Wealth, Information Acquisition, and Portfolio Choice

Wealth,Information Acquisition,and Portfolio Choice

Joe¨l Peress

INSEAD

I solve(with an approximation)a Grossman-Stiglitz economy under general prefer-

ences,thus allowing for wealth effects.Because information generates increasing returns,decreasing absolute risk aversion,in conjunction with the availability of costly information,is sufficient to explain why wealthier households invest a larger fraction of their wealth in risky assets.One no longer needs to resort to decreasing relative risk aversion,an empirically questionable assumption.Furthermore,I show how to distinguish empirically between these two explanations.Finally,I find that the availability of costly information exacerbates wealth inequalities.

The effect of wealth on households’demand for risky assets has long been studied,starting with the works of Cohn et al.(1975)and Friend and Blume(1975).They document that the fraction of wealth households invest in stocks increases with their wealth.Several recent studies using different datasets and estimation techniques confirm their observation.1 One common explanation for the observed pattern of portfolio shares is that relative risk aversion decreases with wealth[e.g.,Cohn et al.(1975)]. Moreover,some authors[e.g.,Morin and Suarez(1983)]use portfolio data to elicit households’preferences and conclude from the observation of shares that relative risk aversion is decreasing.However,abstracting from portfolio data,there is not much evidence in favor of decreasing relative risk aversion.Several studies reject this hypothesis using data that contains information about attitudes toward risk such as farm data,survey data,or experimental data.2Here,I suggest an alternative explanation for I am particularly grateful to my dissertation advisors Lars Peter Hansen(chairman),Pierre-Andre′Chiappori,and Pietro Veronesi for their guidance and support.I would like to acknowledge helpful comments from Antonio Bernardo,Bernard Dumas,Luigi Guiso,Harald Hau,John Heaton,Josef Perktold,Joao Rato,Guy Saidenberg,Jose′Scheinkman,Olivier Vigneron,Robert Verrecchia,Annette Vissing-J?rgensen,and seminar participants at the University of Chicago,the EFA meeting in London, Delta,Crest,Essec,HEC,INSEAD,London School of Economics,Banca d’Italia,AFFI meeting in Namur,and the SED meeting in Stockholm.I thank the University of Chicago,the French government, and the European Commission for their financial support.Address correspondence to Joe¨l Peress, INSEAD,Department of Finance,Boulevard de Constance,77305Fontainebleau Cedex,France,or e-mail:joel.peress@2e90cf40680203d8cf2f2445.

1These studies estimate the elasticity of portfolio shares with respect to wealth to be around0.1,where portfolio shares refer to the fraction of financial wealth invested in risky assets,both directly and indirectly,conditional on holding some risky assets.I review the evidence in detail in Section1.

2In addition,Arrow(1971)makes a theoretical argument in favor of increasing relative risk aversion.The empirical studies are reviewed in Section1.Section6also rules out alternative explanations for portfolio shares based on fixed entry costs and psychological biases.

The Review of Financial Studies Vol.17,No.3a2004The Society for Financial Studies;all rights reserved. DOI:10.1093/rfs/hhg056Advance Access publication October15,2003

The Review of Financial Studies/v17n32004

the observed pattern of portfolio shares and wealth.This explanation only requires absolute risk aversion to be decreasing with wealth,an assump-tion that is supported by all empirical studies.(In particular,the model reconciles the common assumption that relative risk aversion is constant with the observed pattern of portfolio shares.)

In addition to decreasing absolute risk aversion,the explanation offered in this article relies on the possibility to acquire,at a cost,information about stocks.Though they are not directly observable,there is evidence that differences in information do matter to investors’decisions and that these differences are related to households’measurable characteristics such as wealth.Several surveys in Europe and the United States document the importance of information for stock ownership.3For example, Alessie,Hochguertel,and Van Soest(2002)use data from a Dutch survey that includes a measure of interest in financial matters and find that this variable has a significant and positive effect on portfolio shares.More-over,Donkers and Van Soest(1999)show that this financial interest variable is strongly positively correlated to income.In the same spirit, Lewellen,Lease,and Schlarbaum(1977)report that the money spent by investors on financial periodicals,investment research services,and professional counseling increases with both income and education.On another front,research in accounting shows that small trades react less to earnings news than large trades do,suggesting that wealthier investors (i.e.,investors who place large orders)process the news and adjust their orders faster than poorer investors.4

This article explains the cross-sectional pattern of stockholdings and wealth by endogenous differences in information.For that purpose,I model explicitly how investors acquire information.I show that though they do not have lower relative risk aversion,wealthier investors hold a larger fraction of their wealth in stocks.5The reason is that the value of information increases with the amount to be invested,whereas its cost does not.This implies that agents with more to invest acquire more information.Consequently they purchase even more stocks and hold a larger portfolio share.Thus they do so not because they are relatively less 3For Europe,see Alessie,Hochguertel and Van Soest(2002),Borsch-Supan and Eymann(2002),Guiso and Jappelli(2002).For the United States,see King and Leape(1987).

4See Cready(1988)and Lee(1992).In addition,some articles argue that costly information processing explains some puzzling phenomena in finance such as the‘‘home equity bias’’[French and Poterba(1991), Kang and Stulz(1994),Coval and Moskowitz(1999)]and the‘‘weekend effect’’[Miller(1988)and Lakonishok and Maberly(1990)]and others provide evidence on the role of financial education and social interactions for stock ownership[Bernheim and Garret(1996),Chiteji and Stafford(1999), Weisbenner(1999),Bernheim,Garret,and Maki(2001),Huberman(2001),Duflo and Saez(2002)].

5Therefore one should be cautious when infering the determinants of relative risk aversion from portfolio shares.What looks like decreasing relative risk aversion(increasing portfolio shares)may in fact be the result of decreasing absolute risk aversion combined with information purchase.This applies not only to wealth,as the article shows,but also to other determinants of risk aversion such as age or education. 880

Wealth,Information Acquisition,and Portfolio Choice

risk averse,but because the stock is less risky to them.Importantly,this result does not rely on any form of increasing returns to scale embedded in technology or preferences:it is obtained in spite of a strictly convex informa-tion acquisition cost and prevails when relative risk aversion is increasing.

The model builds on Grossman and Stiglitz(1980)and Verrecchia (1982).In Grossman and Stiglitz(1980),traders may purchase private information about the payoff of a stock,which they use to trade competi-tively in the market.Their information gets revealed by the equilibrium price,but only partially because there is some noise in the system.In Verrecchia(1982),traders are allowed to choose continuously the preci-sion of their private signal.A key assumption of these rational expecta-tions models with asymmetric information is that agents have constant absolute risk aversion utility(CARA or exponential).Hence these models ignore the role of wealth,though it is an important determinant of stock-holdings.To capture wealth effects,I solve the model under general preferences.6A closed-form solution is derived by making a small risk approximation.The point of the article is that,as long as absolute risk aversion decreases with wealth,there will be increasing returns to acquir-ing private information even though it gets revealed by public signals.7 Finally,I study the link between wealth inequality and stock prices. Because information generates increasing returns,the demand for stocks is a convex function of wealth.Hence the more unequal the distribution of wealth,the higher the stock price.Conversely,wealthier investors achieve a higher expected return,a higher variance,and a higher Sharpe ratio on their portfolio.Consequently,the distribution of final wealth as measured by expected wealth or by certainty equivalent is more unequal than the distribution of initial wealth.This fact also suggests a simple way of discriminating the information model from the decreasing relative risk aversion model:in the former,the Sharpe ratio on an agent’s portfolio increases with her wealth,whereas in the latter,it is 2e90cf40680203d8cf2f2445ing a comprehensive dataset on Swedish households,Massa and Simonov (2003)report that Sharpe ratios increase with financial wealth,in accor-dance with the information model.More research is needed to confirm these results.

The remainder of the article is organized as follows.Section1reviews the evidence on the relations between wealth,portfolio shares,relative risk aversion,and information acquisition.Section2describes the economy. Section3defines the equilibrium concept.Section4solves the model:the 6However,it should be noted that the model presented here is static and hence does not capture hedging demands.This important feature of portfolio choice is considered in dynamic models with CARA preferences.

7The idea of increasing returns to information is not new,but to my knowledge,it has not been modeled in a setup where private information gets partially revealed by public signals,as in the stock market[Wilson (1975)and Arrow(1987)].

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equilibrium is characterized and the relation between wealth and portfolio shares is described.Section5studies the effect of information acquisition on wealth and return inequality.Finally,Section6addresses some empiri-cal issues:I calibrate the model to U.S.data,show how to discriminate the information acquisition model from the decreasing risk aversion model using micro data,and finally,discuss alternative explanations based on fixed entry costs and psychological biases.Section7concludes and suggests some applications.Proofs and robustness checks are in the appendix. 1.Evidence

In this section I review the evidence on the relations between wealth, portfolio shares,relative risk aversion,and information acquisition.

1.1Wealth and portfolio shares

This article is motivated by the observation that the share of wealth households invest in stocks increases with their wealth,so let me now be more precise about how portfolio shares are measured.First,stocks refer to equity that is held both directly and indirectly through mutual funds.

Second,depending on how housing is treated(whether it is excluded, included as a riskless asset,included as a risky asset,priced at market value,or priced at owner’s equity value),different studies reach different conclusions about the effect of wealth on portfolio shares of risky assets.

However,virtually all agree that the fraction of financial wealth invested in stocks(i.e.,total wealth excluding housing,capitalized labor,private businesses,social security,and pension incomes)increases with financial wealth.Third,portfolio shares of stocks are computed conditional on owning some stocks.Accordingly,the purpose of this article is to explain the fraction of financial wealth households invest in risky assets,both directly and indirectly,conditional on being a stockholder.

Several recent articles estimate the elasticity of portfolio shares with respect to wealth to be around0.1.The ones mentioned below use differ-ent datasets and econometric techniques,but all conform with the three points made above and,in particular,separate the share choice from the participation decision.Vissing-J?rgensen(2002)uses the Panel Study of Income Dynamics and finds estimates of0.09,0.12,and0.10,depending on the specification of the model.8Bertaut and Starr-McCluer(2002)use several waves of the Survey of Consumer Finance and find estimates of

0.17,0.04,and0.06.Finally,Perraudin and S?rensen(2000)use the1983

Survey of Consumer Finance and find an estimate of0.09.Other articles 8For example,in Table2,Vissing-J?rgensen(2002)reports regression coefficients on wealth and wealth squared equal to0.0011andà0.00000149,which imply an elasticity of0.12using the average wealth of $74,810.

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Wealth,Information Acquisition,and Portfolio Choice

report elasticities but differ either in their measures of wealth or do not condition on participation.9An often-cited explanation for the observed positive elasticity is that relative risk aversion is decreasing with wealth. As the next section shows,this hypothesis does not hold in the data.

1.2Wealth and relative risk aversion

In contrast to decreasing absolute risk aversion,there is not much support for decreasing relative risk aversion outside portfolio data.The evidence instead points to increasing or constant relative risk aversion in environ-ments where information cannot be acquired.10First,studies in agricultural economics use data on farmers who allocate their land across crops of different risks,the same way an investor allocates her wealth across securities.Saha,Shumway,and Talpaz(1994)and Bar-Shira,Just,and Zilberman(1997)find a clear pattern of decreasing absolute risk aversion and increasing relative risk aversion using different estimation techniques and datasets.

Second,surveys have been designed to elicit the respondents’risk aver-sion by asking questions about hypothetical lotteries.Barsky et al.(1997) offered the respondents of the Health and Retirement Study gambles involving new jobs and found that relative risk aversion rises and then falls with wealth.Similarly,Guiso and Paiella(2001)asked the respon-dents of the Bank of Italy Survey of Household Income and Wealth for the maximum price they would be willing to pay to participate in a lottery. The answers show that absolute risk aversion is a decreasing function of wealth,while relative risk aversion is an increasing function.Furthermore, when portfolio shares of risky assets are regressed on the measure of risk aversion,wealth,and other demographic variables,the coefficient on risk aversion is significantly negative and the coefficient on wealth is signifi-cantly positive,suggesting that wealth plays a role not captured by risk aversion.11

Finally,experimental studies provide some interesting insights on risk aversion.Gordon,Paradis,and Rorke(1972),Binswanger(1981),and 9For example,King and Leape(1998)use net worth as their measure of wealth and Heaton and Lucas (2000)do not condition on stock ownership in their regressions of portfolio shares on financial wealth (Table IX).

10An exception is Ogaki and Zhang(2001),but this study focuses on households close to their subsistence level.

11Studies in other fields strengthen the case against decreasing relative risk aversion.Szpiro(1986)uses aggregate data on property and liability insurance in the United States from1951to1975and finds that relative risk-aversion is constant.Wolf and Pohlman(1983)examine the bids of a U.S.bond dealer who gets most of his income from a fixed share of the profits he 2e90cf40680203d8cf2f2445bining this information with the dealer’s returns forecasts,they find that absolute risk aversion is decreasing and that relative risk aversion is constant or slightly increasing.A?¨t-Sahalia and Lo(2000)and Jackwerth(2000)use options prices to estimate the risk-neutral and subjective distributions of the S&P500index(a measure of aggregate wealth)from which they infer a representative investor’s risk aversion.They find that relative risk aversion is a nonmonotonic function of wealth.

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Quizon,Binswanger,and Machina(1984)offered subjects(MBA students or Indian villagers)gambles with real prizes.The result is that the fraction of wealth they play declines as their wealth increases,pointing to increas-ing relative risk aversion.This pattern is,however,in sharp contrast with U.S.households’portfolio data.The model presented here provides a way to reconcile these conflicting observations.Indeed,in these experimental studies,subjects have to choose among gambles with known odds and information cannot be acquired,in contrast to the real world.Hence an interpretation is that relative risk aversion is really increasing,but that the returns to scale generated by information acquisition are so powerful that they overturn the tendency for portfolio shares to decrease with wealth into a tendency to increase.Next I review the relation between wealth and information.

1.3Wealth and information acquisition

The evidence on the effect of wealth on information relies mainly on surveys.Lewellen,Lease,and Schlarbaum(1977)asked a sample of customers of a large U.S.retail broker how much they spent on financial periodicals,investment research services,and professional counseling.

They find that information expenditures increase very significantly with income.In the same spirit,Donkers and Van Soest(1999)use data from a Dutch survey which contains information on interest in financial matters and show that it is strongly positively correlated to income.I now turn to the model.

2.The Economy

The model is in the spirit of Grossman and Stiglitz(1980)and Verrecchia (1982).There are three periods,a planning period(t?0),a trading period (t?1),and a consumption period(t?2).Agents receive public informa-tion and may purchase private information about the payoff of a stock, which they use to trade competitively in the market.Some noise prevents the equilibrium price from fully revealing agents’private information.

2.1Investment opportunities

Two assets are traded competitively in the market,a riskless asset(the bond)and a risky asset(the stock).The stock represents the equity market as a whole,which investors attempt to time.12Unfortunately there exists in general no closed-form solution for the equilibrium in this economy when absolute risk aversion is not constant because the demand for risky 12For concreteness,the stock may be viewed as a share of a mutual fund.Most mutual funds are specialized in equity or bonds.In1998there were more than7,000mutual funds in the United States;hybrid funds accounted for only7%of all funds and managed only9%of the industry’s assets according to the Investment Company Institute.

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assets is no longer a linear function of the expected payoff.For this reason,I resort to a local approximation to compute the equilibrium when the stock has small risk.Specifically,I create a continuum of economies,each with a different set of fundamentals and hence a different portfolio problem.Each economy is indexed by a parameter,z ,that scales the variables representing risks,payoffs,and trading costs.In particular,the stock’s expected payoff and variance are both proportional to z so that the mean to variance ratio is constant across the continuum of economies.The model will then be solved in closed form by driving z toward zero.13The riskless asset is in perfectly elastic supply and has a net rate of return of r f z .The risky asset has a price P and a random payoff P that is log-normally distributed.Let p z be the ‘‘growth rate’’of P :

p z ln P :

With the stock price acting as a public signal,one more source of risk is needed to preserve the incentives to purchase private information.This role is played by the supply of stocks emanating from noise traders.14Let u represent the net supply of stocks (i.e.,the total number of shares plus the supply from noise traders).By assumption,u and p are jointly normally distributed and independent and the mean and variance of u z and p z are linear in z :ln P u $N & E ep Tz E eu T , s 2p z 00s 2u

=z ':2.2Information structure

Agents may spend time and resources gathering information about the stock market,i.e.,about the stock’s payoff P .For example,they may read newspapers,listen to radio and TV reports,surf the Web,participate in seminars,subscribe to newsletters,join investment clubs,or hire a financial advisor.Agent j may purchase a signal S j about the payoff of the stock P ,

S j ?ln P t?j ,e1T

where {?j }is independent of P ,u ,and across agents.Let x j denote the precision of agent j ’s signal.I assume that ?j is normally distributed:

?j $N 0,z x j

:13

The scaling factor z has the flavor of the time increment dt in a continuous-time model.14Noise or liquidity traders are a group of agents who trade for reasons not explicitely modeled.For example,these agents may have access to a private investment opportunity such as human capital,durables,or nontraded assets.Alternatively,they could make common random errors in their forecasts of the stock’s payoff.

Wealth,Information Acquisition,and Portfolio Choice

885

The signal costs C(x j)z dollars,where C is increasing and strictly convex in the precision level.Specifically,I assume that

Ce0T?0,C0eáT!0,C00eáT>0on?0,1 and lim

x!1

C0exT?t1: These assumptions ensure the existence of an interior solution.They capture the idea that each extra piece of information is more costly than the previous one;for example,because they are correlated.Allowing for a nonconvex cost function would only strengthen the point of the article, that wealthier investors acquire more information.For example,the specification C(x)?x c for c>1satisfies the assumptions.Agency pro-blems(not modeled here)preclude investors from sharing or selling their private information.

Finally,in a rational expectations equilibrium,agents know that the equilibrium price P contains some information about the risky payoff P and they will use it as an informative signal.F j denotes investor j’s information set:F j?{S j,P}if investor j acquires a private signal and F j?{P}if she does not.E j(áj F j)and E j(á)refer respectively to period1 and period0expectations,by investor j,where the private signal S j is distributed with precision x j.

2.3Investors

There is a continuum of heterogeneous agents in number normalized to one.Their objective is to maximize expected utility from final wealth,W2, where their preferences are represented by the utility function U.I assume that absolute risk aversion is decreasing with wealth,or equivalently,that its inverse,absolute risk tolerance is increasing:

teW2T àU0eW2T

U00eW2T

is increasing with W2:

For convenience,I assume further that lim W

2!0

teW2T?0and

lim W

2!1

teW2T?1,but these limit conditions are not necessary to the

results.Importantly,there is no assumption about relative risk aversion:it may be increasing,decreasing,or constant.For example,preferences could

display constant relative risk aversion[under CRRA,UeW2T?W1àa2

1àa and

teW2T?W2

a ].

In general,agents may differ in their risk aversion,initial endowments, and cost of information.Here,I assume the only source of heterogeneity across agents is their initial endowments in stocks and bonds.Let W0j be agent j’s total endowment in stocks and bonds(i.e.,the number of stocks plus the number of bonds she initially owns)and let a0j be the fraction of that endowment held in the form of stocks.From these definitions,it follows that the number of stocks and bonds initially owned are a0j W0j The Review of Financial Studies/v17n32004

886

and (1àa 0j )W 0j .15Let G be the cumulative joint distribution function of W 0j and a 0j on a compact set ?W 0,W 0 ?a 0,a 0 .

A measure of agents’aggregate risk tolerance ,n ,will help characterize the equilibrium.Let n Z j

t eW 0j TdG eW 0j ,a 0j T:

The choice variables of an agent are the precision of her private signal,x j ,and the fraction of wealth she allocates to the risky asset,a j (i.e.,the value of her stockholdings divided by the value of her endowment).

2.4Timing

The timing is depicted in Figure 1.There are three periods.Period 0is the planning period:the agent chooses how much information to acquire,if any [she chooses x j and pays C (x j )z ].The second period (t ?1)is the trading period.The investor observes her private S j with the precision x j she chose in the previous period.At the same time,markets open and she observes the equilibrium price.She uses the public and private signals to compute E j (ln P j F j )and V j (ln P j F j )and then chooses her portfolio share of stocks,a j .In the third period (t ?2),the agent consumes the proceeds from her investments,W 2j .

3.Equilibrium Concept

3.1Individual maximization

The investor’s problem must be solved in two stages,working from the trading period to the planning period.In the trading period (t ?1),she observes P and S j (where x j ,the precision of S j ,is inherited from the first 15The exogenous variables W 0and a 0approximate at the order zero in z an agent’s initial wealth and initial portfolio share,which are endogenous variables.Indeed,as shown in Theorem 1,the stock price is P ?exp(pz )%1tpz in equilibrium.

Consume W 2t = 0t = 2t = 1

Figure 1

Timing

Wealth,Information Acquisition,and Portfolio Choice

887

period)and then forms her portfolio taking P,r f and C(x j)as given:

max a j E j?UeW2jTj F j subject to

W1j?eP a0jt1àa0jTW0j

W2j?W1je1tr p j zTàCex jTz

r p j z?a j

PàP

P

àr f z

tr f z:

e2T

8

>>>

<

>>>

:

Note that agents may borrow at rate r f z and short stocks if they wish.W1j is the investor’s wealth in period1,i.e.,her endowed portfolio valued at the observed equilibrium price.r p j z is the net return on investor j’s port-folio(excluding the cost of information).Call v(S j,x j,W1j;P)the value function for this problem.

In the planning period(t?0),the agent chooses the precision of her private signal in order to maximize her expected utility averaging over all the possible realizations of S j and P and taking C(á)as given:

max

x j!0

E j?veS j,x j,W1j;PT :e3T

3.2Market aggregation

The gains from private information depend on how much gets revealed by the public signal P.Call i the aggregate precision or informativeness of the price implied by aggregating individual precision choice:

i

Z

j

x j t j dGeW0j,a0jT:e4T

Equivalently its inverse is a measure of the noisiness of the price.Private precisions are weighed by risk tolerance because investors transmit their information through their demand for stocks,which is proportional to their risk tolerance.Individual decisions both depend on and determine the aggregate variable i.We are now ready for the formal definition of an equilibrium.

3.3Definition of an equilibrium

A rational expectations equilibrium is given by two demand functions a j and x j,a price function P of P and u,and a scalar i such that

1.x j?x(W0j,a0j;i)and a j?a(S j,x j,W0j,a0j;P,i)solve the maxi-

mization problem of an investor taking P and i as given[Equations(2) and(3)].

2.P clears the market for the risky asset:

Z j aeS j,x j,W0j,a0j;P,iT

W1j

P

dGeW0j,a0jT?u:

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3.The informativeness of the price i implied by aggregating

individual precision choices equals the level assumed in the

investor’s maximization problem:

i?

Z

j

xeW0j,a0j;iTteW0jTdGeW0j,a0jT:

4.Description of the Equilibrium

For clarity,I will break the presentation of the equilibrium into two parts, but the equilibrium is completely characterized by both parts.Theorem1 describes the equilibrium in the trading period(i.e.,gives the price and demand for stocks for a given level of aggregate information)and Theorem2describes the equilibrium in the planning period(i.e.,the information acquisition decision).Theorem3characterizes the level of information and states the unicity of the equilibrium.Lemma4shows the implications for portfolio shares.

4.1Existence and characterization of the equilibrium

Theorem1(price and demand for stocks).Assume the scaling factor z is small.Assume information decision have been made(i.e.,i and x j are given).

There exists a log-linear rational expectations equilibrium.

The equilibrium price is given by

ln P?pz,where ptr f?p0eiTtp peiTepàmuT,e5T

h0eiT

1

s2

p

t

i2

s2

u

,hei,xT h0eiTtx,"h h

i,

i

n

,

p0 1

"h

EepT

s2

p

t

iEeuT

s2

u

t

1

2

,p p

1à

1

"h s2

p

,and m

1

i

:e6T

The optimal portfolio share of stocks for an investor j with a signal of precision x j(possibly equal to zero)is given by

a j?teW1jT

W1j

E jep z j

F jTàeptr fTzt1

2

V jep z j F jT

V jep z j F jT

?teW1jT

W1j

EepT

s2

p

t

iEeuT

s2

u

t

i2

s2

u

epàmuTtx j

S j

z

t

1

2

àeptr fThei,x jT

!

:e7T

The price function calls for a few remarks.First,the equilibrium price depends on the log-payoff p and the net supply of stocks u.u enters the Wealth,Information Acquisition,and Portfolio Choice

889

price equation,although it is independent of p because it determines the value of stocks to be held,and hence the total risk investors have to bear in equilibrium.p appears directly in the price function,though it is not known by any agent,because individual signals S j are aggregated and collapse to their mean ln P p z.

Second,observing the price is equivalent to observing pàmu,which

acts as a noisy signal for p with noiseàmu.For given s2

u ,the parameter

m 1/i measures the noisiness of the price signal.The smaller the noise m (the bigger i),the more informative the price.The function h0(i)is the precision of the public signal.Similarly the function h(i,x)is the total precision of an investor’s signal using both private and public signals(the precisions simply add up).i/n is a measure of the average private informa-tion,so"h is the average total precision in the market.

Third,it is insightful to decompose the random part of the price,p,in

two components:p??p0ti

2

u "

epàmuTti"p t?àu" àr f:The first

term captures the signal extraction problem.It is a weighted average of the priors(contained in p0)and of the public and private signals.The second term reflects the discount on the price demanded by risk-averse investors to compensate them for the risk in P.The discount is increasing in the net supply of stocks u,the market risk aversion1/n,and the amount of risk per stock,1="h,the average investor has to bear in equilibrium.Two extreme cases are of interest.If m is equal to zero(i?1),then there is no noise and the price reveals the true p.There is no risk in this economy and the price function,p,reduces to(pàr f)so that the two assets have the same net return,r f z.On the other hand,if m is infinite(i?0),then the price contains no information about p.The price function p becomes

EepTt1

2s2

p

àr fàs2p u

n

.The price coefficient p p is increasing in i,while

p0might be increasing or decreasing in i depending on the range of param-eters considered.

Finally,the fraction of her wealth an investor with a signal of precision

x j allocates to stocks can be written as a j?a x?0tteW1jT

W1j x jeS j

z

àpàr fT.In

other words,her portfolio share equals the optimal share had she been uninformed,plus the stock’s premium as predicted by her private signal, scaled by precision and relative risk aversion.

The proof of the theorem is presented in the appendix,so I only outline its key steps.First,guess that the price function p is linear in p and u. Second,solve the portfolio problem for an investor who observes P and S j (set x j to zero if the investor did not acquire a signal).Because of the normality assumption,the signal extraction problem yields an estimate of the stock’s payoff E j(p z j F j),which is linear in p and S j,and a precision

hei,x jT?z

V jep z j F jT,which neither depends on p nor S j.In addition,approx-

imating the Euler equation at the order1in z implies that the demand for

stocks is simply a j?teW1jT

W1j Eep z j F jTt1

2

V jep z j F jTàeptr fTz

Vep z j F jT

,which in turn is

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linear in p and S j.16Third,when summing up individual demands for stocks,apply the law of large numbers for independent,but not identically distributed random variables,and the individual signals all collapse to their conditional mean,p z.Hence the value of aggregate demand is linear in p and p,and equating it to the supply u will yield an equilibrium price linear in p and u as guessed.The information decision will not affect the linearity of the price since it is made ex ante,that is,before P is observed (of course,it will affect the coefficients of the price equation through i). The next theorem describes the information choice.

Theorem2(demand for information).Assume the scaling factor z is small.

There exists a wealth threshold W?0eiTsuch that only agents with initial

wealth above W?

0eiTacquire information.

Their optimal precision level,x j?x(W0j),is characterized by the first-order condition

C0ex jT?1

2

teW0jTw0ex j;iTe8Tand by the second-order condition

C00ex jTà1

2

teW0jTw00ex j;iT!0,e9Twhere w is an increasing and convex function of x as shown in the appendix.

Depending on the value of her initial endowment(but irrespective of how it is split between stocks and bonds),an agent will choose to acquire information or not.In fact,information will only pay off for agents who

are wealthy enough.The wealth threshold W?

0(derived in Appendix B)is

defined as the level of wealth that makes an investor indifferent between

acquiring information and remaining uninformed.The value of W?

0deter-

mines whether all investors are informed,whether none are,or whether informed and uninformed investors coexist in equilibrium.

The precision informed agents choose is then given by Equation(8). The function w is defined in equation Equation(13)in Appendix B.w measures the squared Sharpe ratio an investor expects in the planning period given that she will receive some information in the trading period. For an informed investor,w is an increasing and convex function of x.Its derivative,w0,is increasing and concave.Equation(8)is illustrated by 16The approximation does not amount to assuming quadratic preferences since the expansion is done around different wealth levels.Instead,preferences are modeled as an envelope of quadratic functions. An alternative specification of the model is to posit up front that the demand for stocks is given by this equation.

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Figure 2and states that,at the optimum,the gain from a small increase in precision is exactly offset by its extra cost.It shows that the optimal precision level is increasing with absolute risk tolerance,which by assump-tion is increasing with wealth.Thus wealthier investors acquire more information.The reason is that investors with greater absolute risk toler-ance purchase a larger number of stocks and hence find information more valuable.Putting it differently,there are increasing returns to informa-tion:the cost of achieving a given precision is independent of the scale of the investment (i.e.,of the amount invested),whereas its benefit is increas-ing with the scale.Note that this increasing returns to scale property is obtained in spite of a strictly convex cost function.Figure 3depicts the wealth-precision relationship.

The other properties of the optimal precision,x j ,are the following.First,x j is finite so no investor has an arbitrage opportunity.Second,x j is decreasing in the marginal cost of information and in risk aversion (a less risk-averse investor will buy more stocks and hence will find informa-tion more valuable).Third,x j is decreasing in the informativeness of the price,i (greater informativeness implies that prices are more revealing and

Precision x Figure 2

The optimal precision choice C 0(x )(dashed curve)and 12t (W 0)w 0(x )(top solid curve for a rich investor and bottom solid curve for a poor investor).Rich investors acquire information of precision at the intersection of C 0(x )and 12t (W 0)w 0(x ).Poor investors do not acquire information.The picture is drawn for C(x)?0.07x 2t0.01x ,t (W 0)?W 0,where W 0equals 0.3for the rich investor and 0.15for the poor investor,E (p )?1,s 2p ?1,E (u )?0.01,s 2u ?0:02,n ?1,m ?100,and A ?1.25.

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consequently decreases the incentives to acquire private information).17These results correspond to Lemma 2and Corollaries 1to 4in Verrecchia (1982)in the case of CARA preferences.The next theorem characterizes the level of information in equilibrium.

Theorem 3(equilibrium level of information and unicity).Assume the scaling factor z is small.

In equilibrium ,price informativeness i solves

i ?Z 0

W ?0ei Tx eW 0j ;i Tt eW 0j TdG eW 0j ,a 0j T:e10T

Assume s 2p 2.There exist a unique log-linear equilibrium .

Equation (10)characterizes the aggregate level of private information in equilibrium,i .It follows directly from the definition that was given in 17Strictly speaking,this statement requires that s 2p 2.The problem is that investors care about the expected instantaneous return which involves the payoff P ,whereas information is about the logarithm of the payoff ln P p z .For that reason,the expected return carries a volatility term that complicates the derivations.The upper bound on s 2p ensures that this term does not become too big.

Wo P r e c i s i o n Figure 3

The optimal precision for different levels of wealth under constant (solid curve)decreasing (dotted curve),and increasing (dashed curve)relative risk aversion For wealth levels below W ?0,no information is acquired.The picture is drawn for C (x )?0.07x 2t0.01x ,t eW 0T?W b 0,where b ?0.8,1,and 1.2,E (p )?1,s 2p ?1,E (u )?0.01,s 2u ?0:02,n ?1,and m ?100.Wealth,Information Acquisition,and Portfolio Choice

893

Equation(4).I have only managed to prove that Equation(10)admits a

unique solution under the assumption that s2

p 218.In this case,the

equilibrium is unique within the class of log-linear equilibria.The next section puts the results from Theorems1and2together to study the effect of wealth on portfolio decisions.

4.2Wealth and portfolio shares

Let?t be the elasticity of absolute risk tolerance with respect to wealth,?c the elasticity of marginal cost with respect to precision,and?a the elasti-city of portfolio share with respect to wealth:

?t W0t0eW0T

teW0T

,?c

xC00exT

C0exT

,and?a

W0a0eW0T

aeW0T

:

By assumption?c>0and?t!0.In the definition of?a,a is the uncon-ditional portfolio share,that is,the share of her wealth W0,an investor allocates to stocks,averaging over the possible realizations of all the random variables p,u,and?j.(Alternatively,one could average over the idiosyncratic shocks?j only and consider the shares conditional on the economy-wide shocks p and u.)

Lemma4(wealth and portfolio shares).

For an uninformed investor,

?a??tà1

For a well-informed investor(i.e.,an investor with large precision x j),

?a%?tà1t?t ?c :

Under CRRA preferences,for any informed investor,

?a?1?c :

The lemma shows that the pattern of shares increasing with wealth (?a>0)may hold even if relative risk aversion is not decreasing.This is the case under CRRA(?t?1),regardless of the cost function,and under increasing relative risk aversion(?t<1),provided the cost function is not

too convexe?c

?t à1Tà1T.Figure4illustrates the lemma.

The mechanism through which information acquisition operates on the demand for stocks is again the following:under decreasing absolute risk 18See the previous footnote.

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aversion,wealthier investors purchase more stocks for a given precision level [Equation (7)].Having a riskier portfolio makes information more valuable for these investors,so they acquire more private information.Finally,a higher precision induces investors to hold even more stocks.Thus wealth has a double effect on the demand for stocks:a traditional direct effect and an indirect effect through the demand for information.Under decreasing relative risk aversion,both effects work in the same direction,making portfolio shares increasing with wealth.Under increas-ing relative risk aversion,the direct effect is reversed so the net effect is ambiguous.It depends on the shape of absolute risk aversion relative to that of the cost function.If C is not too convex,then a small increase in wealth will lead to a large increase in private information that will over-turn the increase in relative risk aversion.

The following example illustrates the theorem.Let C (x )?x c for c >1.Differentiating C yields ?c ?c à1.Such a function can reconcile the observed pattern of shares with any increasing relative risk aversion

utility:it suffices to choose c <11à?t .For example,if ?t !12and the cost

function is quadratic,then portfolio shares increase with wealth in spite of

Wo S h a r e Figure 4

Portfolio share of stocks for different levels of initial wealth under constant (solid curve),decreasing (dotted curve),and increasing (dashed curve)relative risk aversion Only investors with wealth above W ?0acquire information.The picture is drawn for C (x )?0.07x 2t0.01x ,t eW 0T?W b 0,where b ?0.8,1,and 1.2,E (p )?1,s 2p ?1,E (u )?0.01,s 2u ?0:02,n ?1and m ?100.

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increasing relative risk aversion.Conversely,suppose preferences are CRRA and all investors are informed,then the observed share elasticity of0.1implies a cost elasticity of10and hence a cost function in x11.The next section studies the connection between the stock market and wealth inequalities.

5.The Stock Market and Wealth Inequality

In Section3,I showed that wealthier households acquire more informa-tion and hence that the demand for stocks is a convex function of wealth.

This means that if a dollar is transferred from a poor to a rich investor, the demand for stocks of the rich will increase by more than the demand of the poor will fall,resulting in a rise in aggregate demand.Consequently the price will increase.In short,the more unequal the distribution of wealth(keeping the average wealth constant),the smaller the equity premium.Interestingly,this is the case regardless of relative risk aver-sion.19The provision of information through prices also increases with wealth inequality(again keeping the average wealth constant).

In Section6,the model is calibrated to U.S.data and the effects on the equity premium are shown to be quantitatively significant for plausible parameter values.

So far I have studied the effect of wealth inequality on the stock price, but I can also look at the reverse causality,that is,at the link from stocks to wealth inequality.It is a well-known fact that wealth is unevenly distributed.In the United States,for example,the top decile of all house-holds own82.9%of all financial wealth in the nation[Wolff(1998)].While several factors may explain these differences[see Quadrini and Rios-Rull (1997)for a review],the model focuses on the role played by the avail-ability of costly information about assets.The model shows how informa-tion generates increasing returns which magnify wealth inequality: wealthier agents acquire more information and more stocks and achieve

a higher expected return,a higher variance,and a higher Sharpe ratio on

their portfolio.It follows that the distribution of final wealth as measured by expected wealth or by certainty equivalent is more unequal than the distribution of initial wealth.Arrow(1987)makes this point,albeit in a partial equilibrium setting.

Formally,recall that r p j z is the net return on investor j’s portfolio (before accounting for the information cost)and let r pe j z r p j zàr f z be the associated excess 2e90cf40680203d8cf2f2445ing Equation(7)and integrating over all

19In contrast,in a standard frictionless symmetric information economy with CRRA preferences,the distribution of wealth has no implication on the equity premium.This is no longer the case under different assumptions on preferences[e.g.,Gollier(2001)]or if frictions such as entry costs or market incompleteness[e.g.,Constantinides and Duffie(1996),Heaton and Lucas(1996)]are introduced.

896

the random variables yields,

E er pe j z T?

t eW 0j T

W 0j

w ex j Tz ,V er pe j z T?

t eW 0j TW 0j

2

w ex j Tz ,

and

E er pe j z T

???????????????V er pe j z T

q ??????????????w ex j Tz q :e11T

Recall that investor j ’s private precision x j is increasing in her wealth W 0j

and that the function w is increasing in x j for an informed investor.These results are illustrated by Figure 5.The next section addresses some empiri-cal issues raised by the model.6.Empirical Issues

In this section I calibrate the model.Then I show how to discriminate among different models of portfolio choice.

43

2

1

Wo

E x p e c t e d u t i l i t y

Wo

E x p e c t e d r e t u r

n

Wo

V a r i a n c e o f r e t u r

n

Wo

S h a r p e r a t i o

Figure 5

The relation between initial wealth and expected utility (top left panel),expected return (top right panel),variance of return (bottom left panel),and Sharpe ratio (bottom right panel)

In the right panels,solid curves do not include the cost of information,dashed curves do.Rich investors acquire information while poor investors do not.The graphs are drawn for log utility,C (x )?0.07x 2t

0.01x ,E (p )?1,s 2p ?1,E (u )?0.01,s 2

u ?0:02,n ?1,m ?100,r f ?0.03,z ?0.01,and a 0?0.

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897

6.1Calibration

The model shows that the ability to acquire information explains,both qualitatively and quantitatively,why richer households invest a larger fraction of their wealth in risky assets.A consequence,pointed out in the previous section,is that the distribution of wealth has an impact on the moments of asset returns.To assess whether the effects on returns are quantitatively important for plausible parameter values,I calibrate the model to U.S.stock market data.Starting from a benchmark economy where no information is acquired,I increase wealth inequalities and examine the consequence on the equity premium and its variance.

I begin by describing the benchmark economy.In this economy,no

household collects information (the wealth is below the threshold W ?0).I

assume that they have CRRA preferences (with a baseline coefficient of relative risk aversion a ?5)so that the distribution of wealth has no effect on asset returns.In 1995,69.3million households owned equity in the United States.On average,they had $74,810in financial wealth with 55%invested in stocks.20It follows that the aggregate level of financial wealth was $5,184billion and that aggregate risk tolerance n was $1,037billion (for a ?5).

I now turn to the assets in the economy.The riskless interest rate is set to 3%per year (the scaling factor z is set to 1).The parameters of the distributions of p and u are chosen so that,in the benchmark economy,the equity premium,variance,and portfolio shares match their historical values.Over the 1889–1978period,the average annual equity premium was 6.18%,its standard deviation was 18%,and its variance was 3.24%.In the benchmark economy,the average portfolio share invested in stocks is E ea T?q t12s 2p 2p ,where q E eln P P Tàr f z E ep TàE ep Tàr f ?eE eu Tn à12Ts 2p is the equity premium.Therefore s 2p ?q 12?0:0275and E eu Tn ?aE ea T?2:750.The variance of the equity premium is v var eln P P Tàr f z var ep àp T?s 2p es 2p s 2u 2t1T,implying that s 2u 2?12p ev 2p à1T?6:539.Note that positive s 2u imposes a lower bound on a :a >1eq t1T?4:38.E (p )is irrelevant and normalized to one.

Next,I describe the distribution of wealth.I assume wealth is evenly distributed among households in the benchmark economy (under CRRA preferences and no information acquisition,the distribution of wealth is irrelevant).The goal here is to analyze the economy when wealth becomes unequally distributed.For simplicity,I assume the distribution of wealth is bimodal:the economy is populated by two groups of agents,the rich 20The number of households holding equity is reported by Poterba (1998).Average financial wealth and portfolio shares are based on the 1994Panel Study of Income Dynamics (PSID)and are measured conditional on having positive financial wealth and positive stockholdings,respectively [Vissing-J?rgensen (2002)].Wolff (1998)reports a larger number for financial wealth using the 1995Survey of Consumer Finances (SCF),but his measure includes business equity.

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and the poor.The rich,in proportion M,have wealth W rich while the poor, in proportion1àM,have wealth W poor.I simulate the economy for different combinations of the fraction of rich,M,and the fraction of aggregate wealth they own.Importantly,M,W rich,and W poor are varied in such a way that aggregate wealth remains constant,so as to capture only the effects of inequality.In practice,financial wealth is very unevenly distributed.For example,Wolff(1998)reports that the top decile of U.S. households owned82.9%of all financial wealth in1995.However,this figure overestimates the relevant number for this calibration because Wolff conditions neither on positive wealth nor on positive stockholdings and includes business equity in his definition of financial wealth(which is mostly concentrated in the hands of the very wealthy).

Finally,I specify the information acquisition technology.I assume C(x)?x ctdx.A large enough d ensures that no information is acquired in the benchmark economy21and that only the rich acquire information in

the unequal economies,that is,W rich>W?

0>W poor.The parameter c is

chosen to match the average share elasticity in the United States.Because

the share elasticity equals zero for uninformed investors and1

cà1for(well)

informed investors(recall that preferences are CRRA),the average elas-

ticity is M

cà1.For example,when M?0.2,an average share elasticity of0.1

(see Section1)implies c?3.

I simulate the economy for a relative risk aversion of5and7,an average share elasticity of0.05and0.1,and wealth distributions such that the top 5%,10%,and20%of the population own25%,50%,or75%of aggregate wealth.The results are reported in Table1.As expected,the equity premium and its variance are lower in unequal economies where informa-tion is collected.The more unequal the economy,the greater the effect on returns.This can be seen either by move along the lines(e.g.,10%of the population owns25%,50%,and75%of aggregate wealth)or up the columns(e.g.,50%of aggregate wealth is owned by20%,10%,or5%of the population).The effects are quantitatively important,especially close to the benchmark economy.For example,with a relative risk aversion of5 and an average share elasticity of0.1,the equity premium decreases by 43%(from6.18%to3.53%)and its variance by39%(from3.24%to1.99%) relative to the benchmark economy when10%of the investors own25%of financial wealth.Furthermore,the effect of inequalities is enhanced by lower risk aversion and higher share elasticity.Indeed,less risk-averse investors acquire more information because they purchase more shares [Equation(8)].A greater share elasticity implies a lower coefficient c in the information cost function and therefore that the optimal precision is more sensitive to differences in wealth.

21Formally,W?

0ei?0T>74,810,which implies that d>74;810es2pT2

2a

à1

s2p

tEeu2T

n2

à1

4

á

?220when a?5.The

value of d,beyond this threshold,has little impact on the results reported in Table1.

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