Precautionary Portfolio Behavior from a Life-Cycle Perspective

B oof Goavernrodrs of the Federal Reserve System International Finance Discussion Papers Number 542 February 1996 PRECAUTIONARY PORTFOLIO BEHAVIOR FROM A LIFE-CYCLE PERSPECTIVE Carol C. Bertaut and Michael Haliassos Note: International Finance Discussion Papers are preliminary materials circulated to stimulate discussion and critical comment. References in publications to International Finance Discussion Papers (other than an acknowledgement that the writer has had access to unpublished material) should be cleared with the author or authors.

ABSTRACT The literature on asset accumulation by households draws a sharp distinction between “short-run” precautionary motives to buffer annual consumption from annual labor income shocks, and “long-run” life cycle considerations under labor income certainty. However, empirical estimates of the persistence of shocks to annual incomes imply that households are subject to considerable career uncertainty. We study long-run precautionary motives for life-cycle wealth accumulation and portfolio choice. We compute optimal portfolios under three sources of uncertainty (stock returns, incomes, and lifespan), and explore the separate contributions of several key factors for mean and median asset holdings, including education, risk aversion, household heterogeneity, utility from bequests, time preference, and variance and serial correlation of income shocks. Numerical solutions for households in three education groups are compared with data from the most recent and comprehensive source,the 1992 Survey of Consumer Finances.

PRECAUTIONARY PORTFOLIO BEHAVIOR FROM A LIFE-CYCLE PERSPECTIVE Carol C, Bertaut and Michael Haliassosl I. Introduction Traditional models of life-cycle motives for wealth accumulation abstract from nondiversifiable labor income risk faced by households, and have tended to underpredict aggregate wealth in the United Statesl and to overpredict “typical” behavior as captured by median wealth.2Following Leland (1968) and Sandmo (1970), Kimball (1990) showed that such risk necessitates additional (“precautionary”)wealth holding to buffer consumption from income shocks when utility exhibits “prudence” in the form of a positive third derivative. Modelsofprecautionary motiveswhichemploy thecommonly usedconstant-relative-riskaversion (CRRA) utilityfunctionrequire numericalsolutions.3Computational studiesby Hubbard andJudd (1987), Skinner (1988), andZeldes (1989) among others, haveabstracted from portfolio choice and shown that ifthe objectiveofhouseholds isto buffer annualconsumption from shocks to annual income, substantial wealth holding is required. In order to reduce unrealistically high predicted levelsof wealthto observed levels, Deaton (1991) and Carroll (1992) proposed models 1 The authorsare respectively: Economist, Division of International Finance, Board of Governors of the Federal Reserve System; and Assistant Professor, Department of Economics, University of Cyprus. We are grateful to two referees and to Stavros Zenios for constructive comments on an earlier draft. Zoe Chimonidou and Matthew Field provided excellent research assistance. We would liketo thank, without implicating,Michael Brennan, SteffenPischke, NikitasPittis, Aris Spanos,andMartha Starr-McCluer for helpful discussions. Haliassos’research was sup~rted by a grant fromtheResearchCommitteeofthe UniversityofCyprus.ThefirstdraftofthispaperwaswrittenontwosidesoftheAtlanticwithoutasingle faxortelephonecall,thankstotheavailabilityofInternet.Thispaperrepresentstheviewsoftheauthors andshouldnotbeinterpretedasreflectingthoseoftheBoardofGovernorsoftheFederalReserve System or other members of its staff.

which postulate considerable impatience in the form of a high rate of time preference. Hubbard et. al. (1995) introduce instead institutional factors such as asset-based, means-tested benefits to accountfor limited wealth holding by those with less than high school education. Hubbard, et.al. (1994) approximate observed wealth levels without assuming impatience by building a ninetyperiod model which incorporates not only income risk, but also uncertainty as to medical expenditures and lifespan. The few existing econometric tests of precautionary motives yield mixed results.q This paper explores the implications of an alternative assumption about household lifecycleobjectives, andextends theanalysisto discuss not only wealth holding butalso theportfolio choice between riskless and risky assets. Empirical evidence on the frequency and extent of portfolio adjustments by households is hard to reconcile with the assumption that they continuously rebalance their portfolios to buffer annual consumption from shocks to annual incomes.sWe postulatethat householdschoose portfolios soasto buffer consumption over longer periods from shocks to income over correspondingly long horizons, and we study the effects of such long-run precautionary motives on portfolio choice by young households.dAs seen below, the observed persistence of shocks to annual incomes implies that such households are subject toconsiderable “career uncertainty”, and this is likely to have quantitatively important influence on their choice of portfolios for retirement and bequests.’ Although households probably decide part of riskless asset holdings (e.g., checking and savings accounts) with other, shorter-run motives in mind (e.g., transactions motives), we investigate what proportion of their risky and riskless asset holding can be accounted for by this life-cycleprecautionary motive alone. For this purpose, predictions from different variantsof the 2

model are compared to the most recent and comprehensive data on household portfolios, namely the 1992 Surve)~of Consumer Finances (SCF). The long-run nature of our model allows us to compute optimal portfolios under three sources of uncertainty (stock returns, incomes, and lifespan), to consider both representative agent models and effects of population heterogeneity, and to explore the separate contributions of several factors, including education, risk aversion, time preference, variance and serial correlation of income shocks, and utility from bequests. Predictions are compared to data on mean and median pofifolios, by education group. This gives us two important perspectives on predictive performance in view of the limited incidence of stockholding analyzed in Haliassos and Bertaut (1995). Section II describes mean and median portfolios by age and education group, using the 1992SCF. Section 111describes two variants of the basic expected-utility model. Section IV first describes calibration of income processes and two departures from certainty models. It then traces the effects of differences in the income processes, starting with the process relevant for the least educated households and gradually transforming it into that applicable to their most educatedcounterparts. SectionVcomparesempiricalwealth-to-income andstock-to-income ratios with those implied by the models. Section VI concludes. The Data Appendix gives definitions of variables and SCF codes. II. Portfolio Holdings of U.S. Households We divide total household financial net worth from the 1992SCF into risky “stocks”and “riskless assets”. For most of our discussion, we include under stocks shares of publicly traded stocks, shares in stock mutual funds, and other “directly held” stocks in IRAs and Keogh plans. 3

We also consider a broader definition that includes stocks held in trusts, managed investment accounts, and defined-contribution pension plans. Our riskless assets include checking, saving, money market, and call accounts, CDs, saving and other bonds, and the cash value of life insurance. We subtract from riskless assets credit card balances, consumer loans, and other non-real estate loans. When we consider our broader definition of stocks, we also include in financial net worth assets in managed accounts and defined contribution pension plans. Labor income includes wage and salary income, income derived from a professional business or practice, unemployment and worker’scompensation payments, and income from Social Security and other pensions. We estimated after-tax labor income for non-retirement sources from information on each household’stax filing status and adjusted gross income (see Appendix 1). Table 1 shows mean and median holdings of directly heId stocks and total financial net worth by education and age. For all age groups, both mean holdings of stocks (co1.1) and mean financial net worth (co]. 2) increase with education. For example, households aged 30-39 with less than a high school education have a mean stock portfolio of $100, about 8 percent of their total financial net worth (co1.3). For those the same age and a high school degree, the mean stock portfolio increases to about $1,500, or 12percent of total financial net worth, and to nearly $20,000, or 56 percent, for those with a college degree. Among older households without a college degree, the share of stocks in financial net worth is larger, reaching about half the portfolio of those atage 40-59.In contrast, stocks comprise more thanhalf the portfolio ofyoung college graduates. Although the leveI of stocks is higher among older college-educated households, the share is Iower.H 4

In large part, mean stock portfolios for all education levels and ages reflect holdings by each group as a whole, but not typical behavior. Only 7 percent of households with iess than a high school education actually hold stocks; this fraction increases to 23 percent for households who have completed high school, and to 48 percent for those with a college degree. Columns 4, 5 and 6 show median stock portfolios, financial net worth, and stock shares. For households with less than a college education, the median portfolio contains no stocks, nor does the portfolio of the median young, college-educated household. For older college-educated households, the median portfolio share in stocks is larger, reaching 11 percent for those aged 60-69, and then drops to about 1 percent. Table 2 gives the same breakdown of information as Table 1, but uses the broader definition of stocks and total financial net wofih that includes indirectly held assets. This increases both the mean amount and portfolio share of stocks for almost all age and education levels.For householdsaged 30-39without acollegedegree, stockholding nowcomprises between 20 and 30percent of the portfolio, and almost60 percent for young college-educated households However, adding indirectly held stocks makes little difference to the median portfolio unless the household has a college education. As in Table 1,the median portfolio still contains stocks only for these households, and the share of their median portfolio in stocks is considerably larger, reaching over 30 percent for those aged 50 to 69. III. The Basic Models We solve numerous variants of two basic models. For each calibrated model, we program analytically derived first-order conditions, identities, and constraints as a system of nonlinear 5

equations, and compute solutions for all endogenous variables using Matlab’s modified Newton method with line search to ensure global convergence. Suppose thata household livesfor three twenty-year periods, and receives exogenous real incomefrom employment, unemployment, or retirement, Y,,inperiod t.The household is subject to three sources of uncertainty: income risk, stock returns, and lifespan uncertainty. We follow the standard practice in the precautionary saving literature of assuming zero correlation between nondiversifiable income risk and asset returns. At the end of each of the first two periods, after all current va~iablesare realized, the household chooses its holdings of stocks and riskless assets for the next period (and implicitly consumption) with a view to their cumulative returns and to present values of incomes over the twenty years. The algebraic sum of households’ stock and bond holdings is defined as their financial net worth (“wealth”). In the third period, households repay any accumulated debts. In the model without bequests, they consume the remainder of their labor and interest income and asset holdings. Formally, the household solves: Max E. [u(cl) +(1+5)-1U(C2+)(1-n)(l +5)-U2(C3)]5,>0,o~n<l (1) NI,BI,NZ,BZ where N,denotes number of stocks held between periods fand ?+/, B,nominal riskless holdings, n is the probability of premature death, and 5 is the rate of time preference. The utility function is of the form l-y -1 U(CJ= c’ (2) l-y where ~ is the (constant) degree of relative risk aversion. The constraints are:

S1 BI Y1-Nl — -— (3) = c1 PI PI BIIZ S2 B1-B2 . Y +(Nld2+—p)(1-t~ +(N1-NQ)~ + ~ (4) C2 . 2 2 2 2 C3= Y,+N2[;+q(l-tJ] +‘2[1+:(1-tJ] (5) 3 3 Ct2 0 v t. (6) S is the nominal stock price, d real dividends per share, 1 the nominal rate of interest on the riskless asset, P the price of the good, and t~the tax rate on interest and dividend income. We abstract from capital gains taxation for simplicity.9Short sales constraints on stocks are never binding in this model. If households need to borrow, they will do so at the (lower) riskless rate rather than at the risky rate which also has higher expected value. The absence of borrowing constraints, in the form of either quantity constraints or of a wedgebetween borrowing and lending rates isa standard assumption in life-cycle models, which is still debated in the literature. It is employed here to investigate how far a standard life cycle model can go in accounting for portfolios when only the assumptions of continuous rebalancing and no incomerisk arerelaxed.Nevertheless, householdsdo notengage in unboundedborrowing. As also noted by Aiyagari (1994), constraint (6) ensures finiteness of net worth, since consumption must occur even in adverse labor income and/or stock return states. Under CRRA utility, such constraints are superfluous, since U’(0) is unbounded, ensuring positive optimal consumption. Moreover, finite networth is notassociated with infinite borrowing combined with 7

infinite stockholding. Such a combination would involve bankruptcy in bad stock return states and would not be chosen by CRRA households. In thesecond model variant,we introduce utility from bequests, G. According to the 1992 SCF, 20.8% of households received or expected to receive a sizeable bequest, while 49.5% said they planned to leave a bequest.1°Third-period utility becomes: l-y -1 + ~ G1-Y-l (7) u(c~, @ = (1-A) C3 l-y l-y “ Bequests are uncertain because of income and stockholding risk. The parameter k controls the choice between last-period consumption and bequest. If households care about bequests (M), they behave so as to leave a bequest in any state of the world. The institutional requirement of nonnegative intentional bequests is met endogenously. Although we impose unconditionally positive bequests for 1>0, we understate bequest motives by eliminating the probability of premature death and household concern over the size of accidental bequests. We found that predictions move closer to the data, without the heavy informational requirements usually imposed by infinite-horizon dynastic models.]1 In the benchmark model without bequests, risk aversion is set at 3, as is usual in this literature, the probability of premature death at zero, and the rate of time preference at 3.13%, equal to the mean riskless rate estimated by Siegel (1992) over the period 1800-199012and close to the valueof 3% assumed in Hubbard et.al. (1994, 1995).Sensitivity of solutions to the values of these parameters is examined below. Annual stock returns can take a high or low value with equal probability, matching the first two moments of the long-run empirical return distribution estimated by Mehra and Prescott 8

(1985).13Following Haliassos and Lyon (1994), we compute the expected valueand the standard deviation of 20-year holding returns using the binomial process for annual returns. Finally, we choose a “high” and a “low” 20-year return that match these two moments. Expected dividend yields are set to about half the expected total pre-tax return on equity, which is consistent with the historical findings of Schwert (1990).14The twenty-year riskless rate is the Mehra-Prescott mean annual riskless rate compounded over twenty years. We consider the case of no correlation between income and stock returns, which gives us four second-period states.isIn the absence of retirement income risk, there areeight third-period states (provided that the household survives). IV. Precautionary Motives from a Longer-run Perspective IV.1 Calibration of Income Processes Households choose portfolios to buffer longer-run measures of consumption from shocks to the present values of their income over twenty-year periods, computed using the riskless rate at which they are able to borrow and lend. In the first version of the model, the representative household is faced with no income uncertainty and is guaranteed the average population income at each age.*bIn the second, income is uncertain in the second period, but its expected value equals the level guaranteed in the certainty version. In the third, we consider a population of heterogeneoushouseholds generatedbydifferent incomeshockrealizationsduringthefirstperiod. Our procedure ensures that average household incomes are equal to those guaranteed to the representative household under certainty and to their expected values under uncertainty.’7We account for serial correlation of annual income shocks not only within the second period of life, 9

but also across the first and the second period. The second and third versions are solved both without and with bequest motives. We distinguish between three groups based on education of the household head: (i) less than high school education (LTHS), (ii) high-school education but no college degree (HS), and (iii) college degree or more (COL). This classification is usually motivated by similarities of long-run income prospects (e.g., Hubbard et.al., 1994; 1995),but it may also be relevant for the degree of financial sophistication. In the versions with income risk, annual incomes for each education group i and year tin a twenty-year period of life are given by (8) where Y*isthecorresponding incomevaluein thecertainty version ofthe model.The logarithms of shocks U and V,denoted by lower-case letters, are assumed to follow the processes estimated by Hubbard et. al. (1994). While the logarithm of shock u exhibits considerable persistence for all education groups, the variances of transitory and persistent shocks decline with education: 0.955uLm~g-+1eLm~J,eL~~$-i.i.d. N(O,O.033),VLm~J-N(O,O.04) ‘LmS,t = UHSJ = 0.946uH~~-l+e~s~, eH~-Ji.i.d. N(O,O.025),vH~-$N(O,O.021) (9) 0.955uCoQ-+1ecou, ecou -i.id. N(O,O.016),VCOQ-N(0,0.014) ‘COL4 = IV.1.1 Representative-household Model with Income Risk In this version, first-period income isset equalto the average income for therelevant ageeducation group. By construction, multiplicative income shocks are equal to unity, and serial correlation in the logarithms does not imply second-period effects of first-period shocks. 10

We derive the implicationsof annual incomeshocks forthepresent valueof incomes over the second twenty-year period. We first draw randomly 200,000 realizations of shocks e and v for each education group from the corresponding lognormal distributions in (9), and use them to construct 10,000draws of 20-year income sequences. Using the resulting 10,000 realizations of present values, we compute the expected present value and its standard deviation by category. High and low income values are set to mean plus or minus one standard deviation respectively. Introductionoflognormallydistributed incomeshocksincreasesmeanincomesabovetheir value under certainty. In order to eliminate this side-effect, annual income values used above are adjusted as follows: Yi*=exp[lnY~-0.5( 0~,t+0~)] (lo) where Y’,f is the income value under certainty for education group i in year ~,and u~,t andU; aretheunconditional variancesofthelogarithmsofpersistent andtransitory shocksrespectively.19 Dropping the education group subscript, their sum in the first year of the period is o:+ u:. Its value in subsequent years can be obtained using the recursive formula U:J=U:J+.l 2(?-1)~: (11) p Comparison of good and bad income states suggests that shocks to annual incomes generate substantial uncertainty for a representative household even with regard to twenty-year present values. Since we are used to thinking about annual incomes rather than present values, Table 3 presents “equivalent”annual incomes which would yield the corresponding twenty-year present value if received each year. This redefinition of units is useful for interpretation and 11

harmless to results. It has been shown that scaling up or down incomes in all states of the world and time periods results in consumption levels and asset holdings which are also scaled by the same factor (Bar-Ilan, 1991). Table 3 shows that all education groups experience their peak incomes in the absence of shocks, or their peak expected incomes in the presence of shocks, during the second period of life. For college graduates, incomes in the third period are higher than in the first, though this is not true for the other two groups. Those with less than high school education, unlike others, are faced with a prospect of lower income than in the first period of life. IV.1.2 From a Representative Household to a Population of Households Households with the same education and income process generally experience different income realizations. Nondiversifiable income risk generates a population of heterogeneous households by the end of the first period, and their average (median) asset holdings need not be the same as those of a representative household which earns average (median) income. In this set of calibrations, we explore the importance of any such differences. Following the aboveprocedure, we compute 10,000present valuerealizations forthefirst period of life and break them up into ten categories (with equal ranges). We economize on computational effort by solving ten problems, where first-period income is set equal to the average for that income category. Predictions for each education group are computed by weighting predictions for each category by its relative frequency. The corresponding weighted average of incomes is equal to average income among all 10,000households, to the income of the representative agent in version 2, and to that under certainty in version 1. 12

The major computational economy comes in calibrating the risk households face conditional on first-period income and in viewof serially correlated shocks.Instead of generating 10,000present value realizations for each of 10,000households, we do so for each of 10income categories. Suppose there are 50 households in a category. By replicating each household two hundred times, we come up with 10,000 households while preserving the relative frequency of incomes within the category. Randomly assigning shocks to each, allowing for serial correlation, and correcting for unwanted mean effects, we generate 10,000present value realizations.20Their mean and variance describe second-period income uncertainty, conditional on being in that category of first-period income. IV. 2 Portfolio E#ects of Differences in Income Processes Table 3 and equation (9) show that income processes differ across education groups both in terms of no-shoch levels and in terms of stochastic properties. In this subsection, we illustrate the effects of such differences in the context of the benchmark model with income risk but no bequests. Comparisons are not influenced by differences in random draws, since the same draws of eand v are used as in the original calibration. The direction and relative importance of effects are shown in Table 4 (COIS4.-6). We first solve the model using the income process for households with less than high school education (step 1).In step 2, we use no-shock income levels of high-school graduates, but shocks applying to the least educated households. In step 3, we set the variance of persistent shocks to that of high-school graduates. In step 4, serial correlation is also adjusted. In step 5 the variance of transitory shocks is adjusted to replicate the process for high-school graduates. The 13

sequence is repeated for transforming high-school graduates to college graduates, with one additional step. Step 7 changes the tax rate on dividend and interest income, t~,from 1570to 30%, the rate assumed to apply to college-educated households. College no-shock income profiles have the largest effects, substantially encouraging borrowing and stockholding. Smaller variance of the (logarithm of) persistent shocks, et, (step 3 or 8) encourages borrowing, lowers W/Y, but also increases stockholding. This is a generalization of a finding by Kimball (1993) in the context of an atemporal model: a meanpreserving reduction in the size of background income risk makes risky assets more desirable because of the property of CRRA utility functions termed “standardrisk aversion”. However, the reduction in variance of temporary shocks, v,, associated with a higher education level (step 5 or 10) has negligible portfolio effects. A reduction in serial correlation (step 4), encourages both borrowing and stockholding, with a negative overall effect on W/Y. Step 9, which increases serial correlation, has the opposite effects. Finally, an increase in tax rates (step 7) reduces wealth by encouraging borrowing, while also encouraging stockholding. Increased borrowing arises because interest taxation is proportional, resulting in larger tax “refunds” from interest payments on loans when tax rates on interest increase. The positive effect on stockholding is attributable to the reduction in riskiness of after-tax stock returns when tax rates are higher.21 IV. 3 Comparison of Model Predictions Table 5 reports predicted ratios of average (median) wealth to average (median) income, for a subset of the models we have solved. Table 6 reports the corresponding ratios for stocks. Solutions for various degrees of risk aversion within the range of 2 to 10, which Mehra and 14

Prescott consider appropriate for representative agent models, are presented. Regardless of education, higher risk aversion lowers optimal stockholding (S/Y), but discourages borrowing by more, lowering current consumption to boost consumption in bad states. This results in higher wealth-to-income ratios (Fig. 1). The representative agent model which ignores income risk predicts positive stock to income ratios but also substantial borrowing which actually makes financial net worth negative (co1.1).If age-earnings profiles areguaranteed to the representative household in each education group and there are no bequest motives, then it should undertake considerable borrowing in the middle of its working life to finance consumption and to purchase stocks. In column 2, income shocks are allowed only in the second period.22The representativeagent nature of the model underestimates the importance of income risk, since shocks during the first period are not incorporated to differentiate households. Wealth to income ratios increase by 8to 16percentage points (pp), andthe magnitudeof theeffect increases with risk aversion. Stock to income ratios are somewhat reduced because of background income risk. In unreported calibrations, we varied the rate of time preference from .1% to 7.590. Reduced concern aboutthefuture makesequitypremia less importantanddiscourages investment in stock. Current consumption increases and is financed both through increased borrowing and reduced stockholding. The schedules for S/Y, B/Y, and W/Y against time preference are essentially linear, negatively sloped, and steeper for college-educated households. The prediction that it is optimal for all groups to be net borrowers is robust to time preference. We also introduced perceived probabilities of premature death at the end of the second period ranging from 5 to 75 percent. Optimal wealth-to-income ratios are reduced. Since households are less 15

likely to survive to retirement, when they would consume and leave bequests, they find equity premia less appealing and borrow to boost current consumption.2q In column 3, we allow for first-period income shocks and for persistence of the serially comelated shocks into the second period, We now average across solutions for heterogeneous households based on first period incomes. Wealth to income ratios are further increased by 3 or 4 pp in all education groups. Stock to income ratios are either unaffected or further reduced by about 1 to 2 pp. Substantial effects are obtained by introducing bequest motives in a representative agent model (co1.4), more so for larger degrees of risk aversion and higher education levels.For ahigh school dropout with risk aversion of 2, the wealth to income ratio rises above the population model without bequests by 5 pp and above the representative agent model with income risk by 9 pp.At the opposite end, the corresponding changes for college graduates with risk aversion of 10are 24 and 28 pp respectively. Stock to income ratios increase, as households accumulate to provide for future consumption and bequests, essentially returning to levels implied by the certainty model. Generalization to a population model with bequests (co]. 5) raises wealth to income ratios by a further 3 to 4 pp, while lowering stock to income ratios by 1to 2 pp. Population models allow computation of predicted ratios of median wealth and median stocks to median incomes. Again, introduction of bequest motives has substantial positiveeffects on the former, especially for college graduates, which range from 9 to 30 pp (columns 6 and 7). Effects on the latter are also positive but small. All in all, the cumulative effect of allowing for long-run income risk, bequests, and population heterogeneity is to increase predicted ratios of average wealth to average income by 16

20to 44 pp, while reducing stock to income ratios by between 1 and 7 pp relative to the representative agent model with no income risk. For each education level, increases in W/Y are larger and reductions in S/Y smaller the higherthe degree of risk aversion. For any risk aversion, effects on W/Yare largest for college graduates and smallest for high school graduates. Effects on optimal S/Y are somewhat more pronounced among high school dropouts than among others. V. A Comparison of Observed Portfolios with Model Predictions Since we are focusing on longer-run motives, our objective is not to match model predictions to the data, but to examine the extent to which the level and composition of asset holdings depart from long-run objectives incorporated in the models. This also defines the direction in which short-run (e.g., transactions) and other motives must operate if they are to account for actual portfolios. We focus on the extent to which the model accounts for (i) average behavior, (ii) typical behavior, and (iii) household heterogeneity in the age cell of 30 to 39 years for each education group. For (i), which is our main focus, the metric used is the ratio of average wealth (or stocks) to average labor income in the cell, which is equivalent to the ratio of total wealth (or stocks) to total income.2qSince assetholdingsare often skewed, medians represent typical behaviormore closely than means. So, for (ii) we use ratios of medians. An interesting puzzle in (iii) is the limited incidence of stockholding among US households. We have addressed this issue extensively in Haliassos and Bertaut (1995). The discussion here explores instead whether longrun models are versatile enough to generate the variety of nonzero portfolio combinations 17

observed in household data (e.g., negative net worth combined with stockholding, or positive holdings of both assets, etc.). For comparison with model predictions, COIS.1 and 2 of Table 7 give U.S. household meanfinancial net worth and mean stocks scaled by mean after-tax labor income (W/Y and S/Y), by age and education level. Columns 5 and 6 use the broader definitions of stocks .’~dfinancial net worth. For all education levels, W/Y (COIS.1and 5) is higher among older groups. For any given age, the ratio increases with education. For older households with at least a high school education,mean financial networth isseveraltimes annual labor income.The ratios of S/Y (COIS. 2 and 6) also show considerable variation byeducation level. For most households with less than high school education, the mean level of stocks never amounts to more than 40 percent of labor income. S/Y is especially small for young households, reflecting the low level of stocks held. Households with high school or college education have a substantially higher S/Y ratio. Older households in these education groups hold a mean stock portfolio that is greater than their mean after-tax labor income. Including indirectly held stocks increases this ratio considerably for these households. Columns 3, 4, 7, and 8 show ratios of median wealth or stocks to median incomes. Average Household Behavior Table 6 shows that for the two more educated categories, models which recognize life cycle precautionary motives yield ratios of stocks to income within the bands of data averages in Table 7. In the case of college graduates, risk aversion of 4 generates ratios between .35 and .45 when long-run uncertainty is recognized. For high school graduates, the empirical band is between .04 and .13, and risk aversion of 7 generates such ratios in the models with uncertainty. 18

This ranking in terms of risk aversion is consistent with responses on attitudes towards financial risk taking in the Survey which suggest that risk aversion declines with education (Table 8). Thus, models which postulate that households decide asset holdings with a view to buffering consumption from shocks to longer run income measures can reasonably account for average stockholding among the two more educated categories. With regard to riskless assets, Tables 5 and 7 suggest that life-cycle precautionary motives, especially in conjunction with heterogeneity and bequest motives, can explain a significant part of the discrepancy between a life cycle model under certainty and empirical observations. However, part ofriskless holdingsby thetwo moreeducated groups are attributable to shorter-run or other motives. For high school graduates, the long-run models accountfor about one third of the wealth to income ratio. For college graduates, they imply that long-run motives alone and the nature of the income process would justify substantial borrowing, to the point of making financial net worth negative early in life. Households with less than high school education differ. As comparison of Tables 5 and 7 suggests, their entire wealth to income ratio can be accounted for by the model with bequests and population heterogeneity at degree of risk aversion of 6. However, their limited average stockholding cannot be explained for degrees of risk aversion between 2 and 12. Had we only looked at aggregate wealth, we would have concluded that the model matches their behavior exactly. This raises some interesting research questions regarding the predictive ability of other models which approximate wealth to income ratios but have not yet been extended to analysis of portfolio composition. 19

Limited average stockholding in this education group is primarily attributable to the very small proportion of stockholders, both in absoluteterms and by comparison to other groups. This is true not only in 1992, but also in 1984PSZD(Mankiw and Zeldes, 1991) and in 1983 SCF data (Haliassos and Bertaut, 1995). Haliassos and Bertaut showed that an expected-utility life cycle model without frictions (such as information costs or other sources of inertia) has trouble accountingforthelimited incidenceofstockholding.For low-educationhouseholds, amongwhom stockholding incidence is minimal, this inherent limitation of the model distorts its prediction of average behavior, but not so for the other two categories. b. Typical Househoid Behuvior and Household Heterogeneity Median direct stockholding is zero for each age-educationgroup, except for college graduates over 40. The finding that not only (he incidence but also the average level of stockholding by high school dropouts is below predictions gives further support to the view that inertia, information, and sophistication are important factors for stockholding behavior. If we focus on median wealth and its relation to median income, we compare observed ratios (Table 7, COIS.3 and 7) to those predicted by our population models (Table 5, COIS6. and 7). Our model with bequests explains about one half of the empirical ratio for high school dropouts if we constrain ourselves not to consider risk aversion in excess of 10.For high school graduates, the same model can fully account for the observedratio for risk aversion of 9. However, the model yields consistently negative ratios for college graduates. Although the typical (median) young household in theSCF has positive riskless holdings and no stocks, there is considerable heterogeneity within each age-education group. In all three 20

education groups, some young households (between 10 and 25 percent) have negative financial net worth, and a fraction of these are stockholders under the broader asset definitions. The proportion of these stockholder households who borrow is slightly higher for college-educated households. Our population models yield (unreported) solutions for various categories consistent with the existence of such households and with their increased importance among households with college education. Even for college-educated households, for whom our models uniformly predict negative ratios of average wealth to average income, the population model with bequests generates positive wealth-to-income ratios for some income categories and degrees of risk aversion. The higher the degree of risk aversion, the larger is the number of such categories. Thus, there is nothing in these models to prevent them from predicting the variety of nonzero portfolio combinations observed in the data. Matching relative frequencies may require more elaborate models and/or greater precision in defining first-period income categories. We intend to pursue this in future research. VI. Concluding Remarks We explore the implications of a bold assumption abou household portfolio behavior, namely that household wealth and portfolios around the middle [ fworking life are chosen so as to buffer consumption over long periods of time from shocks to corresponding long-run income measures, i.e. “career uncertainty”. Using numerical computation, we show how the predictions of an otherwise standard life-cycle model with no income risk can be brought closer to empirical observations by (i) incorporating long-run income risk, (ii) introducing bequest motives, and (iii) 21

abandoning the representative-agent assumption to consider the effects of income shocks in creating a population of heterogeneous households. We distinguish between three education groups and use empirically estimated income processes for each to calibrate the extent of career uncertainty it faces. Dataon portfolios come from the most comprehensive and most recent source, the 1992Survey ofConsumer Finances. The simple career risk model can account for all ofstockholding among the two more educated income groups at plausible degrees of risk aversion and other parameter values, but leaves part ofriskless asset holding to reexplained with referenceto other considerations. Itcan also explain all of wealth holding by high-school dropouts without invoking a large degree of impatience or institutionalfactors.Itoverpredicts stockholding,primarily because itsincidenceis extremely low among this group, and standard expected-utility models need to be otherwise augmented to account for this fact, as we have shown elsewhere. The sensitivity of conclusions to parameter values and model variants is explored extensively. Based on these first results from setups which do not impose any frictions, market imperfections or special factors, the life-cycle model with career risk appears as a useful benchmark for future research. The model and the calibration methods introduced here should prove quite versatile and computationally tractable in allowing for finer distinctions among risky assets, investigatingcorrelations between incomes and asset returns, considering more education and/or income categories, exploring additional dimensions of household heterogeneity, matching population frequencies, introducing important frictions, and exploring the separate contributions of various factors. 22

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Data Appendix Numbers refer to variables in the /992 Surveyof Consumer Finances 1. Stocks (standard definition): dollar value of shares of publicly traded stocks (X3915) plus shares in stock mutual funds (X3822) plus 1/2of shares in combination funds (X3830) plus stocks in IRAs and Keogh plans(= X361O+ X3620 + X3630 if X3631=2) plus 1/2IRAs split between stocksand bonds or stocks and money market accounts (= l/2*(X3610 + X3620 + X3630) if X3631=5 or X3631=6) + 1/3 of mixed stock/bond/money market accounts (= l/3*(X3610 + X3620 + X3630) if X3631=4). 2. Stocks (broad definition) = Stocks (standard definition) plus dollar value of stocks in trusts or managed accounts (=X3942 ifX3947=1) plus 1/2(trustsandmanagedaccountssplitbetweenstocksand bondsor money market (= l/2*X3942 if X3947=5) + 1/3 other diversified accounts (= l/3*X3942 if X3947=6 orX3947=-7) plus stocks indefined contribution pensionplansplus 1/2ofdefinedcontribution plans split be~weenstocks and interest-earning assets (= X4226 if X4234=lor l/2*X4226 if X4234=3) + (X4326 if X4334=1 or l/2*X4326 if X4334=3) + (X4426 if X4434=1 or l/2*X4426 if X4434=3). 3. Financial net worth (standard definition): dollar value of transactions, savings, and money market accounts (= X3506 + X351O+ X3514 + X3518 + X3522 + X3526 + X3529 + X3706 + X3711 + X3716 + X3718 + X3804 + X3807 + X381O+ X3813 + X3816 +X3818+X3930)plus certificates of deposit (X3721) plus IRA/Keogh accounts (X361O+ X3620 + X3630) plus directly held mutual funds (X3822 + X3824 + X3826 + X3828 + X3830) plus saving bonds (X3902) plus other directly held bonds (X3910 +X3906 +X3908 +X7634 +X7633)pluscash valueofwhole lifeinsurance (X4006)plusotherfinancial assets (X4018 + X4022 if 61< X4020 <66 or 72 S X4020 S 74 + X4026 if 61< X4024 S 66 or 72< X4024 <74 + X4030 if 61< X4028 <66 or 72< X4028 < 74) minus credit card balances (X427+ X413 + X421 + X430 + X424 + X7575) minus installment and other non-real estate loans (X2218 + X2318 + X2418 + X2424 + X2519 + X2619 + X2625 + (X2723 if X271Onot=67) + (X2740 ifX2727 not=67) + (X2823 if X281Onot=67) + (X2840 if X2827 not=67) + (X2923 if X291Onot=67) + (X2940 if X2927 not=67) + X7824 + X7847 + X7870 + X7924 + X7947 + X7970 + X1044 + Xl215 + X1219 + Xl 108 +Xl 119+ Xl 130+X4229 + X4329 +X4429 + X4829 + X4929 + X5029 + X401O+X4032 + X3932). 4. Financial net worth (broad definition): financial net worth (standard definition) plus dollar value of assetsin managed or trust accounts (X3942) + amounts in defined-contribution pension plansthat can be borrowed against or from which household can make a withdrawal (X4226 if (X4216=1 or 2) and (X4227=1orX4231=1))+ (X4326 if(X4316=1 or2)and(X4327=1 orX4331=1))+ (X4426 if(X4416=1 or 2)and (X4427=1 or X4431= 1))+ (X4826 if(X4816=1 or 2) and (?.$827=1or X4831=1)) + (X4926 if (X4916=1 or 2) and (X4927=1 or X9831=1)) + (X5026 if (Xj#16=l or 2) and (X5027=1 or X5031=1)). 5. Riskless assets(standard definition) = financial net worth (standard definition) - stocks (standard definition). 6. Riskless assets (broad definition) = financial net worth (broad definition) - stocks (broad definition).

7. Householdlaborincome for 1991: household income derived from wages and salaries (X5702) plus income from professional business or practice (X5704) plus unemployment or worker’s compensation (X5716) plus income from Social Security or pensions (X5722). 8. After-tax laborincome: Fornon-retirement sources ofincome, theaverage tax rate was imputed from information on household adjusted gross income (AGI) (=X5751, or X7651and X7652) and tax filing status (X5746). The average tax ratesfor AGI class and filing status were calculated from the Statistics of income-1991, individual Income Tax Relurns, Table 1.2. AGI classes were: under $1,000; $1,000 to under $5,000; $5,000 to under $10,000; $10,000 to under $15,000; $15,000 to under $20,000; $20,000 to under $25,000; $25,000 to under $30,000; $30,000 to under $40,000; $40,000 to under $50,000; $50,000 to under $75,000; $75,000 to under $1OO,OOO$;100,000 to under $200,000; $200,000 tounder $500,000; $500,000tounder$1,000,000;and$1,000,000ormore. Filingstatuswassingle,marriedfiling ajointreturn,ormarriedfilingseparatereturns. Formarriedcouplesfiling separate returns, a weighted average tax rate (reflecting each spouse’sAGI) was constructed. Total after-tax income = income from retirement sources (X5722) + (1-tax rate)*(X5702 + X5704 + X5716). 9. Age of household head: recoded from X8022 as 5-year spreads between 20 and over 85 for income calibrations, and as 10-year spreads between 20 and over 80 for portfolio estimates. 10. Education of household head: recoded from X5901, X5902, and X5904 as(i) no high schooldegree (ii) high schooldegree orequivalency certificate but no college degree and (iii)college degree or higher. The final data set included 3,906 respondents. All variables were weighted with XWGT to produce population averages. 27

Appendix II Table A1.Calibrated IncomeProcessesforPopulationModels, byEducationGroup Equivalent[ncomesinPeriodandState Education: LessthanHighSchool JncomeCategory frequency Y1 Y2High Y2Low Y3 1 0.2297 8778.3 ~1903.6 7780.4 13633 ~ 0.5224 13947.1 30467.8 11026.2 13633 3 0.1917 20907.4 40316 15200 13633 4 0.0426 28279.0 50289 19551 13633 5 0.0102 35624.3 60404 23780 13633 6 0.0023 42741.5 68235 27777 13633 7 0.0008 50821.6 67703 28423 13633 8 0.0002 58421.8 55836 25324 13633 9 0 .- -- 10 0,0001 77412.6 125043 61187 13633 HighSchool IncomeCategory 1 0.0629 13202.] 33605,2 15174.8 22032 ~ 0.3941 19806.9 43289 20081 22032 3 0.3429 27444.9 53199 25383 22032 4 0.1429 35971.5 63547 30573 22032 5 0.0398 44459.3 73243 35293 22032 6 0.0127 52907.I 81405 40183 22032 7 0.0032 61325.9 94176 46884 22032 8 0.0009 68909.3 100510 51582 22032 9 0.0005 79660.9 123572 66474 22032 10 0.0001 92801.6 165371 95669 22032 ~oilegeorMore ncomeCategory 1 0.1154 23738.1 70964 36698 49663 ~ 0.4486 33606.2 88944 45850 49663 3 0.3114 45252.4 109620 57380 49663 4 0.0956 57920.9 ]Q969] 69023 49663 5 0.0219 70548.4 146945 79755 49663 6 0.0058 83099.4 164253 89567 49663 7 0,001 96697.8 178094 102826 49663 8 0.0002 105650.0 158597 91883 49663 9 0 .- .- -- -- 10 0.0001 142906.0 246917 149563 49663 rst-periodincomesforeacheducationlevelarederivedfrom10,000randomdrawswhich e usedtoconstruct10incomecategories. Second-periodincomesforeachcategoryare :neratedfromadditionalrandomdrawsandallowforpersistenteffectsoffirst-periodshocks tothesecondperiod. equency: fractionof 10,000householdsinsimulatedpopulationwhichbelongtothat tegorybasedonfirst-periodincome. 1

Table 1. Average and Median Dollar Holdings of Directly Held Stocks and Household Financial Net Worth for U.S. Households, by Age and Level of Education of Household Head Average Portfolio Median Portfolio Stocks Financial Stock Stocks Financial Stock Age Net Worth Share Net Worth Share (1) (2) (3) (4) (5) (6) Education: Iessthan high school degree 20- 29years o 1977 0.00 0 3240 0.00 30-39 years 100 1191 0.08 0 3400 0.00 40-49 years 6701 16335 0.41 0 3510 0.00 50-59 years 3073 15040 0.20 0 3510 0.00 60-69 years 2310 23025 0.10 0 3710 0.00 70-79 years 1227 40885 0.03 0 6410 0.00 80+ years 28971 63886 0.45 0 4610 0.00 Education: high school degree 20-29 years 443 5152 0.09 0 3270 0.00 30-39 years 1531 12322 0.12 0 3510 0.00 40-49 years 19421 34698 0.56 0 4610 0.00 50-59 years 27416 64977 0,42 0 1s010 O.OO 60-69 years 30513 92990 0.33 0 2:-’10 0.00 70-79 years 37663 123407 0.31 0 32590 0.00 80+ years 41244 87524 0.47 0 9060 0.00 Education:collegedegree 20- 29years 10610 18812 0.56 0 3111 0.00 30-39 years 19515 34972 0.56 0 8410 0.00 40-49 years 35604 97077 0.37 300 203IO 0.01 50-59 years 73453 202835 0.36 2500 66461 0.04 60-69 years 105302 259167 0.41 8500 77600 0.1I 70-79 years 73308 226~99 0.32 1000 87410 0.01 80+years 269942 526430 0.51 1000 127710 0.01 Data: 1992Survejlof ConsumerFinances Directly held stocks include shares of publicly traded stocks, shares in mutual stock funds, and stocks in IRAs andKeoghs. Directlyheld household financial net worth includes directly held stocks, checking, saving, money market, and call accounts, CDs, saving and other bonds, and the cash value of life insurance, minus balances on credit cards, consumer loans, and other non-real estate loans. 29

Table2, AverageandMedianDollarHoldingsofDirectlyandIndirectlyHeldStocksandHousehold FinancialNetWorthforU.S.Households, byAgeand Leve]ofEducationofHouseholdHead AveragePortfolio MedianPortfolio Age Stocks Financial Stock Stocks Financial Stock NetWorth Share NetWorth Share (1) (~) (3) (4) (5) (6) Education: Iessthanhighschooldegree 20-29 216 3106 0.07 0 3510 0.00 30-39 314 1411 0.2L o 3510 0.00 40-49 6854 16792 0.41 0 3510 0.00 50-59 3431 15063 0.23 0 3510 0.00 60-69 2394 23039 0.10 0 3710 0.00 70-79 3019 46620 0.06 0 6410 0.00 . 80+ 29305 66055 0.44 0 4610 0.00 Education:highschooldegree 20-~9 1043 6468 0.16 0 3370 0.00 30-39 4321 15760 o.~7 o 3610 0.00 40-49 24278 41204 0.59 0 5540 0.00 50-59 41131 76739 0.54 0 18510 0.00 60-69 33149 98974 0.33 0 29010 0.00 70-79 38614 127756 0.30 0 34060 0.00 80+ 43738 93979 0.47 0 9060 0.00 Education:college degree 20-~9 11567 22322 0.52 0 3170 0.00 30-39 25030 42792 0.58 250 9510 0.03 40-49 47792 116736 0.41 6000 28310 0,30 50-59 123783 ~58840 0.48 24000 86910 0.36 60-69 118998 285405 0.42 25000 89910 0.32 70-79 83792 245403 0.34 1250 93010 0,01 80+ 316030 705758 0.45 1000 ~13410 0.01 Data: 1992SurveyofConsUmet-Finances. DirectlyandindirectlyheldstocksincludeallstockslistedinTableI, piusstocksheldindefined contributionpensionplans,trusts,andmanagedinvestmentaccounts. Directlyandindirectlyheldhouseholdfinancialnetworthincludesallassets]istedin Tablel, plusassets held in defined contribution pension plans, trusts, and managed investment accounts. 30

Table 3. Calibrated Income Processes for Representative Agent Models, by Education Group Equivalent Income in Period and State Income Certainty Education: Y1 Y2 Y3 Less than High School 15019 21570 13633 High School 25920 37583 22032 College or More 39483 75527 49663 Income Risk YI Y2 Y3 Education: High Low Less than High School 15019 30088.5 13219.5 13633 High School 25920 48691 26219 22032 College or More 39483 96010 55338 49663 Income values are “equivalent”incomes which, if received each year over a twenty-year period, would yield present values of income equal to those derived in the actual calibrations. Income Certainty: Income values for periods 1,2, and 3 are based on mean incomes in the 1992SCFfor the age-education group specified. Income Risk: High and low income states inperiod 2 are derived on the basis of random draws of income shocks, and random draws of the corresponding income sequences, as described in the text. Incomes are adjusted to restore means to their certainty levels. Income definitions are given in the data appendix. 31

Table 4. Effects of Differences in Income Processes Across Education Groups on Portfolio Allocation I I I First-Period Portfolio Solutions Changes in Portfolio Step Changesto income process Solutions from Previous Step WN SN BN AWIY ASIY ABN (1) (2) (3) (4) (5) (6) 1 incomeprocess forLTHS -0.320 0.363 -0.683 -- -- -- 2 from step l, change non-shock -0.311 0.362 -0.672 0.009 -0.001 0.011 income level to HS 3 from step 2, change varianceof -0.335 0.374 -0.709 -0.024 0.012 -0.036 persistent shockse toHS 4 from step 3. change serial -0.343 0.378 -0.721 -0.009 0,004 -0.013 correlation of shocks to HS 5 from step 4, change transitory -0.343 0.378 -0.721 0.000 0.000 0.000 shocks to HS V = income process for HS 6 from step 5, change no-shock -0.663 0.464 -1.126 -0.319 0.086 -0.405 income level to COL 7 from step 6. change tax rate on -0.739 0.465 -1.204 -0.076 0.001 -0.077 dividends and interest income 8 from step 7, change variance -0.784 0.484 -1.268 -0.044 0.020 -0.064 of persistent shocks e to COL 9 from step 8, change serial -0.774 0.480 -1.254 0.009 -0.004 0.014 correlationshocks to COL 10 fromstep 9, change transitory -0.775 0.480 -1.255 -0.001 0.000 -0.001 shocks t)to COL = income process for COL Notes: LTHS: less than high school education; HS: high school degree; COL: College degree. W/Y: average wealth/average income; S/Y: average stocks/average income; B/Y: average bonds/average income. The table shows the effects on portfolio allocation of gradually converting the income process for the representative agent household with less than high school education into the process for the collegeeducated household. Mean incomes are computed from the 1992SCF. Income definitions are given in the data appendix. The calibration method is analogous to that used in Table 3, 32

Table5: Compa [sonofModelPredictionsofFirst-PeriodWealthto IncomeRatios, by EducationandDegreeofRiskAversion Certain IncomeRisk, IncomeRisk, IncomeRisk, Income, NoBequests Bequests PopulationModels Degreeof NoBequests WN WN MedW/MealY Risk WN RepAgent Population RepAgent Population No Bequests Aversion RepAgent (3) Bequests (1) (2) (4) (5) (6) (7) Education:lessthanhighscl lo]degree 2 -0.58 -0.45 -0.41 -0.36 -0.32 -0.45 -0.36 3 -0.45 -0.32 -0.28 -0.19 -0.15 -0.32 -0.19 4 -0.37 -0.24 -0.20 -0.08 -0.05 -0.23 -0.08 5 -0.33 -0.18 -0.14 -0.01 0.02 -0.18 -0.01 6 -0.29 -0.15 -0.10 0.03 0.07 -0.14 0.04 7 -0.27 -0.12 -0.07 0.07 0.10 -0.11 0.08 J 8 -0.25 -0.09 -0.05 0.10 0.13 -0.09 0.10 9 -0.24 -0.08 -0.04 0.12 0.15 -0.07 0.12 10 -0.22 -0.06 -0.02 0.13 0.17 -0.05 0.14 Education: highschooldegree 2 -0.55 -0.47 -0.44 -0.38 -0,35 -0.42 -0.33 3 -0.43 -0.34 -0,31 -0.21 -0.18 -()>~9 -0.16 4 -0.35 -0.26 -0.23 -0.10 -0.08 -0.21 -0.06 5 -0.31 -0.21 -0.18 -0.04 -0.01 -0.16 0.01 6 -0.27 -0.17 -0.14 0.01 0.04 -0.12 0.05 7 -0.25 -0.14 -0.11 0.05 0.07 -0.09 0.09 8 -0.23 -0,12 -0.09 0.07 0.10 -0.07 0.11 9 -0.22 -0.11 -0.07 0.09 0.12 -0.06 0.13 10 -0.21 -0.09 -0.06 0.11 0.14 -0.04 0.15 Education: collegedegree 2 -1.06 -0.97 -0.93 -0.84 -0.80 -1.09 -0.95 3 -0.87 -0.77 -0.74 -0.58 -0.55 -0.88 -0.67 4 -0.76 -0.66 -0.62 -0.43 -0.40 -0.75 -0.51 5 -0.69 -0.58 -0.54 -0.33 -0,30 -0.67 -0.41 6 -0.64 -0.53 -0,49 -0.27 -0,24 -0.61 -0.34 7 -0.60 -0.49 -0.45 -0.22 -0.19 -0.57 -0.28 8 -0.58 -0.46 -0.42 -0.18 -0.15 -0.53 -0.24 9 -0.56 -0.43 -0.39 -0.15 -0,12 -0.50 -0.21 10 -0.54 -0.41 -0.37 -0.13 -0.10 -0.48 -0.19 Notes: W/Y: Ratioof averagewealthto averageincome. MedW/MealY: Ratioof medianwealthto medianincome. CertainIncome: Baselinemodelwithnosecond-periodincomeshocks. IncomeRisk: Addstransitoryandpersistentshocksto second-periodincomes. RepAgent: Representativeagentmodels. Population: Modelswithpopulationsofheterogenous agents. Bequests: Modelswithbequestmotives;the weightonthebequestmotive~=0.25 (seeequation(7)). Allmodelshaverateoftimepreference6= 3.13PercentPerannum. 33

Table6: ComparisonofModelPredictionsof First-PeriodStocksto IncomeRatios. byEducationandDegreeof RiskAversion Certain IncomeRisk, IncomeRisk. IncomeRisk, Income, NoBequests Bequests PopulationModels Degreeof NoBequests SIY s/Y MedS/MealY Risk siY RepAgent Population Rep Agent Population No Bequests Aversion RepAgent (3) Bequests (1) (2) (4) (5) (6) (7) Education:lessthan highschooldegree 2 0.72 0.63 0.60 0.68 0.65 0.62 0.66 3 0.43 0.36 0.34 0.40 0.38 0.35 0.39 4 0.30 0.25 0,24 0.28 0.27 0.25 0,28 5 oo23 ().1[! 0.18 0.22 0.21 0.19 0.21 6 0.19 0.16 0.15 0.18 0.17 0.15 0.!7 7 0.16 0.13 0.12 0.15 0.14 0.13 0.15 8, 0.14 ().I1 0.11 0.13 0.12 0.11 0.13 9 0.12 0.10 0.10 0.11 0.11 0.10 0.11 10 0.11 0.09 0.09 0.10 0.10 0.09 0.10 Education:highschooldegree 2 0.71 0.65 0.63 0.70 0.68 0.62 0.66 3 0.42 0.38 0.36 0.42 0.40 0.36 0.40 4 0.30 0.26 0.25 0.29 0.28 0.25 0.28 5 0.23 0.20 0.19 o.~3 0.22 0.19 0.21 6 0.19 0.16 0.16 0.18 0.18 0.15 0.17 7 0.16 0.14 0.13 0.15 0.15 0.13 0.15 8 0.14 (-)1.~ 0.11 0.13 0.13 0.11 0.13 9 o.I2 0.10 0.10 0.12 0.11 0.10 0.11 10 0.11 0.09 0.09 0.10 0.10 0.09 0.10 Education: collegedegree 2 0.87 0.80 0.78 0.86 0.84 0.84 0.91 3 0.53 0.48 0.46 0.53 0.51 0.50 0.55 4 0.38 0.34 0.33 0.38 0.36 0.35 0.39 5 0.29 0.26 0.25 0.29 0.28 0.27 0.30 6 0.24 0.21 0.20 0.24 0.23 0.22 0.25 7 0.20 ().18 0.17 0.20 0.19 0.18 0.21 8 0.18 0.15 0.15 0.17 0.17 0.16 0.18 9 0.I5 0.13 0.13 0.15 0.I5 0.14 0.16 10 0.14 0.12 0.12 0.13 0.I3 0.12 0.14 Notes: W/Y: Ratioof averagewealthto averageincome. MedW/MealY: Ratioof medianwealthto medianincome. CertainIncome: Baselinemodelwithnosecond-periodincomeshocks. IncomeRisk: Addstransitoryandpersistentshocksto second-periodincomes. RepAgent: Representativeagentmodels. Population: Modelswithpopulationsofheterogeneousagents. Bequests: Modelswithbequestmotives; the weighton the bequestmotiveL=0.25(seeequation(7)). All modelshaverateof time preference3= 3.13percentoer annum. 34

Table 7. Ratios of Average and Median Financial Net Worth and Stocks to After-tax Labor Income, forU.S. Households, bY Ageand Levelof Education of Household Head Directly Held Assets Directly and Indirectly Held Assets Age wry SN MedW/ MedS/ WbN SbN MedWb/ MedS~ MedY Med Y MedY MedY (1) (2) (3) (4) (5) (6) (7) (8) Education: less than high school degree 20-29 years 0,16 0.00 0.32 0.00 0.25 0.02 0.35 0.00 30-39 years 0.07 0.01 0.27 0.00 0,08 0.02 0.28 0.00 40-49 years 0.88 0.36 0.27 0.00 0.90 0.37 0.27 0.00 50-59 years 0.61 0.12 0.21 0.00 0.61 0.14 0.21 0.00 60-69 years 1,48 0.15 0.31 0.00 1,48 0.15 0.31 0.00 70-79 years 3.54 0.11 0,75 0.00 4.04 0.26 0.75 0.00 80+ years 7.30 3.31 0.70 0.00 7.55 3.35 0.70 0.00 Education: high school degree 20-29 years 0.26 0.02 0.19 0.00 0.33 0$05 0.20 0.00 30-39 years 0.36 0.04 0.13 0.00 0.46 0.13 0.13 0.00 40-49 years 0.90 0.50 0.14 0.00 1.07 0.63 0.17 0.00 50-59 years 1.72 0.73 0.50 0.00 2.04 1.09 0.62 0.00 60-69 years 3,44 1.13 1.43 0.00 3.67 1.23 1.61 0.00 70-79 years 7.34 2.24 2.72 0.00 7.60 2.30 2.84 0.00 80+years 7.84 3.69 0.98 0.00 8.41 3.92 0.98 0.00 Education:collegedegree 20-29 years 0.66 0.37 0.14 0.00 0.78 0.41 0.14 0.00 30-39 years 0.63 0.35 0.21 0.00 0.77 0.45 0.23 0.01 40-49 years 1.51 0.55 0.44 0.01 1.81 0.74 0.62 0.13 50-59 years 2.33 0.84 1.15 0.04 2.97 1.42 1.50 0.38 60-69 years 4.11 1.67 1.94 0.21 4.52 1.89 2.25 0.56 70-79 years 5.97 1.93 3.92 0.04 6.47 2,21 4.17 0.06 80+ years 14.86 7.62 7.98 0.06 19.93 8.92 13.34 0.09 Data: 1992Surve>o:fConsumerFinances. Directlyheldassets: SeenotestoTable1. W: Directlyheldfinancialnetworth, S:Directlyheld stocks. Directlyandindirectlyheld assets: See notes to Table 2. Wb: Directly and indirectly held financial net worth Sb: Directlyandindirectlyheld stocks. Y: After-tax labor income. WN andWb/Y:Ratio of average financial net worth in age-education cell to average after-tax labor income in age-education cell. SN and Sb/Y: Ratio of average stocks to average after-tax labor income. Med W/MealY and Med Wb/MedY: ratio of median financial net worth in age-education cell to median after-tax labor income in age-educationcell. Med S/MealY and Med Sb/MedY: Ratio of median stocks to median after-tax labor income. 35

i Table 8. Declared Willingness to Undertake Financial Risks, by Education Level Household Total Population Less Than High School High School College Response (%) (%) (%) (90) High Risk 14 5 12 23 Average 37 18 36 50 Risk No Risk 48 78 52 27 Source: Computed from the 1992Surve}” ofConsumerFinances. High Risk: Willing toundertakesubstantial risk for substantial return orabove average risk for above average return. Average Risk: Willing to take average risk expecting average return. No Risk: Not willing to undertake any financial risk. 36

s -0 D “9) ‘m ,? ,. B 8 8 1 .,\ 8 4 @ ‘ }1. ‘ .\ 4@ ,,. .‘.\ —.. -— -—.—. -A 4: 8 -\ 41:2 l — ——— -0 ————-.—— .. -\ 40 37

ENDNOTES 1. See, for example, White (1978), Kotlikoff and Summers (1981), Auerbach, Kotlikoff and Skinner (1983), Auerbach and Kotlikoff (1987), Hubbard and Judd (1987). 2. For example, Venti and Wise (1987); Bemheim and Scholz (1993). 3. Analytical results have been derived for utility functions with restrictive properties, such as the exponential which implies zero wealth elasticity of risky investment. 4. Guise, Jappelli, and Terlizzesse (1992) find a smallrole for income variance, while more positive findings for precautionary saving are in Dardanoni (1991), and in Carroll and Samwick (1992). Guise, JappelliandTerlizzesse (1994)isthefirstpaper weareaware ofwhichtestsportfolio predictionsofshortrun precautionary models econometrically, with encouraging results. In a theoretical paper, Aiyagari (1994)arguesthatprecautionary saving is smaller in generalequilibrium, infinitehorizonmodels. 5. Accordingtothe 1992Surve~~of Consumer Fi~zances,90.9% of the populationdidnottradestocksin the course of the survey year and only 3.2% traded more than 4 times. Even among those in the population who stillowned stocksat the time of the interview, 71.8%did not trade any stocksduringthe entire year. The corresponding figures for college-educated households are 80.1~o,7.4Y0,and 63.5V0 respectively 6.Our understandingofoptimalprecautionary portfolio allocation isfairly limited.Effects ofbackground income risk on the demand for risky assets have been derived analytically in atemporal modelsby Pratt and Zeckhauser (1987), Kimball (1991, 1993),and Elmendorf and Kimball (1991), Pratt and Zeckhauser (1987) show that undercertain conditions, if stockholding risk was undesirable in the absence of income risk, it will not bedesirable in its presence. Kimball (1993) derives conditions under which income risk would limit the scale of stockholding. Elmendorf and Kimball (1991) show that the effects of income taxation onthedemand for risky assets are ambiguous, A number of generalequilibrium models specify asset supplies andfocus on explaining asset returns and equity premia. Constantinides and Duffie (1992) explorethe implicationsof persistent and heteroskedastic laborincome shocks for asset pricing.Aiyagari and Gertler (1991) study asset returns, but also find that actual stockholding is too small relative to the amountofgovernmentdebtneededtoclear markets inamodelwithtransactionscosts, aninfinitehorizon, and a continuum of agents. Heaton and Lucas (1992) find that transactions costs and persistent income shocksraisethepredictedequity premium. Weil (1990) shows thatthe representative agentconsumptionbased model underpredicts the equity premium and overpredicts the riskless rate when income risk is ignored for many commonly usedutility functions (excluding exponential utility). 7.SeeHaliassos(1994)fordiscussionandnumerical illustrationsoftheimplicationsofusingdeterministic instead of stochastic models to analyze long-run economic behavior. 8.For alleducation levels,littleemphasis shouldbeplaced on thesizable stockholdings andfinancial net worth of those aged over 80 years. Although all respondents are weighted to match population statistics and these weights produce reliable estimates for broad aggregates, the high mean figures are largely the result of the disproportionate weight of a few especially wealthy households aged 80-85. 38

9. Sincecapitalgainsaretaxedat realizationratherthanaccrual,introductionof capitalgainstaxation complicatestheproblemconsiderably,sincewehaveto keeptrackofthetimeatwhicheachstockwas purchased.SeeHaliassosandLyon(1994)fornumericalsolutionstoamodelincorporatingcapitalgains taxationbutnoincome risk. 10.The mean (received) inheritancebycategory was:$4038forLTHS; $10739forHS; $43884forCOL. 11.While infinite-horizon models have interesting applications, concern with the utility of descendants (who also care about the utility of their offspring) effectively requires households to have information about how possibly unborn descendants feel or behave, and expectations about how their careers and future economic conditions willevolve.There is no obviousempirical counterpart to such well-informed dynasty members, especially since we are looking at households in their twenties and thirties. 12. This is higher than the mean value of 0,8% estimated by Mehra and Prescott (1985) and used for calibrating theriskless rate inthe model.Compared to models which settherate oftime preference equal to theriskless rate, this lowers predicted wealth and makes it more difficult for the model to match the data (see below). 13,These first two moments have received considerable attention in the literature (e.g., on the equitypremium puzzle), and have been the focus of historical studies, such as Mehra and Prescott (1985) and Siegel(1992).Ifconsensus onhigher moments isestablished, these could be incorporated by considering more states ofthe world.Joint consideration of more than two asset return outcomes with more than two labor income outcomes raises the number of states considerably, and use of large-scale optimization methods becomes advisable. Optimization software such as GAMS or MATLAB could be helpful for smaller problems, but high-performance computing (including parallel processing) may be necessary for finer approximations to the range of possible outcomes. 14.The Mehra-Prescott mean annual stock returns and standard deviation for the period 1889-1978are 6.98% and 16.54% respectively. The mean nskless rate was 0.8090.The twenty-year “high”and “low” stockreturnsare5.851241and-0.14075respectively.Thecorrespondingdividend yieldsare2.005997and 0.849247. 15.Someempirical findingsonthe importance ofpositive correlation (“businesscycle risk”)for theissue of whether to hold stocksare discussed in Haliassos and Bertaut (1995). Itscombination with short sales constraints appears to have some explanatory power for the decision not to hold stocks. 16.Incomesundercertainty arederivedfrom age-education profilesinthe 1992SCF.Because ofthesmall number of observations in each age-education cell, we divide the 60 years of data into twelve five-year rangesandcompute average incomesfor each range,usingcorrect population weights.The representative household is then assumed to receive that level of income for each of the five years in the range. In the case of households with less than high school education, six ten-year ranges are used. 17,As can beinferred fromTable 3,very smalldiscrepancies dooccur, because we usestochasticincome draws for a large number of households (at least 10,000) to compute expected values and standard deviations of the income measure. 39

18. When X is lognormally distributed and lnX has mean p and standard deviation u, 1* ~(~ = exp(p+ju ) rather than e“. 19.An analogous adjustment is used in Hubbard et. al. (1994, 1995)who focus on persistent shocksbut abstract from transitory shocks. 20. Generally, we multiply the number of households in a particular category by an integer which will resultin a figure close to 10000.In removing unwanted effects of Iognormal shocks on income means, we apply the same formula (10) to correct annual income levels. The sum of variances needed for 22 correction inthe firstyear is now given by p au~~+o: +a;, where the first varianceistheunconditional variance of persistent income shocks at age 39. The recursive formula now 21. Analogous effects were found in the context of an expected-utility model with more detailed specification of interest, dividend, and capital gains taxation (Haliassos and Lyon, 1994), Whether such resultshingeontheextent towhich thegovernment absorbstheresulting changes inportfolioincomerisk is a subject of ongoing research by Haliassos and Lyon. 22. In end-of-period models, households make choices of portfolios to hold over the second period after observing the realization of first-period income. Since only one household is considered in a representative agent framework, its realized first-period income is set at the per capita income for the education group, allowing no.role for first-period income shocks and their persistence into the second period. 23. Concern about the size of accidental bequest as a result of premature death would be a mitigating factor. 24. Model predictions regarding the average wealth (stocks) to income ratio are verycloseto thoseon the ratio of averages, but empirical wealth (stock) to income ratios are very noisy in our data. Model predictions on ratios of medians virtually coincide with predictions on median ratios. 9 40

International Finance Discussion Papers IFDP Number Titles uthor~ 1996 542 Precautionary Portfolio Behavior from a Life-Cycle CarolC. Bertaut Perspective Michael Haliassos 541 Using Options priCesto Infer PDF’sfor Asset Prices: William R. Melick An Application to Oil Prices During the Gulf Crisis Charles P. Thomas 540 Monetary Policy in the End-Gameto Exchange-Rate Steven B. Kamin Based Stabilizations: The Case of MeXico John H. Rogers 539 Comparing the Welfare Costs and the Initial Dynamics Martin Uribe of Alternative Temporary Stabilization Policies 538 Long Memory in Inflation Expectations: Evidence Joseph E. Gagnon from International Financial Markets 537 Using Measures of Expectations to Identi& the Allan D. Brunner Effects of a Monetary Policy Shock 536 Regime Switching in the Dynamic Relationship Chan Huh between the Federal Funds Rate and Innovations in Nonborrowed Reserves 535 The Risks and Implications of External Financial Edwin M. Truman Shocks: Lessons from Mexico 534 Currency Crashes in Emerging Markets: An Jeffrey A. Frankel Empirical Treatment Andrew K. Rose 533 Regional Patterns in the Law of One Price: Charles Engel The Roles of Geography vs. Currencies John H. Rogers B 532 Aggregate Productivity and the Productivity SusantoBasu of Aggregates JohnG. Femald 531 A Century of Trade Elasticities for Canada, Japan, JaimeMarquez and the United States 530 Modelling Inflation in Australia GordondeBrouwer Neil R. Ericsson . Please address requests for copies to International Finance Discussion Papers, Division of International Finance, Stop 24, Board of Governors of the Federal Reserve System, Washington, DC 2055!. 41

International Finance Discussion Papers IFDP e w thor@ 1995 529 Hyperinflation and Stabilisation: Cagan Marcus Miller Revisited Lei Zhang 528 On the Inverse of the Covariance Matrix in Guy V.G. Stevens Portfolio Analysis 527 International Comparisons of the Levels of Unit Peter Hooper Labor Costs in Manufacturing Elizabeth Vrankovich 526 Uncertainty, Instrument Choice, and the Uniqueness Dale W. Henderson of Nash Equilibrium: Macroeconomicand Ning S. Zhu Macroeconomic Examples 525 Targeting Inflation in the 1990s: Recent Challenges Richard T. Freeman Jonathan L. Willis 524 Economic Development and Intergenerational Murat F. lyigun Economic Mobility 523 Human Capital Accumulation, Fertility and Murat F. Iyigun Growth: A Re-Analysis 522 Excess Returns and Risk at the Long End of the AllanD. Brunner Treasury Market: An EGARCH-M Approach DavidP. Simon 521 The Money Transmission Mechanism in Mexico MartinaCopelman AlejandroM.Werner 520 When is Monetary Policy Effective? JohnAmmer AlIanD. Brunner 519 Central Bank Independence, Inflation and PrakashLoungani Growth in Transition Economies NathanSheets 518 Alternative Approaches to Real Exchange Rates HaliJ. Edison and Real Interest Rates: Three Up and Three Down WilliamR. Melick 517 Product market competition and the impact of VivekGhosal price uncertainty on investment: some evidence PrakashLoungani from U.S. manufacturing industries 516 Block Distributed Methods for Solving JonFaust Multi-country Econometric Models Ralph Tryon 515 Supply-side sources of inflation: evidence Prakash Loungani from OECD countries Phillip Swagel 42