The Cross-Section of Household Preferences

Transcription

1 The Cross-Section of Household Preferences Laurent E. Calvet, John Y. Campbell, FranciscoJ.Gomes,andPaoloSodini 1 July 2017 Preliminary: not for citation 1 Calvet: Department of Finance, EDHEC Business School, 393 Promenade des Anglais, BP 3116, Nice Cedex 3, France, and CEPR; laurent.calvet@edhec.edu. Campbell: Department of Economics, Harvard University, Littauer Center, Cambridge MA 02138, USA, and NBER; john_campbell@harvard.edu. Gomes: London Business School, Regent s Park, London NW1 4SA, UK; fgomes@london.edu. Sodini: Department of Finance, Stockholm School of Economics, Sveavagen 65, P.O. Box 6501, SE Stockholm, Sweden; Paolo.Sodini@hhs.se. We acknowledge hepful comments from Stijn van Nieuwerburgh and seminar participants at ENSAE-CREST, the University of Michigan, and the 2017 Meeting of the American Economic Association. We thank the Sloan Foundation for financial support to John Campbell, and Nikolay Antonov, Zihao Liu, Filipe Pires de Albuquerque, Todor Sapunarov, and Yapei Zhang for able and dedicated research assistance.

2 Abstract This paper estimates the cross-sectional distribution of preferences in a large administrative panel of Swedish households. We consider a life-cycle portfolio choice model, which incorporates risky labor income, safe and risky financial assets inside and outside retirement accounts, and real estate. We study middle-aged households grouped by education, industry of employment, and birth cohort as well as by their accumulated wealth and risky portfolio shares. Our model allows for heterogeneity in risk aversion, the elasticity of intertemporal substitution (EIS), and the rate of time preference. The average values of these parameters are reasonable at 5.8, 1.2, and 4.1% respectively but there is a great deal of cross-sectional variation around these averages, particularly in the EIS and the rate of time preference. Key cross-sectional patterns are negative correlations between initial wealth and both time preference and risk aversion, a negative correlation between time preference and risk aversion, a positive correlation between time preference and the EIS, a weaker negative correlation between risk aversion and the EIS, and a negative correlation between income risk and risk aversion.

3 1 Introduction When households make financial decisions, are their preferences toward time and risk substantially similar, or do they vary cross-sectionally? And if preferences are heterogeneous, how do preference parameters covary with one another and with household attributes such as education and sector of employment? This paper answers these questions using a life-cycle model of saving and portfolio choice fit to high-quality household-level administrative data from Sweden. Modern financial theory distinguishes at least three parameters that govern savings behavior and financial decisions: the rate of time preference, the coeffi cient of relative risk aversion, and the elasticity of intertemporal substitution (EIS). The canonical model of Epstein and Zin (1989, 1991) makes all three parameters constant and invariant to wealth for a given household, while breaking the reciprocal relation between relative risk aversion and the elasticity of intertemporal substitution implied by the older power utility model. We structurally estimate these three preference parameters in the cross-section of Swedish households by embedding Epstein-Zin preferences in a life-cycle model of optimal consumption and portfolio choice decisions in the presence of uninsurable labor income risk and borrowing constraints. To mitigate the effects of idiosyncratic events not captured by the model we carry out our estimation on groups of households who share certain observable features. We first sort households into 468 groups by 3 levels of education, 12 sectors of employment, and 13 birth cohorts. To capture heterogeneity in preferences that is unrelated to these characteristics we further divide each such group by quartiles of initial wealth accumulation in relation to income and by terciles of the initial risky portfolio share, giving us a sample of 5547 composite households that have data available in each year of our sample from 1999 to We allow households age-income profiles to vary with education, and the determinants of income risk (the variances of permanent and transitory income shocks) to vary with both education and the household s sector of employment. These assumptions are standard in the life-cycle literature (Carroll and Samwick 1997, Cocco, Gomes, and Maenhout 2005), It is well known that life-cycle models are much better at jointly matching portfolio allocations and wealth accumulation at mid-life than at younger ages or after retirement. Therefore we estimate the preference parameters by matching the profiles of wealth and portfolio choice between ages 40 and 60. We confine attention to the sub-sample of stockholders, to avoid the need to estimate determinants of non-participation in risky financial markets, and include both housing equity and defined-contribution retirement wealth as components of a composite risky asset. It is a challenging task to identify all three Epstein-Zin preference parameters. In principle, these parameters play different roles with the rate of time preference affecting only the overall slope of the household s planned consumption path, risk aversion governing the 1

4 willingness to hold risky financial assets and the strength of the precautionary savings motive, and the EIS affecting both the overall slope of the planned consumption path and the responsiveness of this slope to changes in background risks and investment opportunities. We observe portfolio choice directly, and the slope of the planned consumption path indirectly through its relation with saving and hence wealth accumulation. However, we require variation in background risks and/or investment opportunities in order to identify the EIS separately from the rate of time preference (Kocherlakota 1990, Svensson 1989). Our model assumes that expected returns on safe and risky assets are constant over time, so we cannot exploit time-variation in the riskless interest rate or the expected risky return to identify the EIS in the manner of Hall (1988). However, our model incorporates variation in background risks. As households approach retirement their human capital diminishes relative to their financial wealth, and this alters their desired portfolio composition and hence the rate of return on the portfolio. A secondary effect is that as households age their mortality rates increase, and this alters the incentive to save or equivalently the effective rate of time discounting. These changes alter wealth accumulation in a way that is mediated by the EIS. Accordingly we are able to identify the EIS from time-variation in the rate of wealth accumulation within each household group. This identification strategy that exploits accelerating or decelerating changes in wealth is a methodological contribution of our paper. Our main empirical findings are as follows. First, we estimate reasonable average levels of each Epstein-Zin preference parameter. Average risk aversion is 5.8, the average EIS is 1.2, and the average rate of time preference is 4.1%. The average level of risk aversion is moderate in part because we treat real estate as a risky investment rather than ignoring it or treating it as a safe asset. The average EIS is close to one but far above the reciprocal of average risk aversion, contrary to the restriction of the power utility model. Second, we estimate considerable heterogeneity in preference parameters across the Swedish population. The cross-sectional standard deviations are 1.3 for risk aversion, 0.8 for the EIS, and 4.7% for the rate of time preference. There is a debate in the asset pricing literature about whether the EIS is less than one, as estimated by Hall (1988), Yogo (2004) and others in time-series data, or greater than one, as assumed by Bansal and Yaron (2004) and a subsequent literature on long-run risk models. We find that EIS is less than one for 42% of households, and even less than the reciprocal of risk aversion for 20% of households, while it is greater than one for 58% of households. This much cross-sectional variation suggests that aggregate results are likely to be sensitive to the way in which households are aggregated and are unlikely to be precise, consistent with large standard errors reported by Calvet and Czellar (2015) in a structural estimation exercise using aggregate data. Third, there are interesting patterns of cross-sectional correlation in preference parameters. Our estimates of the rate of time preference have a negative correlation of about -0.5 with estimated risk aversion and a positive correlation of 0.35 with the estimated EIS, consistent with the view that more patient households are also more risk tolerant and more willing to exploit opportunities for intertemporal substitution. We estimate that risk aver- 2

5 sion is negatively correlated with the EIS, but the correlation is much weaker than would be implied by the power utility model in which one parameter is the reciprocal of the other. Fourth, the initial wealth-income ratio of each group is strongly negatively correlated with the coeffi cient of relative risk aversion (at -0.84) and the rate of time preference (-0.30) that we estimate for the group. The intuition for these results is that initially wealthier cohorts would invest more conservatively than poorer cohorts if they had the same risk aversion, and would accumulate wealth more slowly than poorer cohorts if they had the same rate of time preference; but the first pattern is absent in the data, and the second is not strong enough in the data to be consistent with constant preferences across groups. Fifth, we find that riskier labor income is associated with lower risk aversion across household groups. The reason for this finding is that households with riskier income do not invest as conservatively as they would do if they understood their income risk and had the same risk aversion as households in safer occupations. This pattern is consistent with the hypothesis that risk-tolerant households self-select into risky occupations, but could also result from households failure to understand the investment implications of their income risk exposure. The organization of the paper is as follows. Section 2 reports empirical measures of wealth and the risky share across household groups, and explains how to compute wealth returns. Section 3 discusses the life-cycle model and household labor income processes. Section 4 develops the estimation methodology. Section 5 reports empirical results on the cross-section of household preferences, and section 6 concludes. An appendix discusses some details of the methodology. 2 Measuring Wealth, Risky Share, and Returns In this section, we describe the Swedish household panel, define the key components of wealth, and report the cross-section of household wealth-to-income ratios and risky shares. 2.1 Data Our empirical analysis is based on the Swedish Wealth and Income Registry, a high-quality administrative panel of Swedish households that has been used in earlier research. 2 We define a household as a family living together with the same adult(s) over time. We define the household head as the adult with the highest average income, or, if the average income is the same, the oldest, or, if the other criteria fail, the man in the household. For every 2 See, for instance, Bach, Calvet, and Sodini (2015), Betermier, Calvet, and Sodini (2017), Calvet, Campbell and Sodini (2007, 2009a, 2000b), and Calvet and Sodini (2014). 3

6 household, we observe disaggregated wealth and income over the 1999 to 2007 period. To simplify the analysis, we study households with a head aged 40 to 60. The motivation is that life-cycle models of the type introduced in Section 3 have diffi culties capturing the wealth accumulation of young households and retirees in a parsimonious setting. Understanding the behavior of the young requires the researcher to take into account transfers from relatives, housing purchases, investment in education, and changes in family size. Similarly, the finances of older households are driven by a number of specific factors, such as health shocks and bequest motives. To avoid these complex issues and the identification issues that they pose, we restrict our attention to middle-aged households. We impose several filters on the household panel. We exclude household-year observations in which the head is a student. In each year, we consider households that have at least 3000 Swedish kronor in financial wealth or at least 1000 kronor in non-financial real disposable income. We focus on households that participate in risky asset markets, that is that either hold risky funds or own stocks directly. The resulting panel contains 4.9 million householdyear observations. Recall that our analysis is based on 5547 groups of households sorted by birth cohort, education level, and sector of employment, and then further subdivided by terciles of the risky share and quartiles of the wealth-income ratio. The average group contains just under 150 households, but the distribution of group size is right-skewed, with many small groups and a few much larger ones. The group-level statistics we report in the paper are all sizeweighted in order to reflect the underlying distributions of data and preference parameters at the household level. 2.2 Components of Wealth We consider three broad forms of wealth: liquid financial wealth, real estate equity, and illiquid retirement savings. We define total liquid financial wealth as the market value of holdings in bank accounts, Swedish money market funds, mutual funds, stocks, capital insurance products, derivatives and directly held bonds. Mutual funds include balanced funds and bond funds, as well as equity funds. Our definition of liquid financial wealth excludes durables and definedcontribution (DC) retirement accounts. We define riskless assets as the sum of holdings in bank accounts and Swedish money market funds. Other liquid financial holdings are viewed as risky financial wealth. Real estate consists of primary and secondary residences, rental, commercial and industrial properties, agricultural properties and forestry. The Wealth and Income Registry provides the holdings at the level of each property. The pricing of these properties follows the methodology in Bach, Calvet, and Sodini (2015). We obtain real estate equity by subtracting 4

7 household debt from the gross value of the real estate portfolio, and winsorizing below at zero. Illiquid retirement savings result from contributions to DC retirement accounts and the investment returns on these contributions. We impute contributions using a detailed reconstruction of the administrative rules governing DC retirement plans for different groups of households, reported in the online appendix, and we impute investment returns by assuming that all DC retirement wealth is invested in a global equity index. These imputations provide the pre-tax retirement wealth of every household. Let τ denote the tax rate on withdrawals. The after-tax retirement wealth of each household is obtained by multiplying pre-tax retirement wealth by 1 τ. This definition takes into account that only the after-tax value can be used for consumption. In the remainder of the paper, retirement wealth will always refer to after-tax amounts. The value of total net wealth is defined as W h,t = LW h,t + RE h,t + RW h,t, where LW h,t, RE h,t, and RW h,t, and denote, respectively, the value of the liquid financial wealth, real estate equity, and (after-tax) retirement wealth held by the household. The household s risky portfolio share, α h,t, is the share of risky wealth in total net wealth. As discussed above, we can separate liquid financial wealth into safe and risky components, and in this version of the paper we assume that both real estate equity and retirement wealth are fully invested in risky assets. This enables us to calculate α h,t as one minus the ratio of safe liquid financial wealth to total wealth. 2.3 The Cross-Section of the Household Wealth-to-Income Ratio and the Risky Share We now consider the cross-section of the wealth-to-income ratio and risky share at the start of our sample period in The life-cycle model of Section 3 will condition on the initial wealth-income ratio and will seek to explain the cross-sectional variation of the risky share, as well as the evolution of the wealth-income ratio over our sample period. The top panel of Table 1 shows the variation in average risky portfolio shares and wealthincome ratios across groups with each level of education and sector of employment, averaging across cohorts and the subdivisions by risky share and wealth-income ratio. For the purpose of computing these summary statistics, households in each group are treated as a single composite household that owns all wealth and receives all income of the group, and groups are weighted by the number of households they contain. Average risky shares vary in a narrow range from 79% to 90%, while wealth-income ratios vary more widely from 2.6 to 4.5. Within each sector, risky portfolio shares are somewhat higher and wealth-income 5

8 ratios are considerably higher for more educated households, particularly those with higher education. There is also variation across sectors that we show below has some relation to income risk by sector. The averages in the top panel of Table 1 conceal a great deal of dispersion across disaggregated groups of households. This is shown by the bottom panel of Table I, which reports the standard deviations of the risky portfolio share and wealth-income ratio across all the groups with a given education level and working in a given sector. The standard deviations of the risky share vary from 13% to almost 30%, while the standard deviations of the wealth-income ratio are comparable in magnitude to the averages reported in the top panel. Across all 5547 groups, the initial risky share has a mean of 0.82 with a standard deviation of 0.15 while the initial wealth-income ratio has a mean of 3.3 with a standard deviation of 3.1. Figure 1 plots the cross-sectional distributions, which are left-skewed for the risky share and right-skewed for the wealth-income ratio, as must be the case given the upper bound of one on the risky share and the lower bound of zero on the wealth-income ratio. This cross-sectional variation in wealth and asset allocation immediately suggests that it will be diffi cult to account for Swedish household behavior without allowing for a great deal of cross-sectional variation in preferences. 3 Life-Cycle Model and Income Process In this section, we present the life-cycle model of saving and portfolio choice used to estimate household preferences. We also discuss the estimation of household labor income processes and the choice of calibration parameters. 3.1 Life-Cycle Model We consider a standard life-cycle model, very similar to the one in Cocco, Gomes and Maenhout (2005) Preferences Households have finite lives and Epstein-Zin utility over a single consumption good. The utility function V t is specified by the coeffi cient of relative risk aversion γ, the elasticity of intertemporal substitution ψ, and the time preference parameter δ. It satisfies the recursion V t = [ (1 δ)c 1 1/ψ t + δ ( ) E t p t,t+1 V 1 γ (1 1/ψ)/(1 γ) ] 1 1/ψ t+1, (1) 6

9 where p t,t+1 denotes the probability that a household is alive at age t+1 conditional on being alive at age t. Utility, consumption, and the preference parameters γ, ψ, and δ all vary across households but we suppress the household index h in equation (1) for notational simplicity. The age-specific probability of survival, p t,t+1, is obtained from Sweden s life table. Capturing the wealth accumulation of young households poses several problems for lifecycle models which do not include housing purchases, transfers from relatives, investments in education, or changes in family size. In addition it is well-known that such models predict an extremely high equity share at early ages which is hard to reconcile with our data. For this reason, we focus on the stage of the life-cycle during which households are accumulating retirement saving. We initialize the model at age 40 and endow households with the same initial wealth level as the one they actually have in the data. We follow the standard notational convention in life-cycle models and let age in the model, t, start at 1, so that t is effective age minus 39. Each period corresponds to one year and agents live for a maximum of T = 61 periods (corresponding to age 100). Matching the behavior of retirees is often hard for these models, particularly without introducing health shocks and bequest motives. For this reason in our estimation we only consider the model-implied behavior for ages 40 to 60 years. Our model includes no bequest motive, because it would be diffi cult to separately identify the discount factor and the bequest motive using our sample of households in the 40 to 60 age group, and we prefer not to add one more weakly identified parameter. Our estimates of the time discount factor can be viewed as having an upward bias due to the absence of a bequest motive in the model Budget Constraint, Financial Assets, and Labor Income Before retirement households supply labor inelastically. The stochastic process of the household labor income, L h,t, is described in Section 3.3. All households retire at age 65, as was typically the case in Sweden during our sample period, and we set retirement earnings equal to a constant replacement ratio of the last working-life permanent income. Consistent with the discussion in section 2, total wealth, W h,t, consists of the all the assets held by the household. For tractability, we assume in the model that total wealth is invested every period in a one-period riskless asset (bond) and a composite risky asset. Initial wealth W h,1 is calibrated from the data; it is set equal to the average wealth of a household with a 40-year-old head with similar characteristics as h. The household chooses the consumption level C h,t and risky portfolio share α h,t every period, subject to borrowing and short-sales constraints that imply 0 α h,t 1. Household wealth satisfies the budget constraint W h,t+1 = (R f + α h,t R e t+1)(w h,t + L h,t C h,t ), (2) where R e t+1 is the return on a risky asset in excess of the risk-free rate R f. This excess 7

10 return has a constant mean µ and a white-noise shock η t : where η t N(0, σ 2 η). R e t = µ + η t, (3) 3.2 Composite Risky Asset Definition Consistent with the discussion in section 2.2, total wealth W h,t is the sum of financial wealth, real estate equity, and retirement savings. The inclusion of real estate wealth in the model is consistent with common practice in life-cycle models (Hubbard, Skinner and Zeldes 1984, Castaneda, Diaz-Gimenez and Rios-Rull 2003, De Nardi 2004, Gomes and Michaelides 2005). We treat both real estate equity and retirement wealth as components of a single composite risky asset. The excess rate of return on the composite risky asset relative to the Swedish T-bill is given by Rt+1 e = (1 φ t ψ t )Rt+1 S + φ t Rt+1 RE + ψ t Rt+1 RW, (4) where φ t and ψ t denote, respectively, the shares of real estate equity and risky retirement wealth in aggregate risky wealth, and Rt+1, S Rt+1, RE Rt+1 RW, denote, respectively, the excess returns on the risky components of liquid financial wealth, real estate equity, and retirement wealth. We assume that the excess return on liquid financial wealth, Rt+1, S is equal to the return on a diversified stock market index, the MSCI world index, plus idiosyncratic noise: Rt+1 S = Rt+1+ε W I S t+1. The idiosyncratic noise is caused by imperfectly diversified investing, as in Calvet, Campbell, and Sodini (2007). The computation of the housing equity return, Rt+1, RE takes into account that households use mortgage debt to buy housing, implying that they have a levered position in residential real estate. We compute the excess return on this position, Rt+1, RE as R RE t+1 = RH t+1 λ t R M t+1 1 λ t, (5) where R H t+1 is the excess return on housing, R M t+1 is the mortgage spread (the cost of mortgage borrowing in excess of the riskless interest rate), and λ t is the loan-to-value (LTV) ratio. 3 We 3 This approach is similar to Gomes and Michaelides (2005), where housing is implicitly viewed as a weighted average of equity and the riskless asset. We explicitly state our assumptions about housing returns below. 8

11 assume that the return on housing is the sum of a housing index, R HI t+1, and an idiosyncratic component due to underdiversification. The return on real estate equity is therefore Rt+1 RE = RHI t+1 λ t Rt+1 M + ε RE 1 λ t t+1, where ε RE t+1 is the idiosyncratic component of real estate equity. We assume for simplicity that the rate of return on retirement wealth, Rt+1 RW, is equal to the world index return: Rt+1 RW = Rt+1. W I Thus, there is no idiosyncratic risk in retirement savings. This hypothesis captures that retirement savings are usually allocated to mutual funds are therefore much better diversified than other forms of wealth. The weight of risky retirement wealth, ψ t, is computed as explained in Section 2. In the model, we assume that the rates of return on liquid financial wealth and real estate are taxed at a flat rate of 25%. The rate of return on retirement wealth requires no adjustment: all tax adjustments are in the calculation of wealth itself. The above specification allows households to bear idiosyncratic risk, which is consistent with the empirical evidence in Calvet, Campbell, and Sodini (2007) for financial wealth and Bach, Calvet, and Sodini (2015) for other forms of wealth. The idiosyncratic return on the composite asset is (1 ψ t φ t )ε S t+1 + φ t ε RE t+1, where ε S t+1 denotes the idiosyncratic return on liquid financial wealth and ε RE t+1 the idiosyncratic return on real estate. Recall that we assume that there is no idiosyncratic risk on retirement wealth. If we assume that ε LW t+1 and ε RE t+1 are uncorrelated, the idiosyncratic variance of the composite asset is (1 ψ t φ t ) 2 V ar(ε S t+1) + φ 2 t V ar(ε RE t+1), (6) which we compute by following the methodology in Bach, Calvet, and Sodini (2015) Calibrated Parameters Table 2 Panel A reports the assumptions we make about financial asset returns. We assume that the real return on cash is constant at 1.60% (the average realized real cash return over the period). We proxy the expected excess return on equity by the mean return of the MSCI World Index in kronor in excess of the one month Swedish T-bill over the period (a longer sample period chosen to reduce noise in the estimated mean). We account for transaction costs by subtracting the average management fee on the equity funds held by the households in our calibration sample (1.42%), times the proportion of risky financial assets held in mutual funds (72%). This gives us a net-of-fee financial excess return of 3.69%. 9

12 We measure the volatility of financial asset returns, accounting for household portfolio underdiversification, as in the method of Calvet, Campbell, and Sodini (2007). That is, we estimate the variance-covariance matrix Σ of the excess return of all the stocks and funds held by Swedish households over the 1983 to 2007 period. We then use the panel data to calculate the vector of asset shares ω h,t in the risky financial portfolio of each household in the calibration sample. The product σ 2 h,t = ω h,t Σω h,t estimates the total variance of each household risky portfolio, and includes both systematic and idiosyncratic risk exposures. We estimate the standard deviation of household financial risky portfolios by the pooled cross-sectional average of σ h,t in our calibration sample, which is equal to 21.45%. The implied Sharpe ratio of the financial risky portfolio is 4.73/21.45 = 0.22 before fees and 3.69/21.45 = 0.17 after fees, considerably lower than the Sharpe ratio implied by costless investment in a global equity index. Table 2 Panel B reports our assumptions on real estate equity. We calculate the loan-tovalue ratio λ t at the household level as the ratio of total debt to residential real estate value. Since we can only observe λ t and the share φ t from 1999 to 2007, we set them equal to their pooled cross-sectional averages in our data: 46.1% for the leverage ratio and 47.07% for the share of housing equity. There is only modest cross-sectional variation in these ratios across groups, and we ignore this variation in our analysis. Table 2 Panel C reports the assumptions we make about excess returns on housing and total wealth. We calculate the spread R M t+1, which appears in equation (5), as the difference between the interest rate on newly issued Swedish mortgages and the yield on a Swedish one-month Treasury bill. The time series is available quarterly from 1996 and is based on a volume weighted average of the mortgage rates at all maturities. The resulting average spread is 1.53% in annual units. The measurement of the return on gross real estate wealth, Rt+1, H is challenging. The average yearly excess return on the Swedish index of one- or two-dwelling buildings is 7.16% for the 1983 to 2007 period. If we use this value for Rt+1 H it implies an even higher average return Rt+1 RE of 12% on levered real estate equity. It seems implausible that households expect such a high return on housing. As an alternative, we estimate the expected excess return on housing by assuming that its Sharpe ratio is equal to the Sharpe ratio of 0.17 we estimate for the stock market. We estimate the sample standard deviation of the excess return on housing, σ H, over the period as 9.99%. With a Sharpe ratio of 0.17, the implied average excess return on housing Rt+1 H is 1.70% and the implied average return Rt+1 RE on levered real estate is 1.84%. The implied average excess return on all risky assets Rt+1 e is 2.92%. In Table 2 panel D, we estimate the second moments of the components of risky wealth and the variance of the composite risky asset, as we now explain. We set the mortgage spread Rt+1 M and the leverage ratio λ t equal to their sample means R M and λ, and use (5) to relate the variance of the excess return on the levered position in residential real estate Rt+1 RE 10

13 to the variance of the underlying housing return: V ar ( ) Rt+1 RE V ar ( ) R H = t+1. (7) (1 λ) 2 The covariance of Rt+1 RE with the excess return on financial equity Rt S is related to the covariance of housing with financial equity by Cov ( ) Rt+1, S Rt+1 RE Cov ( R W I = t+1, Rt+1 HI 1 λ ). (8) The standard deviation of levered housing equity is 18.54%, about twice the standard deviation of housing returns. The sample correlation of the Swedish housing index and the MSCI world index is close to zero (-5.7%) over the 1983 to 2007 period, and we set this correlation equal to zero in simulations. We impute the variance of the composite risky asset by taking the variance of (4) and setting φ t and ψ t equal to their respective sample means φ and ψ: V ar ( ) ( ) Rt+1 e 2 ( ) = 1 φ ψ V ar R S 2 ( ) t+1 + φ V ar R RE t+1 [ + 2(1 φ) ψ 2] V ar ( Rt+1) W I. Note that to derive this equation, we use the properties that Rt+1 S = R W I and Cov(Rt+1, S Rt+1) RE = 0. t+1+ε S t+1, R RW (9) t+1, t+1 = R W I Overall, the composite risky asset has a mean excess return of 2.92% and a standard deviation of 12.68% in annual units. The implied Sharpe ratio is 2.92/12.68 = 0.23, which slightly exceeds the historical Sharpe ratio of the stock market and the assumed Sharpe ratio of real estate due to diversification. 3.3 Labor Income Life-Cycle Income Profile We consider the labor income specification used in Cocco, Gomes, and Maenhout (2005): log(l h,t ) = a h + b x h,t + ν h,t + ε h,t, (10) where L h,t denotes real income for household h in year t, a h is a household fixed effect, x h,t is a vector of characteristics, ν h,t is a permanent random component of income, and ε h,t is a temporary income shock distributed as N (0, σ 2 ε,h ). The random variable ν h,t follows a random walk, ν h,t = ν h,t 1 + ξ h,t, (11) 11

14 where ξ h,t N (0, σ 2 ξ,h ) is the permanent income shock in period t. The shocks ε h,t and ξ h,t are Gaussian white noise and are uncorrelated with each other at all leads and lags. The vector of characteristics x h,t contains age dummies and the number of children in the family. 4 For the estimation of labor income processes, we use the Swedish Income Registry data set compiled by Statistics Sweden over the 1983 to 2007 period. The full registry contains 36.4 million household-year observations. In order to estimate income risk by business sector, we exclude households for which the sector of employment is not available. We obtain a panel of 29.2 million household-year observations, to which we will refer as the labor income panel. We estimate the income process from consecutive observations of household yearly income data, excluding the first and last year of labor income to avoid measuring annual income earned over less than 12 months. We classify households by the head s age and education level. Specifically, since the vast majority of Swedish residents retire at 65, we consider two age groups: (i) non-retired households less than 65, and (ii) retired households that are at least 65. We consider three levels of educational attainment: (i) basic or missing education, (ii) high school education, and (iii) post-high school education. We assume that the parameter vector b depends on the level of education and estimate it separately for active and retired households. For active households younger than 65, we estimate b by running pooled regressions of specification (10) for each of the three education groups. As in Cocco, Gomes, and Maenhout (2005), the vector of explanatory variables x h,t includes age dummies. We then regress the estimated age dummies on a third-degree polynomial in age and use the fitted third-degree polynomial in our life-cycle model. For retired households that are at least 65, we estimate specification (10) for each education group, excluding age variables from the vector of explanatory variables, to estimate income replacement ratios in retirement. Figure 2 illustrates the estimated age dummies over the life-cycle, the replacement ratios, and the fitted polynomials of the three education groups. When estimating permanent and transitory income risk σ 2 ξ,h and σ2 ε,h, we consider 12 employment sectors within each education group, so that the estimation is conducted on 12 3 = 36 sector-education groups. Within each group, we follow the procedure of Carroll and Samwick (1997) by estimating the variances of cumulative income growth innovations at the household level over non-overlapping intervals, and using the estimates to infer the variances of permanent and transitory income shocks. In Table 3, we report the standard deviations of the permanent and transitory components of income risk, σ ξ,h and σ ε,h, for each sector-education group. There are intuitive differences across sectors, with relatively little income risk in the public sector and in mining and quarrying, electricity, gas, and water supply, and relatively high income risk in wholesale and retail trade, hotels and restaurants, and real estate activities. As in Low, Meghir and Pistaferri (2010) we find that in most sectors educated households face larger transitory shocks, whereas permanent shocks are more evenly distributed across education levels. These 4 The household head and the number of adults in the household are constant over time by construction. 12

15 results contrast with earlier studies showing that in the United States, more educated people have lower transitory income risk and higher persistent income risk, or put slightly differently, that low-education people have layoff risk and high-education people have career risk. The explanation is likely due to the fact that in Sweden, uneducated workers face lower unemployment risk and enjoy higher replacement ratios than in many other countries, while educated workers face relatively high income losses when they do become unemployed. 5 Across sectors, a comparison of Table 1 with Table 3 shows a slight tendency for initial risky portfolio shares to be lower in sectors with risky labor income (such as wholesale and retail trade, hotels and restaurants, and real estate activities as compared with mining and quarrying, electricity, gas and water supply or the public sector) and a more noticeable tendency for initial wealth-income ratios to be higher in risky sectors. These patterns are intuitive given that labor income risk discourages investment in risky financial assets and encourages precautionary saving. Table 4 documents these effects by regressing the risky share and the initial wealth-income ratio on dummies for high school and post-high school education, and on either total income volatility or the separate volatilities of temporary and permanent income shocks. The first two columns of the table show that post-high school education raises the risky share while the volatility of permanent income shocks lowers it. The third and fourth columns show that initial wealth-income ratios increase with education and with the volatility of transitory income shocks, consistent with the view that wealth is accumulated earlier in the life-cycle in part as a buffer stock against such temporary shocks. When we estimate our model we will ask what these facts imply for the underlying distribution of preferences across households with higher or lower education working in riskier or safer sectors Correlation Between Labor Income and Stock Returns We follow Campbell, Cocco, Gomes and Maenhout (2001) and decompose the permanent shock ξ h,t into a group-level shock κ g,t, common to all households in group g, and an idiosyncratic shock ω h,t : ξ h,t = κ g,t + ω h,t. (12) 5 The Swedish labor market has the following features. First, it is easy for companies to downsize divisions, but extremely diffi cult for them to lay off single individuals unless they have a high managerial position. Second, companies that need to downsize typically restructure their organizations by bargaining with unions. Third, unions are nationwide organizations that span large areas of employment and pay generous unemployment benefits. Fourth, the pay cut due to unemployment is larger for better paid jobs. After an initial grace period, an unemployed person will be required to enter a retraining program or will be assigned a low-paying job by a state agency. All these features imply that unemployment is slightly more likely and entails a more severe proportional income loss for workers with higher levels of education. See Brown, Fang, and Gomes (2010) for related research on the relation between education and income risk. 13

16 The idiosyncratic shock ω h,t is uncorrelated across different households h and also uncorrelated with the group-level shock κ g,t. We allow for the possibility that the group-level income shock co-moves with risky asset returns. Let ρ gη denote the correlation between group-level labor income shocks κ g,t+1 and lagged risky asset returns R e t. Risky returns are lagged one year, following Campbell, Cocco, Gomes, and Maenhout (2001), to capture a delayed response of income to macroeconomic shocks that move stock prices immediately. We estimate ρ gη as follows. Consider the household-level income growth innovation: u h,t = log(l h,t ) log(l h,t 1 ) b (x h,t x h,t 1 ) = κ g,t + ω h,t + ε h,t ε h,t 1. (13) The average of u h,t across households in group g coincides with the group-level shock: ū g,t = κ g,t. (14) We compute the covariance between the group-level shock and each wealth class over the period, as reported in Table II. We then impute the covariance of the group-level shock and the composite risky asset return by applying: Cov (R e t, κ g,t+1 ) = ( 1 φ ψ ) Cov ( R S t, κ g,t+1 ) + φ 1 λ Cov ( R H t, κ gt+1 ) +ψcov ( R RW t, κ g,t+1 ). The implied correlations, Corr (R e t, κ g,t+1 ), are very small in all groups, with an average across groups of 0.09 and a maximum value of However, there is a much higher average correlation between housing returns and group-level labor income shocks, which implies an average overall correlation between the risky excess return and group-level income shocks of 35%, as Table 2 shows. This correlation plays an important role in our model, because it helps to choke off household demand for risky assets even at moderate levels of risk aversion. (15) 4 Estimation Method and Identification This section explains our procedure for estimating and identifying the preference parameters. Using the parameters from Tables 2 and 3 as inputs, we solve the life-cycle model for different groups of households which similar characteristics. We treat each group as a large household with uniform preference parameters. We consider a very large number of groups so that we can indeed identify the cross-sectional distribution of preferences, but do not estimate the model for each individual household for two main reasons. First, by grouping households into bins we hope to eliminate, or at least significantly decrease, the impact of idiosyncratic events that they might face and which we are not capturing in our model. Second, this allows us to derive properties for our estimator relying on cross-sectional asymptotics. 14

17 4.1 Identification We have three separate preference parameters to identify: the subjective discount factor (δ), the elasticity of intertemporal substitution (EIS, ψ) and the coeffi cient of relative risk aversion (γ). In a model with incomplete markets all three parameters affect both portfolio shares and wealth accumulation making their identification non-trivial. The main challenge comes from separately identifying the discount factor and the EIS, as we discuss next The Identification Challenge The Euler equation for the return on the optimal portfolio is given by [ ( ) 1 ( ) ] 1 Ct+1 ψ V ψ γ t+1 1 = E t δ p t,t+1 Rt+1 P C t µ(v t+1 ) where R P t+1 = αr e t+1 + (1 α)r f, and µ(v t+1 ) denotes the certainty equivalent of V t+1. 6 Taking logs of both sides and making the usual assumption of joint lognormality we obtain 0 = ln(δ p t,t+1 ) 1 ( ) 1 ψ E tg t+1 + ψ γ E t ṽ t+1 + E t rt+1 P + 1 2ψ 2 σ2 g + 1 ( ) ψ γ σ 2 ṽ σ2 r + 1 ( ) ( ) 1 1 ψ ψ γ σ gṽ + ψ γ σṽr + 1 ψ σ gr, where lower cases letters denote logs of upper case letters, g t+1 ln(c t+1 /C t ) and Ṽt+1 = V t+1 /µ(v t+1 ). We solve for E t g t+1 and obtain: E t g t+1 = ψ [ ln(δ p t,t+1 ) + E t rt+1] P + (1 γψ) Et ṽ t+1 (16) [ ( + 1 2ψ σ2 g + ψ ) 2 ( ) ] ( ) ψ γ σ 2 ṽ + σ 2 r + 2 ψ γ σṽr + ψ γ σ gṽ + σ gr Naturally we could also have derived an expression for expected log consumption growth using the Euler equations for the individual assets R f and Rt+1, e but Campbell and Viceira (1999) show that the optimal consumption-wealth ratio is, to a first-order approximation, driven by the trade-off between the endogenous expected return on invested wealth and the discount rate, exactly the first term in equation (16). Their results are obtained in an infinite horizon model without labor income but Gomes and Michaelides (2005) reach 6 With labor income risk and a utility function that satisfies u (0) = the agent will always keep some precautionary savings and therefore, even in the presence of borrowing constraints the Euler equation always holds with equality. We also have short-sales constraints on holdings of the safe asset and the risky asset but these do not bind in the data. 15

18 the same conclusion numerically in a life-cycle model that is almost identical to the one considered here. This equation highlights the identification problem, since ψ and ln(δ) appear multiplicatively in that first term. So in principle one can change δ and ψ in offsetting ways and obtain the same implications from the model. Furthermore, the impact of ψ on savings is either positive or negative, depending on the sign of ln(δ p t,t+1 ) + E t r P t+1. So we could find one solution with a low δ and a high ψ and a very different solution with a high δ and a very low ψ, both delivering the same wealth accumulation within the model. Equation (16) however also reveals how, in theory, we can obtain a separate identification of the EIS and the discount factor. One solution would be to exploit the remaining terms in the expression. Unfortunately, as just mentioned, in these models savings behavior is largely determined by the first term, plus the precautionary savings behavior which is mostly determined by γ. Our identification is therefore based instead on exploiting variation in ln(δ p t,t+1 ) + E t r P t+1. Even though we do not have exogenous variation in expected returns, we have endogenous variation driven primarily by changes in the optimal portfolio of the agent over time as human capital diminishes relative to financial wealth, and secondarily by increasing mortality risk. To illustrate our identification strategy we consider an example with γ = 6, δ = 0.98, and a representative initial wealth to income ratio. Ignoring all the other terms in equation (16), since as we just argued their impact on expected consumption growth is second order, our model simulations imply that at typical ages within our sample 7, E t g t+1 = ψ [ ln(δ p t,t+1 ) + E t r P t+1 ] { ( 2.17% %)ψ = 2.32% ψ for age 43 ( 2.37% %)ψ = 1.62% ψ for age 50. The expected consumption growth rate varies across these two ages by 0.70% ψ, of which 0.50% comes from the change in expected portfolio return and 0.20% from the effect of increasing mortality risk. Using our model we can convert this into implied differences in wealth accumulation. For these particular parameter values we obtain a yearly growth rate of the wealth-income ratio of 1.2% ψ at age 43, and 0.7% ψ at age 50. These results indicate that the impact of changes in the EIS on wealth accumulation varies in a non-trivial way with age, even over a short time span such as the one considered in our analysis. Our identification strategy builds on this result. 7 This equation is an approximation since the value of E t r P t+1 depends on ψ through the wealth accumulation channel. To simplify the exposition we are using the average value E t r P t+1 for ψ between 0.1 and 2. The calculation for the growth rate of the wealth-income ratio takes this feedback effect into account since it is computed from the simulations in the model. 16

19 4.1.2 Identification Strategy The identification strategy is based on three moments that pin down relative risk aversion, the discount factor, and the EIS. We motivate and demonstrate this procedure by running a series of regressions based on simulated data from the model, where we regress different moments against the preference parameters that were used to generate those same simulations, drawing the preference parameters from a plausible range and conditioning on a representative value of the initial wealth-income ratio. The first set of regressions is α it = k 0 + k γ γ i + k ψ ψ i + k δ δ i + e it, t = 1,..., 9 where i denotes household, and t denotes age, with t = 1 corresponding to age 40, the first one used in our estimation. 8 Panel A of Table 5 reports the results from the unrestricted regression and from a regression on the risk aversion coeffi cient alone. Taking the unrestricted specification first, the highest R 2 is naturally for the t = 1 regression. At this initial age we match the agent s wealth almost exactly, with the only source of cross-sectional heterogeneity coming from that single year s income shock. As a result we can explain 92% of the cross-sectional variation in the portfolio share by regressing α i1 on the multiple preference parameters, but the R 2 naturally decreases as we increase t and further shocks come into play. More importantly, since initial wealth is given at t = 1, ψ and δ are almost irrelevant for determining the optimal portfolio share, as they have not yet influenced wealth accumulation. In fact, the R 2 from a regression with γ as the only explanatory variable is 90%, almost identical to the one from the unrestricted model. 9 This shows that α i1 is an ideal moment for identifying relative risk aversion. The second set of regressions that we consider is ( ) W = k 0 + k γ γ Y i + k ψ ψ i + k δ δ i + e it, t = 1,..., 9 it and the results are shown in Panel B of Table 5 for an unrestricted specification and several restricted specifications that include only one preference parameter at a time. Again, the R 2 of the unrestricted model naturally decreases as we increase t and further shocks determine the agents decisions, in this case wealth accumulation. The R 2 statistics of 8 We truncate the left-hand side variable such that α it [0.1, 0.9] to avoid making the results particularly sensitive to the choice of grid for the preference parameters. All household groups in our sample have a risky share in this interval so this also restricts the regressions to the relevant range. 9 In theory the time discount factor still has an impact on the optimal portfolio share through two different channels. First it affects the discount rate for future labor income. Second, given that we have Epstein-Zin preferences both the time discount factor and the elasticity of intertemporal substitution also have a direct impact on the agents risk preferences. These channels are reflected in the non-zero difference between the unrestricted and restricted R 2 statistics, but as we can see their effect is very small quantitatively, explaining less than 2% of the cross-sectional variation in portfolio allocations. 17