Wealth and portfolio composition: Theory and …darp.lse.ac.uk/papersdb/King-Leape_(JPE_98).pdfJournal of Public Economics 69 (1998) 155–193 Wealth and portfolio composition: Theory

Journal of Public Economics 69 (1998) 155–193

Wealth and portfolio composition: Theory and evidencea b ,*Mervyn A. King , Jonathan I. Leape

aDeputy Governor, Bank of England, London, UKbLondon School of Economics, Houghton Street, London WC2A 2AE, UK

Abstract

We examine a survey of 6010 U.S. households and estimate a model for householdportfolio allocation. We extend the conventional portfolio choice model in order to capturethe observed incompleteness of household portfolios. We show that the empirical spe-cification of the joint discrete and continuous choice that characterizes household portfoliobehavior is a switching regressions model with endogenous switching. We go on to examinethe impact of taxes on portfolio composition, using detailed survey data to calculateprecisely the marginal tax rate facing each household. Finally, we estimate wealthelasticities of demand for a range of assets and liabilities. 1998 Elsevier Science S.A.

Keywords: Portfolio choice; Wealth; Taxation; Incomplete portfolios; Wealth elasticities;Switching regressions model

JEL classification: G11; H31

1. Introduction

Empirical studies of household savings behavior have focused primarily on thetotal level of savings rather than its allocation among different types of assets.Although there is a substantial theoretical literature on portfolio behavior, there arefew empirical studies of the composition of household portfolios using individualdata. For example, Blume and Friend (1975), Feldstein (1976), and Projector andWeiss (1966) used the 1962 Federal Reserve Board data, and Uhler and Cragg

*Corresponding author. Tel.: 144 171 955 7505; fax: 144 171 430 1769; e-mail:[email protected]

0047-2727/98/$19.00 1998 Elsevier Science S.A. All rights reserved.PI I : S0047-2727( 98 )00027-9

156 M.A. King, J.I. Leape / Journal of Public Economics 69 (1998) 155 –193

(1971) used data from the 1960–62 Michigan Surveys of Consumer Finances. Inthis paper, we examine a survey of 6010 U.S. households and estimate a model forthe allocation of total net worth among different assets.

The paper has three main aims. The first is to extend the conventional portfoliochoice model in order to capture the observed differences in portfolio compositionamong households. In the finance literature, the model has been tested using dataon security prices rather than household asset demands. Our survey data show thatmost households hold only a subset of the available assets. Hence we analyze amodel in which investors choose to hold incomplete portfolios. We show that theempirical specification of the joint discrete and continuous choice that character-izes household portfolio behavior is a switching regressions model with endogen-ous switching. This involves estimating equations both for the probability ofowning particular combinations of assets (the discrete choice), and for the assetdemand system conditional upon ownership (the continuous choice).

The second aim is to examine the impact of taxes on portfolio composition. Thesurvey contains a great deal of information on taxable incomes and deductionswhich enable us to calculate rather precisely the marginal tax rate facing eachhousehold. Much of the interest in tax reform, and the debate over the merits ofincome or consumption as the personal tax base, concerns the distortions createdby the present tax system in the composition of household savings. For example,tax ‘‘preferences’’ for certain assets, such as housing, are widely believed to haveled to overinvestment at the expense of other, possibly more productive assets.Proposals for reform often include the elimination of deductions for a range oftax-sheltered investments. Yet there is very little empirical evidence on the effectof taxes on portfolio composition; only Feldstein (1976) has examined this issuewith U.S. data.

The third aim is to estimate wealth elasticities of demand for a range of assetsand liabilities. Although much attention has been paid to estimating the demandfor money, less effort has been devoted to estimating the demand for other assets(primarily because of lack of data). An assumption frequently made in somemacroeconomic literature is that households exhibit constant relative risk aversion.This implies unit wealth elasticities of demand. We test this hypothesis below. Achange in wealth alters both the number of households that own an asset and thedemand for the asset conditional upon ownership. We shall estimate wealthelasticities at both the intensive and extensive margins. We shall also estimate anaggregate elasticity for the household sector as a whole, and test the hypothesisthat the impact of a change in wealth depends upon how that change is distributedamong households. Because the financial structure of the private sector’s net worthis an important determinant of real decisions, such as corporate investment, themacroeconomic consequences of a change in total household net worth willdepend upon the magnitude of the wealth elasticities of demand for differentassets. Although our results are based on microdata, they have macroeconomicimplications.

M.A. King, J.I. Leape / Journal of Public Economics 69 (1998) 155 –193 157

In Section 2, we describe the pattern of household portfolio composition thatemerges from the survey and present some summary statistics. In Section 3, wederive a specification for the portfolio decision and discuss an appropriateestimation procedure. The estimation and results are described in Section 4.Estimates of the effects of taxes on portfolio composition are presented in Section5 and the wealth elasticities in Section 6.

2. Portfolio composition of the sample

Our data are taken from the 1978 Survey of Consumer Financial Decisionsconducted by SRI International. The survey is a stratified random sample of 6010U.S. households. It includes detailed information on household characteristics,such as age, education, region and area of residence, marital status and familycomposition, occupation, race, and housing. It also includes information on familyincome as well as portfolio composition and net worth. The asset data refer tomarket values in May and June 1978 and the income data to the calendar year1977. The survey heavily oversampled high income families and therefore

1provides a rich source of information on household portfolio behavior. No fewerthan 2430 households (40.4% of the sample) reported net worth in excess of$100 000 in 1978. The sample contains 204 millionaires and the largest value ofreported net worth is $73 million.

The survey is remarkable for the information it provides on the asset holdings ofeach of the 6010 households. The dollar holdings of over one hundred differentspecific assets and liabilities were recorded for each household. These weregrouped 3by SRI into 23 different assets and 13 types of liability. Table 1 showsthese 36 categories. They include different types of bank and savings account,money market funds, bonds of various types, stocks, mutual funds, convertiblesecurities, owner-occupied housing, real estate and other tangibles, IRA-Keogh

2accounts, tax shelters and life insurance. For purposes of estimation, we havegrouped these thirty-six categories into eleven aggregate asset and liabilityclassifications. These are shown as the main headings in Table 1. The survey dataexclude social security and certain types of private pension wealth, ordinary

3consumer durables, and the expected value of future inheritances. Table 1 alsoshows the percentage of the total net worth of the sample that is held in each of the

1The sample has two parts. The ‘‘NFO’’ file consists of 3804 households with incomes of less than$30 000. The ‘‘Gallup’’ file consists of 2206 households with incomes of $30 000 or more. Weightingfactors for each household are provided on the tape that allow us to gross-up sample totals to providepopulation estimates.

2The recorded data on the value of partnerships and unquoted businesses of which the household wasthe principal owner were imputed using sales figures. Because of the arbitrary nature of the imputationwe excluded ‘‘own business value’’ from our analysis.

3Durables such as boats, planes, works of art, etc. are included in the category ‘‘other assets’’.


Table 1Asset classifications

Mean asset Percentage ofholding total wealth(in dollars)

I. CHECKING ACCOUNTS 2419 1.11. Checking accounts 2419 1.1

II. LIQUID SAVINGS 11 236 5.02. Savings accounts 10 336 4.63. Credit union share accounts 835 0.44. Money market funds 65 0.0

III. LESS LIQUID SAVINGS 8520 3.95. Savings certificates 6589 3.06. U.S. Savings Bonds 913 0.47. Money market instruments 1018 0.5

IV. CONTRACTUAL SAVINGS 14 459 6.58. Pension or retirement 10 485 4.7

plan account9. Single-premium annuities 811 0.4

(excluding IRA’s)10. Cash value of life insurance 3163 1.4

V. EQUITY 52 102 23.311. Stocks 49 833 22.312. Stock mutual funds 2269 1.0

VI. TAXABLE BONDS 5217 2.413. Corporate bonds 3059 1.414. Federal government bonds 2158 1.0

VII. MUNICIPAL BONDS 6860 3.115. Municipal bonds 6860 3.1

VIII. HOMES 74 342 33.316. Value of primary residence 70 293 31.517. Value of secondary residence 4049 1.8

IX. OTHER ASSETS 86 087 38.618. Tax shelters 1910 0.9

and equipment leases19. Closely-held stock 19 020 8.520. Convertible securities, 9452 4.2

REITS, boats, and planes21. Investment real estate 49 982 22.422. Tangibles 4022 1.8

(marketable art, gold, etc.)23. Other assets 1701 0.8

X. HOME MORTGAGES 18 529 8.324. First mortgage 8281 3.7

on primary residence


Table 1. Continued

Mean asset Percentage ofholding total wealth(in dollars)

25. Second mortgage 257 0.1on primary residence

26. Mortgage on second home 869 0.427. Home improvement loan 9122 4.1

XI. OTHER LIABILITIES 19 524 8.728. Personal loans 1813 0.829. Cash value loans 956 0.430. Revolving bank card account 564 0.3

Liabilities against:31. Tax shelters 456 0.232. Closely-held stock 1549 0.733. Convertible securities, 306 0.1

REITS, boats and planes34. Investment real estate 13 700 6.135. Tangibles 20 0.036. Other assets 160 0.1

NET WORTH 223 188 100.0

Source: Own calculations based on SRI survey.

assets. Most of the wealth of households is held in the form of homes, stocks, andinvestment real estate. The implausibly low percentage accounted for by homemortgages is due to a high degree of nonreporting—close to fifty percent—of thevalue of first mortgages on primary residences. This is analyzed further below.

Table 2 presents summary statistics on the portfolio composition of householdsin the sample. The mean level of net worth in 1978 was almost a quarter of amillion dollars. This oversampling of the rich is a great advantage for the analysisof portfolio composition. Not surprisingly, certain assets are held by householdswith above average wealth levels, notably corporate equity, bonds and ‘‘otherassets’’ (tax shelters, etc.). Of the 6010 households only 593 owned municipalbonds but the mean net worth of this group was almost a million dollars.

The SRI survey is particularly well suited to a study of the effects of taxes onhousehold portfolio behavior. It provides a detailed account of the sources ofincome in each household and of tax-deductible expenses. Twenty-two sources ofincome and twenty-one types of expense are distinguished. This information,combined with the demographic data provided for each household and the specificinformation on the household’s tax filing status, made it possible to derive accurateestimates of the marginal tax rates faced by each household. We computed thesetax rates using the TAXSIM program developed at the National Bureau ofEconomic Research, which generated tax liabilities for each household using both

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Table 2Portfolios of sample of 6010 U.S. households

Checking Less liquid Contractual Contractual Corporate Taxable Municipal Owner-occupied Other Home Other Net

accounts savings liquid savings equity bonds bonds housing assets mortgages liabilities worth

savings

Number

holding 5558 5438 3255 4596 2915 632 593 5051 2610 3341 3714 6010

the asset

Percentage

of total 92.5 90.5 54.2 76.5 48.5 10.5 9.9 84.0 43.4 55.6 61.8 100.0

sample

Mean asset,†share 4.9 16.8 12.9 17.1 18.1 12.3 10.5 82.1 40.4 61.1 21.8 100.0*

(percent)

Mean net

worth* 229 339 217 071 247 083 238 482 392 030 744 202 969 941 258 006 399 743 200 349 221 984 223 188

($)

*Conditional on ownership of the asset.†Percent of net worth (reported only for households with positive net worth).Source: Own calculations based on SRI survey.


4the federal and state tax code relevant for the appropriate tax year. Theinformation available to construct marginal tax rates is, to our knowledge, moredetailed than that used in any previous study of the incentive effects of taxation.

It is clear from rows 1 and 2 of Table 2 that most households owned incompleteportfolios. In only two cases, checking accounts and liquid savings, was an assetheld by more than 90% of the sample. At the other extreme the proportions of thesample owning municipal bonds and taxable bonds were 9.9 and 10.5% respective-ly. Almost one-half of the sample owned corporate equity and 84 percent werehome-owners. In fact, for the complete classification no household held more than23 out of the possible 36 assets and liabilities, and for the aggregated classificationless than one percent of households held a complete portfolio with all eleven assetsand liabilities. The modal number of assets owned was eight for the completeclassification and seven for the aggregated classification, accounting for 12.0 and19.6% of the sample respectively. The median number owned was eight and sixfor the two classifications. Given that the mean net worth of the sample was almosta quarter of a million dollars, it is noteworthy that the number of assets held wasso small. It is important, therefore, to take into account the phenomenon ofincomplete ownership when estimating the demand for individual assets; thisproblem is the focus of Section 3.

The survey provides indicators of ownership for each asset that are independentof the dollar holdings reported. Some households do not report dollar values forevery asset they own and the existence of independent responses indicatingownership and dollar holdings enables us to measure the extent of this nonreport-ing (although not, of course, under-reporting). The additional information allowsus to obtain more efficient estimates by exploiting techniques to deal with theproblem of missing data. Of the 6010 households, 2048 provided a complete set ofdollar values for the assets that they owned. In Section 4, we describe how data forboth the sample of 2048 households with complete responses and the full sampleof 6010 households may be used in estimation. Although SRI used the ownershipindicators to impute values in cases of nonreporting, we were able to separate outthe imputed values and used only the raw data actually reported by the surveyrespondents in our analysis.

To test the quality of the survey data, we compared the estimated populationholdings for each asset (constructed by using the grossing-up factors described infootnote 1) with the estimated aggregates shown in the year-end balance sheetspublished by the Federal Reserve Board. The comparison is of necessity inexact.The two sources use different classifications of assets and refer to different dates inthe year. Moreover, the FRB estimates for the household sector are a residual inthe balance sheet calculations. It was possible to make a direct comparison forseven of our eleven categories of asset and liability, and an approximatecomparison for the remaining assets. For most of the assets the survey population

4For a description of TAXSIM see Feldstein and Frisch (1977) and Feenberg and Rosen (1983).


totals are quite close to the FRB figures. The largest discrepancies were for taxablebonds and ‘‘other assets’’ where the survey data were only one-half and two-thirdsof the balance sheet totals respectively. Overall, it seems reasonable to concludefrom these comparisons that the survey data used here are worthy of study.

We turn now to the specification and estimation of a model of householdportfolio composition that allows for incomplete portfolios.

3. A model of incomplete portfolios

It is clear from the survey data that most households own only a subset of thepossible array of assets that they could hold. It is therefore necessary to modelboth the probability of ownership of each asset and the demand for the assetconditional upon ownership. The problems created by incomplete ownership aretwofold. First, the existence of zero holdings for some assets will have ‘‘spillover’’effects on the demand for the remaining assets. Conditional upon the values ofobservable characteristics, the demand for an asset will depend upon the particularcombination of other assets in the portfolio. Second, there is the more familiarproblem that estimates based on observations with positive holdings of the assetare subject to sample selection bias. Previous empirical studies of householdportfolio behavior have not tackled these problems, especially the first, satisfactori-ly. Ignoring such problems results in a misspecification of the model and biasedestimates of the parameters of interest.

Previous theoretical discussions of incomplete portfolios have not analyzed thejoint discrete and continuous choice of which combinations to own and, con-ditional upon ownership, what fraction of net worth to invest in each asset.Brennan (1975) and Goldsmith (1976) both analyze the optimal number ofsecurities in a portfolio (taken to be a continuous variable) in the presence oftransaction costs. Optimal asset demands for a given combination have beenexamined by Levy (1978) and Mayshar (1981). Only Mayshar (1979) examines

5the choice of which assets to own.There are several reasons for the existence of incomplete portfolios. Two of the

most important are the following. First, the existence of transaction costs,interpreted broadly to include the ‘‘holding costs’’ in both money and time ofmonitoring and managing a portfolio, means that the optimal portfolio may containonly a small number of assets. Second, it is possible that the optimal portfolio mayimply negative holdings of certain assets, and these may be infeasible because ofconstraints on short sales (Auerbach and King, 1983). To prevent tax arbitrage, therevenue authorities adopt rules, such as the asymmetric treatment of gains andlosses and of interest receipts and interest payments, that lead to nonlinear budget

5Leape, 1987 analyzes the effects of taxes on portfolio choice in a model incorporating the jointdiscrete and continuous choice.


constraints. These constraints may be very similar in practice to nonnegativityconstraints on asset holdings. In other instances it may be very difficult orimpossible to hold certain assets, such as checking accounts, pension rights andhousing, in negative quantities. We shall therefore examine the optimal portfolio ofa household with both positive holding costs for each asset and nonnegativity

6constraints on asset holdings.The specification we derive involves estimating equations both for the probabili-

ty of ownership of an asset and also for its demand conditional upon ownership. Ifthe incompleteness of portfolios results from either the existence of holding costsor an optimization problem subject to nonnegativity constraints on holdings ofindividual assets, then the demand system is a switching regressions model with asmany regimes as there are distinct combinations of assets that may be owned. It isimportant to note that the extension of the univariate Tobit model to themultivariate case does not represent the behavior of an individual investoroptimizing subject to holding costs or nonnegativity constraints. This is demon-

7strated below. The way in which the asset demand function changes according tothe regime is shown to be a particular form of structural shift. These shiftsrepresent the ‘‘spillover’’ effects of the absence of certain assets in the optimumportfolio on the demands for those assets that do appear.

To examine the optimal portfolio we consider an investor whose preferences canbe represented by the following utility function which is assumed to be additivelyseparable over time.

T

V 5 E E U(C, t) dt, (1)0

U is assumed to be a strictly concave function of a single consumption good, C,and time (or age), denoted by t. T is either the investor’s maximum life span(uncertainty about date of death may be incorporated in the time-dependency ofU ) or a longer planning horizon which allows for a bequest motive. The twoconstraints are the initial condition for wealth and the budget constraint.

W(0) 5 W , (2)0

JdW 5O N (t) dP 1 Y(t) dt 2 C(t) dt, (3)j51 j j

6A further reason might be the existence of a number of ‘‘mutual funds’’ (less than the number ofassets and liabilities distinguished in our survey) which in themselves are sufficient for a portfoliooptimum. Given the nature of the eleven assets and liabilities used in this study (see Table 1) theexistence of such mutual funds is improbable. The number of funds that would be required dependsupon the stochastic process generating asset returns.

7For further discussion of this point see King (1984).


where N (t) is the number of units of asset j held at time t. P (t) is the price of onej j

unit of asset j at time t. Y(t) is the nonstochastic flow of labor income. C(t) is theconsumption flow at time (t).

The budget constraint given by (3) embodies two strong assumptions. First, theonly source of uncertainty is assumed to be that concerning returns on the J riskyassets. Second, investors are assumed to be able to trade continuously with no

8costs of buying or selling securities.If one further assumption is added, namely that asset prices follow a continuous

time Markov process and do not exhibit jumps, then investors choose their currentportfolio to maximize V subject to Eqs. (2) and (3) as if they were optimizing overthe mean and variance of the portfolio given the level of current wealth (Merton,1971, 1973, 1982). The stochastic process generating asset prices is not required tobe stationary, but it must change in a way that is either random or time-dependent

9in a nonstochastic manner. With this assumption on the way in which exogenousshocks affect asset returns, we may analyze portfolio decisions as if investorsmaximized the following function defined over the mean and variance of theirportfolios

h h h h 2F 5 f (m , (s ) ), (4)

J h h hO e 5 W 1 b , (5)j51 j

hsubject to the budget constraint where m denotes the mean return on the portfolioh 2 h hof household h, (s ) the variance of the portfolio, e the demand for asset j, Wj

hnet worth, and b the amount borrowed at the nonstochastic interest rate R. For themoment we shall assume the existence of a riskless asset and return later to amodel with no riskless asset.

The mean return on the portfolio is

h h h h hm 5O e m (1 2 t ) 2 Rb (1 2 t ), (6)j j j j b

hwhere m is the mean pretax return per dollar invested in asset j, t is investor h’sj jheffective tax rate on the return from asset j, and t is the tax rate against whichb

interest payments may be deducted. An important determinant of portfoliocomposition is the way in which effective tax rates vary across both assets andhouseholds.

The variance of the portfolio is

8It is important to distinguish between transaction costs of trades and ‘‘holding costs’’ of differentassets. The latter are discussed below; the former we shall ignore.

9If preferences are restricted such that U belongs to the HARA (hyperbolic absolute risk aversion)class, then the one-period portfolio problem equivalence result holds for a more general nonstationarystochastic process (Merton, 1971).


h 2 h h h h(s ) 5O O e e C (1 2 t )(1 2 t ), (7)i j i j ij i j

where C is the covariance of the per dollar returns on assets i and j. Short salesij

constraints will be assumed to exist for all risky assets

he $ 0 ; j, h. (8)j

Maximizing (4) subject to (5) and (8) yields a set of first-order conditions togetherwith the complementary slackness conditions corresponding to the nonnegativity

10constraints. Once it is known which assets are owned in positive amounts in theoptimal portfolio then the first-order conditions can be inverted to yield a

11conditional asset demand system. The values of asset demands conditional uponthe combination of assets that is owned is given by

h h h h he 5 A A T V (m 2 Ru ), (9)k k k k k k] ]]

Jwhere k denotes which of the possible (2 21) combinations of positive assetholdings characterize the optimal portfolio. The subscript k denotes that the matrixor vector includes only those rows and columns of the original matrix which

hcorrespond to the assets contained in combination k. A is equal to the inverse of12the investor’s degree of absolute risk aversion. V is the inverse of the covariance

h hmatrix C, T is a diagonal matrix with 1/(12t) as the j element of the leadingh th h hdiagonal, and u is a vector with j element equal to (12t ) /(12t ).b j

It is convenient to assume that the differences in the tax treatment of assets canbe captured by the following specification for the effective tax rate on asset j

ht 5 d t , j 5 1,...,J, b, (10)j

hThe value of t is the statutory marginal tax rate facing household h.Eq. (10) encompasses most of the important differences in effective tax rates

across assets (for example, the tax-exempt nature of pension funds and the failure13to tax the imputed income of home-owners). With this assumption, the demand

for asset j, conditional upon ownership, as a proportion of net worth (denoted byp) may be written as

10 hThe second-order conditions will be satisfied if f is a concave function of the equity holdings anda suitable constraint qualification holds.

11This model and its relationship to the econometric literature on multivariate tobit and relatedmodels is discussed in more detail in King (1984).

12 hIn the continuous time model underlying (4), A is equivalent to the inverse of the degree ofabsolute risk aversion of the derived utility of wealth function. This function is defined below in (17).

13One exception is that the effective tax rate on equity, which depends upon the treatment of capitalgains, varies with the frequency of trading because the tax is based on realizations. The tradingfrequency may, in turn, be a function of household characteristics.


hh 1 2 d tA 1 bh k] ]]] ]]]p 5 O m 2 R v , (11)S D F Gj h h iek i h ijS DW 1 2 d t 1 2 d tj i

k thwhere v denotes the ij element of the matrix V . Using the approximationij k

ln(11x)5x, we have

h h h hln p 5 a (k, t ) 2 ln r 1 d t , (12)j j j

hwhere r is the degree of relative risk aversion and a (k, t) summarizes the way inj

which demands depend upon the particular combination of assets in the house-hold’s portfolio. Risk aversion depends upon household wealth (human andnonhuman), age and other observable characteristics. We assume that

h h hln r 5 X b 1 e , (13)hwhere X is a vector of observable household characteristics (including net worth),

and e accounts for unobservable differences in risk aversion. The SRI surveyasked households to describe their degree of aversion to risk on a scale from oneto five. The responses are used as an explanatory variable in the estimation of the

14model discussed in Section 4.15Finally, we take a linear approximation of a and assume thatj

ha (k, t) 5 a 1 b t . (14)j j j

Combining (12), (13) and (14) yields the following equation for asset demandsconditional upon ownership

h h h hln p 5 a 1 g t 1 X b 1 u , (15)j jk j j

h h hwhere g 5b 1d , and u 5e 1h (where h represents measurement and optimi-j j j j j2zation error) is assumed to be distributed N(0, s ).

The demand system (15) is a switching regressions model with endogenousswitching. The advantage of deriving an explicit model is that the assumption ofutility maximization restricts the structural shift between regimes to a rather simpleform. The intercept in (15) depends upon the particular combination of assets(other than j) that are present in the household’s portfolio. The functional form of(15) is independent of the regime and demands are a linear function of the sameset of explanatory variables in each regime with the constant term varyingaccording to the combination of assets owned. This form of shift is easilymodelled in terms of dummy variables, d , denoting the combination of assetsk

14The estimated coefficient on the self-reported degree of risk aversion must, of course, beinterpreted with caution.

15We ignore here the effect of the combination of assets owned on the coefficient of the tax rate inorder to reduce the problems of collinearity discussed in Section 4.


J21other than j in the portfolio. There are N5(2 21) such combinations. Hencethe equation we shall estimate may be written as

Nh h h h hln p 5O a d 1 g t 1 X b 1 u , j 5 1,...,J, (16)j k51 jk k j j

Eq. (16) gives asset demands conditional upon ownership. To determine whichcombination of assets a household owns it is not sufficient to examine thefirst-order conditions. Rather, the levels of utility corresponding to each combina-tion must be compared. This means that the household faces a discrete choiceproblem with as many alternatives as there are distinct combinations of assets,

Jwhich (ignoring the null portfolio) is 2 21. For combination k, define themaximum level of utility (as a function of wealth, the distribution of asset returnsand other household characteristics) that can be attained with that combination ofassets as

h h h h h h 2*V 5V(X , t , e , m , C ) 5max f (m , (s ) ). (17)k k ke u j[k] j

*The conditional demands that characterize the optimum at V are given by (9).k

The probability that combination k characterizes the optimum portfolio is thus

J* *Pr(k) 5 Pr(V 5max V ; i 5 1,...,2 2 1). (18)k ii

The structure of portfolios varies across households because of differences inobservable and unobservable household characteristics. The observables includethe tax rate, wealth and other socioeconomic characteristics recorded in the survey.

hThe unobservables include the differences in risk aversion captured by e , anddifferences in beliefs about the distribution of asset returns. In this model taxes andheterogeneous expectations play a crucial role in producing incomplete portfolios.With homogeneous beliefs about the distribution of asset returns and in theabsence of taxes, the constraints on short sales would never be binding. Theseparation theorem implies that with a riskless asset the two mutual funds thatspan the set of available returns are the riskless asset itself and a risky portfolio.All investors own the same risky portfolio and this can be chosen to be the marketportfolio. By construction it has positive amounts of the risky assets. A portfoliooptimum is characterized by nonnegative holdings of the two mutual funds, and so

16no investor would wish to sell short any of the risky assets. In the presence oftaxes or of heterogeneity in beliefs about the distribution of asset returns, investorswill no longer wish to hold the same risky portfolio, and there will be

17opportunities for trade to exploit differences in tax rates (or beliefs). At theoptimum some investors may wish to engage in short sales. With constraints on

16We assume that none of the risky assets are redundant in the sense that its returns are a linearcombination of the returns on other assets.

17See also Tobin (1958) early discussion of the effect of taxes on optimal portfolios.


short sales the result will be that some, or more likely many, investors will holdincomplete portfolios. Another factor leading to incomplete portfolios is holdingcosts, and we analyze these below.

JTo model the household’s discrete choice among the 2 21 mutually exclusivealternative portfolios would involve estimating the joint distribution of theunobservables (differences in (i) risk aversion, (ii) beliefs about the distribution ofasset returns and (iii) holding costs), and would be computationally infeasiblegiven the number of alternatives and assets examined here. We shall, therefore,estimate reduced form equations for the probability of owning a particularcombination of assets, and also for the probability of owning a owning a givenasset j. The latter probability is given by

Pr(own j) 5O Pr(k). (19)k u j[k

We assume that the probabilities in (18) and (19) can be described as if they weregenerated by a probit model, but we provide no rigorous justification for theimplied assumption of normally distributed errors. Estimates of probit models for(18) and (19) are described in Section 4.

One strong restriction implied by the model in (16) is that b, the coefficientvector of household characteristics, is the same in all asset demand equations. Thisimplies that the wealth elasticities of conditional demands are the same for allrisky assets. As we shall see in Section 6, the null hypothesis that wealthelasticities are the same for all assets is rejected by the data.

To generalize the model we relax two of the assumptions made above. The firstis the existence of a riskless asset. When all assets are risky, the wealth elasticitiesof conditional demands vary across assets. This is because the separation theoremnow implies that the two mutual funds that span the set of returns both consist ofparticular combinations of risky assets. We shall regard (16) with b replaced by bj

18as a linear approximation to the underlying demands.The second assumption is that of zero holding costs of monitoring and

managing a portfolio. We can incorporate holding costs in the model in a limitedfashion. Suppose that such costs comprise two components: a fixed cost that isindependent of the size of the holding and a variable cost proportional to theamount owned. As far as the variable costs are concerned, the first-orderconditions and the demand system (9) are unchanged with the single exception

h h hthat u is now equal to (12t )(11c ) /(12t ), where c is the variable holdingb j j j

cost per dollar for asset j. Assuming that investors face the same cost schedule theinfluence of the holding costs is absorbed into the coefficients of the ownershipcombination dummy variables in (16). The fixed costs have no effect on theconditional demand system, but both types of cost influence the decision as towhich assets will appear in the optimal portfolio and affect the coefficients of the

18Bodie (1982) has argued that short-term Treasury Bills are reasonably close to a risklessinvestment.


reduced form probit models (18) and (19). The existence of holding costs can,therefore, also lead to incomplete portfolios with the conditional demand systemgiven by (16). Transaction costs of the more conventional kind defined over tradesrather than the size of holdings are equivalent to holding costs in a one-periodmodel. In a continuous time model, however, trading costs lead to infrequentrevisions of the portfolio and thus undermine the equivalence of the mean-variancemodel and the life-cycle model with continuous trading.

4. Econometric estimation and results

With either nonnegativity constraints or holding costs of the type analyzedabove, the asset demand system conditional upon ownership is given by aswitching regressions model with as many regimes as there are distinct combina-tions of assets. The way in which the demand system changes according to theregime represents the ‘‘spillover’’ effects of the absence of certain assets in theoptimal portfolio on the demands for those assets that do appear. It would beimpossible to estimate separate functional forms for each regime because with

j2 21 regimes even a large data set such as that used here does not provideadequate degrees of freedom. To derive a model for which estimation is feasiblewe have tried to exploit the economic structure of the consumer’s programmingproblem. This led to a specification in which the shift in functional form acrossregimes was rather straightforward in that only the intercept varied with regime.The choice of the optimal combination of assets is made by a comparison of the

jutility levels associated with the 2 21 discrete alternative combinations. Empiri-cally we shall model this choice by estimating reduced form probits for the

19probability of owning particular combinations of assets.Several problems arise in the estimation of the demand system (16). First, since

the demand equation for each asset is estimated using observations on only thosehouseholds with positive holdings of the asset, there is potential sample selectionbias. To correct for this, we use a standard two-stage procedure to yield consistentestimates. We first estimate reduced form probit equations for the ownershipprobabilities of each asset and then include the estimated hazard as an additional

20regressor in the demand system (Heckman, 1976, 1979).

19This is not entirely satisfactory because when all assets are risky the demand for asset j conditionalupon ownership of combination k is

h h h h 21p 5 a (k, t ) 1 b (k, t )(r ) ,j j j

where a and b are functions of the means, covariances, and tax rates. The logarithmic transformationj j

of the dependent variable was made to reduce the otherwise severe heteroskedasticity. The influence ofthe combination of assets owned was limited to the constant term as in (16) to reduce to manageableproportions the number of coefficients to be estimated.

20For a very small number of assets the likelihood function corresponding to the Kuhn–Tuckerconditions for the investor’s programming problem can be evaluated using approximations to themultivariate normal distribution (Wales and Woodland, 1983). But for the number of assets which it iseconomically interesting to examine this approach is infeasible.


Second, only a subset of the full sample, namely 2048 households, reported thevalue of holdings for each asset which they owned. Provided nonreporting is eitherrandom or, more generally, ‘‘ignorable’’ (Griliches, 1984), then estimation on thesample of 2048 households yields consistent estimates. Nonreporting is ‘‘ignor-able’’ if it is a function of the explanatory variables in (16) but not of the errorterm u . More efficient estimates can, however, be obtained by using thej

information contained in the remaining observations. Nonreporting of dollar valuesleads to the problem that reported wealth understates true net worth if values ofassets are not reported, and overstates it if liabilities are not reported. Net worth isan important explanatory variable in the model, and we wish to test the hypothesisthat risk aversion, as given by (13), is a function of net worth. Hence we mustallow for the effect of nonreporting in estimation. One approach would be to treatall instances of nonreporting as missing observations on net worth and use existingmethods for dealing with missing data (Dagenais, 1973; Gourieroux and Monfort,1981 and Conniffe, 1983). This would, however, be very inefficient becausenonreporting is usually of the form where a household fails to report values foronly one or two of its assets, and reported net worth contains valuable informationon holdings of other assets which could be used in estimation. The values of networth are incomplete rather than missing. In Appendix A therefore we describe atwo-step method for dealing with the problem of nonreporting and which exploitsthe information contained in recorded net worth.

Third, because of the differential tax treatment of assets, the marginal tax rate isendogenous to the choice of portfolio. Under a nonlinear tax schedule, such as thatin the U.S., a household’s marginal tax rate depends upon the composition of itsportfolio. For example, a household can lower its marginal tax rate for a givenlevel of net worth by investing in municipal bonds rather than taxable bonds. Todeal with this we estimate (16) by instrumental variables. FIML estimation isinfeasible here because of the number of dimensions (eleven assets) over which thenonlinear budget set is defined. We calculate the marginal tax rate that eachhousehold would face at an exogenously given hypothetical portfolio using theactual nonlinear tax schedule applicable to the household. With this constructed taxrate as an instrument for the actual marginal tax rate consistent estimates of the

21parameter vector may be obtained. To construct the exogenous portfolio we haveassumed that households hold all of their wealth in short-term government

21The theoretical model implies that the function of observable characteristics which determineswhether a household owns an asset is not the same as the function which describes demandsconditional upon ownership. Hence we estimate a probit ownership model and demand systemseparately rather than a tobit model. Because we provide no rigorous justification for the probitspecification, the assumption of joint normality of the errors in the ownership and conditional demandequations must be regarded as a maintained hypothesis.


securities (3-month Treasury Bills), the return on which is taxed as ordinary22income.

Fourth, the theoretical specification implies that the intercept in (16) consists ofa sum of dummy variables, each one corresponding to the ownership of aparticular combination of assets other than that in the dependent variable. In

J21principle the required number of dummy variables is 2 21 (J is the number ofassets), which for J511 is equal to 1023. It would clearly be infeasible to estimatea demand system with that number of dummy variables in each equation.Moreover, some combinations are not held by any household in the sample, andhence some of the dummy variables would be perfectly collinear. We have,therefore, aggregated the assets into four groups for the purpose of defining theownership dummy variables in (16). The four groups are (1) checking accountsand liquid savings, (2) equity, municipal bonds, taxable bonds and other assets, (3)homes, contractual and less liquid savings, and (4) home mortgages and otherliabilities. The criterion for selecting the groups was that the correlation coefficientbetween the ownership dummies of any pair of assets within a group be greaterthan 0.75 and that between any pair of assets in different groups be less than 0.75.This criterion was sufficient to determine both the number and composition of ouraggregate groups with one exception. Membership of contractual savings schemeswas more highly correlated with positive holdings of liabilities than with holdingsof assets, but we chose on a priori grounds to group contractual savings withhomes and less liquid savings (which had correlation coefficients of 0.54 and 0.50with membership of contractual savings schemes). Grouping assets in this waymeans that in each of the eleven conditional demand equations there are amaximum of seven ownership dummy variables. In cases where the number ofhouseholds owning a particular combination was very small (well below 1 percentof the sample) the dummy corresponding to that combination was omitted to avoidproblems of collinearity. The ownership dummies used in each equation are shownin Table 4.

Fifth, the ownership combination dummy variables in (16) are endogenous. Theoptimal portfolio choice determines both the combination of assets that are ownedand the demands conditional upon ownership. Hence the dummy variables thatenter into (16) are endogenous. Several estimation methods can be used to obtainconsistent estimates in the presence of endogenous dummy variables (Heckman,1978; Dubin and McFadden, 1984). We use the reduced form method in whichprobit models are estimated for each ownership combination and then ordinary

22The nonlinearity of the tax schedule means that the instrument is a nonlinear function of theexogenous variables. In this case instrumental variables estimation is not equivalent to two-stage leastsquares. For further discussion of estimation with nonlinear budget constraints see Hausman, 1981,1982.


least squares is applied to (16) with predicted probabilities replacing the ownershipcombination dummies. Coefficient estimates of the asset demand equations andheteroskedastic-consistent standard errors are shown in Table 4.

The estimated reduced-form probit equations are shown in Table 3 and theestimated asset demand equations in Table 4. The explanatory variables in theprobit equations include terms in total net worth, current employment income, themarginal tax rate, the age of the head of household, marital status, occupation,education, employment status, and the subjective perception of aversion to risk.The explanatory variables in the demand equations include all of these variablesplus the relevant set of dummy variables corresponding to the ownership ofdifferent combinations of assets. Although the asset shares in the demandequations must satisfy an adding-up constraint, this constraint is not imposed inthe estimation procedure because the dependent variable (the asset share) is inlogarithmic form. One equation may therefore be regarded as redundant.

We focus on the estimated tax and wealth effects in Section 5 Section 6, andsummarize briefly here the results for the other explanatory variables. Income fromemployment has (as predicted by the model in which net worth is the relevantvariable) little effect on asset ownership or demands except for homes and homemortgages where the consumption services provided are likely to be correlatedwith employment income. Age is an important determinant of ownership. The

23results in Table 4 reveal a pronounced quadratic relationship. In the conditionaldemand equations age effects are significant only for the first four assets, where aquadratic relationship is again evident. The most significant effect of marriageappears to be on the size of checking accounts. The effects of occupation and, inparticular, education suggest that information costs may be an important deter-minant of portfolio behavior. Both occupation and education significantly affectthe probability of owning corporate equities, taxable bonds, municipal bonds, and‘‘other assets,’’ the four asset categories for which we would expect informationcosts to be highest. Consistent with this explanation is the finding that neithereducation nor occupation play a significant role in the conditional demandequations. Being employed has a positive effect on the probability of owning mostassets (except for equity and bonds). It appears, however, to have no effect on thelevel of conditional demands. Risk averse households are more likely both to ownand to invest a higher proportion of net worth in contractual savings and taxablebonds. They invest correspondingly less in the more unusual and risky assets thatare included in the ‘‘other’’ assets and liabilities categories.

The coefficients on the ownership combination dummies would generally beexpected to be negative, either because the assets are substitutes or because thegreater the number of assets in the portfolio the smaller is likely to be the share ofany one asset. Positive coefficients would arise when assets were complements. Of

23The interest rate was taken to be the mean of the quarterly rates for 1977 published in The FederalReserve Bulletin, a figure of 5.27 percent per annum.

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173Table 3Probit model for ownership of assets (asymptotic standard errors in parentheses)

aExplanatory variable Checking Liquid Less Contractual Corporate Taxable Municipal Homes Other Home Other

account savings liquid savings equity bonds bonds assets mortgages liabilities

savings

Constant .866 .885 21.599 22.704 22.266 23.172 23.001 25.635 21.315 24.951 2.718

(.315) (.302) (.231) (.246) (.271) (.516) (.477) (.387) (.256) (.294) (.228)

ln (net worth) .145 .397 .375 .399 2.119 .169 2.084 1.146 2.058 1.049 .112

(.057) (.051) (.052) (.051) (.063) (.126) (.109) (.105) (.060) (.070) (.045)2 21[ln (net worth)] 310 2.107 2.283 2.198 2.221 .413 .146 .333 2.453 .347 2.749 2.106

(.049) (.043) (.041) (.043) (.053) (.085) (.075) (.091) (.049) (.053) (.037)26Income from employment310 2.747 21.931 21.046 .034 .975 1.248 1.560 3.109 .093 3.518 .047

(.978) (.760) (.645) (.837) (.905) (.719) (.719) (1.317) (.758) (.833) (.682)

Marginal tax rate .544 1.139 .544 .864 .310 2.133 .057 21.107 .491 .309 .734

(.202) (.183) (.129) (.153) (.149) (.165) (.167) (.232) (.137) (.143) (.134)21Age310 2.128 2.562 2.119 .646 .335 2.154 .151 .671 .008 .779 .344

(.124) (.124) (.082) (.090) (.090) (.126) (.135) (.111) (.088) (.096) (.088)2 23(Age) 310 .149 .568 .173 2.750 2.267 .270 2.048 2.717 2.119 21.074 2.574

(.121) (.120) (.080) (.087) (.088) (.118) (.127) (.011) (.087) (.096) (.087)

Married .130 .208 .117 .124 2.049 2.110 2.308 .641 .062 .408 .244

(.082) (.077) (.055) (.061) (.060) (.076) (.078) (.083) (.059) (.059) (.057)

Occupation: Managerial 2.051 2.171 2.076 2.065 .290 .035 .152 2.164 .244 .134 .137

(.074) (.070) (.047) (.055) (.050) (.070) (.071) (.078) (.048) (.050) (.049)

Occupation: Professional 2.114 2.063 2.078 .123 .172 .189 .164 2.137 .257 .137 .014

(.086) (.083) (.054) (.066) (.058) (.075) (.078) (.086) (.056) (.058) (.056)

Education: College or above .300 .072 .062 .085 .505 .472 .399 2.163 .082 .164 .007

(.067) (.062) (.042) (.048) (.045) (.061) (.062) (.066) (.043) (.045) (.044)

Employed .161 .202 .072 .306 2.181 2.226 2.254 .091 2.073 2.007 .162

(.079) (.075) (.054) (.058) (.059) (.078) (.080) (.085) (.058) (.058) (.055)

Risk averse 2.003 2.067 .067 .223 2.047 2.100 .004 .078 2.051 .041 .020

(.068) (.065) (.043) (.051) (.048) (.058) (.060) (.075) (.046) (.048) (.046)

No. holding the asset 5246 5149 3144 4342 2819 621 581 4875 2514 3185 3394

No. not holding the asset 365 462 2467 1269 2792 4990 5030 736 3097 2426 2217

Chi-square (12) 83.9 195.4 293.7 941.3 1752.5 769.4 784.9 1748.0 1304.8 1678.6 918.6

aDummy variables take the value unity when the description applies to the household, zero otherwise. Individual variables refer to the head of a household.

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Table 4Asset demand equations (asymptotic standard errors in parentheses)

aExplanatory variable Checking Liquid Less Contractual Corporate Taxable Municipal Homes Other Home Other

account savings liquid savings equity bonds bonds assets mortgages liabilities

savings

Constant 23.177 0.362 10.693 23.645 3.647 11.780 26.348 4.505 4.012 5.814 1.718

(0.902) (0.765) (8.652) (1.974) (1.888) (9.864) (8.396) (0.936) (1.405) (2.386) (1.812)

ln (net worth) 20.525 20.498 22.089 20.577 21.255 21.289 20.916 20.727 21.805 21.245 22.153

(0.128) (0.194) (1.199) (0.251) (0.423 (1.389) (1.207) (0.188) (0.340) (0.394) (0.321)2 21[ln (net worth)] 310 0.052 0.002 0.808 0.046 0.813 0.031 0.555 20.015 1.082 0.178 1.205

(0.087) (0.131) (0.651) (0.140) (0.260) (0.738) (0.619) (0.104) (0.198) (0.252) (0.189)26Income from employment310 1.556 2.482 4.199 0.382 0.721 22.915 21.191 1.533 20.109 2.021 1.331

(1.081) (1.164) (2.942) (1.428) (0.756) (2.066) (1.752) (0.615) (0.579) (0.863) (1.356)

Marginal tax rate 1.819 0.473 21.446 0.743 20.139 20.260 20.794 20.049 20.585 0.734 0.267

(0.335) (0.375) (1.302) (0.389) (0.279) (0.667) (0.608) (0.093) (0.277) (0.211) (0.538)21Age310 0.424 1.215 1.545 0.898 20.223 20.033 0.089 20.001 0.175 20.113 0.296

(0.170) (0.210) (0.419) (0.380) (0.320) (0.496) (0.679) (0.102) (0.259) (0.324) (0.392)2 23(Age) 310 20.401 21.296 21.599 20.541 0.357 20.126 0.070 20.085 20.254 20.282 20.427

(0.210) (0.239) (0.580) (0.458) (0.385) (0.646) (0.729) (0.130) (0.339) (0.418) (0.560)

Married 0.456 0.130 20.284 20.431 20.154 20.142 20.410 0.057 20.223 0.117 0.214

(0.119) (0.124) (0.364) (0.155) (0.205) (0.485) (0.446) (0.068) (0.173) (0.159) (0.223)

Occupation: Managerial 0.485 0.321 0.412 20.287 20.030 20.099 0.229 20.017 0.062 0.229 0.235

(0.075) (0.082) (0.239) (0.122) (0.133) (0.259) (0.297) (0.047) (0.118) (0.090) (0.138)

Occupation: Professional 0.307 0.306 0.398 0.155 20.082 20.426 0.123 0.051 20.042 0.183 0.414

(0.088) (0.072) (0.227) (0.109) (0.115) (0.354) (0.279) (0.036) (0.117) (0.082) (0.096)

Education: College or above 0.770 0.183 20.120 0.060 0.285 21.128 0.213 20.009 20.169 0.191 0.068

(0.188) (0.077) (0.222) (0.116) (0.154) (0.728) (0.539) (0.040) (0.079) (0.105) (0.129)

Employed 0.218 20.260 20.226 0.329 20.014 0.120 20.746 20.041 20.223 20.206 20.287

(0.128) (0.098) (0.204) (0.152) (0.163) (0.510) (0.415) (0.031) (0.132) (0.103) (0.189)

Risk averse 20.153 20.041 20.270 0.303 20.131 0.575 20.158 20.022 20.247 20.056 20.339

(0.045) (0.055) (0.187) (0.083) (0.078) (0.208) (0.131) (0.021) (0.067) (0.055) (0.076)

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Ownership dummies:

O1COM3 20.131 20.372

(0.402) (0.440)

O1COM23 0.705 0.844

(0.563) (0.666)

O1COM34 20.917 22.753

(0.221) (0.246)

O1COM234 22.737 23.206

(0.442) (0.513)

O2COM13 20.334 0.582 6.882 3.234

(2.637) (7.937) (6.540) (2.171)

O2COM134 20.824 1.570 7.709 3.299

(2.677) (8.518) (6.958) (2.151)

O3COM1 22.177 23.097 20.623

(0.738) (0.703) (0.189)

O3COM12 2.300 23.074 0.877

(1.035) (1.071) (0.345)

O3COM14 23.784 0.287 20.757

(0.536) (0.388) (0.226)

O3COM124 23.437 1.926 0.859

(1.074) (0.910) (0.323)

O4COM13 20.035 2.623

(1.416) (1.568)

O4COM123 1.074 3.982

(1.767) (2.246)

MILLS RATIO 6.931 22.838 25.644 2.730 20.780 23.143 0.227 20.070 21.170 0.540 20.193

(Inverse) (2.799) (1.206) (4.369) (0.847) (0.418) (1.948) (1.547) (0.189) (0.499) (0.559) (1.268)

R2squared 0.366 0.239 0.223 0.211 0.106 0.140 0.070 0.625 0.075 0.629 0.205

Degrees of freedom 5086 4852 3026 3506 2281 477 438 4724 2106 1637 3230

Standard error of the equation 1.179 1.279 1.695 1.270 1.471 1.281 1.255 0.485 1.286 0.875 1.602

aDummy variables take the value unity when the description applies to the household, zero otherwise. Individual variables refer to the head of a household.Note: The dummy variables of the form O1COM23 take the value unit if given that it contains an asset in group 1, the household’s portfolio also contains assets fromgroups 2 and 3; and zero otherwise.


the fifteen ownership combination coefficients that are significant at the 5 percentlevel, eleven are negative and four are positive. In the cases of positive coefficientsthe results suggest complementarily between less liquid assets and ownershipgroup 2 (equity and bonds), contractual savings and group 4 (liabilities), andbetween homes and group 2.

5. Effects of taxes on portfolio composition

The way in which taxes affect portfolio choices has been at the center of debatesabout the merits of fundamental reform of the tax system. A switch to either acomprehensive income tax or a consumption tax would eliminate the differentialtax treatment of assets and consequent distortions associated with the currentsystem. Very little empirical evidence has been brought to bear on this issue. TheSRI survey is well suited to an examination of the problem because of the verydetailed information on the components of taxable incomes and deductionsrecorded for each household (see Section 2 above).

In Table 5 are shown the marginal tax rates for those households owning eachof our eleven categories of assets and liabilities. We show both the unweightedaverage for holders of each asset, and also the average marginal tax rate weightedby households’ shares of the total holdings in the sample. Not surprisingly, theweighted average marginal tax rates are higher than the unweighted means. For thesample as a whole the unweighted average marginal tax rate is 27.1% in 1977 and

Table 5Marginal tax rates of clienteles

Asset Number holding Average marginal taxthe asset rate of clientele %

Unweighted Weighted

1. Checking accounts 5558 27.8 35.02. Liquid savings 5438 28.1 35.13. Less liquid savings 3255 28.1 32.34. Contractual savings 4596 29.9 39.85. Corporate equity 2915 32.7 49.96. Taxable bonds 632 36.1 45.97. Municipal bonds 593 37.5 50.58. Owner-occupied housing 5051 28.4 34.89. Other assets 2610 33.6 45.6

10. Home mortgages 3341 31.4 33.111. Other liabilities 3714 30.1 45.1Total net worth 6010 27.1 42.7

Note: The weighted average marginal tax rates are weighted by the household’s share of the totalholdings of the asset in the sample.Source: Own calculations based on SRI survey.


the weighted (by shares of net worth) average marginal tax rate is 42.7%. Thereare larger differences in marginal tax rate among the holders of different types ofasset. These are particularly pronounced for the weighted averages. For checkingaccounts, liquid and less liquid savings, owner-occupied housing and homemortgages, the marginal tax rates are noticeably less than the weighted averagemarginal tax rate for the whole sample. The two highest marginal tax rates are forholders of municipal bonds (which are tax-exempt) and corporate equity, wherethe weighted averages are 50.5 and 49.9% respectively.

There are two ways in which taxes affect portfolio composition. The first is thatdifferences in effective tax rates among assets and households may lead toportfolio specialization in order that households can exploit their comparativeadvantage in the production of posttax income streams (Auerbach and King,1983). Households with lower marginal tax rates will hold more of the taxedsecurities, such as liquid and less liquid savings, and those facing higher marginaltax rates will hold more of the tax-privileged securities, such as municipal bondsand equity. The second effect is that taxes alter the trade-off between risk andreturn. The impact of this on the demands for risky assets is theoreticallyambiguous (see Feldstein, 1976 for a survey of the literature).

The evidence from Table 5 bears out the predictions of the theory ofcomparative advantage. But this is not the whole story. Taxable bonds, forinstance, are owned by households with very high marginal tax rates whichcontradicts the pure comparative advantage model. Moreover, Tables 3 and 4 showrather mixed results. The ownership probabilities are generally positively related tothe tax rate which suggests that taxes increase the demand for risky assets. But inTable 4 the tax rate coefficients are insignificantly different from zero in mostcases.

One potential difficulty with the estimated tax coefficients is a possibleidentification problem associated with including both the tax rate and currentincome as explanatory variables. The tax rate is a known nonlinear transformationof income and other exogenous variables, and if the way in which income entersinto the true behavioral model is unknown the separate effects of the tax rate maynot be identified. This problem is unlikely to be serious in our case for tworeasons. First, the income variable that we use in the behavioral model isemployment not taxable income. As noted earlier, employment income is just oneof twenty-two sources of income, twenty-one types of expense and a range ofdemographic variables that are arguments in the function we use to estimatetaxable income and hence the marginal tax rate. Second, the marginal tax ratedepends not only on taxable income but also on the household’s tax filing status,state of residence and the relevant tax year. We use observations on all of thesevariables. Indeed, the correlation coefficient between the marginal tax rate andemployment income was only 0.53, and that between the tax rate and thelogarithm of net worth was 0.37.

It appears that tax rates are more important in determining the probability of


ownership of an asset than its share in net worth conditional upon ownership. It isperhaps surprising that our econometric estimates do not corroborate the clienteleeffects apparent from Table 5. The failure of our estimation procedure, using aseemingly accurate measure of the marginal tax rate, to identify more pronouncedtax effects suggests, contrary to much of the recent literature, that taxes do notplay a decisive role in explaining the differences in portfolio composition acrosshouseholds.

6. Wealth elasticities

The model estimated above allows us to calculate the wealth elasticities ofdemand for each asset, at both the intensive and extensive margins, as a functionof the estimated coefficients and values of the exogenous variables. A change inwealth has two effects. First, it alters the probability that a household owns anasset. Secondly, conditional upon ownership it changes the demand for each asset.The expected demand for an asset is the product of the ownership probability, p ,jand the conditional demand, e , and we define the total wealth elasticity as thej

elasticity of the expected demand with respect to wealth. This is given by(dropping household superscripts)

d(p e )W j j]] ]]E 5 . (20)j p e dWj j

This expression may be written as the sum of the intensive and extensiveelasticities superscripts)

dp deW Wj j O D]] ]]E 5 1 5 E 1 E , (21)j j jp dW e dWj j

O Dwhere E and E are the ownership and demand elasticities, respectively.j j

The ownership probability model is

Zg j

p 5 E f(u) du, (22)j

2`

where f() is the standard normal density, Z is the vector of values of theexplanatory variables included in the probit model, and g the vector of coefficientsj

for which our estimates are given in Table 3. The specification we have usedimplies that

2 9Zg 5 b ln W 1 b (ln W ) 1 Z9g . (23)j 1j 2j j

Hence the wealth elasticity of the ownership probability is


OE 5 (b 1 2b ln W )h , (24)j ij 2j j

where h is the value of the hazard function for asset j (the ratio of the density of fj

to the ownership probability for asset j) and depends upon household characteris-24tics, including wealth. The asset demand equations shown in Table 4 also include

2ln w and (ln w) as explanatory variables and if their coefficients are denoted by aj

and a , respectively, then the conditional demand wealth elasticity isj

≠ ln ejD ]]E 5 5 1 1 a 1 2a ln W. (25)j 1j 2j≠ ln W

Hence the total wealth elasticity is

E 5 1 1 (a 1 b h ) 1 2(a 1 b h )ln W. (26)j 1j 1j j 2j 2j j

This expression shows that the estimated model allows a reasonably flexiblespecification for the wealth elasticities. They depend upon the level of wealth itselfboth directly (though the linear dependence on ln W ) and indirectly through thevalue of the hazard h . Household characteristics other than wealth enter via theirj

influence on the hazard. Because the hazard appears in (26), the wealth elasticity isa nonlinear function of the estimated parameters. If all four of the wealthcoefficients are zero then the wealth elasticity is unity. This is useful because anatural null hypothesis to test is that the wealth elasticity is equal to unity. Ifhousehold preference orderings exhibit constant relative risk aversion then thewealth elasticities of all assets will be equal to unity. It is clear, however, that thehypothesis of constant relative risk aversion is strongly rejected by the data,contrary to the findings of Blume and Friend (1975). Only in the case of taxablebonds are all four of the estimated wealth coefficients insignificantly different fromzero at the 1% level. One caveat to this conclusion is that neither we nor Blumeand Friend included social security wealth in the measure of net worth. Given thehigh wealth levels of our sample this may not be of great importance. Anadditional caveat is that we ignore the possibility that households see through the‘‘institutional veil,’’ and condition their portfolios on the assets in which theirdefined contribution pension plans and IRA/Keogh accounts are invested.

For the conditional demand equations Table 6 shows the value of the chi-squarestatistic to test the null hypothesis that the coefficients on both wealth terms arezero. The critical values of chi-square with two degrees of freedom are 5.1 at the5% level and 9.21 at the 1% level. The hypothesis of constant relative risk

25aversion can be rejected for all assets except equity and bonds at the 1% level.Table 7 presents point estimates of the ownership, demand, and total wealth

elasticities for each asset evaluated for two representative households. The first is a

24Life cycle aspects of portfolio composition are discussed further in King and Leape (1987).25For the probit model where f() is the standard normal density the hazard is equal to the inverse of

Mills’ ratio.


Table 6Test statistic for constant relative risk aversion

Asset Chi-squarestatistic

1. Checking account 109.002. Liquid savings 83.403. Less liquid savings 35.584. Contractual savings 43.725. Equity 8.126. Taxable bonds 3.987. Municipal bonds 0.608. Home 929.569. Other assets 19.46

10. Home mortgages 256.2211. Other liabilities 64.20

Source: Own calculations.

Table 7Wealth elasticities for two representative households

Asset Predicted elasticity evaluated at the Predicted elasticity evaluated atpopulation mean wealth of $57 408 the sample mean wealth of $223 188

Ownership Demand Total Ownership Demand Total

Checkingaccount .001 .541 .542 2.003 .555 .552

Liquidsavings .006 .504 .510 2.007 .504 .498

Less liquidsavings .087 2.063 .025 .049 .157 .206

Contractualsavings .046 .481 .527 .023 .493 .516

Equity .323 .778 1.101 .412 .999 1.411Taxablebonds .605 2.250 .355 .673 2.242 .431

Municipalbonds .589 .789 1.378 .746 .939 1.685

Owner-occupiedhousing .138 .351 .489 .108 .312 .420

Otherassets .338 .540 .908 .421 .864 1.285

Homemortgages .067 2.019 .048 2.073 .029 2.043

Otherliabilities 2.014 .377 .363 2.032 .704 .672

Note: Totals may not equal sum of components because of rounding errors.Source: Own calculations.


household with wealth equal to the estimated population mean net worth of$57 408, and the second is one with wealth equal to the sample mean net worth of

26$223 188. In both cases the households are assumed to have values of otherhousehold characteristics equal to the sample mean. The ownership elasticities aresmall for most assets but for corporate equity, taxable bonds, municipal bonds andother assets, an increase in wealth has a sizeable positive impact on the probabilityof ownership.

In Fig. 1a–1k we plot the predicted ownership probabilities as a function ofwealth implied by the probit estimates reported in Table 3. The probabilities areevaluated at sample means for all household characteristics other than wealth. Theplots show clearly that the relationship between ownership and wealth differsacross assets. Of particular interest are the sigmoid-shaped curves for equity and‘‘other assets.’’

The demand elasticities also vary significantly across assets. At the samplemean level of net worth the demand elasticities for equity and municipal bonds arevery close to unity. For checking accounts (a large component of money demand)and liquid savings the elasticities are around one-half, in contrast to thepresumption in much of the macroeconomic literature that they are close to zero.The elasticity for contractual savings is also around one-half and that for housingabout one-third. The only negative elasticity is that for taxable bonds and herenone of the estimated wealth coefficients were significantly different from zero. Atthe opposite end of the spectrum, the largest total elasticities are for municipalbonds, corporate equity and other assets, and at the sample mean value for networth all three exceed unity.

For both types of liability the elasticities are less than unity. From this and thebalance sheet constraint that net worth equals assets minus liabilities we maydeduce that the wealth elasticity of the nine assets taken as a group is less thanunity. Both assets and liabilities are measured as positive amounts. Let A denotetotal assets, L denotes total liabilities, E and E the wealth elasticities of assetsA L

and liabilities (averages of the individual asset and liability elasticities weighted bythe shares of each asset (liability) in total assets (liabilities)). Then the conditionthat W5A2L implies that

W W] ]S DE 5 E 1 2 1 . (27)A L A A

Since 0#W /A#1, if both liability elasticities are less than unity, then E ,1 andL

by (27) E ,1. It is not surprising, therefore, that for most of the assets theA27elasticities in Table 7 are less than unity.

26We do not present a test statistic for the asset demand system as a whole because the number ofobservations used in estimation varied across assets.

27The population mean value of net worth was estimated using the sample weights described in note1.


Fig. 1. Effect of wealth on probability of ownership.


Fig. 1. (continued)


Fig. 1. (continued)


Fig. 1. (continued)


Fig. 1. (continued)


Fig. 1. (continued)

Because the magnitude of the wealth elasticities depends upon wealth and otherhousehold characteristics, the aggregate effect of a change in wealth depends uponhow that change is distributed among the population. Table 7 shows that the

28elasticities are different for households with different levels of net worth. InTable 8 we show the aggregate wealth elasticity for each asset under theassumption that the proportionate change in wealth is uniform among the sample.The predicted aggregate demand for asset j is

h ha 5O p e . (28)j h j j

The aggregate wealth elasticity is

O D O DE 5O (E 1 E )s 5 E 1 E , (29)aj h jh jh jh aj aj

where s is the predicted share of household h’s demand for asset j in thejhh h h haggregate demand for the asset (p e /S p e ). The first three columns of Tablej j h j j

28Because we are using cross-section data, we are unable to distinguish between permanent andtransitory shocks to wealth which, in the presence of conventional transaction costs, would havedifferent effects on asset demands. Transaction costs are, however, unlikely to be of major importancefor wealthy households such as those in our sample.


Table 8Aggregate wealth elasticities

Asset Sample Population

Ownership Demand Total Ownership Demand Total

Checkingaccount 2.002 .570 .568 .000 .534 .534

Liquidsavings 2.006 .508 .502 .002 .505 .507

Less liquidsavings .263 2.179 .083 .252 2.231 .021

Contractualsavings .008 .504 .512 .023 .495 .518

Equity .038 1.541 1.579 .200 1.120 1.320Taxable

bonds .718 2.307 .411 .749 2.309 .441Municipal

bonds .378 1.172 1.550 .558 1.025 1.584Owner-occupied

housing .067 .252 .320 .118 .255 .373Other

assets .079 1.531 1.610 .249 .975 1.244Home

mortgages 2.005 .004 2.001 .073 2.035 .039Other

liabilities 2.070 1.312 1.241 2.022 .537 .515

Note: Totals may not equal sum of components because of rounding errors.Source: Own calculations.

8 show the aggregate ownership, demand and total elasticities for the full sampleof households with positive net worth. For those households which did not reportthe dollar values of all asset holdings, the predicted level of net worth (asdescribed in the discussion of missing data in Section 4) was used to calculate theelasticities.

The broad pattern of elasticities shown in Table 7 holds also for the aggregateelasticities in Table 8. Taking into account the variation in wealth and othercharacteristics among the sample leads to somewhat larger elasticities for equity,‘‘other assets,’’ and liabilities other than mortgages. Because the sample is drawnheavily from upper-income groups, we calculate also aggregate elasticities for thepopulation as a whole where the individual elasticities are weighted not only bypredicted asset shares but also by population weights. The population estimates ofthe aggregate elasticities are displayed in the final three columns of Table 8. Theseare our best estimates of the elasticities relevant for macroeconomic behavior. Thelargest elasticity is for municipal bonds, followed by equity and other assets. The


smallest elasticities are for owner-occupied housing, home mortgages and less29liquid savings.

7. Conclusions

Examining the pattern of household portfolio composition that emerges from asurvey of 6010 U.S. households, we have found that the vast majority ofhouseholds hold only a subset of the available assets. This has implications forboth the theoretical modelling of portfolio behavior and the econometric estima-tion of asset demand equations that have not been dealt with satisfactorily inearlier studies. We have shown that if the incomplete portfolios are the result of anoptimization subject to holding costs and constraints on short sales then the assetdemand system is a switching regressions model with as many regimes as there aredistinct possible portfolios. The specification we derived involved equations bothfor the probability of owning each asset and combination of assets and for theasset demand system conditional upon ownership. An estimation procedure wasdeveloped to respond to the problems caused by incomplete portfolios, nonreport-ing of dollar holdings, and the endogeneity of the tax rate and of the portfoliocomposition dummies.

We have used this model to examine the impact of taxes on portfoliocomposition. The information provided by the survey on taxable incomes anddeductions has enabled us to calculate the marginal tax rate facing each household.Our results show that tax rates are a significant determinant of asset ownership but,surprisingly, not of the share of net worth invested in the asset. Though far fromconclusive, this finding suggests that the magnitude of the distortion induced bycapital taxation on household portfolio choices may be less than previouslythought.

We have estimated wealth elasticities of demand for a range of assets andliabilities at both the intensive and extensive margins. The presumption that thewealth elasticity of demand for money is zero and that for other assets is unity wasshown to be unjustified. For corporate equity, municipal bonds and ‘‘other assets,’’the estimated wealth elasticities were greater than unity. Other assets, such aswealth in contractual savings schemes and short-term financial assets, hadelasticities of around one-half. These estimates suggest that changes in totalhousehold net worth will change the structure of household balance sheets andconsequently affect real decisions, such as corporate investment, in the economy.

Finally, this study suggests that the differences in portfolio composition acrosshouseholds cannot be fully explained within the framework of the conventional

29The condition that the aggregate wealth elasticity is independent of the distribution of wealth isa 5b 5b 50. Only in the case of taxable bonds would this null hypothesis fail to be rejected at the2j 1j 2j

1 percent level.


portfolio choice model. The households in our sample, though wealthy, own asurprisingly small number of assets and liabilities and this lack of diversificationwas found to be important when estimating asset demand equations. Given that themean net worth of the sample was almost a quarter of a million dollars in 1978, itis hard to imagine that transaction costs, as traditionally defined, played a decisiverole in producing incomplete portfolios.

One alternative explanation is that some assets produce joint products. Forexample, housing produces both a stream of returns on the investment and aconsumption flow of housing services. Other assets may be held as part of a laborcontract, either on the part of senior managers or in own businesses, in which casethe desire to diversify portfolio risks may be constrained by contracts that preserveincentives for managers. Still others may offer liquidity, as in the case of checkingaccounts or liquid savings, or insurance, as in the case of contractual savings. Thejoint nature of the services provided by some assets means that the separation ofthe optimal consumption plan and the optimal portfolio allocation no longer holds.Differences in consumption patterns across households could therefore lead toincomplete portfolios.

A second and more compelling explanation of this phenomenon is the existenceof significant holding costs in the management of a portfolio that increase with thenumber of assets owned. In particular, our results suggest that a major factoraffecting household portfolio behavior may be the costs of acquiring andprocessing the information required to make decisions about how best to allocateresources across different assets. We would expect such costs to vary amonghouseholds and, in particular, with observable variables such as the level ofeducational attainment and occupation. Our results show that education andoccupation are important determinants of the ownership decision for corporateequities, taxable bonds, municipal bonds, and ‘‘other assets’’—the four types ofassets for which we would expect information costs to be highest. The existence ofsuch household-specific information costs casts doubt on the traditional assump-tion of homogeneous expectations and suggests the need for greater attention inboth theoretical and empirical work on portfolio behavior to the process and costs

30of acquiring information.

Acknowledgements

This study was funded by the National Science Foundation (grant no. SES-8410896) and the National Bureau of Economic Research. The views expressedare those of the authors and should not be attributed to the NSF nor to the NBER.For helpful discussions and comments, we thank Dan Feenberg, Ben Friedman,

30See King and Leape (1987).


Roger Gordon, Zvi Griliches, Jerry Hausman, Adam Jaffe, Dan McFadden, ArielPakes, Jim Poterba, Jim Powell and David Wise.

Appendix A

We describe here a two-step procedure for dealing with the problem ofnonreporting of asset values. At the first stage we estimate an equation for thelogarithm of net worth as a function of exogenous and observable householdcharacteristics using data for the sample of 2048 households which report valuesfor all assets owned.

ln W 5 Xb 1 e. (A.1)

As explanatory variables we include a piece-wise linear function of age (using themethod proposed in King and Dicks-Mireaux, 1982), income from employment,marital status, education, occupation, household composition, religion, pensionstatus, and a farm family dummy. The sample selection bias induced by using onlyobservations with positive values of net worth was corrected by a standardtwo-step procedure (Heckman, 1976, 1979). The parameter estimates, can be usedto construct estimates of the logarithm of net worth for households withincomplete responses. In fact, if we stopped at this point and used X as an estimateof ln w in (16) for nonreporting households, then this procedure would beequivalent to a Dagenais estimator for the missing data problem (Dagenais, 1973).But because net worth is not missing but only incompletely recorded we use asecond step which improves efficiency. We assume that we may approximate the

Rrelationship between recorded and true net worth (denoted by W and W,respectively) by

JRln W 5 ln W 2 O a d 1 u, (A.2)j j

j51

where

d 51 if asset j is owned and value of holding is not reportedj

50 otherwise.

ˆWe obtain estimates of the coefficients a by regressingj

R ˆln W 2 Xb,

on the nonreporting dummies using observations on households where the value ofat least one holding is not reported ((A.2) holds exactly for households whichreport all holdings). Our estimate of net worth for households with incompletereporting is given by


JRˆ ˆln W 5 ln W 1Oa d , (A.3)j j

j51

If nonreporting is ‘‘ignorable’’ (Griliches, 1984), then the nonreporting dummyvariables are exogenous and (A.2) can be estimated by OLS. Where this conditionis not satisfied instrumental variables estimation can be employed. In theestimation of both the reduced form probits and the asset demand system (16), weuse predicted net worth as given by (A.3).

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