the model can also explain low stock market participation rates in the population as a whole. Furthermore, if households have even slightly different perceived probabilities of disaster, the model can also account for the heterogeneity in asset allocations observed in the data. 23

24 Table 1 Benchmark Parameters Parameter Symbol Benchmark value Preferences Subjective discount factor β 0.98 Coefficient of relative risk aversion γ 5 Labor income Standard deviation of shocks to permanent income ς ψ 0.09 Standard deviation of shocks to transitory income ς θ 0.18 Correlation between shocks to permanent income and stock ρ Nor 0 returns in the normal state Replacement ratio λ 0.75 Asset returns Risk-free rate R f 1 2% Equity premium R e R f 6% Standard deviation of stock returns ς e 18% Disaster Unconditional probability of stock market crash π MktCras h 10.1 % Unconditional probability of macroeconomic contraction π MacroContr 3.8% Joint probability of stock market crash and macroeconomic contraction π BothDis 2.6% The table reports parameters used in the benchmark calibration of the life-cycle consumption and portfolio-choice model with stochastic labor income and disasters. The household has power utility over a composite consumption good. I calibrate the probability of disaster (a stock market crash, a macroeconomic contraction or both) to match the ratio of the number of disasters to the number of normal years in the Barro and Ursua (2009) dataset. I calibrate the joint distribution of stock returns and permanent income shocks during disasters to match the empirical distribution of stock market crashes and macroeconomic contractions from the Barro and Ursua (2009) dataset. I solve and simulate the models with and without disasters at an annual frequency. 24

25 Table 2 Labor Income Profiles: Fixed Effects Regression Dependent variable: No College College ln(real income) Coefficient p-value Coefficient p-value Age Age²/ Age³/ Family size Married Single Widowed Divorced Constant Family fixed effects? Yes Yes Adjusted R² n 70,329 19,545 Replacement rate (λ) The data are from the Panel Study of Income Dynamics from 1977 to Age is defined as the age of the head of household. Married, single, widowed and divorced are indicator variables for marital status of the head. The omitted category is separated. Family size is the number of members of the household. P-values are calculated using heteroscedasticity and autocorrelation consistent errors, clustered by household. 25

26 Table 3 Standard deviation of the permanent and transitory labor income components No College College Permanent shock, ς Ψ Transitory shock, ς Θ The data are from the Panel Study of Income Dynamics from 1977 to The variance of the residual from the regression presented in Table 2 is decomposed into its permanent and temporary components based on the method of Carroll and Samwick (1997) (details in Appendix B). All estimates are statistically significant at 5%, using heteroscedasticity robust standard errors. Table 4 Correlation between the aggregate component of permanent labor income and excess stock returns: normal state No College College Coefficient p-value Coefficient p-value Correlation The correlation between the innovations to excess stock returns and the permanent labor income shocks in normal times is calibrated using data on labor income from PSID and returns on the value-weighted NYSE index from CRSP between 1977 and The correlation coefficient is obtained by regressing Δln(Y t ) on demeaned excess returns (details in Appendix B). P-values are calculated using heteroscedasticity robust standard errors. 26

27 Table 5 Summary Statistics of Disaster Data Total number of observations 3037 years Stock market crashes 232 events Avg. duration stock-market crash 3.2 years P(market crash) 10.11% Macro contractions 100 events Avg. duration macro contraction 4 years P(macro contraction) 3.79% Matched market crash and macro contraction 71 events Avg. duration of matched events 4.9 years P(matched market crash and macro contraction) 2.64% Stock market crashes featuring no depression 161 events Avg. stock return Avg. duration of stock market crash 2.9 years Avg. macro contraction 0.01 Avg. duration of macro contraction 1.5 years P(crash and no contraction) 7.47% Depressions featuring no stock market crash 29 events Avg. stock return Avg. duration of stock market crash 1.7 years Avg. macro contraction 0.17 Avg. duration of macro contraction 3.8 years P(contraction and no crash) 1.15% Matched market crashes and macro contractions 71 events Avg. stock return Avg. duration of stock market crash 3.9 years Avg. macro contraction 0.23 Avg. duration of macro contraction 5.2 years P(crash and contraction) 2.64% 27

28 Table 6 Simulated portfolio share in stocks over the life-cycle Panel A: No disasters Mean Percentiles of stockholdings distribution Age 5th 50th 95th % 100% 100% 100% % 100% 100% 100% % 91% 100% 100% % 71% 94% 100% % 57% 83% 100% % 53% 76% 100% % 51% 78% 100% % 49% 75% 100% Panel B: Disasters Mean Percentiles of stockholdings distribution Age 5th 50th 95th % 81% 96% 100% % 37% 49% 77% % 28% 33% 43% % 24% 29% 35% % 30% 38% 52% % 32% 45% 76% % 30% 47% 96% % 30% 57% 100% Panel C: Independent Disasters (no correlation between stock market crashes and labor income contractions) Mean Percentiles of stockholdings distribution Age 5th 50th 95th % 100% 100% 100% % 75% 98% 100% % 44% 70% 100% % 36% 52% 94% % 31% 43% 77% % 30% 42% 83% % 28% 45% 98% % 27% 52% 100% 28

29 Table 7 Effect of different perceived probabilities of a coincident disasters on median portfolio allocation by age group Perceived probability of coincident disaster Age 0% 1% 2.64% 5% 6% 7% % 100% 96% 20% 20% 20% % 93% 49% 4% 0% 0% % 61% 33% 10% 3% 0% % 47% 29% 11% 5% 2% % 42% 38% 33% 31% 30% % 43% 45% 47% 47% 47% % 45% 47% 49% 50% 49% % 54% 57% 63% 64% 63% 29

30 Thousands of 2005 US $ 80 Figure 1: Age profiles by education group College (age dummies) No college (age dummies) Age College (3rd degree polynomial) No college (3rd degree polynomial) Labor income processes estimated from PSID for households without a college degree (benchmark group), and college graduates. For each group, the figure plots estimated age dummies and a fitted 3rd degree polynomial. The histogram above depicts the joint distribution of stock market crashes and macroeconomic contractions from long-term data up to 2006 on 30 countries. Source: Barro and Ursua (2009). 30