Using Financial Assets to Hedge Labor Income Risks: Estimating the Benefits

Transcription

1 Using Financial Assets to Hedge Labor Income Risks: Estimating te Benefits Steven J. Davis Graduate Scool of Business University of Cicago and NBER Paul Willen Department of Economics Princeton University First version: July 1998 Tis version: Marc 23, 2000 JEL Nos: G11, D91, D52, J30 We tank Jeremy Nalewaik for excellent researc assistance. Davis gratefully acknowledges researc support from te Graduate Scool of Business at te University of Cicago. risksv2s.tex Pone: (773) Fax: (773) Pone: (609)

2 Abstract We caracterize te covariance structure between asset returns and labor income socks for syntetic persons defined in terms of sex, education and birt coort. Te correlation of income socks wit bot aggregate and own-industry equity returns tends to rise wit educational attainment and, surprisingly, is negative for several sex-education groups. We ten develop a tractable equilibrium life-cycle model wit incomplete markets. We implement te model using te estimated covariance structure and oter data in order to evaluate te portfolio coice and welfare implications of edging wit financial assets. Tere are large equilibrium risk-saring 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 demograpic groups. Te edging motive as significant consequences for te structure of te optimal portfolio. 1

3 1 Introduction Among te most important economic risks confronting ouseolds is te uncertain nature of labor income. New financial assets create new opportunities to sare tis risk, and so do financial innovations tat facilitate better use of existing assets. Tese observations prompt several questions regarding te edging role of financial assets: How does te edging motive affect optimal portfolio structure? How large are te welfare gains from using financial assets to edge labor income risk? Wo benefits in equilibrium from financial innovations tat expand opportunities to sare tis risk? To address tese sorts of questions, we develop a framework tat can be tailored to particular economies and financial innovations, implemented using available data, and readily solved to compute equilibrium outcomes. Te framework is a dynamic equilibrium excange economy wit incomplete markets, normal random variables and exponential preferences. Willen (1999b) sows ow to address tese questions in a two-period version of te model. We extend is framework to encompass overlapping generations and a flexible life-cycle structure for earnings and consumption. We implement te framework using U.S. data from 1965 to 1994 on security returns and on labor income for syntetic persons defined in terms of birt coort, educational attainment and sex. We quantify te risk-saring benefits associated wit two types of financial innovations. First, in te spirit of Siller (1993), we consider te introduction of new financial assets wit payoffs tied to measures of aggregate, sectoral or group-level outcomes. Second, we consider mutual funds constructed from existing securities in suc a way as to mirror te industry distribution of employment for eac syntetic person at eac point in time. If te industry-level components of returns to equity and uman capital are correlated, tese mutual funds expand risksaring 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 rigt, we caracterize te covariance between returns on financial assets and returns on uman capital. Rater surprisingly, few previous studies investigate tis covariance despite its obvious relevance to individual portfolio optimization, mutual fund structuring, pension fund management, savings beavior 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 teories of consumption empasize te role of borrowing and lending, wic is also important in our analysis, but sidestep te pure edging function of financial assets. Asset-pricing studies tat consider a role for labor income risk typically restrict attention to igly aggregated labor income measures or caracterize individual income uncertainty in ways tat preclude cross-sectional variation in te 2

4 Tis neglect is also surprising in anoter respect: Sifts in relative earnings among individuals and demograpic groups ave been te focus of extraordinary researc and policy attention in recent years. Levy and Murnane (1992) and Katz and Autor (1999) review an extensive body of work in tis area. Neiter review devotes any attention to te idea tat financial assets migt serve to edge income risks. For eac sex-education group, we estimate te covariance between financial asset returns and labor income socks. Te correlation between aggregate equity returns and labor income socks ranges from -.25 over most of te life-cycle for te least educated men to.25 or more for college-educated women. For bot men and women, te correlation between income socks and equity returns tends to rise wit educational attainment. Te correlation between income socks and own-industry equity returns also tends to rise wit educational attainment. In fact, for men wit less tan a college education and certain educational groups of women, labor income socks covary negatively wit own-industry equity returns. Artificial assets wit payoffs tied to job creation and destruction rates are igly correlated wit te income socks of less-educated men. 2 Tree assets te S&P 500 plus te job creation and destruction assets jointly account for between 27 and 40 percent of group-level earnings risk for less-educated men Given te stocastic properties of labor income, including its covariance structure wit financial asset returns, we calculate equilibrium asset prices and optimal consumption and portfolio allocations for eac (syntetic) individual. Tese calculations require an assumption about te asset coice menu over wic individuals optimize. Te available evidence suggests tat neiter individuals nor pension fund managers employ sopisticated portfolio strategies to edge earnings risk. 3 Based edging portfolio. Heaton and Lucas (1997) consider ow equity returns covary wit labor income and proprietary business income. Tey report a small negative correlation of equity returns wit aggregate wage income and a larger, but still modest, positive correlation wit aggregate proprietary business income. Tey pursue te implications of tis covariance structure for cross-sectional variation in portfolio coice and for asset pricing. 2 Te job creation and job destruction rates are calculated from establisment-level employment canges, as described in Davis, Haltiwanger and Scu (1996). Many researcers and government agencies trougout te world ave begun to construct and report statistics of tis sort. See Davis and Haltiwanger (1999) for a review of tis work. Te U.S. Bureau of Labor Statistics plans to begin publising official job destruction statistics of tis type for most sectors of te U.S. economy in te near future (Spletzer, 1997). 3 Most U.S. ouseolds ave small financial asset oldings and even smaller oldings of equity securities. See, for example, Poterba and Samwick (1997) and Flavin and Yamasita (1998). Empirical work on portfolio allocation beavior devotes little attention to edging labor income risk. See, in addition, Poterba et al. (1996), Papke (1998) and Sunden and Surette (1998). Professional financial advisors place little weigt on edging labor income risk (Canner, Mankiw and Weil, 1998). 3

5 on tis evidence, we proceed under te assumption tat individuals (or teir portfolio managers) allocate financial wealt between a riskless asset and a broad-based equity portfolio. 4 Tis assumption yields a bencmark dynamic equilibrium, against wic we evaluate te effects of financial innovations. Relative to tis bencmark, tere are large welfare benefits from introducing securities tat allow full saring 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, te equilibrium welfare gains for college-educated men amount to nearly 27,000 per person in 1998 dollars. Bigger socks, greater risk aversion and lower discount rates imply larger welfare gains. Te size of equilibrium welfare gains also depends on te covariance structure among earnings socks. For example, college-educated women reap relatively large equilibrium gains from te introduction of a full asset menu for saring group-level risks, because teir earnings socks ave low correlation wit per capita earnings socks. Te full menu experiment involves te introduction of many new assets, but sometimes a single asset can substantially improve welfare for certain groups. As an example, among men wo did not obtain post-secondary scooling, te equilibrium benefit of trading te job destruction asset exceeds 1,800 dollars per person. Traditional approaces to portfolio coice typically ignore te covariance between labor income socks and equity returns. Our empirical results and teoretical analysis imply tat tis approac can misstate te ouseold s optimal portfolio allocation and te welfare effects of olding equities. We sow tat te 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 te S&P 500. For less-tan-ig-scool educated men, labor income risk is equivalent to a 16,000 dollar sort position in te S&P 500. Te paper proceeds as follows. Section 2 constructs and examines labor income data for syntetic persons. We caracterize systematic and stocastic aspects of labor 4 Our teoretical framework easily accommodates oter assumptions about portfolio coice beavior, including coice over a large asset menu, but we require some assumption or estimate regarding portfolio coice beavior in order to calculate te initial equilibrium and te equilibrium response to financial innovation. 5 Attanasio and Davis (1996) sow tat canges in te coort-education structure of pre-tax ourly wages among men drove large canges in te between-group distribution of ouseold consumption during te 1980s. Tey also calculate large, but unrealized, welfare gains from saring tis type of group-level earnings risk. We assess te extent to wic financial assets can be used to acieve some of te potential between-group risk-saring gains uncovered by Attanasio and Davis. However, te metodology we develop in tis paper is more general; it can be applied to panel data on countries, groups or individuals. 4

6 income variation for eac syntetic person, and we calculate te size of socks to te present value of lifetime earnings (as a function of age, education and sex) implied by our caracterization of te labor income process. Section 3 describes te data on financial asset returns and our procedures for constructing new financial assets and group-specific mutual funds. Section 4 investigates te covariance between uman capital returns and financial asset returns. Section 5 sets fort a dynamic equilibrium life-cycle framework wit incomplete markets. We sow ow to calculate asset prices, consumption pats and portfolio allocations. We also explain ow to calculate te welfare effects of new risk-saring opportunities in terms of te equilibrium response to financial innovation. Section 6 draws on our caracterization of labor income beavior, portfolio coice menus and oter data to estimate te initial dynamic equilibrium. Section 7 ten quantifies te equilibrium effects of financial innovations tat take te form of expanded portfolio coice menus. Section 8 concludes. 2 Labor Income 2.1 Syntetic Panel Data Te Annual Demograpic Files of te Marc Current Population Survey (CPS) contain individual data on pre-tax labor earnings in te previous calendar year. Using te 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 birt coort. By following particular groups over time as tey age, we obtain longitudinal data on syntetic persons. For eac sex, we group persons into four educational attainment categories: less tan ig scool, ig scool, some college but no degree, and a college education (including persons wit post-graduate scooling). 6 Witin eac sex-education group, we consider tree-year coorts indexed by te birt year of te middle coort members. For example, te 1956 coort contains persons born between 1955 and We also measure coort age based on midpoints, so tat te 1956 coort is 34 years old in We exclude observations for wic any coort members are less tan 24 or more tan 59 years old. We exclude coorts wit fewer tan seven annual observations. After imposing tese restrictions and allowing for two lags in te specification 6 We construct time-consistent categories across te break in te CPS education codes following te recommendations of Jaeger (1997). 7 More precisely, because te CPS records age at te Marc survey date, te 1956 coort contains persons born between Marc of 1955 and February of

7 of te labor income process, our sample runs from calendar years 1965 to 1994 for coorts born between 1911 and Te panel contains mean annual earnings and mean log earnings for syntetic persons wo are between 27 and 58 years old. Te sample selection criteria and our procedures for grouping by sex, education and birt coort yield panel data for = 144 syntetic persons. 8 After using one observation to compute lags in our preferred specification for te income process, tere are between 6 and 31 consecutive annual observations for eac syntetic person. To compute earnings for syntetic persons, we use CPS data on wage and salary workers in te private and public sectors. We exclude unincorporated self-employed persons from te earnings calculations, but we include self-employment and farm income for persons wo are mainly wage and salary workers. We also exclude persons wo were students or in te military at least part year. 9 We express earnings in 1982 dollars using te GDP deflator for personal consumption expenditures. Table 1 reports summary information about te sample sizes in te cells tat underlie te syntetic panel data. Mean sample sizes number in te undreds, and no cell as fewer tan 55 observations. 2.2 Income Processes for Syntetic Persons To estimate te risk-saring potential of financial assets, we must first identify and caracterize te stocastic component of earnings variation for eac (syntetic) person. Given a model for labor income, we can identify socks, calculate teir effects on te present value of lifetime earnings, and investigate ow te earnings socks covary wit asset returns. We caracterize te labor income process presently and take up te covariance between earnings socks 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 te specification vary freely across sex-education groups. For eac sex-education group, we pool over birt coorts, allow for coort fixed effects, and estimate te earnings specification by nonlinear least squares. We experimented wit several specifications in te 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. Tese specifications 8 In any given calendar year, 10 or 11 (9 in 1994) birt coorts in eac sex-education group satisfy te age restriction, wic corresponds to 80 or 88 (72 in 1994) syntetic persons. 9 In addition, we exclude persons wo report an ourly wage less tan 75 percent of te federal minimum. We andle top-coded earnings observations in te same manner as Katz and Murpy (1992). 6

8 deliver similar results for te covariance between income socks and asset returns, but te stationary processes imply muc smaller effects of income socks on te present value of lifetime earnings. Wen fed into our teoretical model, te stationary processes also imply small consumption responses to income socks, wereas te integrated processes imply empirically plausible consumption responses. On tis basis, we prefer te integrated processes. Te two integrated processes we considered deliver similar results, and we encefort restrict attention to te ARIMA(0,1,2) process. 10 We include a (second-order) tird-order age polynomial in te (log) eanings specifications. Differencing reduces te order of te age polynomial by one and sweeps out te coort fixed effects. Tus, te fitted earnings specification for a particular sex-education group is yt = α 0 + α 1 age t + α 2 (age2 ) t + ɛ t, ɛ t = η t + ψ 1 η t 1 + ψ 2 η t 2, were yt denotes mean annual earnings for coort at t, teα s are coefficients in te age polynomial, ɛ is a moving-average residual, η t is te earnings innovation at t, and te ψ s are moving-average coefficients. Table 2 reports te estimated MA parameters and oter statistics. Based on te sign and size of te moving-average coefficients, earnings socks sow greater persistence for men tan for women at eac education level. Men wit a ig scool education sow te most persistent response to earnings socks, and women wit some college or less tan a ig scool education sow te least persistent responses. Earnings innovations for women are larger in percentage terms tan for comparably educated men but smaller in absolute terms. 2.3 Socks to te Present Value of Lifetime Earnings Te potential welfare gains from improved risk saring depend partly on te size of labor income socks, as measured by te impact of a typical income innovation on te present value of lifetime earnings. To address tis matter, table 2 reports ow innovations to current labor income affect te expected present value of remaining lifetime earnings. Te reported present value multiplier equals te cumulative impact of a unit earnings innovation assuming a one percent annual discount rate and retirement after age MaCurdy (1982) provides a useful discussion of ow to specify and estimate stocastic specifications for individual earnings in longitudinal data. Based on standard time-series diagnostics, e arrives at te ARIMA(0,1,2) process as te preferred specification in is analysis of annual earnings. 7

9 At 35 years of age, te present value multipliers range from 13 to 20 for men and from 6 to 14 for women. Tese values imply large effects of a typical income sock on lifetime earnings. For example, based on te specification for 35-year old men wit a ig scool education, a unit standard deviation innovation in mean annual earnings canges te expected present value of lifetime earnings by 15, 566(= ) in 1982 dollars. Te corresponding figure for college-educated men is about 31,000 dollars, but it is only about 4,500 dollars for women wo do not ave a college degree. Tese large differences across sex-education groups (and over te life cycle) in te impact of an income sock on te present value of lifetime earnings translate into large differences in optimal edging portfolios, as we sow in Section 5. 3 Financial Asset Returns 3.1 Equity Returns and Artificial Securities To measure returns on a broad equity index, we use te annual rate of return on te S&P 500, as reported by Datastream. We also construct returns for several artificial securities tat reflect movements in output and oter economic aggregates. For output, we use real GDP in 1992 dollars from te National Income and Product Accounts (NIPA). It is convenient for our purposes to ortogonalize returns on te GDP asset wit respect to returns on te S&P 500 index. Te ortogonalized GDP asset pays off te residual in an OLS regression of GDP on te rate of return on te S&P 500 index. We construct (ortogonalized) returns for oter artificial securities in te same manner using te following data: Consumption: real personal consumption expenditures in 1992 dollars. Employment: civilian employment among persons 16 years and older. Unemployment: te civilian unemployment rate. Job Creation and Destruction: te gross annual rates of job creation and job destruction in te manufacturing sector, as described in Davis, Haltiwanger and Scu (1996). We update te DHS data troug 1993 and extend tem 11 Te present value multipliers for te annual earnings specification are easily calculated from te recursion, PVM(age) =PVM(age +1)+(1+r) 58 age (1 ψ 1 ψ 2 )forage 56, were r denotes te annual discount rate, PVM(58) = 1, and PVM(57) = 1 + (1 + r) 1 (1 ψ 1 ). For convenience, we calculate te present value multipliers in te same way for te log earnings specification; i.e., ignoring te slope in te expected lifetime earnings profile. 8

10 back to 1965 using te metodology described in te 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 wit individual-level data on industry of employment. To link te firm and worker data at te industry level, we prepared a concordance between te Standard Industrial Classification (SIC) used in firm-level data on equity returns and te Census Industrial Classification (CIC) used in te CPS. Appendix A describes te concordance and lists our 62 industry categories. Given te concordance, we construct returns on group-specific mutual funds in tree steps. First, we acquired montly data on firm-level equity returns (inclusive of dividends and oter distributions) from te Center for Researc on Security Prices (CRSP). From tese data, we compute annual value-weigted equity returns for te industry categories. Second, using te CPS data described in section 2.1, we calculate te industry sares of earnings and ours for eac syntetic person in eac year from 1967 to Te earnings-weigted calculations draw on data for wage and salary workers only, wereas te ours-weigted calculations include self-employed persons. Tird, we compute te weigted mean equity return for eac syntetic person in eac year using te industry-level returns from step one and te industry weigts from step two. Depending on sex and education, we can assign an industry-level equity return to about percent of observed ours and earnings. A preliminary analysis gave igly similar results for te ours-weigted and earnings-weigted mutual fund returns. Te reported results make use of ours-weigted returns. Wen we introduce te group-specific mutual funds into our teoretical model, we require a covariance matrix for mutual fund returns. We estimate tis covariance matrix in two steps. First, we estimate te covariance matrix of industry-level annual returns using CRSP-based data from 1965 to We assume tat tis covariance matrix is stationary over time. Second, we combine te covariance matrix of industrylevel returns wit te CPS-based industry weigts to compute te implied covariance between eac pair of mutual funds in eac year. Tis procedure yields a time-varying covariance matrix for te group-specific mutual funds. 9

11 4 Covariation between Labor Income Socks and Asset Returns In tis section, we caracterize te covariance structure between labor income socks and various asset returns. We first caracterize te covariance for te S&P 500 as a function of sex, education and age. We ten consider artificial financial assets based on aggregate outcomes. We turn last to te covariance between group-level labor income socks and group-specific mutual funds. For risk-saring purposes, we care about two aspects of te covariance structure. First, we seek assets wit non-zero covariances for at least some group s income socks, so tat persons can edge by adopting a long or sort position. Second, we seek assets wit eterogeneity across people in te covariances. Absent suc eterogeneity, an asset in zero net supply does not enable te saring of income risk among persons wit equal risk tolerances. Of course, suc an asset is still useful for allocating labor income risk toward persons wit greater risk tolerance. 4.1 Te Covariance Structure for a Broad Equity Index To estimate te covariance structure for te equity index, we fit regressions of te following form for eac sex-education group: η t = β 0 x t + β 1 age t x t + β 2 age 2 t x t + β 3 age 3 t x t, were η t is te earnings innovation estimated in Table 2, and x t is te rate of return on te S&P 500 in year t. Weestimateteβ parameters by least squares to caracterize te covariance structure as a smoot function of age. As in Table 2, te regression for eac group contains 331 annual observations on 18 syntetic individuals for te period. Table 3 reports R 2 values for regressions of earnings innovations on te S&P 500 in te top row of eac panel. Te R 2 values are small for all groups, not more tan 3 percent in most cases and always less tan 10 percent. Te small R 2 values imply tat a broad equity index affords very limited scope for workers to edge earnings risk or sare risk. Figures 1 and 2 display te covariance as a function of sex, education and age for te annual earnings specification. Te upper left panels pertain to te regressions on te S&P 500. We construct tese figures as follows. First, before running te regressions, we transform te asset return to ave unit variance. Second, we evaluate te regression function at eac age and divide troug by te standard deviation of te labor income sock at tat age to obtain a correlation coefficient. 10

12 According to figures 1 and 2, te S&P 500 is a good candidate for risk saring in te sense tat its correlation differs a lot across people. Te correlation wit income socks is around -.25 for men wo did not finis ig scool. In contrast, te correlation is positive for college-educated workers, ranging from 0 to.3 over te working life for men and generally exceeding.2 for women. So, even toug returns on te S&P 500 account for a very modest portion of variation in earnings socks, te eterogeneity in correlations across groups suggests some potential as a risk-saring tool. Te correlation structure for aggregate equity returns fits reasonably well wit a large body of researc on te demand for labor. Empirical studies of labor demand in te modern economy consistently find tat more skilled (i.e., educated) workers are relatively complementary to pysical capital, te use of advanced tecnologies and researc and development activity. 12 In ligt of tis evidence, and to te extent tat equity value derives from residual claims on firms pysical capital, intellectual property and tecnological know-ow, one migt anticipate tat te correlation between aggregate equity returns and earnings socks rises wit education. Figures 1 and 2 largely support tat 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 eac new asset and eac group, we fit regressions of te form: η t = β 0 x t + β 1 age t x t + β 2 age 2 t x t + β 3 age 3 t x t + β 4 z t + β 5 age t z t + β 6 age 2 t z t + β 7 age 3 t z t, were z t denotes te return on te new asset. As before, we transform z to ave unit variance. We also ortogonalize te vector [z, agez, age 2 z, age 3 z] wit respect to [x, agex, age 2 x, age 3 x]. Table 3 sows tat te job creation and destruction assets outperform te oter artificial assets according to an R 2 metric, especially for less educated men. 13 Indeed, te fraction of labor income risk accounted for by te S&P 500 plus te job destruction asset ranges from 18-28% for men wo do not attend college. Figures 1 and 2 display te correlation functions for te creation and destruction assets. Increases in manufacturing job destruction are bad news for everyone, but 12 See Capter 3 in Hamermes (1993) for a review of evidence on static labor demand relationsips. Goldin and Katz (1996) discuss recent work and istorical evidence on capital-skill and tecnologyskill complementarity. 13 Te sample period ends in 1993 for regressions tat include te job creation and destruction assets, but te sample difference does not account for te superior fit. 11

13 muc worse news for people wit low education. Te correlation is about -.5 for men wo did not finis ig scool and about -.4 for men wit a ig scool education. In contrast, te correlation is near zero for college-educated men and women. Tis sarp eterogeneity in te correlations and te ig explanatory power of te destruction asset (for some persons) make it a promising tool for risk-saring purposes. For te job creation asset, te correlations wit earnings socks are uniformly positive for men and negatively ordered by education. For women, te correlations wit te creation asset are also negatively related to educational attainment, but te correlations are mostly negative for more educated women. Heterogeneity in its correlation structure makes te job creation asset a promising tool for risk-saring purposes. 4.3 Group-Specific Mutual Funds Based on a sample tat begins in 1967 rater tan 1965, Table 3 also reports results for group-specific mutual funds. Adding te mutual funds to te baseline specification wit only te S&P 500 provides a very small improvement in fit. For more educated women, owever, te mutual funds add a few percentage points to te fit of regressions tat include te creation and destruction assets. Figures 1 and 2 display te correlations functions for te group-specific mutual funds. To our initial surprise, te mutual fund returns display a negative correlation wit income socks for all persons wit less tan a college education. In oter words, for six of te eigt sex-education groups, a long equity position in te worker s own industry acts to edge group-level income risk. Tis surprising finding runs directly counter to te view tat industry-specific fluctuations in equity returns are mainly driven by factor-neutral demand and tecnology disturbances. Tat view implies a positive witin-industry correlation between equity returns and uman capital returns. Te negative correlation in te data migt arise because movements in equity returns are dominated by rent-sifting between firms and teir workers, or because factor-biased tecnology socks cause capital and labor returns to move in opposite directions. Upon reflection, we view te latter ypotesis, in particular, as quite plausible, especially in ligt of te important role tat as emerged for factor-biased tecnology socks in leading explanations for relative wage movements across sex, education and experience groups. 14 We also note tat te pattern of correlations between income socks and own-industry returns sows te same type of relationsip to educational attainment as we found for te broad 14 Autor and Katz (1999) review te relevant literature. 12

14 equity index. 4.4 Multiple Assets Four risky assets S&P 500, group-specific mutual fund, job creaton, job destruction account for percent of group-level income risk for less-educated men and 5-21 percent for oter groups. Most of te explanatory power for less educated men comes from te creation and destruction assets. Mutual funds play a nontrivial role for more educated women. Based on te annual earnings measure, figure 3 sows te fraction of stocastic variability in group-level income risk jointly accounted for by tese four assets over te life cycle. For less educated men, te four assets account for rougly one-fift to one-tird of group-level income risk trougout te life cycle. For oter groups, tese assets account for less tan 15 percent of group-level income risk during most or all of te life cycle. 5 A life-cycle framework wit incomplete markets Following Willen (1999b), we adopt a dynamic exponential-normal framework tat delivers closed-form solutions. Most oter analyses of dynamic equilibrium models wit incomplete markets assume constant relative risk aversion and rely eavily on numerical solution metods. 15 Tose analyses typically require a very small number of agent types. In contrast, our framework remains tractable wit an essentially arbitrary number of agents and assets, overlapping generations, and parameters tat vary wit age, person and time. A simple example elps to appreciate some of te issues tat arise in seeking to quantify te welfare effects of financial innovation. Example 1 Te economy contains tree agents and lasts five periods (t =0, 1, 2, 3, 4). For eac agent, expected earnings equal five dollars in periods 1, 2 and 3. For agents 1 and 3, earnings follow a random walk; wereas, for agent 2, earnings are wite noise. All agents retire after t =3and ave no labor income at t =4. All earnings innovations ave unit variance. A riskless asset is freely traded. Risk aversion is lowest for agent 1 and igest for agent 3. Starting from a bencmark in wic agents can trade only te riskless asset, introduce a risky asset wit unit variance. 15 See, for example, Telmer (1993), Heaton and Lucas (1996), Constantinides, Donaldson and Mera (1998), Storesletten, Telmer and Yaron (1998) and Judd, Kubler and Scmedders (1998). 13

15 Te correlation between earnings innovations and te risky asset return is zero for agent 1 and 0.5 for agents 2 and 3. Panel I in Table 4 summarizes te situation. It is not obvious ow te risk-saring benefits afforded by te new asset are distributed among te agents. It seems plausible tat agents 2 and 3 benefit, because te new asset covaries wit teir labor income socks. Agent 3 presumably as a stronger edging demand because of greater risk aversion and more persistent earnings socks. Agent 1 as low risk aversion, so e may gain because bot agents 2 and 3 want to take sort positions. In te analysis below, we sow ow to solve tis problem and muc more ricer ones Description of te model and assumptions Tere is one consumption good per period and T +1periods,t =0,..., T.Tereare H agent types, =1,..., H, witnt individuals of type in period t. An agent of type enters te workforce at t, retires at Tr and dies at T.Letn t be te fraction of type- agents at time t, andletn t be te H-dimensional vector of n t values. For any H-dimensional vector, q, q t = N tq denotes te per capita mean of q. A consumption pat is a random vector, C = { c } T t. t=t Condition 1 Agents ave exponential utility, U ( C ) T ( =E t 1 (δ ) t A t=t were A is te coefficient of absolute risk aversion. ) exp ( A c t Let A be te H-dimensional vector of absolute risk aversion coefficients. In an abuse of notation, let A 1 be te H-dimensional vector of reciprocals of te risk aversion coefficients. An endowment pat is a random vector, { } ỹt T. Let Y t=t t be te H-dimensional vector of period-t endowments. ) Condition 2 Individual endowments follow ARIMA processes, in wic η t te income innovation for individual at time t. denotes Let ψi be te i t coefficient in te moving average representation of te endowment (ỹ ) (ỹ ) process for individual. Tatis,E t t+i Et 1 t+i = ψ i η t. 16 For tose wo can t wait, te results are in panel III of Table 4. 14

16 Tere are J + 1 financial assets: J risky assets wit gross one-period rates of return, R j,t, and a riskless asset tat pays gross rate of return, R 0,t, wit certainty. 17 Let R t be te J-dimensional vector of period-t risky asset returns, and let ER t be te ( corresponding ) vector of expected excess returns wit representative element, E t 1 Rj,t R 0,t. Persons enter te world wit no financial assets. In eac period of life except te last, an agent invests ωj,t units of te consumption good in risky asset j, j = 1, 2,...,J,andω0,t units in te riskless asset. Let ω t denote te J-dimensional vector of investments in te risky assets, and let ω denote te portfolio allocation pat (risky and riskless assets) for individual. Tere are no limits on ω oter tan te budget constraint, so tat unlimited sort sales are possible. Denote te vector of period-t income innovations and asset returns by η 1 t. Φ t = Condition 3 Te distribution of time-t asset returns and income innovations, conditional on information available at t 1, is jointly normal for all t. Tat is, η H t R 1,t. R J,t Φ t N (E (Φ t ), S t information at t 1) [ ] wit partitioned covariance matrix, S t = Ξ t β t β t Σ t. t, Condition 3 encompasses any ARIMA process for dividends wit i.i.d. normal innovations. To simplify several expressions, we introduce an operator for te discounted expected value of an arbitrary time series: ) T PDV t ({z s } T s=t+1 = s=t+1 1 Π s i=t+1 R E t ( z s ). 0,i ) In tis notation, PDV t ({1} T s=t is te present discounted value of an annuity tat pays one dollar eac period from t to T, inclusive. Te annuitization factor, a t,equals 17 Trougout te rest of te paper, te return on an asset means te gross one-period rate of return, unless oterwise noted. 15

17 ( ) te reciprocal of PDV t {1} T s=t. Similarly, define te present value multiplier T Ψ t = s=t 1 Π s i=t R ψs t, 0,i wic equals te revision to te present value of lifetime income implied by a unit labor income innovation. Some additional notation is needed to andle long-lived risky assets. Let d t be a J-dimensional dividend process wit representative element d j,t. A long-lived asset wit ex-dividend price, P j,t, is a claim to te stream of future dividends, {d j,s } T s=t+1. Let x t be te J-dimensional vector of innovations in te ARMA dividend process, and let λ t be te present value multiplier on dividend innovations. For example, λ j,t is te impact of a unit innovation in te j t dividend process on te present value of future dividends on asset j. LetˆΣ t =var( x t ) denote te covariance matrix of dividend innovations, and let ˆβ t =cov(ηt, x t) be te covariance between labor income innovations and dividend innovations. Te following condition plays an important role in our derivation of analytic solutions for portfolio allocations and equilibrium asset returns. Condition 4 Te price of risk, Σ 1/2 t ER t, and te scaled covariance between asset returns and income innovations, Σ 1/2 t β t, are nonstocastic. If te covariance matrix of asset returns is nonstocastic, ten Condition 4 means tat expected excess returns and te covariance between income innovations and asset returns are also nonstocastic. One furter condition is inessential to our approac, but it simplifies te analysis and welfare calculations. Condition 5 Te riskless asset is elastically supplied at an exogenous interest rate. It is possible to endogenize te risk-free rate in our setting (and accommodate forces tat drive nonstocastic interest rate variation). However, Condition 5 allows us to streamline te development below and sidestep several issues tat ave a minimal bearing on te questions addressed in tis paper. 18 Now consider a perfectly competitive economy wit te price of te consumption good normalized to one, and denote asset returns for all t by R. Tebudget set for 18 Willen (1999b) sows ow to endogenize te risk-free rate in a two-period version of te model, and Willen (1999a) discusses sufficient conditions for a nonstocastic, endogenously determined interest rate in a related model. 16

18 agent is B (R) = { {c } t,ω0,t, T ω c t + ω 0,t + 1 ω t = y t + R 0,tω0,t 1 + R t ω t 1, t t=t and given ω t 1 = 0, ω 0,t 1 =0. ( Definition of Equilibrium: An equilibrium R, ( C, ω ) ) is a time pat for H asset returns, consumption and portfolio allocations for eac suc tat: } 1. Eac individual optimally cooses in er budget set: ( C, ω ) B (R) U ( C ) U ( C ) ; ( C, ω ) B (R) and 2. Te goods market clears at all dates: c t + ω 0,t = ỹ t + ω 0,t 1R 0,t for all t, wit ω 0,T 0; 3. Risky asset markets clear at all dates: ω t = 0 for all t. 5.2 Individual Consumption and Portfolio Allocation Tere are several natural concepts of wealt in tis framework. Financial wealt at t is It = ( ( R 0,t ω0,t 1 + R tωt 1), uman capital is Y {y } ) t =ỹt T +PDV r t s,and s=t+1 simple wealt is Wt = It + Y t. We sall sow tat te welfare effects of a financial innovation can be calculated using a concept tat we call generalized wealt. Definition 1 Generalized wealt is GW t = ỹ t }{{} (1) current labor income ( {y } ) T + PDV r t s + R s=t+1 0,t ω0,t 1 + R t ω t 1 }{{}}{{} (2) PDV of future labor income (3) financial wealt ( {ER } ) ( T {(1/A ) +PDV t s ω s 1 PDV s=t+1 t s ln R0,s δ } ) T s=t+1 }{{}}{{} (4) PDV of future excess returns (5) riskles rate arbitrage ( {(A PDV t s /2 ) var ( )} Ws T ) s=t+1 }{{} (6) PDV of precautionary savings Te sum of terms (1) and (3) measure liquid resources. Term (2) is te value of expected future labor income, discounted at te riskless rate. Tese first tree wealt components are standard, but generalized wealt includes tree more terms tat influence a person s sense of wealtiness. Term (4) captures te value of excess expected returns on te portion of wealt invested in risky assets. Term (5) reflects any difference between te subjective discount rate and te risk-free rate.oter tings equal, a iger discount rate implies tat te individual feels wealtier and consumes more. 17

19 Term (6) reflects te precautionary saving motive, wic plays an important role in our analysis. As te variance of simple wealt (or consumption) rises, an individual feels less wealty and, ence, consumes less. Te definition of generalized wealt may seem ad oc, but it greatly simplifies te analysis, as te following proposition sows. Proposition 1 (Individual optimization) Given conditions 1, 2, 3, 4 and a nonstocastic rate of return on te riskless asset: Risky asset oldings in te optimal portfolio are given by ω t = 1 Σ 1 J 1 A t+1 ER t+1 Ψ t+1 Σ 1 t+1 β t+1 (1) t+1 were A t = a t A and Ψ t is te present-value multiplier on income innovations. Te expected excess return on te optimal portfolio, ER t ω t 1 = ( ) ( ) ( ) ( ) ( 1/A t Σ 1/2 t ER t Σ 1/2 t ER t Σ 1/2 t ER t and te variance of simple wealt along te optimal pat, var ( ) ( ) Wt = Ψ 2 ( ) t var ( η t β are nonstocastic. t Σ 1 t β t Σ 1/2 t β t ) + ( A t ) 2 ER t Σ 1 t ER t (2) Consumption is proportional to generalized wealt along te optimal pat: Proof: See Appendix c t = a t GW t. Te first part of tis proposition states tat risky asset allocations contain two components: one tat reflects a desire to exploit te excess return on risky assets, and anoter tat reflects a desire to edge uncertain labor income. Te magnitude of te first component declines in risk aversion, wile te magnitude of te second rises wit te persistence of income socks and teir covariance wit asset returns. Te second part of te proposition is useful in calculating generalized wealt. Te tird part states tat consumption equals te properly annuitized value of generalized wealt. Tus we ave converted a complex dynamic programming problem into a simple annuitization and discounting exercise tat any MBA can solve So states te autor wo as yet to teac MBA students. ) 18

20 To understand te portfolio decision, consider a simple case wit one risky asset tat as constant variance. Te consumption Euler equation for te risky asset is ( ) [ ( ) ] 1/A E t 1 R1,t R 0,t =cov t 1 ( c t, R ) 1,t. In words, an individual sets te covariance between consumption and te risky return equal to te excess return on te risky asset, scaled by risk tolerance. Now substitute te solution for consumption from Proposition 1 on te rigt side, noting tat simple wealt (Wt ) contains te only stocastic components of generalized wealt: ( 1/a t A ) [ ( ) ] ( E t 1 R1,t R 0,t =cov t 1 W t, R ) 1,t (3) Tis equation is almost identical to te Euler equation in a two-period model. Te only difference is tat absolute risk aversion is multiplied by te marginal propensity to consume out of generalized wealt. Tis product, wic 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 wic to spread gains and losses, and dynamic risk aversion rises. Consequently, as a person ages, te optimal oldings of risky assets tend to decline. 20 Substituting for Wt in (3) and solving yields te risky asset allocation (in units of te consumption good), ω1t 1 ( ) ] β = [E A t R1,t+1 R 0,t+1 Ψ 1,t+1 t+1 σ2 t+1, 1 σ1 2 }{{}}{{} Hedging portfolio Risk premium exploitation portfolio were σ1 2 is te variance of te risky asset rate of return. Note tat te risk premium exploitation portfolio is te same for everyone up to a scaling factor, a result tat generalizes to te case of multiple risky assets. Given a undred assets and a tousand agents, we can compute te portfolio weigts for one agent, ten scale up or down for everyone else to obtain te individual premium exploitation prortfolios. 21 Te edging portfolio depends on te slope coefficient (β1 /σ2 1 ) in a regression of te individual s earnings socks on te risky asset return and on te present value multiplier (Ψ t )on 20 Tis analysis elps rationalize te conventional wisdom of financial planners wo recommend tat investors reduce risky asset oldings as tey age. Properly interpreted, te logic of our analysis applies to te level of risky assets, not te proportion of financial assets eld in risky form. However, as a person ages, financial wealt grows relative to uman capital and simple wealt, so a decline in te level of risky financial assets implies an even faster decline in teir proportion of financial wealt. 21 Tis is te market portfolio in te CAPM two-fund separation teorem. 19

21 earnings socks. A larger regression coefficient or more persistent socks raises te magnitude of te edging demand. Tis expression for asset oldings leads to a useful decomposition for te variance of generalized wealt. Consider again te case of a single risky asset. Along te optimal pat, te time-t innovation to simple and generalized wealt can be written Ψ t ( [ ]) ηt (β1t/σ 1) 2 R1,t E t 1 R1,t + 1 ] [E A t σ1 2 t 1 R1,t R 0,t ][ R1,t E t 1 R1,t Te first term is proportional to te residual in a regression of income innovations on te risky asset return; tus it is ortogonal to te second term, wic is proportional to te excess return. Hence, we can decompose te variance of wealt into te scaled variance of undiversifiable income socks and te variance of te premium exploitation portfolio. Te following proposition generalizes tis result to te multi-asset case. Proposition 2 (Variance Decomposition) Given conditions 1, 2, 3, 4 and a nonstocastic rate of return on te riskless asset, te variance of an individual s wealt can be decomposed into two pieces: var ( ) ( ) GWt = Ψ 2 ( ) ) t var ( η t β t Σ 1 t β t }{{} undiversifiable part of idiosyncratic risk + ( 1 ) A 2 ER tσ 1 ER t t }{{} premium exploitation risk Proof: It is easy to see tat var ( ) ( GWt =var Ψ t η t + ) R t ω t 1. Substituting in optimal oldings of te risky asset from Proposition 1 and reorganizing gives te decomposition t 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 te Capital Asset Pricing Model to old. Our model differs from te standard CAPM, because individuals ave idiosyncratic risk tat is not (fully) tradable. Consider again te case of a single risky asset wit constant variance, and assume one agent of eac type. Summing te risky asset allocations over individuals, H ω1t = =1 H 1 A =1 t+1 σ2 1 (E t R1,t+1 R 0,t+1 ) H =1 β Ψ 1,t+1 t+1 σ Te following derviation is similar to te one for te exponential-normal model wit two periods. See, for example, Demange and Laroque(1995). 20

22 Since H =1 ω 1t = 0 in equilibrium, we ave we can solve to obtain ( 1 E t R1,t+1 R 0,t+1 = H H 1 A =1 t+1 ) 1 ( 1 H H Ψ t+1 β 1,t+1 =1 ) (4) In words, te excess return on te risky asset equals te armonic mean of dynamic absolute risk aversion times te average covariance between te risky return on te financial asset and te sock to uman capital. Tis result as te flavor of te traditional consumption CAPM, and te intuition is also te same. A risky asset tat pays off wen times are bad offers a iger return, because it plays a more valuable insurance function. We can write te covariance between te asset return and te labor income sock in terms of exogenous variables and parameters as follows: β 1,t+1 =cov t( R 1,t+1, η t+1 )= 1 P 1t cov( P 1,t+1 + d 1,t+1, η t+1 )= Λ 1 P 1t cov( x 1,t+1, η t+1 ), since te stocastic component of P 1,t+1 + d 1,t+1 equals te present value of te dividend innovation, Λ 1 x 1,t+1. Substituting into (4), and recalling tat cov( x 1,t+1, η t+1 ) ˆβ t+1 we obtain ( ) 1 ( ) H 1 H Ψ ˆβ t+1 1,t+1 ER 1,t+1 E t R1,t+1 R 0,t+1 = Λ 1 P 1,t+1 1 H A =1 t+1 Tis result extends straigtforwardly to te case of many assets and an arbitrary distribution of agent types: Proposition 3 (Equilibrium Excess Returns) Given conditions 1 troug 5, te equilibrium expected excess returns can be written 1 H =1 ER jt = Λ j P jt A t ˆβ jt Ψ t, j =1,...,J, (5) were A t = ( ) N 1 t A 1 t is te armonic mean of dynamic absolute risk aversion, and ˆβ jt Ψ t is te per capita mean of te covariance between dividend innovations on te jt asset and socks to te value of uman capital. Furtermore, Σ 1/2 t ER t = A t ˆβ t Ψ t and Σ 1/2 t β t = 1/2 ˆΣ t ˆβ t (6) We can also calculate equilibrium generalized wealt entirely in terms of exogenous parameters. 21

23 Proposition 4 (Equilibrium generalized wealt) Given conditions 1 trog 5,, equilibrium generalized wealt for individual at time t can be written ( {(1/A ) GWt = Wt PDV t s ln R0,s δ } ) ( {(A T PDV s=t+1 t s /2 )( ) Ψ 2 ) s var ( η } T s ( {(A +PDV t s /2 )( Ψ sβs ( ) ) A s/a ( s Ψs β s Σ 1 s Ψ s βs ( ) A s/as) } ) T Ψs β s s=t+1 (7) Proof: See Appendix s=t+1 ) 5.4 Computing te Welfare Effects of Financial Innovations We measure te welfare effect of a financial innovation in terms of its consumptionequivalent effect on utility. Definition 2 θ is te uniform variation in te consumption good at eac state and date tat leaves te consumer indifferent between consumption pats C and C : θ U (C + θ) =U (C ) Given our assumptions, uniform variation is simple to evaluate. Proposition 5 Consider two equilibria wit different portfolio coice menus and consumption profiles C and C for person. Under conditions 1, 2 and 3, te uniform variation ( ) θ = c 0 c A ln PDV t {1} T s=t ( ) PDV t {1} T s=t Proof: See Appendix We can now prove te main result tat we use to calculate welfare effects: Corollary 1 Consider a financial innovation tat alters te portfolio coice menu. Under conditions 1, 2, 3 and 4, te present discounted value of te uniform variation for individual at time t is ( {θ Θ t =PDV } T ) s = GWt GWt, s=t+1 is gen- were GWt is generalized wealt before te financial innovation, and GWt eralized wealt after te financial innovation. 22