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We thank Jeremy Nalewaik for excellent research assistance. Davis gratefully ... attention to the idea that financial assets might serve to hedge income risks. ... For less-than-high-school educated men, labor income risk is equivalent to ..... agents can trade only the riskless asset, introduce a risky asset with unit variance. 15.
Using Financial Assets to Hedge Labor Income Risks: Estimating the Benefits ∗ Steven J. Davis † Graduate School of Business University of Chicago

Paul Willen ‡ Department of Economics Princeton University

and NBER

First version: July 1998 This version: March 23, 2000

JEL Nos: G11, D91, D52, J30



We thank Jeremy Nalewaik for excellent research assistance. Davis gratefully acknowledges research support from the Graduate School of Business at the University of Chicago. riskshv2s.tex † Phone: (773) 702-7312. Fax: (773) 702-0458. E-mail: [email protected] ‡ Phone: (609) 258-4032. E-mail: [email protected]

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Abstract We characterize the covariance structure between asset returns and labor income shocks for synthetic persons defined in terms of sex, education and birth cohort. The correlation of income shocks with both aggregate and own-industry equity returns tends to rise with educational attainment and, surprisingly, is negative for several sex-education groups. We then develop a tractable equilibrium life-cycle model with incomplete markets. We implement the model using the estimated covariance structure and other data in order to evaluate the portfolio choice and welfare implications of hedging with financial assets. There are large equilibrium risk-sharing gains from trading a ”full menu” of group-level assets, exceeding 15,000 dollars in present value for many persons, and a single asset can generate sizable gains for certain demographic groups. The hedging motive has significant consequences for the structure of the optimal portfolio.

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Introduction

Among the most important economic risks confronting households is the uncertain nature of labor income. New financial assets create new opportunities to share this risk, and so do financial innovations that facilitate better use of existing assets. These observations prompt several questions regarding the hedging role of financial assets: How does the hedging motive affect optimal portfolio structure? How large are the welfare gains from using financial assets to hedge labor income risk? Who benefits in equilibrium from financial innovations that expand opportunities to share this risk? To address these sorts of questions, we develop a framework that can be tailored to particular economies and financial innovations, implemented using available data, and readily solved to compute equilibrium outcomes. The framework is a dynamic equilibrium exchange economy with incomplete markets, normal random variables and exponential preferences. Willen (1999b) shows how to address these questions in a two-period version of the model. We extend his framework to encompass overlapping generations and a flexible life-cycle structure for earnings and consumption. We implement the framework using U.S. data from 1965 to 1994 on security returns and on labor income for synthetic persons defined in terms of birth cohort, educational attainment and sex. We quantify the risk-sharing benefits associated with two types of financial innovations. First, in the spirit of Shiller (1993), we consider the introduction of new financial assets with payoffs tied to measures of aggregate, sectoral or group-level outcomes. Second, we consider mutual funds constructed from existing securities in such a way as to mirror the industry distribution of employment for each synthetic person at each point in time. If the industry-level components of returns to equity and human capital are correlated, these mutual funds expand risksharing opportunities relative to portfolios defined over a riskless asset and a broad equity index only. As an input to our welfare analysis of financial innovation and an interesting topic in its own right, we characterize the covariance between returns on financial assets and returns on human capital. Rather surprisingly, few previous studies investigate this covariance despite its obvious relevance to individual portfolio optimization, mutual fund structuring, pension fund management, savings behavior and asset pricing.1 1

Labor income risk plays an important role in work on precautionary savings (e.g., Carroll, 1997) and in some recent work on asset pricing (e.g., Heaton and Lucas, 1996). Precautionary theories of consumption emphasize the role of borrowing and lending, which is also important in our analysis, but sidestep the pure hedging function of financial assets. Asset-pricing studies that consider a role for labor income risk typically restrict attention to highly aggregated labor income measures or characterize individual income uncertainty in ways that preclude cross-sectional variation in the

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This neglect is also surprising in another respect: Shifts in relative earnings among individuals and demographic groups have been the focus of extraordinary research and policy attention in recent years. Levy and Murnane (1992) and Katz and Autor (1999) review an extensive body of work in this area. Neither review devotes any attention to the idea that financial assets might serve to hedge income risks. For each sex-education group, we estimate the covariance between financial asset returns and labor income shocks. The correlation between aggregate equity returns and labor income shocks ranges from -.25 over most of the life-cycle for the least educated men to .25 or more for college-educated women. For both men and women, the correlation between income shocks and equity returns tends to rise with educational attainment. The correlation between income shocks and own-industry equity returns also tends to rise with educational attainment. In fact, for men with less than a college education and certain educational groups of women, labor income shocks covary negatively with own-industry equity returns. Artificial assets with payoffs tied to job creation and destruction rates are highly correlated with the income shocks of less-educated men.2 Three assets – the S&P 500 plus the job creation and destruction assets – jointly account for between 27 and 40 percent of group-level earnings risk for less-educated men Given the stochastic properties of labor income, including its covariance structure with financial asset returns, we calculate equilibrium asset prices and optimal consumption and portfolio allocations for each (synthetic) individual. These calculations require an assumption about the asset choice menu over which individuals optimize. The available evidence suggests that neither individuals nor pension fund managers employ sophisticated portfolio strategies to hedge earnings risk. 3 Based hedging portfolio. Heaton and Lucas (1997) consider how equity returns covary with labor income and proprietary business income. They report a small negative correlation of equity returns with aggregate wage income and a larger, but still modest, positive correlation with aggregate proprietary business income. They pursue the implications of this covariance structure for cross-sectional variation in portfolio choice and for asset pricing. 2 The job creation and job destruction rates are calculated from establishment-level employment changes, as described in Davis, Haltiwanger and Schuh (1996). Many researchers and government agencies throughout the world have begun to construct and report statistics of this sort. See Davis and Haltiwanger (1999) for a review of this work. The U.S. Bureau of Labor Statistics plans to begin publishing official job destruction statistics of this type for most sectors of the U.S. economy in the near future (Spletzer, 1997). 3 Most U.S. households have small financial asset holdings and even smaller holdings of equity securities. See, for example, Poterba and Samwick (1997) and Flavin and Yamashita (1998). Empirical work on portfolio allocation behavior devotes little attention to hedging labor income risk. See, in addition, Poterba et al. (1996), Papke (1998) and Sunden and Surette (1998). Professional financial advisors place little weight on hedging labor income risk (Canner, Mankiw and Weil, 1998).

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on this evidence, we proceed under the assumption that individuals (or their portfolio managers) allocate financial wealth between a riskless asset and a broad-based equity portfolio. 4 This assumption yields a benchmark dynamic equilibrium, against which we evaluate the effects of financial innovations. Relative to this benchmark, there are large welfare benefits from introducing securities that allow full sharing of group-level earnings risk. 5 As an example, assuming an annual discount rate of 2.5 percent and a relative risk aversion of 3, the equilibrium welfare gains for college-educated men amount to nearly 27,000 per person in 1998 dollars. Bigger shocks, greater risk aversion and lower discount rates imply larger welfare gains. The size of equilibrium welfare gains also depends on the covariance structure among earnings shocks. For example, college-educated women reap relatively large equilibrium gains from the introduction of a full asset menu for sharing group-level risks, because their earnings shocks have low correlation with per capita earnings shocks. The “full menu” experiment involves the introduction of many new assets, but sometimes a single asset can substantially improve welfare for certain groups. As an example, among men who did not obtain post-secondary schooling, the equilibrium benefit of trading the job destruction asset exceeds 1,800 dollars per person. Traditional approaches to portfolio choice typically ignore the covariance between labor income shocks and equity returns. Our empirical results and theoretical analysis imply that this approach can misstate the household’s optimal portfolio allocation and the welfare effects of holding equities. We show that the effect of labor income on one’s portfolio is equivalent to a significant position in equity markets. For college educated men, labor income risk is equivalent to a 50,000 dollar long position in the S&P 500. For less-than-high-school educated men, labor income risk is equivalent to a 16,000 dollar short position in the S&P 500. The paper proceeds as follows. Section 2 constructs and examines labor income data for synthetic persons. We characterize systematic and stochastic aspects of labor 4

Our theoretical framework easily accommodates other assumptions about portfolio choice behavior, including choice over a large asset menu, but we require some assumption or estimate regarding portfolio choice behavior in order to calculate the initial equilibrium and the equilibrium response to financial innovation. 5 Attanasio and Davis (1996) show that changes in the cohort-education structure of pre-tax hourly wages among men drove large changes in the between-group distribution of household consumption during the 1980s. They also calculate large, but unrealized, welfare gains from sharing this type of group-level earnings risk. We assess the extent to which financial assets can be used to achieve some of the potential between-group risk-sharing gains uncovered by Attanasio and Davis. However, the methodology we develop in this paper is more general; it can be applied to panel data on countries, groups or individuals.

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income variation for each synthetic person, and we calculate the size of shocks to the present value of lifetime earnings (as a function of age, education and sex) implied by our characterization of the labor income process. Section 3 describes the data on financial asset returns and our procedures for constructing new financial assets and group-specific mutual funds. Section 4 investigates the covariance between human capital returns and financial asset returns. Section 5 sets forth a dynamic equilibrium life-cycle framework with incomplete markets. We show how to calculate asset prices, consumption paths and portfolio allocations. We also explain how to calculate the welfare effects of new risk-sharing opportunities in terms of the equilibrium response to financial innovation. Section 6 draws on our characterization of labor income behavior, portfolio choice menus and other data to estimate the initial dynamic equilibrium. Section 7 then quantifies the equilibrium effects of financial innovations that take the form of expanded portfolio choice menus. Section 8 concludes.

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Labor Income

2.1

Synthetic Panel Data

The Annual Demographic Files of the March Current Population Survey (CPS) contain individual data on pre-tax labor earnings in the previous calendar year. Using the CPS, we construct panel data on mean annual earnings and mean log earnings from 1963 to 1994 for groups of individuals defined by sex, educational attainment and birth cohort. By following particular groups over time as they age, we obtain longitudinal data on synthetic persons. For each sex, we group persons into four educational attainment categories: less than high school, high school, some college but no degree, and a college education (including persons with post-graduate schooling). 6 Within each sex-education group, we consider three-year cohorts indexed by the birth year of the middle cohort members. For example, the 1956 cohort contains persons born between 1955 and 1957. 7 We also measure cohort age based on midpoints, so that the 1956 cohort is 34 years old in 1990. We exclude observations for which any cohort members are less than 24 or more than 59 years old. We exclude cohorts with fewer than seven annual observations. After imposing these restrictions and allowing for two lags in the specification 6

We construct time-consistent categories across the 1991-92 break in the CPS education codes following the recommendations of Jaeger (1997). 7 More precisely, because the CPS records age at the March survey date, the 1956 cohort contains persons born between March of 1955 and February of 1958.

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of the labor income process, our sample runs from calendar years 1965 to 1994 for cohorts born between 1911 and 1962. The panel contains mean annual earnings and mean log earnings for synthetic persons who are between 27 and 58 years old. The sample selection criteria and our procedures for grouping by sex, education and birth cohort yield panel data for 2 × 4 × 18 = 144 synthetic persons. 8 After using one observation to compute lags in our preferred specification for the income process, there are between 6 and 31 consecutive annual observations for each synthetic person. To compute earnings for synthetic persons, we use CPS data on wage and salary workers in the private and public sectors. We exclude unincorporated self-employed persons from the earnings calculations, but we include self-employment and farm income for persons who are mainly wage and salary workers. We also exclude persons who were students or in the military at least part year. 9 We express earnings in 1982 dollars using the GDP deflator for personal consumption expenditures. Table 1 reports summary information about the sample sizes in the cells that underlie the synthetic panel data. Mean sample sizes number in the hundreds, and no cell has fewer than 55 observations.

2.2

Income Processes for Synthetic Persons

To estimate the risk-sharing potential of financial assets, we must first identify and characterize the stochastic component of earnings variation for each (synthetic) person. Given a model for labor income, we can identify shocks, calculate their effects on the present value of lifetime earnings, and investigate how the earnings shocks covary with asset returns. We characterize the labor income process presently and take up the covariance between earnings shocks and asset returns in section 4. We model labor income as an ARIMA process, augmented by a polynomial in age to capture systematic life-cycle variation. We let the specification vary freely across sex-education groups. For each sex-education group, we pool over birth cohorts, allow for cohort fixed effects, and estimate the earnings specification by nonlinear least squares. We experimented with several specifications in the ARIMA class: (i) stationary AR processes fit to levels (ii) an integrated MA(2) process fit to first differences and (iii) an integrated ARMA(1,1) process fit to first differences. These specifications 8

In any given calendar year, 10 or 11 (9 in 1994) birth cohorts in each sex-education group satisfy the age restriction, which corresponds to 80 or 88 (72 in 1994) synthetic persons. 9 In addition, we exclude persons who report an hourly wage less than 75 percent of the federal minimum. We handle top-coded earnings observations in the same manner as Katz and Murphy (1992).

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deliver similar results for the covariance between income shocks and asset returns, but the stationary processes imply much smaller effects of income shocks on the present value of lifetime earnings. When fed into our theoretical model, the stationary processes also imply small consumption responses to income shocks, whereas the integrated processes imply empirically plausible consumption responses. On this basis, we prefer the integrated processes. The two integrated processes we considered deliver similar results, and we henceforth restrict attention to the ARIMA(0,1,2) process. 10 We include a (second-order) third-order age polynomial in the (log) eanings specifications. Differencing reduces the order of the age polynomial by one and sweeps out the cohort fixed effects. Thus, the fitted earnings specification for a particular sex-education group is h

∆yth = α0h + α1h ageht + α2h (age2 )t + ht ,

h h ht = ηth + ψ1h ηt−1 + ψ2h ηt−2 ,

where yth denotes mean annual earnings for cohort h at t, the α’s are coefficients in the age polynomial,  is a moving-average residual, ηt is the earnings innovation at t, and the ψ’s are moving-average coefficients. Table 2 reports the estimated MA parameters and other statistics. Based on the sign and size of the moving-average coefficients, earnings shocks show greater persistence for men than for women at each education level. Men with a high school education show the most persistent response to earnings shocks, and women with some college or less than a high school education show the least persistent responses. Earnings innovations for women are larger in percentage terms than for comparably educated men but smaller in absolute terms.

2.3

Shocks to the Present Value of Lifetime Earnings

The potential welfare gains from improved risk sharing depend partly on the size of labor income shocks, as measured by the impact of a typical income innovation on the present value of lifetime earnings. To address this matter, table 2 reports how innovations to current labor income affect the expected present value of remaining lifetime earnings. The reported “present value multiplier” equals the cumulative impact of a unit earnings innovation assuming a one percent annual discount rate and retirement after age 58. 11 10

MaCurdy (1982) provides a useful discussion of how to specify and estimate stochastic specifications for individual earnings in longitudinal data. Based on standard time-series diagnostics, he arrives at the ARIMA(0,1,2) process as the preferred specification in his analysis of annual earnings.

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At 35 years of age, the present value multipliers range from 13 to 20 for men and from 6 to 14 for women. These values imply large effects of a “typical” income shock on lifetime earnings. For example, based on the specification for 35-year old men with a high school education, a unit standard deviation innovation in mean annual earnings changes the expected present value of lifetime earnings by 15, 566(= 18.8×828) in 1982 dollars. The corresponding figure for college-educated men is about 31,000 dollars, but it is only about 4,500 dollars for women who do not have a college degree. These large differences across sex-education groups (and over the life cycle) in the impact of an income shock on the present value of lifetime earnings translate into large differences in optimal hedging portfolios, as we show in Section 5.

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Financial Asset Returns

3.1

Equity Returns and Artificial Securities

To measure returns on a broad equity index, we use the annual rate of return on the S&P 500, as reported by Datastream. We also construct returns for several artificial securities that reflect movements in output and other economic aggregates. For output, we use real GDP in 1992 dollars from the National Income and Product Accounts (NIPA). It is convenient for our purposes to orthogonalize “returns” on the GDP asset with respect to returns on the S&P 500 index. The orthogonalized GDP asset pays off the residual in an OLS regression of GDP on the rate of return on the S&P 500 index. We construct (orthogonalized) returns for other artificial securities in the same manner using the following data: • Consumption: real personal consumption expenditures in 1992 dollars. • Employment: civilian employment among persons 16 years and older. • Unemployment: the civilian unemployment rate. • Job Creation and Destruction: the gross annual rates of job creation and job destruction in the manufacturing sector, as described in Davis, Haltiwanger and Schuh (1996). We update the DHS data through 1993 and extend them 11

The present value multipliers for the annual earnings specification are easily calculated from the recursion, P V M (age) = P V M (age + 1) + (1 + r)58−age (1 − ψ1 − ψ2 ) for age ≤ 56, where r denotes the annual discount rate, P V M (58) = 1, and P V M (57) = 1 + (1 + r)−1 (1 − ψ1 ). For convenience, we calculate the present value multipliers in the same way for the log earnings specification; i.e., ignoring the slope in the expected lifetime earnings profile.

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back to 1965 using the methodology described in the appendix to Davis and Haltiwanger (1999).

3.2

Group-Specific Mutual Funds

We construct returns on group-specific mutual funds by combining firm-level data on equity returns with individual-level data on industry of employment. To link the firm and worker data at the industry level, we prepared a concordance between the Standard Industrial Classification (SIC) used in firm-level data on equity returns and the Census Industrial Classification (CIC) used in the CPS. Appendix A describes the concordance and lists our 62 industry categories. Given the concordance, we construct returns on group-specific mutual funds in three steps. First, we acquired monthly data on firm-level equity returns (inclusive of dividends and other distributions) from the Center for Research on Security Prices (CRSP). From these data, we compute annual value-weighted equity returns for the industry categories. Second, using the CPS data described in section 2.1, we calculate the industry shares of earnings and hours for each synthetic person in each year from 1967 to 1994. The earnings-weighted calculations draw on data for wage and salary workers only, whereas the hours-weighted calculations include self-employed persons. Third, we compute the weighted mean equity return for each synthetic person in each year using the industry-level returns from step one and the industry weights from step two. Depending on sex and education, we can assign an industry-level equity return to about 80-95 percent of observed hours and earnings. A preliminary analysis gave highly similar results for the hours-weighted and earnings-weighted mutual fund returns. The reported results make use of hours-weighted returns. When we introduce the group-specific mutual funds into our theoretical model, we require a covariance matrix for mutual fund returns. We estimate this covariance matrix in two steps. First, we estimate the covariance matrix of industry-level annual returns using CRSP-based data from 1965 to 1996. We assume that this covariance matrix is stationary over time. Second, we combine the covariance matrix of industrylevel returns with the CPS-based industry weights to compute the implied covariance between each pair of mutual funds in each year. This procedure yields a time-varying covariance matrix for the group-specific mutual funds.

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4

Covariation between Labor Income Shocks and Asset Returns

In this section, we characterize the covariance structure between labor income shocks and various asset returns. We first characterize the covariance for the S&P 500 as a function of sex, education and age. We then consider artificial financial assets based on aggregate outcomes. We turn last to the covariance between group-level labor income shocks and group-specific mutual funds. For risk-sharing purposes, we care about two aspects of the covariance structure. First, we seek assets with non-zero covariances for at least some group’s income shocks, so that persons can hedge by adopting a long or short position. Second, we seek assets with heterogeneity across people in the covariances. Absent such heterogeneity, an asset in zero net supply does not enable the sharing of income risk among persons with equal risk tolerances. Of course, such an asset is still useful for allocating labor income risk toward persons with greater risk tolerance.

4.1

The Covariance Structure for a Broad Equity Index

To estimate the covariance structure for the equity index, we fit regressions of the following form for each sex-education group: ηt = β0 xt + β1 aget xt + β2 age2t xt + β3 age3t xt , where ηt is the earnings innovation estimated in Table 2, and xt is the rate of return on the S&P 500 in year t. We estimate the β parameters by least squares to characterize the covariance structure as a smooth function of age. As in Table 2, the regression for each group contains 331 annual observations on 18 synthetic individuals for the 1965-1994 period. Table 3 reports R2 values for regressions of earnings innovations on the S&P 500 in the top row of each panel. The R2 values are small for all groups, not more than 3 percent in most cases and always less than 10 percent. The small R2 values imply that a broad equity index affords very limited scope for workers to hedge earnings risk or share risk. Figures 1 and 2 display the covariance as a function of sex, education and age for the annual earnings specification. The upper left panels pertain to the regressions on the S&P 500. We construct these figures as follows. First, before running the regressions, we transform the asset return to have unit variance. Second, we evaluate the regression function at each age and divide through by the standard deviation of the labor income shock at that age to obtain a correlation coefficient. 10

According to figures 1 and 2, the S&P 500 is a good candidate for risk sharing in the sense that its correlation differs a lot across people. The correlation with income shocks is around -.25 for men who did not finish high school. In contrast, the correlation is positive for college-educated workers, ranging from 0 to .3 over the working life for men and generally exceeding .2 for women. So, even though returns on the S&P 500 account for a very modest portion of variation in earnings shocks, the heterogeneity in correlations across groups suggests some potential as a risk-sharing tool. The correlation structure for aggregate equity returns fits reasonably well with a large body of research on the demand for labor. Empirical studies of labor demand in the modern economy consistently find that more skilled (i.e., educated) workers are relatively complementary to physical capital, the use of advanced technologies and research and development activity. 12 In light of this evidence, and to the extent that equity value derives from residual claims on firms’ physical capital, intellectual property and technological know-how, one might anticipate that the correlation between aggregate equity returns and earnings shocks rises with education. Figures 1 and 2 largely support that view.

4.2

Artificial Financial Assets

Table 3 also reports regression results for artificial financial assets based on aggregate outcomes, as described in section 3.1. For each new asset and each group, we fit regressions of the form: ηt = β0 xt + β1 aget xt + β2 age2t xt + β3 age3t xt + β4 zt + β5 aget zt + β6 age2t zt + β7 age3t zt , where zt denotes the return on the new asset. As before, we transform z to have unit variance. We also orthogonalize the vector [z, agez, age2 z, age3 z] with respect to [x, agex, age2 x, age3 x]. Table 3 shows that the job creation and destruction assets outperform the other artificial assets according to an R2 metric, especially for less educated men. 13 Indeed, the fraction of labor income risk accounted for by the S&P 500 plus the job destruction asset ranges from 18-28% for men who do not attend college. Figures 1 and 2 display the correlation functions for the creation and destruction assets. Increases in manufacturing job destruction are bad news for everyone, but 12

See Chapter 3 in Hamermesh (1993) for a review of evidence on static labor demand relationships. Goldin and Katz (1996) discuss recent work and historical evidence on capital-skill and technologyskill complementarity. 13 The sample period ends in 1993 for regressions that include the job creation and destruction assets, but the sample difference does not account for the superior fit.

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much worse news for people with low education. The correlation is about -.5 for men who did not finish high school and about -.4 for men with a high school education. In contrast, the correlation is near zero for college-educated men and women. This sharp heterogeneity in the correlations and the high explanatory power of the destruction asset (for some persons) make it a promising tool for risk-sharing purposes. For the job creation asset, the correlations with earnings shocks are uniformly positive for men and negatively ordered by education. For women, the correlations with the creation asset are also negatively related to educational attainment, but the correlations are mostly negative for more educated women. Heterogeneity in its correlation structure makes the job creation asset a promising tool for risk-sharing purposes.

4.3

Group-Specific Mutual Funds

Based on a sample that begins in 1967 rather than 1965, Table 3 also reports results for group-specific mutual funds. Adding the mutual funds to the baseline specification with only the S&P 500 provides a very small improvement in fit. For more educated women, however, the mutual funds add a few percentage points to the fit of regressions that include the creation and destruction assets. Figures 1 and 2 display the correlations functions for the group-specific mutual funds. To our initial surprise, the mutual fund returns display a negative correlation with income shocks for all persons with less than a college education. In other words, for six of the eight sex-education groups, a long equity position in the worker’s own industry acts to hedge group-level income risk. This surprising finding runs directly counter to the view that industry-specific fluctuations in equity returns are mainly driven by factor-neutral demand and technology disturbances. That view implies a positive within-industry correlation between equity returns and human capital returns. The negative correlation in the data might arise because movements in equity returns are dominated by rent-shifting between firms and their workers, or because factor-biased technology shocks cause capital and labor returns to move in opposite directions. Upon reflection, we view the latter hypothesis, in particular, as quite plausible, especially in light of the important role that has emerged for factor-biased technology shocks in leading explanations for relative wage movements across sex, education and experience groups.14 We also note that the pattern of correlations between income shocks and own-industry returns shows the same type of relationship to educational attainment as we found for the broad 14

Autor and Katz (1999) review the relevant literature.

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equity index.

4.4

Multiple Assets

Four risky assets – S&P 500, group-specific mutual fund, job creaton, job destruction – account for 25-40 percent of group-level income risk for less-educated men and 5-21 percent for other groups. Most of the explanatory power for less educated men comes from the creation and destruction assets. Mutual funds play a nontrivial role for more educated women. Based on the annual earnings measure, figure 3 shows the fraction of stochastic variability in group-level income risk jointly accounted for by these four assets over the life cycle. For less educated men, the four assets account for roughly one-fifth to one-third of group-level income risk throughout the life cycle. For other groups, these assets account for less than 15 percent of group-level income risk during most or all of the life cycle.

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A life-cycle framework with incomplete markets

Following Willen (1999b), we adopt a dynamic exponential-normal framework that delivers closed-form solutions. Most other analyses of dynamic equilibrium models with incomplete markets assume constant relative risk aversion and rely heavily on numerical solution methods. 15 Those analyses typically require a very small number of agent types. In contrast, our framework remains tractable with an essentially arbitrary number of agents and assets, overlapping generations, and parameters that vary with age, person and time. A simple example helps to appreciate some of the issues that arise in seeking to quantify the welfare effects of financial innovation. Example 1 The economy contains three agents and lasts five periods (t = 0, 1, 2, 3, 4). For each agent, expected earnings equal five dollars in periods 1, 2 and 3. For agents 1 and 3, earnings follow a random walk; whereas, for agent 2, earnings are white noise. All agents retire after t = 3 and have no labor income at t = 4. All earnings innovations have unit variance. A riskless asset is freely traded. Risk aversion is lowest for agent 1 and highest for agent 3. Starting from a benchmark in which agents can trade only the riskless asset, introduce a risky asset with unit variance. 15

See, for example, Telmer (1993), Heaton and Lucas (1996), Constantinides, Donaldson and Mehra (1998), Storesletten, Telmer and Yaron (1998) and Judd, Kubler and Schmedders (1998).

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The correlation between earnings innovations and the risky asset return is zero for agent 1 and 0.5 for agents 2 and 3. Panel I in Table 4 summarizes the situation. It is not obvious how the risk-sharing benefits afforded by the new asset are distributed among the agents. It seems plausible that agents 2 and 3 benefit, because the new asset covaries with their labor income shocks. Agent 3 presumably has a stronger hedging demand because of greater risk aversion and more persistent earnings shocks. Agent 1 has low risk aversion, so he may gain because both agents 2 and 3 want to take short positions. In the analysis below, we show how to solve this problem and much more richer ones.16

5.1

Description of the model and assumptions

There is one consumption good per period and T + 1 periods, t = 0, ..., T . There are H agent types, h = 1, ..., H, with Nth individuals of type h in period t. An agent of type h enters the workforce at th , retires at Trh and dies at T h . Let nht be the fraction of type-h agents at time t, and let Nt be the H-dimensional vector of nht values. For any H-dimensional vector, q, q t = N0t q denotes the per capita mean of q.  T h A consumption path is a random vector, Ch = c˜ht t=th . Condition 1 Agents have exponential utility,   h   T X   −1 h h  U h Ch = Eth  exp −A (δh )t c t Ah h t=t

where Ah is the coefficient of absolute risk aversion. Let A be the H-dimensional vector of absolute risk aversion coefficients. In an abuse of notation, let A−1 be the H-dimensional vector of reciprocals of the risk aversion coefficients.  T h An endowment path is a random vector, y˜th t=th . Let Yt be the H-dimensional vector of period-t endowments. Condition 2 Individual endowments follow ARIMA processes, in which ηeth denotes the income innovation for individual h at time t. Let ψih be the ith coefficient in the moving average representation of the endowment   h h − Et−1 y˜t+i = ψih ηeth . process for individual h. That is, Et y˜t+i 16

For those who can’t wait, the results are in panel III of Table 4.

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There are J + 1 financial assets: J risky assets with gross one-period rates of ˜ j,t , and a riskless asset that pays gross rate of return, R0,t , with certainty.17 return, R Let Rt be the J-dimensional vector of period-t risky asset returns, and let ERt be the vector of expected excess returns with representative element,  corresponding  ˜ j,t − R0,t . Et−1 R Persons enter the world with no financial assets. In each period of life except h the last, an agent invests ωj,t units of the consumption good in risky asset j, j = h 1, 2, . . . , J, and ω0,t units in the riskless asset. Let ω ht denote the J-dimensional vector of investments in the risky assets, and let ω h denote the portfolio allocation path (risky and riskless assets) for individual h. There are no limits on ω other than the budget constraint, so that unlimited short sales are possible. Denote the vector of period-t income innovations and asset returns by   η˜t1   ..   .    η˜H    Φt =  t  ˜  R1,t      ..   . ˜ J,t R Condition 3 The distribution of time-t asset returns and income innovations, conditional on information available at t − 1, is jointly normal for all t. That is, Φt ∼ N (E (Φt ) , St |information at t − 1) ∀t, # " Ξt β t with partitioned covariance matrix, St = . β 0t Σt Condition 3 encompasses any ARIMA process for dividends with i.i.d. normal innovations. To simplify several expressions, we introduce an operator for the discounted expected value of an arbitrary time series: T   X T PDVt {zs }s=t+1 =

1

Πsi=t+1 R0,i s=t+1

Et (˜ zs ) .

  In this notation, PDVt {1}Ts=t is the present discounted value of an annuity that pays one dollar each period from t to T , inclusive. The annuitization factor, aht , equals 17

Throughout the rest of the paper, the “return” on an asset means the gross one-period rate of return, unless otherwise noted.

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  h the reciprocal of PDVt {1}Ts=t . Similarly, define the present value multiplier h

Ψht

=

T X

1

Πsi=t R0,i s=t

h ψs−t ,

which equals the revision to the present value of lifetime income implied by a unit labor income innovation. ˜ t be a Some additional notation is needed to handle long-lived risky assets. Let d J-dimensional dividend process with representative element d˜j,t. A long-lived asset with ex-dividend price, Pj,t, is a claim to the stream of future dividends, {dj,s }Ts=t+1 . ˜ t be the J-dimensional vector of innovations in the ARMA dividend process, Let x and let λt be the present value multiplier on dividend innovations. For example, λj,t is the impact of a unit innovation in the j th dividend process on the present ˆ t = var (˜ value of future dividends on asset j. Let Σ xt ) denote the covariance matrix h h ˆ = cov (η , x of dividend innovations, and let β t ˜ t ) be the covariance between labor t income innovations and dividend innovations. The following condition plays an important role in our derivation of analytic solutions for portfolio allocations and equilibrium asset returns. −1/2

Condition 4 The ”price of risk”, Σt ERt , and the scaled covariance between asset −1/2 returns and income innovations, Σt β ht , are nonstochastic. If the covariance matrix of asset returns is nonstochastic, then Condition 4 means that expected excess returns and the covariance between income innovations and asset returns are also nonstochastic. One further condition is inessential to our approach, but it simplifies the analysis and welfare calculations. Condition 5 The riskless asset is elastically supplied at an exogenous interest rate. It is possible to endogenize the risk-free rate in our setting (and accommodate forces that drive nonstochastic interest rate variation). However, Condition 5 allows us to streamline the development below and sidestep several issues that have a minimal bearing on the questions addressed in this paper.18 Now consider a perfectly competitive economy with the price of the consumption good normalized to one, and denote asset returns for all t by R. The budget set for 18

Willen (1999b) shows how to endogenize the risk-free rate in a two-period version of the model, and Willen (1999a) discusses sufficient conditions for a nonstochastic, endogenously determined interest rate in a related model.

16

agent h is

( T h  h h B h (R) = ct , ω0,t , ωth t=th

h h + 10 ω ht = yth + R0,t ω0,t−1 + R0t ω ht−1 , ch + ω0,t 3 t h and given ω hth −1 = 0, ω0,t h −1 = 0.

)

   Definition of Equilibrium: An equilibrium R, Ch , ω h h∈H is a time path for asset returns, consumption and portfolio allocations for each h such that:  1. Each individual optimally chooses in her budget set: Ch , ω h ∈ B h (R) and    Ch0, ω h0 ∈ B h (R) ⇒ U Ch0 ≤ U Ch ; 2. The goods market clears at all dates: P h for all t, with h ω0,T ≡ 0;

P

3. Risky asset markets clear at all dates:

5.2

˜ht hc

P h

+

P h

h ω0,t =

P h

y˜th +

P h

h ω0,t−1 R0,t

ω ht = 0 for all t.

Individual Consumption and Portfolio Allocation

There are several natural concepts of wealth in this framework. Financial  h wealth  at  h h 0 h h h h Tr t is It = R0,t ω0,t−1 + Rt ω t−1 , human capital is Yt = y˜t + PDVt ys s=t+1 , and simple wealth is Wth = Ith + Yth . We shall show that the welfare effects of a financial innovation can be calculated using a concept that we call generalized wealth.

Definition 1 Generalized wealth is  h  Tr GWth = y˜th + PDVt ysh s=t+1 |{z} | {z } (1) current labor income

h + R0,t ω0,t−1 + R0 ω h | {z t t−1}

(2) PDV of future labor income

(3) financial wealth

  T h   h  0 h h h T 1/As ln R0,s δ s=t+1 + PDVt ERs ω s−1 s=t+1 − PDVt | {z } | {z } (4) PDV of future excess returns

(5) riskles rate arbitrage

− PDVt |



 T h   Ahs /2 var Wsh s=t+1 {z }

(6) PDV of precautionary savings

The sum of terms (1) and (3) measure liquid resources. Term (2) is the value of expected future labor income, discounted at the riskless rate. These first three wealth components are standard, but generalized wealth includes three more terms that influence a person’s “sense of wealthiness.” Term (4) captures the value of excess expected returns on the portion of wealth invested in risky assets. Term (5) reflects any difference between the subjective discount rate and the risk-free rate.Other things equal, a higher discount rate implies that the individual “feels” wealthier and consumes more. 17

Term (6) reflects the precautionary saving motive, which plays an important role in our analysis. As the variance of simple wealth (or consumption) rises, an individual “feels” less wealthy and, hence, consumes less. The definition of generalized wealth may seem ad hoc, but it greatly simplifies the analysis, as the following proposition shows. Proposition 1 (Individual optimization) Given conditions 1, 2, 3, 4 and a nonstochastic rate of return on the riskless asset: • Risky asset holdings in the optimal portfolio are given by ω ht =

J×1

1 h Σ−1 ERt+1 − Ψht+1 Σ−1 t+1 β t+1 Aht+1 t+1

(1)

where Aht = aht Ah and Ψht is the present-value multiplier on income innovations. • The expected excess return on the optimal portfolio, 0    0     −1/2 −1/2 −1/2 −1/2 ERt ω ht−1 = 1/Aht Σt ERt Σt ERt − Σt ERt Σt β ht and the variance of simple wealth along the optimal path,  2    −1 h h −2 var η˜th − β h0 + A var Wth = Ψht Σ β ER0t Σ−1 t t t t t ERt

(2)

are nonstochastic. • Consumption is proportional to generalized wealth along the optimal path: cht = aht GWth . Proof: See Appendix The first part of this proposition states that risky asset allocations contain two components: one that reflects a desire to exploit the excess return on risky assets, and another that reflects a desire to hedge uncertain labor income. The magnitude of the first component declines in risk aversion, while the magnitude of the second rises with the persistence of income shocks and their covariance with asset returns. The second part of the proposition is useful in calculating generalized wealth. The third part states that consumption equals the properly annuitized value of generalized wealth. Thus we have converted a complex dynamic programming problem into a simple annuitization and discounting exercise that any MBA can solve.19 19

So states the author who has yet to teach MBA students.

18

To understand the portfolio decision, consider a simple case with one risky asset that has constant variance. The consumption Euler equation for the risky asset is   i   h h h ˜ ˜ 1/A Et−1 R1,t − R0,t = covt−1 c˜t , R1,t . In words, an individual sets the covariance between consumption and the risky return equal to the excess return on the risky asset, scaled by risk tolerance. Now substitute the solution for consumption from Proposition 1 on the right side, noting that simple wealth (Wth ) contains the only stochastic components of generalized wealth:   i   h ˜ 1,t − R0,t = covt−1 W ˜ h, R ˜ 1,t 1/aht Ah Et−1 R (3) t This equation is almost identical to the Euler equation in a two-period model. The only difference is that absolute risk aversion is multiplied by the marginal propensity to consume out of generalized wealth. This product, which we refer to as dynamic absolute risk aversion, captures a simple source of life-cycle variation in risk aversion. As a person gets older, fewer years remain over which to spread gains and losses, and dynamic risk aversion rises. Consequently, as a person ages, the optimal holdings of risky assets tend to decline.20 Substituting for Wth in (3) and solving yields the risky asset allocation (in units of the consumption good), h ω1t

h h   i β1,t+1 h ˜ = h 2 Et R1,t+1 − R0,t+1 − Ψt+1 2 , σ A σ | {z 1 } | t+1 1 {z }

1

Risk premium exploitation portfolio

Hedging portfolio

where σ12 is the variance of the risky asset rate of return. Note that the “risk premium exploitation portfolio” is the same for everyone up to a scaling factor, a result that generalizes to the case of multiple risky assets. Given a hundred assets and a thousand agents, we can compute the portfolio weights for one agent, then scale up or down for everyone else to obtain the individual premium exploitation prortfolios.21 The hedging portfolio depends on the slope coefficient (β1h /σ12 ) in a regression of the individual’s earnings shocks on the risky asset return and on the present value multiplier (Ψht ) on 20

This analysis helps rationalize the conventional wisdom of financial planners who recommend that investors reduce risky asset holdings as they age. Properly interpreted, the logic of our analysis applies to the level of risky assets, not the proportion of financial assets held in risky form. However, as a person ages, financial wealth grows relative to human capital and simple wealth, so a decline in the level of risky financial assets implies an even faster decline in their proportion of financial wealth. 21 This is the “market portfolio” in the CAPM two-fund separation theorem.

19

earnings shocks. A larger regression coefficient or more persistent shocks raises the magnitude of the hedging demand. This expression for asset holdings leads to a useful decomposition for the variance of generalized wealth. Consider again the case of a single risky asset. Along the optimal path, the time-t innovation to simple and generalized wealth can be written  h i h ih i h ˜ 1,t − Et−1 R ˜ 1,t + 1 ˜ ˜ ˜ Ψht × ηth − (β1t /σ12 ) R − R − E R R E R t−1 1,t 0,t 1,t t−1 1,t Aht σ12 The first term is proportional to the residual in a regression of income innovations on the risky asset return; thus it is orthogonal to the second term, which is proportional to the excess return. Hence, we can decompose the variance of wealth into the scaled variance of undiversifiable income shocks and the variance of the premium exploitation portfolio. The following proposition generalizes this result to the multi-asset case. Proposition 2 (Variance Decomposition) Given conditions 1, 2, 3, 4 and a nonstochastic rate of return on the riskless asset, the variance of an individual’s wealth can be decomposed into two pieces:  2   1 −1 h 0 −1 var GWth = Ψht var η˜th − β h0 Σ β 2 ERt Σt ERt t t t + | {z } Aht undiversifiable part of idiosyncratic risk | {z } premium exploitation risk

  Proof: It is easy to see that var GWth = var Ψht η˜th + R0t ω ht−1 . Substituting in optimal holdings of the risky asset from Proposition 1 and reorganizing gives the decomposition 

5.3

Equilibrium Outcomes

We now calculate equilibrium asset returns and portfolio allocations.22 Since asset returns are normally distributed, we expect some form of the Capital Asset Pricing Model to hold. Our model differs from the standard CAPM, because individuals have idiosyncratic risk that is not (fully) tradable. Consider again the case of a single risky asset with constant variance, and assume one agent of each type. Summing the risky asset allocations over individuals, H X h=1

h ω1t

=

H X

1

h=1

Aht+1 σ1

˜ 1,t+1 − R0,t+1 ) − (Et R 2

H X h=1

22

Ψht+1

h β1,t+1 σ12

The following derviation is similar to the one for the exponential-normal model with two periods. See, for example, Demange and Laroque(1995).

20

Since

PH

h h=1 ω1t

= 0 in equilibrium, we have we can solve to obtain H 1 X 1 H h=1 Aht+1

˜ 1,t+1 − R0,t+1 = Et R

!−1

H 1 X h h Ψ β H h=1 t+1 1,t+1

! (4)

In words, the excess return on the risky asset equals the harmonic mean of dynamic absolute risk aversion times the average covariance between the risky return on the financial asset and the shock to human capital. This result has the flavor of the traditional consumption CAPM, and the intuition is also the same. A risky asset that pays off when times are bad offers a higher return, because it plays a more valuable insurance function. We can write the covariance between the asset return and the labor income shock in terms of exogenous variables and parameters as follows: h ˜ 1,t+1 , η˜h ) = β1,t+1 = covt (R t+1

1 Λ1 h h cov(P˜1,t+1 + d˜1,t+1 , η˜t+1 )= cov(˜ x1,t+1 , η˜t+1 ), P1t P1t

since the stochastic component of P˜1,t+1 + d˜1,t+1 equals the present value of the divih dend innovation, Λ1 x˜1,t+1 . Substituting into (4), and recalling that cov(˜ x1,t+1 , η˜t+1 )≡ h βˆt+1 we obtain ˜ 1,t+1 − R0,t+1 = ER1,t+1 ≡ Et R

H 1 X 1 H h=1 Aht+1

Λ1 P1,t+1

!−1

H 1 X h ˆh Ψ β H h=1 t+1 1,t+1

!

This result extends straightforwardly to the case of many assets and an arbitrary distribution of agent types: Proposition 3 (Equilibrium Excess Returns) Given conditions 1 through 5, the equilibrium expected excess returns can be written ERjt =

Λj ∗ ˆ A β Ψt , Pjt t jt

j = 1, . . . , J,

(5)

−1 where A∗t = N0t A−1 is the harmonic mean of dynamic absolute risk aversion, and t ˆ Ψt is the per capita mean of the covariance between dividend innovations on the β jt jth asset and shocks to the value of human capital. Furthermore, 1/2 ˆ Ψt Σt ERt = A∗t β t

1/2 ˆ ˆ 1/2 β and Σt β t = Σ t t

(6)

We can also calculate equilibrium generalized wealth entirely in terms of exogenous parameters.

21

Proposition 4 (Equilibrium generalized wealth) Given conditions 1 throgh 5,, equilibrium generalized wealth for individual h at time t can be written GWth + PDVt

Wth −PDVt

= n

Ahs /2





1/Ahs



T h ln R0,s δ h s=t+1



−PDVt

n

Ahs /2



2 Ψhs

 0  Ψhs βsh − A∗s /Ahs Ψs β s Ψhs βsh − A∗s /Ahs Ψs β s Σ−1 s

var

η˜sh

oT h

oT h



s=t+1

s=t+1

(7) Proof: See Appendix

5.4



Computing the Welfare Effects of Financial Innovations

We measure the welfare effect of a financial innovation in terms of its consumptionequivalent effect on utility. Definition 2 θh is the uniform variation in the consumption good at each state and date that leaves the consumer indifferent between consumption paths C and C∗ : θh 3 U h (C + θ) = U h (C∗ ) Given our assumptions, uniform variation is simple to evaluate. Proposition 5 Consider two equilibria with different portfolio choice menus and consumption profiles Ch and Ch∗ for person h. Under conditions 1, 2 and 3, the uniform variation    Th {1} h PDV h t s=t 1 h   θh = ch∗ ln  0 − c0 + h Th ∗ A PDV h {1} h t

s=t

Proof: See Appendix We can now prove the main result that we use to calculate welfare effects: Corollary 1 Consider a financial innovation that alters the portfolio choice menu. Under conditions 1, 2, 3 and 4, the present discounted value of the uniform variation for individual h at time t is  T h  Θht = PDV θsh s=t+1 = GWth∗ − GWth , where GWth is generalized wealth before the financial innovation, and GWth∗ is generalized wealth after the financial innovation. 22



Proof: By Propositions 1 and 5, and the fact that the riskless rate is fixed, θh = aht GWth∗ − aht GWth . Since θh is a constant amount in each remaining period of life,  the present discounted value is simply 1/aht ∗ θh = GWth∗ − GWth .  To summarize, we have shown that the welfare effects of a financial innovation can be computed from its impact on generalized wealth. We have also solved for generalized wealth as a function of exogenous parameters. Thus, one could calculate generalized wealth both before and after a financial innovation to determine welfare effects. Instead, we proceed to derive a single expression for the generalized wealth change caused by a financial innovation. To accomplish this, we first analyze the impact of the financial innoation on portfolio allocations and asset returns. As a preliminary, note that new financial assets in zero net supply have no effect on the equilibrium structure of excess returns for existing assets. 23 This fact follows from Proposition 3.24 The empirical implementation considers new assets that are not necessarily orthogonal to existing assets, but it is possible to support the new equilibrium consumption allocations as follows: 1. Hold the same amount of the existing assets as before. 2. Take the new asset and create a portfolio whose returns is the portion of the new asset that is orthogonal to the existing assets. 3. Treat this new portfolio as the new asset and invest using Proposition 1.    Formally, consider an economy E = uh , y h h∈H , (RX ) . Suppose that we intro   duce a set of new assets RZ . Call the new economy E ∗ = uh , y h h∈H , (RX , RZ ) .Let an asterisk denote quantities in the equilibrium of the new economy (e.g. ER∗ denotes excess returns in the equilibrium of economy E ∗ . Let a X or Z subscript p denote objects specific to those assets. Finally, consider the set of assets, RZ = E (RZ ) + RZ − E (RZ |RX ). This implies that RZ p = RZ − ΣZX Σ−1 X RX . 23

Changes that typically do alter the structure of equilibrium excess returns include (i) changes in the set of agents who trade existing risky assets and (ii) changes in the net supply of new or existing risky assets. 24 If the new asset is orthogonal to existing assets, the covariance matrix of asset returns is block diagonal, so the new asset has no effect of the structure of excess returns for existing assets. If a new financial asset lies within the space spanned by the existing assets, it is redundant and also has no effect. Any new financial asset can be written as a linear combination of an orthogonal asset and a redundant asset.

23

Proposition 6 (Effect of new asset on portfolio allocations and asset returns) In equilibrium of economy E ∗ , ER∗X,t = ERX,t and ER∗Z,t = A∗t βZ,t Ψt . Equilibrium allocations of risk can be supported by the following strategy: h∗ h 1. ωX,t = ωX,t h − Σ−1 Zp ,t βZp ,t # # " " IJ RX 0 . Let RY = M . Our solution then Proof: Let M = −ΣZX Σ−1 I R K Z X " # 0 ΣX follows from the fact that ΣY = .  0 ΣZ p

2. ωZh∗p ,t =

1 Σ−1 ERZp ,t Ah Zp ,t

Proposition 7 Suppose we introduce assets Z and we assume that the price of the riskless asset R0,t is held constant for all t. The present discounted value of the uniform variation for individual h is " # 0  ∗ T h  Ahs A∗s A s PDVth ≥ 0 (8) Ψβ Zp ,s − Ψhs βZh p ,s Σ−1 Ψβ Zp ,s − Ψhs βZh p ,s Zp ,s 2 Ahs Ahs s=th +1 Proof: This result follows from equation (7). The wealth term is unchanged because we assumed individuals have no period th − 1 asset holdings. The PDV of variance of income and the riskless rate arbitrage terms are unchanged. Thus we are left with equation (8)  Why is this true? Consider equation (7). Consider an individual h at time th . Her wealth Wth is equal to the present discounted value of future labor earnings. Since we are holding the riskless rate constant, this cannot change. The next two terms depend only on the riskless rate and exogenous parameters, thus they cannot change either. Thus all the change will be in the last term. Consider the last term from the generalized wealth equation for an economy with two assets x and z. The key step is to note that:  0   Ψhs βsh∗ − A∗s /Ahs Ψs βs∗ Σ∗−1 Ψhs βsh∗ − A∗s /Ahs Ψs βs∗ s  0 ∗−1 h h∗   h∗ ∗ ∗ = Ψhs βx,s − A∗s /Ahs Ψs βx,s σx,s Ψs βx,s − A∗s /Ahs Ψs βx,s  0   h h∗ ∗ h ∗ (9) Ψhs βzh∗p ,s − A∗s /Ahs Ψs βz∗p ,s σz∗−1 p ,s Ψs βz p ,s − As /As Ψs βz p ,s The first term on the right-hand side of the equation is the last term from generalized wealth for the economy with only asset x. When we take the difference between generalized wealth before and after the new asset, it will drop out. So the welfare 24

change will just be the present discounted value of the second term on the right hand side. The following proposition shows this in general. We finally consider an important special case. Suppose we have no risky assets at all. Suppose we then introduce what we call a complete menu of assets. It is well-known that under the assumptions we have made, it is not necessary to introduce a “complete menu” of contingent claims to produce a Pareto-optimal allocation. Rather it is sufficient to introduce assets that span individual endowments.25 The following condition is sufficient to guarantee that:  H ˜h ˜h − Condition 6 A complete menu of assets is a set of assets R such that R h=1   ˜ h = q h η˜h where q h is a deterministic constant, for all h and t > 0. E R t t t Under these conditions, we can give a characterization of the welfare benefits of new financial markets. Corollary 2 Suppose we have only a riskless asset. We introduce a complete menu of assets and assume that the riskless rate stays fixed. The uniform variation will be for individual h; " # T h  h ∗ A A s θh = PDVth ≥0 var Ψhs η˜sh − hs ηs Ψs 2 As s=th +1 Proof: According to the definition of a complete menu of markets, βS = Σs . Thus substituting into the expression inside the curly braces in equation (8), we get: h ∗ i0 h ∗ i  ∗ 2 As As A∗s h0 h −1 As h h0 −1 h or A Σ − Σ Σ Σ − Σ N0s Σs Ns − 2 A s s h h h Σs Ns + Σs Σs Σs . One can s s s Ah A s s s s    show that: N0s Σs Ns = var ηs Ψs , Σh0 N = cov ηs Ψs , Ψhs η˜sh and Σh0 Σ−1 Σh = var Ψhs η˜sh which gives our solution. 

5.5

Understanding changes in welfare

We now return to Example 1. What does our analysis tell us? Recall that in our example we introduced a single new risky asset with unit variance to an economy with no risky assets. Thus the new asset is trivially orthogonal to the existing assets. We can write the welfare change for individual h as:  2             θh = PDVth  Ahs /2  A∗s /Ahs Ψβ s − Ψhs βsh (10)   | {z }   | {z }   equilibrium exposure

25



I.e. η˜th − E ηth |Xt = 0 for all h, t > 0.

25

endowed exposure

s=1,2,3

The key to analyzing welfare benefit is understanding the “equilibrium exposure” and “endowed exposure.” Consider an individual h. Note that table 4 shows information about equilibrium and endowed exposure for periods 1 to 4, which are used to calculate portfolios the previous period (i.e. periods 0 to 3). Equilibrium exposure tells us how sensitive her wealth will be to a particular asset return shock. If risk is tradable, then those most willing to bear it should be most sensitive. Thus one’s equilibrium exposure to a traded shock is decreasing in “dynamic risk aversion.” This implies that, holding age constant, the wealth of more risk averse people will be less sensitive to asset return shocks. It also means that, holding risk aversion constant, the wealth of older people will be less sensitive to asset return shocks. The endowed exposure, on the other hand is simply how sensitive individual h’s wealth is to a shock in the absence of any trade in risky assets. The benefit of this asset comes completely from her ability from one level of exposure to another. Thus, if she, for one reason or another, ends up with the same exposure as before, then the asset has provided no benefit at all. The benefit of an asset, therefore, to individual h depends not on how exposed she is to the asset but how exposed she is relative to the economy as a whole. Return to Example 1; we will illustrate this precisely. Table 4 describes the results. We assume that gross return on the riskless asset is 1 (or the interest rate is zero). Panel I shows the basic parameters of the economy. Recall that both agents 1 and 3 have income processes that follow a random walk where agent 2’s income follow a white noise process. Agents 2 and 3 both have correlation of .5 with the new asset. What happens? Panel II shows step by step calculations. Line (1) show the marginal propensity to consume out of wealth. Note that it is increasing with age. Line (2) shows the present value multiplier. For the agent whose process is white noise, this is one all the time; a one dollar shock to income has a one dollar effect on wealth. For the agents whose earnings follow a random walk process a one dollar shock to income in the first period has a three dollar effect on wealth in the first period. If there were retirement, then the present value multiplier would be equal to the reciprocal of the marginal propensity to consume out of wealth. Thus the marginal propensity to consume out of income would be one. Because of retirement, it is always less than one. Line (3) shows the “dynamic absolute risk aversion.” Recall that dynamic absolute risk aversion is the product of absolute risk aversion and the marginal propensity to consume. It reflects the fact that as people get older they have fewer periods in which to spread out shocks to wealth and thus are more sensitive to them. Dynamic absolute risk aversion goes up over the life cycle. Line (4) shows the “endowed exposure” which

26

is the covariance with the asset times the present value multiplier. While both agent 2 and 3 have the same correlation with the asset, agent 3 has much higher “endowed exposure” for the first two periods because of the high persistence of her earnings process. Line (5) shows the average exposure. Line (6) shows the “equilibrium exposure.” This is simply a function of dynamic absolute risk aversion. Agents’ exposure to the new asset is increasing in their absolute risk aversion. The last step is the key. Consider the differences (Line (7). Compare agents 2 and 3. Both had correlation of 0.5 with the new asset. Yet the difference between agent 2’s endowed and desired exposure is small. For agent 3, the difference is much bigger. For agent 1, who has no exposure to the asset to begin with, the difference between desired and equilibrium exposure is quite large, as well. Finally, in interpreting the results the following decomposition is useful. Recall that if we substitute prices back into equation 10, we get: n   o  h h h h h 2 θ = PDVth As /2 ER/As − Ψs βs s=1,2,3

If we expand the quadratic expression, we get:                 2 h h h h h h h 2 θ = PDVth  .5 × ER /As − ERs Ψs βs + .5As Ψs βs | {z } | {z } |  {z }     hedging cost  hedging benefit  risk premium     benefit

       s=1,2,3

The risk premium benefit is the increase in welfare due to one’s ability to exploit the risk premium. The hedging cost is the reduction in the equity premium benefit that came from the fact that one was already exposed to the new asset. This is usually positive – i.e. it is costly to get insurance. However, if one’s covariance with a new asset has the opposite sign from the average, then this will be negative – you will receive money for hedging. Finally, the benefit of hedging is the reduction in the variance of wealth that comes from the introduction of a new asset. Panel III shows the overall welfare effects (multiplying the differences by risk aversion and taking the present discounted value) for each individual. Notice that for agent 2 the hedging cost is greater than the hedging benefit. What’s going on here? Agent 2 shouldn’t hedge at all. In fact she doesn’t. Notice that her equilibrium exposure is actually higher than her endowed exposure in period 1– she is actually investing in her own risks. This illustrates that allowing people to trade their own income will not necessarily make them less exposed to the risks in it; they may (as agent 2 did) become more exposed. 27

6

Empirical Implementation

We implement the theoretical model as follows: (A). Absolute risk aversion (Ah ): We specify relative risk aversion and divide by expected consumption to obtain a value for Ah , absolute risk aversion. We approximate expected consumption by average annual earnings in the person’s sex-education group, multiplied by the percentage of adult life spent working, which we take to be 80 percent. Our benchmark calculations assume a relative risk aversion of 3. (B). Risk-free rate (R0,t ): Our benchmark calculations assume a risk-free real interest rate of 2.5 percent per year. (C). ARIMA processes for earnings (ψth ): We use the estimated moving average coefficients reported in Table 2.  (D). Size of earnings innovations (var η˜th ): We set the variance of earnings innovations to the values reported in Table 2. h (E). Covariance between earnings innovations and asset returns (βj,t ): We use the covariance structure estimated in Section 4 and displayed in Figures 1 and 2.

(F). Demographic structure (Nt ): We let the sex-education-age structure of the population evolve over time in line with the observed CPS employment shares. We further assume that working-age adults (persons covered by our sample) account for two-thirds of aggregate income and consumption. (G). Benchmark Portfolio Choice Menu: Unless otherwise stated, we compute welfare gains relative to a benchmark equilibrium in which persons freely trade the riskless asset and the S&P 500. (H). Timing, Retirement and Death: We calculate the welfare effects of introducing a financial innovation in 1964 (t = 0). The financial innovation is unanticipated, so that asset allocation decisions prior to t = 0 are unaffected by the expanded portfolio choice menu at t = 1. An individual makes the first asset allocation decision the year before entering the workforce. We assume that earnings decline linearly from age 58 to zero at age 65. Age 74 is the last period of life. In this draft, we assume that the new financial assets are no longer open after 1994 (1993 for the creation and destruction assets), an assumption that leads to smaller welfare effects. We concentrate on the 1939 birth cohort, for whom this assumption has a small impact. 28

Table 5 reports the absolute risk aversion levels (times 104 ) for men in the 1939 birth cohort. Absolute risk aversion declines with education, because expected earnings are greater for more educated men. Figure 4 displays the marginal propensity to consume out of income shocks as a function of sex, education and age. These values appear reasonable in light of empirical evidence on consumption responses to income shocks.26 One additional point is in order. We implement the model using the estimated covariance structure between asset returns and group-level earnings shocks. For this reason, we do not account for any within-group risk-sharing benefits of financial assets. Consequently, our welfare calculations miss a portion of the hedging benefits. However, our neglect of within-group risk sharing does not affect the equilbrium structure of asset returns, because within-group hedging demands net to zero.

7

The Risk-Sharing Benefits of Financial Assets

Table 5 helps us bridge the gap between the theory and the data by replicating Table 4 with information from one of our calculated equilibrium comparisons – the introduction of the Job Destruction asset to an economy in which agents can already trade the S&P 500. Table 5 illustrates several important points: • Endowed exposure (line (4)) generally drops over the life cycle as the present value multiplier (line (2)) drops. This is not always true – for college-educated men, notice that endowed exposure rises dramatically up to the age of 45 and then falls until retirement.27 • Average exposure (line (5)) drops over the sample, reflecting changing demographics. As our sample progresses, the percentage of more educated men (with low or negative correlation with Job Destruction) rises dramatically. • Equilibrium exposure drops over the life cycle as dynamic absolute risk aversion (line (2)) rises – which in turn rises because the marginal propensity to consume out of wealth (line (1)) rises. 26

Figure 4 helps explain why household consumption is much more responsive to movements in men’s hourly wages than women’s. The more persistent character of men’s shocks implies a bigger impact on human capital and a larger marginal propensity to consume, as shown in the figure. The greater annual earnings of men is also an important part of the explanation. 27 Note that these numbers are unscaled in contrast to Figure 1. Thus, while correlations for college educated men may be small, absolute covariance are relatively large.

29

• Agents do not always ‘hedge’ in the sense of reducing the absolute value of their exposure to a particular shock. Take college-educated men at age 35 (in 1974). Note that their endowed exposure about 1700 dollars but their equilibrium exposure is more that 3600 dollars. As with the example in Table 4, this is reflected in the fact that the total ‘hedging cost’ is higher than the ‘hedging benefit.’(see Table 7 which replicated Panel III of Table 4 for actual data). Table 6 shows the overall benefits to different members of the 1939 birth cohort of different assets for the base case where relative risk aversion is set to 3 and the real interest rate is set 2.5 percent.28 In general, the best performing assets are Job Destruction, Job Creation and the S&P 500 (and, of course, complete menu). We will concentrate on these assets for the rest of our discussion. To understand the sources of the benefits better, we decompose the benefits for four assets in Table 7. Note that: • The hedging benefits of the S&P 500 are relatively small compared with either Job Creation or Job Destruction, yet the overall benefits are often of the same order of magnitude. For example, for a less-than-high-school educated man, the hedging benefits of Job Creation are almost twice the hedging benefits of the S&P 500, yet the overall benefit is smaller. The key here is that the cost of hedging with the S&P 500 is very low (and in some cases negative), which reflects the fact that average exposure to the S&P 500 is almost zero.29 • The hedging costs of complete menu of group-level securities are high for everyone except less-than-high-school educated and college educated women. The comparison here is particularly stark. Note that the hedging benefits of complete menu are more than twice as big for college educated men as for college educated women. Yet the overall benefits are similar. The key again is the hedging cost – which is actually negative for women. This reflects the fact that women’s earnings and college-educated women’s in particular has very low correlation with other education groups and with the average. In the parlance of this paper, endowed exposure to college-educated women’s earnings risk is quite small for most groups 28

We assume that the S&P 500 is already available for all calculations except complete menu and of course, the S&P 500 itself. 29 This implies an equity premium that is close to zero – which obviously contradicts asset-return data. We consider hedging with a realistic equity premium in Table 9. Keep in mind that while inconsistent with asset return data, our results are consistent with consumption data (not just other equilibrium models) which imply a low equity premium as well.

30

These two points highlight the value of a general equilibrium approach to this problem. Figures 1 and 2 identified high correlations between Job Destruction and Job Creation and earnings of certain groups, but risk-sharing cannot survive on correlation alone. The theoretical treatment above shows that dispersion in correlation is as important as correlation alone. Table 8 shows the benefits of S&P, Job Destruction, Job Creation and complete menu for different levels of risk aversion.30 Note that: • Not surprisingly, benefits are linear in risk aversion.31 • For less-than-high-school and high school educated men, the cocktail of S&P 500, Job Destruction and Job Creation achieves a significant fraction of the benefits of a complete menu of group-level securities. For high-school educated men, the three financial assets achieve more than half the benefits of the complete menu. • For college-educated men and women, the benefits of the complete menu of group level securities are quite dramatic. For a woman with a college education, availability of a complete menu of group-level securities is worth more than a year of average earnings. However, financial assets achieve little of this benefit. For both college educated men and women, the cocktail of S&P 500, Job Destruction and Job Creation achieves less than ten percent of the benefit of the complete menu. Table 8 also shows the benefits of S&P, Job Destruction, Job Creation and complete menu for different levels of riskless rate. The benefits are decreasing in in the riskless rate. This is not hard to understand – the welfare measure is the present discounted value of a time series and raising the riskless rate lowers the the present discounted value. However, the effects, unlike the effects of a different specification of risk aversion are not uniform across groups. For example, with a riskless interest rate of 2.5 percent, the benefits of the S&P 500 are slightly higher for college educated men than less-than-high-school educated men, whereas with a riskless interest rate of 1 percent, they are more or less the same. This reflects different persistence of shocks and different timing of welfare improvements. 30

In this case, the benefits of the the three financial assets are cumulative – i.e. the benefits of Job Creation presume that one can already trade Job Destruction. 31 From Proposition 7, one can see that (1) the ratio of individual absolute risk aversion to the harmonic mean of risk aversion will not change and (2) we can take the first risk aversion term out of the present value operator.

31

Table 9 shows a decomposition of the benefits of the S&P 500 asset with an equity premium of 8 percent. These figures illustrate the usefulness of looking at the covariance between earnings and asset returns. The ‘Risk premium benefit’ line shows the benefit we traditionally ascribe to holding equities – the increase in expected returns less the increase in variance. However, by comparing that with the next line, overall benefit, we see that the traditional approach mis-states the benefits. For lessthan-high school educated men and college educated women, this benefit is mis-stated by more than ten percent. An alternative way to look at this is to consider what endowment of risky assets would be equivalent to labor income for the individual portfolio decision. Tables 10 and 11 show endowed exposure to the S&P 500 and mutual funds respectively in 1998 dollars. This shows that even an individual who holds no risky assets is exposed to equity market risk, often significantly so. For example, for a high school educated man, simply participating in the labor force is equivalent to a 16,000 dollar position in the S&P 500. For college educated men, their labor income risk is equivalent to a 60,000 dollar position in the mutual fund associated with their group.

8

Concluding Remarks

On the purely empirical front, this paper advances our knowledge of the covariance structure between earnings shocks and asset returns. Perhaps most notably, the correlation of earnings shocks with aggregate and own-industry equity returns rises with education. The correlation with aggregate and own-industry returns is negative for several sex-education groups. As we remarked earlier, this covariance structure is relevant for a variety of important topics in economics and finance including portfolio choice behavior, mutual fund structuring, pension fund management, savings behavior and asset pricing. We hope that our work stimulates a much more intensive investigation of the covariance structure and its implications. This paper also develops and implements a flexible, operational framework for addressing several related questions: 1. What is the optimal portfolio structure in the presence of risky labor income and multiple risky assets? 2. How does the optimal portfolio structure vary over the life cycle, and how does it differ among individuals and groups with different income processes and risk tolerances?

32

3. How large are the welfare gains from using financial assets to hedge labor income risks? Which new assets would offer the greatest benefit? 4. How does the structure of covariances among earnings shocks influence the equilibrim distribution of welfare gains from financial innovation? How is the distribution of these welfare gains influenced by heterogeneity in risk aversion? Our analysis suggests that the hedging motive plays a major role in optimal portfolio structure. For exogenous and realistic asset returns, the hedging gains from trading a broad equity index often amount to a large fraction of the gains from exploiting the equity premium. Based on this result, we conjecture that the hedging motive will dominate the premium exploitation motive in empirical implementations that provide a finer characterization of labor income risks and allow for a broad portfolio choice menu. For endogenously determined returns on risky assets in zero net supply, we find large welfare gains from a “full menu” of group-level assets and sizable gains from the opportunity to trade a small number of assets. To place the welfare results in proper perspective, we note three aspects of our empirical implementation that understate the hedging potential of financial assets. First, we ignore international risksharing possibilities in this paper, a potentially important source of welfare gains.32 Second, data limitations prevent us from examining the gains from sharing systematic risks within broad demographic groups. There are probably important systematic components of income risk associated with industry, occupation, region and firm size that we miss in our synthetic panel data. Third, our characterization of labor income probably understates low-frequency stochastic variation in earnings. All of these issues merit attention in future work, and they are easily treated within our framework. It is worthwhile to elaborate slightly on the point about within-group risk sharing. According to our empirical results, the standard deviation of group-level earnings shocks ranges from four to eight percent of mean earnings. In the Panel Study of Income Dynamics and other longitudinal data sets, researchers often find that the standard deviation exceeds twenty percent of mean earnings. Because the welfare gains to hedging rise linearly in the variance of income shocks, this observation implies potential hedging benefits as much as ten or fifteen times bigger than our estimates. Simply put, the gains from hedging labor income risks appear large. Our framework can be extended in several important directions without loss of tractability. First, endogenizing the risk-free rate is straightforward under suitable 32

See, for example, van Wincoop (1994), Lewis (1996) and Baxter and Jermann (1997).

33

restrictions. Endogenizing the risk-free rate requires information about the variance of individual-level earnings innovations, because this variance influences the level of precautionary savings. Second, it is also reasonably straightforward to treat risky assets in positive net supply. This extension is essential for an inquiry into the equity premium puzzle and the risk-free rate puzzle. Investigating these puzzles in our framework requires information about the net supplies of risky human and nonhuman assets. Third, the analytic solutions for consumption and portfolio allocations provided by Proposition 3 greatly simplify the treatment of short-sale constraints on risky assets. At the individual level, optimization subject to short-sale constraints involves a search over candidate solutions, each of which is easily calculated. This approach can handle short-sale constraints in dynamic settings with a large number of assets and a rich covariance structure. At the equilibrium level, short-sale constraints imply a more complicated fixed-point problem. But, here as well, the ability to compute candidate solutions analytically makes it feasible to handle economies with many agents, many assets and a rich covariance structure. Fourth, exogenous restrictions on asset market participation can be incorporated simply by letting the portfolio choice menu differ among agents. This extension introduces no new computational issues. Endogenous restrictions on asset market participation that arise from one-time fixed costs present essentially the same technical issues as short-sale constraints. These are very easily handled at the level of individual optimization, and well within the bounds of feasibility for equilibrium calculations.

34

References [1] Attanasio, Orazio and Steven J. Davis, 1996. Relative wage movements and the distribution of consumption. Journal of Political Economy, 104(6), pp. 12271262. [2] Baxter, Marianne and Urban J. Jermann, 1997, “The International Diversification Puzzle Is Worse than You Think,” American Economic Review, 87, number 1 (March) 170-180. [3] Brainard, William C. and F. Trenery Dolbear. Social risk and financial markets. American Economic Review, 61, 360-370. [4] Canner, Niko, N. Gregory Mankiw, and David N. Weil, 1998, “An Asset Allocation Puzzle,” American Economic Review, 87, no. 1 (March), 181-191. [5] Carroll, Christopher D., 1997, “Buffer-Stock Saving and the Life Cycle/Permanent Income Hypothesis,” Quarterly Journal of Economics, 112, no. 1 (February), 1-55. [6] Davis, Steven J., John C. Haltiwanger and Scott Schuh, 1996, Job Creation and Destruction. Cambridge, Massachusetts: MIT Press. [7] Davis, Steven J. and John Haltiwanger, 1999, “Gross Job Flows.” Forthcoming in Orley Ashenfelter and David Card, editors, Handbook of Labor Economics, Volume 3. Amsterdam: North-Holland Press. [8] Davis, Steven J. and John Haltiwanger, 1999, “On the Driving Forces Behind Fluctuations in Employment and Job Reallocation,” forthcoming in the American Economic Review. [9] Demange, Gabrielle and Guy Laroque, 1995. “Optimality of incomplete markets.” Journal of Economic Theory, 65, 218-232. [10] Flavin, Marjorie and Takashi Yamashita, 1998, “Owner-Occupied Housing and the Composition of the Household Portfolio over the Life Cycle,” NBER Working Paper Number 6389. [11] Goldin, Claudia and Lawrence F. Katz, 1996, “The Origins of Capital-Skill Complementarity,” NBER Working Paper Number 5567. [12] Hamermesh, Daniel S., 1993, Labor Demand. Princeton, New Jersey: Princeton University Press. 35

[13] Heaton, John and Deborah J. Lucas, 1996. Evaluating the effects of incomplete markets on risk sharing and asset pricing. Journal of Political Economy, 104(3), 443-487. [14] Heaton, John and Deborah J. Lucas, 1997, “Portfolio Choice and Asset Returns: The Importance of Entrepreneurial Risk,” Working Paper Number 234, Kellogg Graduate School of Management, Northwestern University. [15] Jaeger, David A. “Reconciling the Old and New Census Bureau Education Questions: Recommendations for Researchers,” Journal of Business and Economic Statistics, 15, no. 3, April 1997, 300-309. [16] Judd, Kenneth L., Felix Kubler and Karl Schmedders, 1998, “Incomplete Asset Markets with Heterogeneous Tastes and Idiosyncratic Income,” the Hoover Institution, Stanford University. [17] Katz, Lawrence F. and David H. Autor, 1999, “Changes in the Wage Structure and Earnings Inequality.” Forthcoming in Orley Ashenfelter and David Card, editors, Handbook of Labor Economics, Volume 3. Amsterdam: North-Holland Press. [18] Katz, Lawrence F. and Kevin M. Murphy, 1992, “Changes in Relative Wages, 1963-87: Supply and Demand Factors,” Quarterly Journal of Economics, 107, 35-78. [19] Levy, Frank and Richard J. Murnane, 1992, “U.S. Earnings Levels and Earnings Inequality: A Review of Recent Trends and Proposed Explanations,” Journal of Economic Literature, 30, no. 3: 1333-81. [20] Lewis, Karen K., 1996, “What Can Explain the Apparent Lack of International Consumption Risk Sharing?” Journal of Political Economy, 104, number 2 (April), 267-297. [21] MaCurdy, Thomas E., 1982, “The Use of Time Series Processes to Model the Error Structure of Earnings in a Longitudinal Data Analysis,” Journal of Econometrics, 18, 83-114. [22] Papke, Leslie, 1998, “How Are Participants Investing Their Accounts in Participant-Directed Individual Account Pension Plans?” American Economic Review, 88, no. 2 (May), 212-216.

36

[23] Poterba, James M. and Andrew A. Samwick, 1997, “Household Portfolio Allocation over the Life Cycle,” NBER Working Paper Number 6185. [24] Poterba, James M., Steven F. Venti and David Wise, 1996, “Individual Financial Decisions in Retirement Savings Plans and the Provision of Resources for Retirement,” NBER Working Paper Number 5762. [25] Shiller, Robert, 1994. Macro Markets (Oxford: Clarendon Press). [26] Spletzer, James R., 1997, “Longitudinal Establishment Microdata at the Bureau of Labor Statistics: Development, Uses, and Access,” Proceedings of the American Statistical Association. [27] Sunden, Annika E., and Brian J. Surette, 1998, “Gender Differences in the Allocation of Assets in Retirement Savings Plans,” American Economic Review, 88, no. 2 (May), 207-211. [28] van Wincoop, Eric, 1994, “Welfare Gains from International Risksharing,” Journal of Monetary Economics, 34, no. 2 (October), 175-200. [29] Willen, Paul S. 1999a. “Welfare, Financial Innovation and Self Insurance in Dynamic Incomplete Markets Models,” Princeton University, May. [30] Willen, Paul, 1999b. “New financial markets: Who gains and who loses?” Manuscript. Princeton University: Princeton.

37

Appendix A. Constructing Group-Specific Mutual Funds Section 3.2 describes the construction of group-specific mutual funds and the calculation of returns and covariances for these synthetic assets. Table A.1 presents the concordance that we prepared to link the Census Industrial Classification system (CIC) used in the CPS to the Standard Industrial Classification system (SIC) used by CRSP. Table A.1 Industry Concordance

38

1

Code 1100 1200 1300 2000 3100.1 3100.2 3200.1 3200.2 3200.3 3300.1 3300.2 3410 3420 3500.1 3500.2 3500.3 3600 3700 3810 3820 3830 3840.1 3840.2 3840.3

Industry Description Agriculture Forestry and Logging Fishing and Hunting Mining Food and Beverages Tobacco Textiles Apparel Leather Lumber Furniture Pulp and Paper Products Printing and Publishing Chemicals Petroleum and Coal Products Rubber and Plastics Stone, Clay and Glass Products Basic (Primary) Metals Fabricated Metals Nonelectrical Machinery Electrical Machinery Automobiles Aircraft and Missiles Other Transport Equipment

1987 SIC 01, 02, 07 08, 241 09 10, 12, 13, 14 20 21 22, 31 23 31 24 (ex. 241) 25 26 27 28 29 30 32 33 34 35 36 371 372, 376 373, 374, 375, 379

3850 3900 4100

38 39 46, 49 (ex. 495)

5000 6100 6200

Professional, Scient. Instruments Miscellaneous Manufacturing Electricity, Gas, Steam and Water Works Construction Wholesale Trade Retail Trade

6310 6320

Eating/Drinking Establishments Hotels and Other Lodging

39

15, 50, 52, 57, 58 70

16, 17 51 53, 54, 55, 56, 59

2

1980 CIC 10, 11, 20, 21 30 31 40-42, 50 100-122 130 132-150 151-152 220-222 230-231 242 160-162 171-172 180-192 200-201 210-212 250-262 270-280 281-291, 300-301, 310-332 340-350 351 292, 352, 362 232, 360, 361, 370 371-382 390-392 422, 460-470, 472 60 500-571 580-640, 1 642-691 641 762-770

Number of Years3

Code 7111 7112 7114

Industry Description Railway Transport Other Passenger Transport Freight Transport by Road

1987 SIC 40, 474 41, 4725 42, 472 (ex. 4725), 473, 478 752 751 44 45 43 48 60, 61 62, 67 63, 64 65, 7349 81 872 737 (ex. 7377) 871, 7336

7116.1 7116.9 7120 7130 7201 7202 8100.1 8100.2 8200 8310 8321 8322 8323 8324

Parking Services Vehicle Rental Services Water Transport Air Transport Postal Service Telecommunications Banking and Credit Agencies Other Financial Institutions Insurance Real Estate Legal Services Accounting and Auditing Computer and Data Processing Engineering, Architectural and Technical Services 8325 Advertising Services 731, 7331 8329.1 Typing, Duplicating, Copying 7334, 7338 8329.2 Business Management and 874 Consulting Services 8329.9 Business Services nec 732, 7334, 7338, 736 including: 7383, 7389, 89 Business Machinery and Equip. 735, 7377 Rental and Leasing 9101 Public Administration part 91-96 9102 National Defense 97 9103 Police and Security Services 7381, 7382 9104 Fire Protection part 91-92 9200 Sanitary Services 495, 7342 9310 Education Services 82 (ex. 823), 833 9320 Research and Scientific Institutes 873

40

1980 CIC 400 401-402 410-411, 432 None None 420 421 412 440-442 700-702 710 711 712,722 841 + govt.4 890 740 882 721 None 732 731, 742

part 900-9315 932 741, 910, govt.6 government7 471 842-862 730, 891

Number of Years

Code Industry Description 9330 Medical, Dental, Health Services 9340 Welfare Services 9350 Business, Professional and Labor Associations 9390 Religious, Political, Social and International Organizations 9400 Recreational and Cultural Services 9510 Repair Services nec 9520 Laundries and Cleaning Services 9530 Private Household Workers 9590 Other Personal Services

1987 SIC 80 83 (ex. 833) 861, 862, 863

1980 CIC 812-840 870, 871, 9228 881

864, 865, 866, 869

880

78, 79, 823, 84 753, 754, 76 721 88 72 (ex. 721), 7335, 7384

800-802, 872 750-760 771 761 772-791

Notes to Table A.1. 1. The 1987 U.S. Standard Industrial Classification codes. There were important changes in the classification of some industry groups during our sample period. Computer programs related to our treatment of these changes are available upon request. 2. The 1980 U.S. Cenus Industrial Classification codes. These codes are used in the CPS for wage years 1982-1990 and 1992-1994. The classification scheme for 1991 is highly similar. Years prior to 1982 used a somewhat different industry classification scheme. Computer programs related to our treatment of changes in the CIC are available upon request. 3. The number of years in the sample period for which we can successfully match the CIC and SIC codes and construct an industry-level equity return index. 4. Includes government lawyers and judges. 5. This category includes state, local and federal government employees except for persons coded in (i) CIC industries 910, 922 and 932 and (ii) Census occupation categories 6, 174, 178-179 and 413-427. 6. Includes state, local and government employees in Census occupation codes 6, 414-415, and 418-427. 7. State, local and government employees in Census occupation code 413, 416 and 417. 8. Includes government employees in Census occupation 174, regardless of whether they are classified in industry code 922.

41

Number of Years

Appendix B: Proofs of Theoretical Propositions This appendix is in rough form. The theoretical model is based on Willen (1998), which contains a fuller discussion and more detailed proofs. Here we provide a somewhat abbreviated treatment. Two basic results underlie our propositions: first, that consumption is affine in  wealth cht = aht Wth − bht , and, second, that the Euler equations have the form derived in the lemma below. The first result is well known (e.g., Caballero, 1990), although its derivation under the conditions of this model is somewhat tedious. The interested reader is referred to Willen (1998). We first prove the lemma that gives the Euler equations governing consumption and portfolio allocations. We then show that this lemma, when combined with the result that consumption is affine in wealth, allows us to prove Proposition 1 on individual optimization and Proposition 2 on equilibrium asset prices. Lemma 1 Assume that returns distributions are non-stochastic. Assume that c˜ht and Rt are jointly normally distributed. The for any h:     ˜ j,t+1 − R0,t+1 = Ah cov c˜h + 1, R ˜ j,t+1 E R (11) t E

c˜ht+1





cht

 1 Ah = var c˜ht+1 + h ln R0,t+1 2 A

(12)

Proof: The intertemporal Euler equation gives us for any h   uhC cht = R0,t+1 δ h E uhC e cht+1     ˜ j,t+1 j = 1, ..., J cht+1 R uhC cht = δ h E uhC e     ˜ j,t+1 + By definition of covariance and Stein’s lemma: uhC cht = δ h E uhC e cht+1 E R      h ˜ j,t+1 for all j. Dividing through by uh ch = R0,t+1 δ h E uh e cht+1 cov e cht+1 , R δ h E uhCC e C t C ct+1 E u h (e ch t+1 ) = Ah we get equation (11). Because e and noting that − E uCC cht+1 is normal h e h C (ct+1 )    (A h )2  h h h h h h exp −A ct = R0,t+1 δ exp −A E e ct+1 + 2 var e ct+1 and we solve for equation (12)

Proof: (Proposition 1) We will show this for T − 1. It is easy to continue for t < T −1. Using Lemma 1 (equation (12)), and substituting in the budget constraint, we get: E

y˜Th



+

h R0,T ω0,T −1

+

J X

   h h h h h ln δ h R0,T E (Rj,T ) ωj,T −1 = cT −1 + A /2 var cT + 1/A

j=1

42

Substitute in the budget constraint for T − 1 ωTh −1 = ITh−1 − solve for chT −1 for the solution at T − 1. 

PJ

j=1

h h ωj,T −1 − cT −1 and

h h h Proof:  (Proposition 3)At T − 1, by  proposition 1, cT = aT GW  Lemma   T . By ˜ j,T − E R ˜ j,T − R0,T = A∗ cov N0 η ˜ j,T . By definition, R ˜ j,T = x˜j,T . ˜T , R 1, E R T Pj,T −1

ˆ . By definition of PT , this implies that Thus ERT = diag (Pt )−1 A∗T β T   ˆT PT −1 = π0 E d˜T − π0 A∗T β

(13)

At T − 2, by Proposition 1, chT = ahT GWTh . By Lemma 1,     ∗ 0 ˜ ˜ ˜ T −1 , Rj,T −1 . E Rj,T −1 − R0,T −1 = AT −1 cov N diag (ΨT −1 ) η By definition,   x˜j,T −1 + Pj,T −1 ˜ j,T ˜ j,T − E R R = Pj,T −2 Λj,T −1x˜j,T −1 = Pj,T −2

(14)

which which implies equation (5). The subsequent equations follow from their definitions  Proof: (Proposition 5) The Euler equation (equation (12)) tells us that exp −Ah cht =  δ h R0,t+1 E exp −Ah c˜ht+1 which implies that: τ   δh 1 1 − h E exp −Ah c˜hτ = − h Πτs=1 exp −Ah cht A A R0,s    h Which implies that U h C h = − A1h PDVth {1}Ts=th exp −Ah ch0 . It is easy to see     h that: U h C h + θ = − A1h PDVth {1}Ts=th exp −Ah ch0 + θ . Setting U h (C + θ) = U h (C ∗ ), we can solve for θ



Proof: (Proposition ??) c˜ht+1 is a linear combination of normally distributed random variables and is thus normally distributed itself. Equation (11) from Lemma 1   h h ˜ j,t+1 ∀j. Substituting in the budget constraint gives us: ERj,t+1 = A cov c˜t+1 , R    P  h ˜ i,t+1 , R ˜ j,t+1 + J cov ω h R ˜ j,t+1 ∀j and re-organizing , we get: ah 1 Ah ERj,t+1 = Ψht cov η˜t+1 ,R i,t i=1 t+1

1 ERt+1 Ah t+1 −1 h h Ψt+1 Σt+1 βt+1

Using our matrix definitions, we can re-write this for any individual as: h Ψht+1 βt+1

+

Σt+1 ωth

which we solve to get:

ωth

=

1

Ah t+1

Σ−1 t+1 ERt+1





=

Proof: (Proposition 7) Substituting in the optimal portfolio solution, we get the excess return terms:  0 −1 h h h ER0s ωs−1 = 1/Ahs ER0s Σ−1 (15) s ERs − Ψs ERs Σs βs 43

And the precautionary savings term is. Ahs /2



var

Wsh



=

Ahs /2



2 Ψhs

var

η˜sh





h βsh0 Σ−1 s βs



+

Ahs /2



(Ahs )2

ER0s Σ−1 s ERs (16)

 0 −1 h h Our solution follows from from the fact that that: .5 1/Ahs ER0s Σ−1 s ERs −Ψs ER s+ s Σs β        0 p p p p  2 h −1 h Ψh β h − 1/ h ER h Ψh β h − 1/ Ah ER .5Ahs Ψhs βsh0 Σ−1 β = .5 A A Σ A s s s s s s s s s s s s s



44

Table 1: CPS Sample Sizes for Sex-Education-Cohort-Year Cells, 1965-94 Sex Men Men Men Men Women Women Women Women

Education Level < High School High School Some College ≥ College < High School High School Some College ≥ College

Mean Cell Size 477 688 403 496 277 650 328 310

45

Minimum Cell Size 188 222 86 90 136 185 55 57

Table 2: Estimated Labor Income Processes, ARIMA(0,1,2) Specification

Annual Earnings MA(1) MA(2) st. error of η PVM (age=35) PVM (age=50) Log Earnings MA(1) MA(2) st. error of η PVM (age=35) PVM (age=50)