Information acquisition and mutual funds∗

Diego Garcıa† Joel M. Vanden

June 24, 2005

Abstract

We explain the size and the existence of the mutual fund industry by generalizing thestandard competitive noisy rational expectations framework with endogenous informationacquisition. Since informed agents optimally choose to open mutual funds in order to selltheir private information, mutual funds are an endogenous feature of our equilibrium. Ourmodel yields novel predictions on price informativeness, optimal fund fees, the equilibriumrisk premium, and the size and competitiveness of the mutual fund industry. In particular,we show that a sufficiently competitive mutual fund sector yields more informative pricesand a lower equity risk premium. Thus, the paper explicitly links the existence of mutualfunds to equilibrium asset prices.

JEL classification: D43, D82, G14.Keywords: partially revealing equilibrium, competitive equilibrium, rational expecta-

tions, markets for information, mutual funds.

∗We would like to thank Shmuel Baruch, Jonathan Berk, Josh Coval, Peter DeMarzo, Espen Eckbo, KenFrench, Steve Ross, Matthew Slaughter, Branko Urosevic and Josef Zechner for comments on an early draft, aswell as seminar participants at MIT, Queens’ University, the Econometric Society Winter meetings (Philadel-phia, 2005), the Winter Finance Workshop (Revelstoke, 2004), the FIRS conference on banking, insurance andintermediation (Capri, 2004), the Arison School of Business (IDC Herzliya), the European Finance Associa-tion meetings (Maastricht, 2004), and the European Econometric Society meetings (Madrid, 2004). The latestversion of this paper can be downloaded from http://diego-garcia.dartmouth.edu.

†Corresponding author. Both authors are from the Tuck School of Business, Dartmouth College, HanoverNH 03755-9000, USA; tel: (603) 646-3615; fax: (603) 646-1308; email: Diego.Garcia@dartmouth.edu.

1 Introduction

Over the past two decades mutual funds have become one of the most popular investmentvehicles for individual investors.1 While mutual funds have received a great deal of attentionin the literature, little has been done to formally explain the existence, the size, or the assetpricing implications of the mutual fund industry.2 We examine these issues by analyzing a noisyrational expectations model of mutual fund formation, which effectively merges the literatureon endogenous information acquisition (Verrecchia, 1982) with the literature on informationsales (Admati and Pfleiderer, 1990). In equilibrium, we show that informed agents are alwaysbetter off by establishing a mutual fund rather than using their private information to tradeon their own accounts. Thus, mutual funds endogenously arise as the optimal strategy ofinformed utility-maximizing agents. This establishes the existence of a mutual fund industry.Second, we endogenously determine the fraction of agents that optimally choose to acquirecostly private information. This establishes the equilibrium size of the mutual fund industry.We thus go beyond the information acquisition decision in order to study the optimal use ofthe acquired information. We finally show that as long as the mutual fund sector is sufficientlycompetitive, the introduction of mutual funds will yield more informative prices and a lowerequity premium. This links the existence of secondary assets, in our case mutual funds, toequilibrium asset pricing behavior.

In our model, investing in a stock via a mutual fund whose trading strategy is based uponprivate information is quite different than owning that same stock directly. In the formercase, the household sector’s perceived payoff per share is the product of the stock’s payoffand the fund manager’s optimal stock demand. Since households and fund managers areasymmetrically informed, this product can have a payoff distribution that is very differentthan the stock’s payoff. Thus, mutual funds increase the span of the market structure, and avery large number of mutual funds might be offered in equilibrium, even when the number ofprimary equities is quite small, as in our model. Since we endogenize the agents’ informationacquisition decisions, our model offers a sharp prediction as to the equilibrium size of the mutualfund industry. Specifically, as the level of competition increases, the equilibrium mutual fundsector becomes “small” with respect to the number of households in the economy. However, atthe same time, the mutual fund sector as a group trades more aggressively, thereby generating

1As reported by the Investment Company Institute, a trade association for the mutual fund industry, 52%of U.S. households owned mutual funds in 2001, up from 5.7% in 1980. Over the same time period, the numberof mutual funds offered by the financial services industry increased from fewer than 600 funds in 1980 to over8,000 funds in 2001.

2Exceptions to this statement include the empirical work of Khorana and Servaes (1999) and Khorana, Ser-vaes, and Tufano (2005). These authors take the mutual fund industry as given and examine the determinantsof mutual fund starts and the determinants of mutual fund size, respectively. In contrast to their work, weshow how mutual funds endogenously arise in equilibrium, i.e., we investigate why some agents in the econ-omy optimally choose to establish mutual funds and why households choose to engage in delegated portfoliomanagement.

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more informative prices. Both of these results are consistent with what we observe in practice.There are about 8,000 U.S. equity mutual funds that serve approximately 100 million U.S.households. Thus, the mutual fund sector is “small” on this dimension. However, mutual fundsas a group manage a significant portion of all invested funds in the economy, and thereforetheir trading strategies play an important role in determining asset prices.

Our model exhibits the following features, all of which correspond to what we tend toobserve today in financial markets: (i) the mutual fund sector is imperfectly competitive; (ii)the stock market is perfectly competitive; and (iii) the fund managers charge proportionalinvestment management fees. Our analysis of the oligopolistic competition in the mutual fundsector allows households to hold multiple mutual funds within their investment portfolios.3

Thus, a fund manager’s fee-setting decision is affected not only by the other managers’ fees,which would naturally occur in an oligopolistic setting, but also by the household sector’sequilibrium demand for mutual funds. The competitive feature (ii) accomplishes two things.First, it allows us to better connect our results to the existing literature, i.e., we can easilycompare our results to those that do not allow for a mutual fund sector. Second, it addsparsimony to our model since we are able to analyze the imperfect fee-setting competitionbetween mutual funds without the additional complications that would arise if we allowed thefund managers to behave strategically with respect to their stock market trades. As for feature(iii), the fund managers in our model are compensated via investment management fees thatare proportional to their funds’ final values. Furthermore, the mutual funds in our model tradein a liquid secondary market, which mirrors the current industry practice of calculating a dailyNAV (net asset value) and allowing fund shareholders to purchase or redeem shares at theprevailing NAV.

We use our model to study the relationship between the informativeness of price, theequilibrium risk premium, the optimal investment management fees, and the size and compet-itiveness of the mutual fund industry. Specifically, we show that the equilibrium stock price inan economy with a sufficiently competitive mutual fund sector will always be more informativethan in the standard model (Verrecchia, 1982). Furthermore, the equilibrium risk premiumdemanded by investors will be lower. We note that competition has two offsetting effects in theeconomies that we study. First, competition reduces the incentives to gather information, bothdirectly through the lower profits that informed agents can earn and through the informationthat is conveyed by price. Second, competition makes fund managers trade more aggressivelyon their private information. We show that the latter effect dominates if the mutual fundsector is sufficiently competitive. This allows us to conclude that mutual funds produce moreinformative prices, which in turn drives down the equity premium. Given that the actual equity

3In the standard oligopoly game, agents value only one unit of the good in question. In the context of ourmutual fund model, this appears to be a strong and undesirable assumption due to the signal diversificationthat arises from investing in multiple funds. For related work on information acquisition and oligopolies, seeVives (1988) and Hwang (1993).

2

risk premium has declined substantially over the past 30 thirty years, which is a time periodthat coincides with increasing use of institutional money management, our model appears tooffer predictions that are consistent with the data.

While the analysis of mutual funds in the financial economics literature has a very longhistory, we know of no other paper that specifically examines mutual fund formation as theoutcome of endogenous information acquisition. Our paper effectively fills this gap. However,given that our paper also offers predictions concerning the fund managers’ optimal fees, wepoint out that there is an overlap between our paper and the literature that analyzes optimalmutual fund contracts.4 The main difference is that our model uses a general equilibriumframework, while most of the existing literature is partial equilibrium in nature. Closelyrelated are the papers by Brennan (1993), Cuoco and Kaniel (2000), Arora and Ou-Yang(2001), Mamaysky and Spiegel (2001), and Ross (2005), who study mutual funds in variousgeneral equilibrium settings. However, our paper appears to be the only one to allow forheterogeneous information among traders, with endogenous information acquisition activities,while allowing for proportional management fees and competition among managers. Our workalso provides a contribution to the literature on information sales, by jointly studying theeffects of competition among information sellers and the incentives to acquire information.5

The remainder of our article is organized as follows. Section 2 outlines all of the model’sprimitives, including the overall structure of the model and our equilibrium definitions. Section3.1 then assumes that mutual funds are available to the household sector and solves for arational expectations equilibrium of the Hellwig (1980) type. Our equilibrium is summarizedin Proposition 1. Section 3.2 endogenizes the contingent fees of the mutual funds (Proposition2) and shows that informed agents always establish mutual funds rather than trade for theirown accounts using their private information. Section 3.3 then endogenizes the fraction ofagents that optimally become informed (Proposition 3) and thus effectively determines theequilibrium size of the mutual fund industry. Section 4 presents several extensions of ourmodel and lastly section 5 offers concluding remarks. All of our proofs are collected in theAppendix.

4For early work in this area, see Ross (1974), Admati and Ross (1985), Bhattacharya and Pfleiderer (1985),Dybvig and Ross (1985a), Dybvig and Ross (1985b), Dybvig and Spatt (1986). Some recent work in the areaincludes Stoughton (1993), Brennan and Chordia (1993), Huberman and Kandel (1993), Heinkel and Stoughton(1994), Dow and Gorton (1997), Admati and Pfleiderer (1997), Carpenter (2000), Rajan and Srivastava (2000),Dybvig, Carpenter, and Farnsworth (2000), Das and Sundaram (2002), Christoffersen and Musto (2002), Berkand Green (2002), Germain (2003), Palomino and Prat (2003), Ou-Yang (2003), Massa (2004a), Massa (2004b).

5The paper is thereby closely related to the early literature on information sales, in particular Admati andPfleiderer (1986), Admati and Pfleiderer (1988), and Admati and Pfleiderer (1990), who studied the problem ofa monopolist seller of information in different market settings. For recent contributions in this area see Fishmanand Hagerty (1995), Simonov (2000), Biais and Germain (2002), Veldkamp (2003), Garcıa and Urosevic (2004),Cespa (2004) and Ross (2005).

3

2 A model of mutual fund formation

We analyze information acquisition and mutual fund formation in a noisy rational expectationssetting by modifying the framework of Grossman and Stiglitz (1980), Hellwig (1980), andVerrecchia (1982). We first outline the main elements of the model, we discuss how informedagents may sell their information via mutual funds, and we present the equilibrium conceptsthat are used in the analysis. These items fully specify our framework for modeling mutualfund formation.

2.1 Basic elements

There are two types of primitive assets available for trading, a riskless asset and a risky asset.The riskless asset pays zero interest and has a perfectly elastic supply. The risky asset, onthe other hand, has a payoff of X, where X has a normal distribution with mean µx andvariance σ2

x, i.e., X ∼ N (µx, σ2x). The per capita supply of the risky asset is U ∼ N (µu, σ2

u)and is interpreted as the presence of noise traders in the economy.6 All random variables inour model are defined on a probability space (Ω,F , P). In addition to the primitive assets, theinvestment opportunity set of the household sector includes mutual funds. The mutual fundsare optimally established in equilibrium and are discussed in greater detail below.

Investors in our model can either remain uninformed (i.e., as “households”) or they canacquire costly private information. If they choose to become informed, they further face thechoice of whether to trade on their own accounts (i.e., as “proprietary traders”) or to offerinvestment management services to the household sector (i.e., as “mutual fund managers”). Inthe latter case, the mutual fund manager earns a contingent investment management fee. Inother words, the manager retains a fraction of the fund’s final value as his compensation. Forsimplicity, we do not allow informed agents to invest with other informed agents. For example,mutual fund managers cannot hold positions in other mutual funds nor in their own accounts,which is consistent with what we tend to observe in practice. As discussed in section 4.2, theseassumptions are purely for tractability reasons, and do not affect any of our conclusions.

Agents in our model have CARA preferences with risk aversion parameter τ . Thus, fora given final payoff Wi, the ith agent has the expected utility E [u(Wi)] = E [− exp(−τWi)].We relax the homogeneous risk aversion assumption in section 4.1, but for now it allows us tofocus on the informational aspects of the model. In addition, we assume that all agents havezero initial wealth, which is without loss of generality due to the standard properties of CARAutility. Agents can acquire private information by paying a fixed cost c > 0. Upon paying c,the ith agent observes the private signal Yi = X + εi, where εi ∼ N (0, σ2

ε ). We let λ denote6Alternatively, one could interpret U as representing endowment shocks rather than noise trading. Our

results remain unchanged under this alternative interpretation.

4

the fraction of agents in our economy that is informed. When this fraction is endogenouslydetermined, we use the notation λ∗ instead of λ. Agents in our model have rational expectationsin the sense of Grossman (1976), i.e., they rationally use the information revealed by price whenforming their posterior beliefs. Thus, even though some agents may remain uninformed by notacquiring private information, the uninformed agents do learn something about the privateinformation of the informed agents by observing the risky asset price. Lastly, we assume thatX, U , and the collection of signal errors εi are mutually independent.7

The sequence of events in our model can be described by using a timeline with threedates. Date 0 is the fund formation stage of the model. At this date we analyze the agents’information acquisition decisions in order to determine the equilibrium fraction of informedagents in our economy. We also analyze the mutual fund managers’ fee setting problemsin order to determine the optimal contingent management fees. Date 1 is the trading stageof the model. Each manager observes Yi and, given his private signal and the informationrevealed by price, the manager chooses his fund’s investment in the risky and riskless assets inorder to maximize his expected utility. Likewise, given the information revealed by price, eachhousehold chooses its investment strategy in order to maximize its expected utility. Finally,date 2 is the payoff stage of the model. At this date the risky asset pays X, the mutual fundsdistribute their payoffs, and all agents consume their final realized wealth levels.

Consistent with our assumption of a perfectly competitive stock market, we consider alarge economy in which there is a continuum of agents. Thus, while the fund managers inour model behave strategically when choosing their contingent fees, they are price takers inthe risky asset market, i.e., their private signals do not individually affect the risky asset’sprice. While this fact enhances the tractability of our analysis, it is apparent that for λ > 0a continuum of mutual funds will exist in equilibrium. Although a household would benefitfrom holding all of these funds, we initially focus on the case in which each household can holdonly m < ∞ mutual funds. Comparative statics with respect to m then allow us to analyzehow mutual fund fees and the equilibrium fraction of mutual funds vary with respect to thehousehold sector’s ability to invest in additional funds. We motivate m as the outcome of acostly search model in which each household identifies an appropriate set of mutual funds forinvestment. Although we do not formally model the search process, we note that the value ofm would be decreasing in the search frictions between the households and the mutual funds.

To be more specific, we focus on a purely symmetric equilibrium by introducing a continuumof identical groups of size n, as in Ross (2005). Each group contains m = λn informed agentsand h = (1− λ)n uninformed households. Thus, if every informed agent establishes a mutualfund and serves all of the households in the group, it is easy to see that each fund servesh = m

(1−λ

λ

)households. In the sequel we use the number of managers m in each group as a

parameter, instead of the total group size n, since it seems more natural to restrict this number7It is possible to allow for correlated signals, but at some cost in terms of notational simplicity.

5

to be an integer (we shall ignore integer problems with h or n). This grouping procedure allowsfor a tractable model for studying imperfect competition in the mutual fund industry withoutexplicitely modelling the matching process between fund managers and households. Note thatalthough we refer to economies with large m as economies with high degrees of competition,our results on large m may also be viewed as economies for which search (or contracting) costsare small.

2.2 Mutual funds

In our economy, an informed agent is allowed to indirectly sell his private information to theuninformed agents within his group by establishing a mutual fund. The ith informed agent(i.e., the ith fund manager) can set up a mutual fund whose payoff, Zi, is given by

Zi = Pi + γi (X − Px) , (1)

where Pi denotes the price of the ith fund, γi is the trading strategy of the ith manager, andPx is the price of the risky stock (whose payoff is X). Without loss of generality, we normalizethe mutual fund’s shares to be equal to 1 unit. Thus, Pi represents the initial amount investedin the ith fund by the households that belong to the ith manager’s group. Given the formof (1), note that we have assumed that Pi remains in the mutual fund and is invested by thefund manager. While this corresponds to current practice in the mutual fund industry, it is aninnocuous assumption (see pp. 906-907 of Admati and Pfleiderer (1990)).

The ith fund manager’s compensation is given by αiZi, where αi is the manager’s contingentfee.8 We assume that the contingent fee is chosen by the fund manager prior to any agentobserving his private signal. Contingent fees in our model are therefore constants, i.e., theydo not directly depend on any of the model’s random variables and they do not convey anyinformation.9 Essentially the fund managers create securities with payoffs that are linearfunctions of their trading strategies, charging proportional fees. In section 4.2 we discussthe possibility of charging both a fixed fee and a proportional fee, allowing for more generalcontracts.

For ease of exposition, we also define the ith manager’s total fee as the product of thecontingent fee and the fund’s price. In other words, the total fee for manager i is given by

8The compensation contract αiZi covers many of the fee structures that are typically used by mutual fundstoday. In particular, we can think of αi as representing the joint effect of any contingent deferred sales chargeand investment management fees.

9We assume that the signal precisions are common knowledge. In the case where agents have heterogeneousprecisions, the absence of this assumption would open up the possibility that the managers might signal theirquality via their contingent fees. In this case, unlike the symmetric model, the contingent fees might beinformative to the household sector. See Huberman and Kandel (1993) and Das and Sundaram (2002) and thereferences cited therein for the effects of signalling in mutual fund markets.

6

αiPi. We motivate the concept of the total fee by examining the typical household’s payofffrom investing in the ith mutual fund. Since the manager keeps αiZi, the net fund payoff tothe household (per unit investment in the mutual fund) is given by

(1− αi)Zi − Pi = (1− αi)γi(X − Px)︸ ︷︷ ︸net risky asset bet

− αiPi︸︷︷︸total fee

. (2)

As equation (2) illustrates, the net fund payoff can be separated into a fixed part and a variablepart. The fixed part, given by αiPi, represents the total fee paid to the fund manager by thehousehold sector. The variable part, on the other hand, represents the household sector’sportion of the mutual fund’s risky asset bet. The household sector pays a net amount equalto αiPi in exchange for the risky exposure (1− αi)γi(X − Px).

In general, households purchase mutual funds for many reasons, including asset diversi-fication, access to possibly superior financial market information, and signal diversification.Obviously, since our economy contains only a single risky asset, we address only the last tworeasons. For example, households in our model can obtain equity market exposure in one oftwo ways – by directly investing in the stock market or by indirectly investing via a mutualfund. If the ith fund manager has private information about X, then his demand for the stock,γi, will depend on his private information. Thus, from the perspective of the household sector,γi is a random variable and the mutual fund’s stock market bet represents a new asset class.This is easy to see by examining (2). While (X − Px) is normally distributed, the quantityγi(X − Px) will not be normally distributed since it involves a product of random variables.Essentially, in a world with asymmetric information, investing in a stock via a mutual fund isquite different than investing in that same stock directly.

2.3 Equilibrium

With αi fixed, the ith manager’s optimization problem at date 1 can be written as

maxγi

E[−e−τ(αiZi−c)

∣∣∣Px, Yi

]; (3)

where Zi is given in (1). Our model reduces to a simplified version of the standard noisyrational expectations equilibrium (Hellwig, 1980; Verrecchia, 1982) if we set αi = 1 for everyinformed agent. In this case, the household sector does not invest in any mutual fund (i.e.,Pi = 0 for every i) and the informed agents use their private information to trade on theirown account. In other words, an informed agent engages in “proprietary trading” for his ownaccount when αi = 1. For αi ∈ (0, 1), the informed agent establishes a mutual fund andmanages money for the household sector, allowing the uninformed to share the returns of theirtrading strategy.

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Turning to the household’s investment problem, let θj denote the number of shares of therisky asset and let φij denote the number of units of the ith mutual fund that are demandedby the jth household. Since there is only 1 unit outstanding of each mutual fund, φij can beinterpreted as the fraction of fund i that is held by household j. Using this notation, we canwrite the optimal investment problem of the jth household at date 1 as

maxθj ,φiji=1,...,m

E[−e−τWj

∣∣Px

]; (4)

where

Wj = θj (X − Px) +m∑

i=1

φij [Zi(1− αi)− Pi] . (5)

Given problems (3) and (4), a rational expectations equilibrium at date 1 is defined as:

(i) a collection of mutual fund trading strategies such that the ith manager’s strategy, γi,solves (3);

(ii) a collection of household trading strategies such that the jth household’s strategy, θj andφiji=1,...,m, solves (4);

(iii) a price function for the stock, Px : Ω → R, and a collection of price functions for themutual funds, Pi : Ω → R for all i, such that all markets clear, i.e.,∫ λ

0γidi +

∫ 1

λθjdj = U ; (6)

h∑j=1

φij = 1; for all i. (7)

Given the normal-exponential setup and the symmetry of our model, we make severalnatural conjectures. All of these conjectures are formally verified to be true in equilibrium.First, we conjecture that the risky asset’s equilibrium price at date 1 is a linear function of X

and U , i.e., we conjecture thatPx = a + bX − dU ; (8)

where the price coefficients a, b, and d are endogenously determined in equilibrium. Weremark that this conjecture implies that individual agents’ signals do not appear on the pricefunction, as in the standard model of Hellwig (1980) and Verrecchia (1982). This is driven bythe structure of the contracting groups discussed section 2.1. Second, we conjecture that theprices of the established mutual funds are uninformative with respect to X. As will becomeapparent in the next section, the price of the funds in equilibrium are given by the covariance

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of their payoffs with those of the risky asset, which makes the prices of the funds independentof both informed agents’ signals and the market clearing stock prices.

Contingent fees are chosen by the managers non-cooperatively after the information ac-quisition stage. Our equilibrium concept is the standard Nash equilibrium, where each fundmanager solves

maxαi

E [u(αi (Pi + γi(X − Px))] ; (9)

taking the fees of the other m−1 managers in their group as given. Note we assume that whensolving (9) each fund manager anticipates the equilibrium price function (8). We denote theoptimal fee of the ith manager as αi and we map the ith manager’s optimal fee choice into afund formation decision. If αi = 1, the ith manager trades for his own (proprietary) account.On the other hand, if αi < 1, the manager establishes a mutual fund and markets his optimalinvestment strategy to the household sector.

At the information acquisition stage we exploit the model’s symmetry and we identify λ∗

as the value of λ that equates the date 0 indirect utilities of a typical fund manager and atypical household. Thus, in equilibrium, neither an informed agent nor an uninformed agenthas an incentive to alter his information acquisition decision. Essentially, we can interpret λ∗

as the unique outcome of a Nash equilibrium in which each agent optimally chooses whetheror not to become informed. If we were to force αi = 1 for all managers, this reduces to theendogenous information acquisition models of Verrecchia (1982) and Diamond (1985).

3 A rational expectations equilibrium with mutual funds

We break the analysis of the model into three parts. First, we discuss the equilibrium priceand trading strategies fixing the fees charged by the managers. We then solve for the Nashequilibrium in the fee setting stage. Finally, we endogenize the equilibrium fraction of informedagents.

3.1 A rational expectations equilibrium at the trading stage

We begin our analysis at date 1 by examining the trading stage of the model. For now, wefix αi ∈ (0, 1] for all i and we take λ ∈ (0, 1) as given. Exploiting the symmetry of themodel, we further conjecture that all managers will charge the same contingent fees αi = α.This conjecture will be verified to hold in equilibrium in section 3.2. Given this setup, thefollowing proposition characterizes the rational expectations equilibrium at the trading stageof our model.

Proposition 1. There exists a noisy rational expectations equilibrium at date 1 with the fol-lowing properties:

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(i) the optimal date 1 risky asset demand of manager i is

γi =E [X|Px, Yi]− Px

αiτ var(X|Px, Yi); (10)

(ii) the optimal date 1 risky asset demand of household j is

θj =[

E [X|Px]− Px

τ var(X|Px)

](1 +

var(X|Px)h σ2

ε

m∑k=1

(1− αk

αk

))− 1

h

m∑k=1

(1− αk) E [γk|Px] ;

(11)

(iii) the optimal date 1 demand of household j for mutual fund i is φij = 1h ;

(iv) the date 1 market clearing risky asset price is given by (8) with price coefficients

d =1 +

λ

ατ2σ2ε σ

2u[

λ

ατσ2ε

+1τ

(1σ2

x

+(

λ

ατσε

)2 1σ2

u

)] ; (12)

and

b =(

λ

ατσ2ε

)d;

a

d=

µx

σx+(

b

d

)µu

σ2u[

τ +(

b

d

)1σ2

u

] ; (13)

where λ = mm+h and α denotes the average contingent fee across all managers in the

economy;

(v) the date 1 equilibrium value of fund i is

Pi =

(1

αiτσ2ε

)[(1− αi

αi

)− 1

h

(1− αi

αi

)2]

[var(X|Px)−1 +

2h σ2

ε

m∑k=1

(1− αk

αk

)− 1

h2 σ2ε

m∑k=1

(1− αk

αk

)2] . (14)

For notational purposes, note that we can write the manager’s demand in (10) in terms ofthe economic primitives by substituting

var(X|Px, Yi) =[

1σ2

x

+1σ2

ε

+b2

d2σ2u

]−1

(15)

and

E [X|Px, Yi] = µx −[µx

σ2ε

+Pxb

d2σ2u

]var(X|Px, Yi) +

[Yi

σ2ε

+Pxb

d2σ2u

]var(X|Px, Yi), (16)

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where Px = E [Px]. Letting σ2ε → ∞ in (15) and (16) produces expressions for E [X|Px] and

var(X|Px), which show up in the expressions for θj and Pi in (11) and (14), respectively.

The above equilibrium has several interesting features. First, while the manager’s demandin (10) has the familiar mean-variance form, it is apparent that the aggressiveness of thefund manager’s trading strategy is decreasing in his contingent fee. As αi increases, the fundmanager trades less aggressively by taking a smaller position (either long or short) in the riskyasset, thereby exposing himself to the same amount of risk for all αi. The contingent fee hasthe effect of reducing the ith manager’s effective risk aversion from τ , which would prevail inthe Hellwig (1980) setting, to αiτ . In turn, due to demand aggregation and market clearing,the coefficients of the equilibrium price function depend on the quantity ατ . For a given λ, thisproduces a larger value for the relative price coefficient b/d, which measures the informationcontent of price. Thus, relative to what would prevail in the Hellwig (1980) setting, the riskyasset price in the presence of mutual funds is more informative. We formally state this resultas a corollary.

Corollary 1. For a fixed λ, the risky asset price in the equilibrium in which the informedagents offer mutual funds to the household sector is more informative than the risky assetprice in the equilibrium in which the informed agents engage only in proprietary trading.

Given the expression for γi in (10), the ith fund’s stock market bet γi(X − Px) is a noisyquadratic function of X. Since it depends on X2, which is always positive, households derivesome benefit from investing in mutual funds.10 However, since the bet γi(X−Px) also dependson the error term εi, investing in a mutual fund exposes the household to additional noise thatis unrelated to fundamentals. Thus, investing in mutual funds involves a trade-off between theprofitability of the informed managers’ investment strategies and the idiosyncratic risk that isassociated with them. Furthermore, households will generally invest in multiple mutual fundsin order to diversify their exposure to the error terms εi.

Given the expression for θj in (11), it is easy to see how the existence of mutual fundsaffects the typical household’s stock market investment. There are two separate effects – a riskaversion effect and a feedback effect. To see this, note that the term in square brackets in (11)is the familiar mean-variance demand that shows up in Hellwig (1980). This is multiplied bya quantity that is greater than 1. We call this the risk aversion effect since we can think ofthe household’s effective risk aversion as being equal to τ divided by this quantity. The riskaversion effect leads to aggressive trading, i.e., the typical household trades more aggressivelythan they otherwise would in an economy without mutual funds. The aggressive trading that

10While Brennan and Cao (1996) analyze a quadratic payoff in the standard rational expectations setting, ourmodel differs from theirs in several respects. First, our quadratic function arises endogenously as the mutualfund’s optimal payoff. Second, our quadratic function is noisy, i.e., it depends on the errors εi. Lastly,our quadratic function depends on αi which implies that the fund managers control the household sector’sexposure to X2.

11

stems from the risk aversion effect is partially offset by the household’s expectation of themanagers’ optimal trades. Indeed, the final term in (11) reveals that the typical householddecreases its long position (or increases its short position) if it believes that mutual fundmanagers are taking long positions in the risky asset. We call this the feedback effect sincethe household’s demand for the risky asset is at least partially determined by the investmentstrategies of the mutual fund managers. Finally, note that as αi → 1 for all i, both the riskaversion effect and the feedback effect vanish. In this case, all managers trade in a proprietaryfashion for their own accounts and our model collapses to the standard Hellwig (1980) setting.

The optimal mutual fund demands in property (iii) of Proposition 1 arise due to efficientrisk sharing among the h households in each group. In particular, due to the symmetricalnature of our model, the jth household holds 1

h of each of the m mutual funds in whichit is allowed to invest. Later in the article (see Proposition 4) we discuss how our tradingstage equilibrium would be altered if households were allowed to be heterogeneous. If the h

households in a group instead had heterogeneous risk-aversion, the jth household’s equilibriumdemand for mutual fund i would be equal to φij = τ−1

j /∑h

k=1 τ−1k .

We note that b and d in (12)-(13) depend only on the average contingent fee in the economy,α, and not on any particular manager’s fee, αi. However, the equilibrium fund value in (14)explicitly depends on the fees of the m managers that comprise a group. These facts allowus to distinguish between strategic behavior in the mutual fund sector and strategic behaviorin the stock market. Specifically, fund managers in our model are price takers with respectto their stock market investments but they behave strategically when setting their contingentfees. They correctly account for the fact that other managers are also offering investmentmanagement services to the household sector. Furthermore, they correctly account for the factthat their contingent fee affects the fund’s payoff via their optimal stock demand, γi.

Lastly, note that the expressions for a, b, and d in (12)-(13) depend only on λ and donot directly depend on m and h. We can therefore scale m and h by the same constantwithout changing the equilibrium stock market level. However, this type of scaling will have asignificant effect on mutual fund values since Pi in (14) depends on m and h individually. Thisdependence will in turn affect the equilibrium mutual fund fees, since those are determined bymaximizing αiPi, which is essentially the objective function of the ith manager. We exploreboth of these features more thoroughly in the next section.

3.2 Optimal fees

We turn now to the analysis of the Nash equilibrium at the fee setting stage. Using (1) wesee that the payoff for the fund manager, αiZi, is of the form αiPi + αiγi(X − Px). From (10)it is immediate that αiγi is independent of αi, so the ith manager’s problem at date 0 in (9)

12

reduces tomax

αi

αiPi ≡ f(αi) (17)

where Pi is given in (14). Obviously, since Pi depends on the entire collection of contingentfees, every manager behaves in a strategic (non-cooperative) manner when choosing his owncontingent fee. The next proposition characterizes the Nash equilibrium of the fee settinggame.

Proposition 2. The symmetric Nash (fee setting) equilibrium at date 0 is characterized byα1 = α2 = · · · = αm ≡ α where α is the unique solution of the cubic equation

k3α3 + k2α

2 + k1α + k0 = 0 (18)

that satisfies α ∈ (1/(1 + 0.5h), 1). The coefficients of the cubic equation are given in the proof.

A remarkable property of the fee setting equilibrium is that α ∈ (1/(1 + 0.5h), 1), i.e.,αi = 1 is never optimal for the ith informed agent in our economy. This implies that everyinformed agent establishes a mutual fund and markets his optimal investment strategy to thehousehold sector. In other words, rather than trade on their own accounts and keep the entireamount of their risky asset bets, the informed agents find it optimal to share their risky assetbets with the household sector in exchange for the total fees. Thus, our model provides onepossible explanation for why mutual funds arise in practice.

In order to gain some intuition about the factors that drive the fees, we can use (14) torewrite the ith manager’s problem in (17) as

maxρi

ρi(1− ρi/h)

var(X|Px)−1 +[

2hσ2

ε

∑mk=1 ρk − 1

h2σ2ε

∑mk=1 ρ2

k

] (19)

where ρi = (1−αi)/αi. The importance of the choice variable ρi can be seen by examining thenet fund payoff per unit that is received by a typical household. In particular, we can expandexpression (2) to get

(1− αi)Zi − Pi =ρi

τσ2ε

Yi(X − Px)︸ ︷︷ ︸Yi−bet

+ q(Px)(X − Px)︸ ︷︷ ︸Px−bet

− αiPi︸︷︷︸total fee

(20)

where q(Px) is a linear function of Px but does not depend on Yi. The above expressionincludes three terms, the Yi-bet, the Px-bet, and the total fees. Since the latter two terms canbe replicated by trading in the risky and riskless assets, the household’s marginal willingnessto pay for the fund’s shares is independent of these two terms. On the other hand, the Yi-bet is measurable with respect to the manager’s information set, but not with respect to thehousehold’s. Furthermore, the quantity ρi controls the household sector’s exposure to the

13

portion of the total bet (i.e., the Yi-bet) that cannot be replicated by trading directly in thestock market. Because of this, we denote ρi as the ith fund’s signal exposure to the risky asset.

Returning to expression (19), it is apparent that several factors influence the fund manager’stotal fee. Since agents are risk-averse, there is an upper bound to the amount of exposure to amanager’s signal they are willing to pay for. Optimal risk-sharing would call for the ith fundto have an exposure ρi = h.11 However, when the fund managers set their fees, they chooseρi ∈ (0, h/2), which is more than 50% less than the household sector’s optimal exposure. Thisis a direct result of each fund manager maintaining some market power within his group. Themanagers extract rents from the household sector by limiting the households’ exposure to therisky asset. This result is analogous to how a monopolist would restrict the demand for hisproduct, as in Admati and Pfleiderer (1988).

The denominator of (19) reveals that there are two additional factors that affect a manager’stotal fee. The first factor in the denominator is var(X|Px)−1, which is the precision of X

given the household sector’s information set. Rather intuitively, a fund manager’s total fee isdecreasing in the household’s precision, suggesting that households are less willing to pay forinvestment management services if they already have good quality information about the riskyasset’s payoff. We also note that even though var(X|Px)−1 is independent of any particularmanager’s fee, it does depend on α, which is the average contingent fee across all managersin the economy. In fact, recalling (15), we see that var(X|Px)−1 is decreasing in α. As Px

becomes more informative, the household sector is less willing to invest through mutual fundsand the total fee that each manager can charge in equilibrium is reduced.

The second factor in the denominator of (19) is given by the term in square brackets.Since this term involves m, it allows us to assess the impact of mutual fund competition onthe ith manager’s total fee. In equilibrium, this term is always positive12 and it thereforeintroduces an additional negative effect on the manager’s total fee. If we increase the numberof fund managers in a group from m to m + 1, the term in square brackets increases and thislowers the equilibrium total fee of the ith manager in the group. While the contingent fee α isalways decreasing in m, we find that the total fee αP (α) is non-monotonic in m and is actuallyincreasing in m for some parameter values. These facts can be explained by noting that thenumber of households that is served by a typical mutual fund is equal to h = m

(1−λ

λ

). Thus,

h is proportional to m, and both affect the denominator of (14), albeit in different directions.We also note that a similar analysis can be performed for the total amount of money, mαP (α),that is spent by a typical household on investment management services.

11This would give the household sector a portfolio that is identical (in terms of payoffs) to the one that theywould choose if they were able to observe the signals themselves.

12This follows from the simple observation that in equilibrium the ith fund manager will always choose ρi

such that ρi(1− ρi/h) > 0. Otherwise his total fee will be negative.

14

3.3 Equilibrium information acquisition

While all of our previous results treated λ as a parameter, we now turn to the problem ofendogenizing this quantity (i.e., we solve for λ∗). The next proposition states the main resultin this section.

Proposition 3. The equilibrium fraction of informed agents is given by the value of λ thatsolves the nonlinear equation

e−τ(f(α)−c)

√1F

= eτλ

(1−λ)f(α)

√1D

; (21)

where f(α) is expression (17) evaluated at α, F = var(X|Yi, Px)−1, and

D = var(X|Px)−1 +2η

σ2ε

(1− α

α

)− η2

σ2ε m

(1− α

α

)2

. (22)

The equilibrium fraction of informed agents satisfies λ∗ ∈ (0, 1].

As illustrated by (21), the solution to our mutual fund model with endogenous informationacquisition is characterized by a single equation for λ that is parameterized by m, σ2

ε , σ2u, σ2

x,c, and τ .13 However, given that α itself is a nonlinear equation in λ, it does not seem possibleto solve (21) in closed-form. To shed some light on our results, we compare our value of λ∗ tothe value that would otherwise arise in the absence of a mutual fund sector. Of course, thislatter case is the one that is analyzed by Verrecchia (1982) and Diamond (1985) and henceour model can be viewed as a direct extension of their work. For example, since the agents inVerrecchia (1982) can only trade on their own account, it is possible none of the agents in hismodel will find it optimal to acquire private information. This situation would arise preciselywhen the cost to acquire information is extremely high. In our notation, this is when c takeson a very large value. In contrast, our model predicts that the corner solution λ∗ = 0 is neverpossible. The intuition behind our claim is that even if it is very costly to acquire information,the total fee of a typical fund manager grows without bound as λ → 0, since the number ofhouseholds he serves grows without bound, i.e. he gets to sell his services to a large numberof uninformed agents. Thus, there always exists a λ∗ > 0 such that the informed agents canrecoup their investment in private information even if c is large. While λ∗ = 0 never arises inour model, it is possible that all agents will become informed, i.e., λ∗ = 1 may occur. Thereason is that private information always has positive value to an agent. Thus, if the cost ofacquiring information is sufficiently low, every agent in our model may optimally decide topurchase a private signal. We note that this is the only case in which no mutual funds are

13In principle, the solution to (18) can be substituted into (21) for α. This produces a single equation (for λ)that drives the entire equilibrium.

15

established in equilibrium and that this corner solution does depend critically on our symmetryassumptions.

We can compare our model to that of Verrecchia (1982) by examining the informativenessof the risky asset price. Corollary 1 addressed this issue for a fixed (i.e., exogenous) value ofλ, but now we can analyze the case of an endogenous λ. We first remark that it might be thecase that the price is more informative without a mutual fund sector. We illustrate this factwith a numerical example. Suppose that σ2

x = σ2u = 1, σ2

ε = 0.6, τ = 3.1, m = 2, and c = 0.12.Using these parameter values, we find that λ∗ = 0.5 and h = 2, i.e., one-half of the agentsacquire private information and each informed agent establishes a mutual fund that serves twohouseholds. In equilibrium, we also find that b/d ≈ 0.502. On the other hand, if we do notallow agents to establish mutual funds, we find that the equilibrium fraction of informed agentsis 1, i.e., all agents become informed. In this case, the equilibrium price coefficients satisfyb/d ≈ 0.538. Since a higher value of b/d implies a lower value of var(X|Px), the risky assetprice in the equilibrium without mutual funds is more informative.

Exploring this topic a bit further, let us return to the expression for b/d in (13). Using thisexpression, it is easy to see that the risky asset price in the mutual fund equilibrium is the moreinformative of the two if λ∗ > αλV , where λV denotes the equilibrium fraction of informedagents when no mutual funds exist.14 For an exogenous value of λ, Corollary 1 showed thatthe risky asset price in the mutual fund equilibrium is always more informative. This effectis driven by the fact that α < 1. However, when λ is endogenously determined, the effectin Corollary 1 might be more than offset by the fact that different fractions of traders willbecome informed in the two equilibria (i.e., with and without mutual funds). In fact, as theabove example demonstrates, the value of λV might be sufficiently large such that the riskyasset price in the mutual fund equilibrium is actually less informative.

While the above discussion indicates that mutual funds may or may not produce a moreinformative price, we partially restore the appeal of Corollary 1 next.

Corollary 2. The risky asset price in the equilibrium in which there is a mutual fund sectoris more informative than the risky asset price in the equilibrium in which there are no mu-tual funds if any of the following parameters are sufficiently large: number of managers m,information gathering costs c, risk-aversion τ , and/or signal error variance σ2

ε .

The corollary establishes that as long as search frictions are low, or equivalently the groupsizes are sufficiently large, so that there is enough competition between the fund managers,then prices will reveal more information in the presence of a mutual fund sector than if allagents were to trade only on their own account. This result can be understood by noting

14An expression for the equilibrium fraction of informed traders in the absence of a mutual fund sectorcan be found in Lemma 3(c) of Diamond (1985). We can express this result using our notation as λV =

τσuσε

q1

C(τ)− σ2

εσ2

xwhere C(τ) ≡ e2τc − 1.

16

that for large m, the average equilibrium contingent fee is very close to the lower bound, i.e.,α ≈ 1/(1+0.5h). One can easily verify that λ∗ ≈ o(m−1) for m large, and thereby α ≈ o(m−2)for large m. Even though the fraction of informed agents is reduced as the group size grows, theeffect on the fees they charge is an other of magnitude larger, and for m large enough we canconclude the condition λ∗ > αλV is always satisfied. Thus, the introduction of a sufficientlylarge set of funds in the economy does lead to a more informative risky asset price. This showsthat a crucial element in our previous numerical example is the imperfect competition in themutual fund sector.

The proposition also establishes that high levels of risk-aversion, signal variance and/orinformation gathering costs will yield more informative prices in the equilibrium with mutualfunds. The rationale for this result stems from the fact that under any of these conditionsthe equilibrium without a mutual fund sector will not yield an equilibrium with informedagents.15 For example, if the risk-aversion is too high, agents will not find it profitable toinvest in gathering information, since they will heavily discount the risky stock market bet.On the other hand, a mutual fund sector can help spread the benefits of informed investingthroughout the members of each group, making it sufficiently profitable for at least some agentsto acquire information.

Our model also allows us to study how the endogenous formation of a mutual fund sectoraffects equilibrium risk premium in the stock market.16 Following the discussions in Cao (1999)and O’Hara (2003), we define the ex-ante risk premium on the stock as

µx − E [Px] =τµu

λvar(X|Px, Yi)−1 + (1− λ)var(X|Px)−1. (23)

Rather intuitively, if we fix the price informativeness, the risk premium is increasing in therisk aversion and the average aggregate supply of the risky stock. More generally, however,the risk premium also depends on the information that is revealed by price, and the relativemasses of informed and uninformed agents. The next corollary ties the previous result on priceinformativeness to the equilibrium risk premium.

Corollary 3. For m large enough, the risk premium on the risky asset is smaller with a mutualfund sector than in the equilibrium in which there are no mutual funds.

This result highlights how secondary assets, in our case mutual funds, can influence theequilibrium prices of primary assets. In particular, it establishes that a sufficiently competitivemutual fund sector will reduce the risk premium demanded by investors. The intuition forthis result stems from our previous discussion on price informativeness. From (23), recall that

15In particular if e2τc > 1 +σ2

xσ2

εwe have λV = 0.

16Comparative predictions concerning trading volume and price volatility follow in straightforward mannerfrom our previous expressions. We leave a thorough study of these issues, as well as a dynamic extension of themodel, for future work.

17

an increase in the conditional precision var(X|Px)−1 reduces the ex-ante risk-premia. Sinceλ tends to zero as m increases, it is straightforward to see that the dominant term in thedenominator of (23) is the one that includes var(X|Px)−1. In Corollary 2 we established thatthis quantity is larger than in the case of no mutual funds as long as m is large enough. Thus, itfollows that the risk premium is lower when the mutual fund sector is sufficiently competitive.

4 Extensions

While the above model relies on several simplifying assumptions (e.g., identical risk aversion,identical signal precisions for the informed agents, etc.), we illustrate below how the modelcan be generalized to accommodate certain types of heterogeneity. In some cases, our modelremains valid with little more than a change of notation. However, as we discuss in detailbelow, other types of heterogeneity present more of a challenge.

4.1 Agent heterogeneity

We first generalize the model in sections 3.1 and 3.2 in order to accommodate different riskaversion coefficients among the agents. We also allow the informed agents to have differentsignal precisions. Maintaining the assumption that each group contains h households andm fund managers, we denote the jth household’s risk aversion by τj for j = 1, . . . , h. For

notational purposes, we let τ =(∑h

k=11τk

)−1, i.e., 1/τ is the average risk tolerance of the

household sector. Likewise, we denote the ith fund manager’s risk aversion by τi and wedenote the ith manager’s signal precision by σ2

i , where i = 1, . . . ,m. Thus, in this case, the ithfund manager observes the private signal Yi = X + εi, where εi ∼ N(0, σ2

i ). While the groupsthemselves are identical to one another, we have altered our symmetric model in order toaccommodate heterogeneity within a group.17 For a fixed λ, the next proposition summarizesthe rational expectations equilibrium at the trading stage and the optimal fees that are chosenby each fund manager.

Proposition 4. Allowing for heterogeneity within a group, the mutual fund equilibrium hasthe following properties:

(i) the equilibrium price of the risky asset is given by (8) where the coefficients satisfy

b

d= λ

(1m

m∑k=1

1αkτkσ

2k

); (24)

17Note that this is not the most general case since we could also allow for heterogeneity across groups (includingperhaps a different size for each group). However, due to the notational complexity that arises in this moregeneral case, we maintain the assumption of identical groups and only allow heterogeneity within a group.

18

(ii) the optimal demand of household j for mutual fund i is φij = τ /τj;

(iii) the optimal contingent fees are given by αi = 1/(1+τiσ2i ρi), and satisfy αi ∈

((1 + 0.5(τi/τ))−1, 1

),

where the collection ρi solves the system of non-linear equations

1− 2τ ρiσ2i =

2τ ρi(1− τσ2i ρi)2

var(X|Px)−1 + 2τ∑m

k=1 ρk − τ2∑m

k=1 σ2kρ

2k

; i = 1, . . . ,m (25)

The equilibrium has a similar flavor to the one described in Proposition 1. For example,due to the heterogeneity, the relative price coefficients b/d now depend on the average tradingaggressiveness across managers in each group, which in turns depends on the contingent feeseach manager charges, as well as their risk-aversion and signal precision.18 From property (ii) ofthe proposition, we note that the heterogeneous risk aversion of the household sector producesa different allocation of mutual funds from that in the homogeneous case. Each household nowholds a fraction of each mutual fund that is equal to the household’s risk tolerance dividendby the average risk tolerance, i.e., unlike the symmetric case, different households now holddifferent fractions. However, the relative holdings remain unchanged – the jth household stillholds identical fractions of all m mutual funds. This can easily be seen by noting that the jthhousehold’s demand for fund i does not explicitly depend on i.

We remark that the optimal fees cannot be solved analytically in the case of heterogeneoussignal precisions for each fund manager. Nonetheless, we do establish that all fund managersfind it optimal to establish mutual funds and offer investment management services to thehousehold sector, i.e. αi < 1. In general, the optimal contingent fees will depend on both therisk-aversion and the signal precision of a manager. When managers have similar risk-aversion,the following corollary yields an intuitive result concerning the cross-sectional relationshipbetween a manager’s information quality and her optimal contingent fee.19

Corollary 4. Suppose that all agents have identical risk aversion. Then the manager withthe highest information precision charges the highest contingent fee, the manager with the nexthighest information precision charges the next highest contingent fee, and so forth.

18Recall that the trading strategy of the ith manager is of the form γi = (1/(αiτiσ2i ))Yi + q(Px), where q(Px)

is a linear function of Px. The quantity 1/(αiτiσ2i ), which is the coefficient on Yi, is a measure of how aggressive

the manager trades with respect to his private signal.19To proof the corollary, note that the nonlinear system of equations in Proposition 4 implies, under the

conditions of the Corollary, thatσ2

k

σ2i

=t (αk)

t (αi)(26)

where the function t (z) is

t (z) =

`1−z

z

´ ˆ1−

`1−z

z

´˜2ˆ1− 2

`1−z

z

´˜Noting that dt(z)

dz< 0, we conclude from expression (26) that σ2

k > σ2i implies that αk < αi. We can therefore

rank the agents according to their information precisions and this ranking will coincide with an otherwiseidentical ranking that is based on the agents’ contingent fees.

19

While there are other related topics for investigation, due to the complexity of the modelwith full heterogeneity we only mention (and do not pursue) these topics here. For example,it would be interesting to examine the information acquisition stage in a setting in whichagents have heterogeneous risk aversion and have the ability to purchase signals with differentprecisions at different costs. Second, it would be interesting to investigate an equilibriumin which two sets of households have access to overlapping collections of mutual funds. Forexample, suppose that one household has access to a collection of m1 mutual funds while asecond household has access to a partially overlapping collection of m2 mutual funds. In thiscase, the households’ equilibrium holdings of the mutual funds do not reduce to the optimal risksharing holdings that are given in Proposition 4. Instead, a system of nonlinear equations mustbe solved in order to obtain the households’ equilibrium mutual fund holdings.20 Moreover,one could accommodate heterogenous groups, say with different number of managers andhouseholds, along the lines of Proposition 4. Equilibrium price coefficients would then dependon average across aggregate group trading aggressiveness. We note that our characterization(25) of the optimal fees for each fund manager will hold in this model with heterogeneousgroups, with only var(X|Px) having a different functional form.

4.2 Discussion

Our grouping procedure is a parsimonious way to capture the idea that there are some frictionsthat prevent households (and managers) from contracting with an infinite number of funds(households). We note that our approach is limited in the sense that if one were to formallymodel a search game we would expect our parameter m to be endogenized. This would alsoaddress the potential integer problems that arise for h in the current setup of our model. Evenwithin our current framework, we note that several modifications to the grouping procedureare possible. For example, rather than exogenously fix m, we could instead impose a constrainton the overall group size, i.e., we could fix m + h and then allow λ∗ to determine m and h

individually. However, if we impose an integer constraint on m, the two models do producedifferent results.21 One could actually eliminate the grouping procedure and focus on a finite-agent model and each fund is held by all agents in equilibrium, which would be a directgeneralization of Admati and Pfleiderer (1990). This type of model does not yield itself toanalytical solutions, since prices depend on each informed agent’s signal, and fund managersinternalize their effect on price informativeness when deciding on the fees they will charge.22

20The same tractability problem arises if the household sector has heterogeneous information precision, i.e. ifif the households happened to be endowed with heterogeneous prior information.

21If we ignore integer problems, this variation of the model is mathematically equivalent to the one that wediscuss in the paper.

22The rational expectations equilibrium essentially reduces to that discussed on pages 479-485 of Hellwig(1980) if we replace the informed agent’s risk aversion τi with his effective risk aversion αiτi. Since the equi-librium at the trading stage does not allow for a closed-form solution, the fee setting stage and the endogenousinformation acquisition analysis cannot be solved analytically.

20

Although it is more challenging to characterize the equilibrium in this case, numerical analysissuggests that the results and the intuition from the body of the paper is robust: a sufficientlycompetitive stock market yields more informative prices and a lower equilibrium risk premium.

We impose two constraints on the fund managers that deserve some comments. By as-sumption, we do not allow an informed agent to establish a mutual fund and trade for hisown account. Accommodating this feature is possible but it results in a model in which onlythe agent’s total stock market exposure is identified in equilibrium. Furthermore, the agent isindifferent as to how this exposure is divided between the mutual fund and his own account.23

Under standard assumptions, this yields a model that is equivalent to the one that we analyze.To close the model we could assume, as in the agency literature, that the agent acts in thebest interest of the principal when indifferent. Also, we restrict fund managers from holdingpositions in other funds, either through their own mutual fund trades or in their own account.This restriction is due to tractability: the equilibrium fund prices will be characterized by anon-linear system when agents with heterogenous portfolios hold funds. Nonetheless our re-sults for large m are robust to this restriction, since one can show that for m large equilibriumprices are arbitrarily close to the prices that we characterize. Rather intuitively as group sizeincreases intra-fund holdings tend to zero and the equilibrium converges to the one in thepaper.

Lastly, our model can be extended by considering what Admati and Pfleiderer (1990) referto as “general pricing schemes”, i.e., contracts that contain both a proportional fee (as in ourmodel) and a fixed fee (e.g., a front-end load). Essentially, given that fund prices are constantin equilibrium, our model is equivalent to one in which fund managers and households contractprior to the trading stage of the model. Expanding the contracting space by allowing for afixed fee component would make the informed agents better off by increasing the returns toinformation acquisition. We mention this extension because it is as tractable as our ownmodel. One can show that the optimal fees in this case achieve first-best risk sharing andthe information sellers use the fixed fee to extract consumer surplus.24 Furthermore, one canverify that our results on price informativeness and the equity risk premium are robust to thisextension.

23We remark that this assumption is crucially dependent on the competitive aspect of the equilibria we study.Strategic or reliability issues, which we assume away, could generate a trade-off for trades inside and outsidethe fund.

24In particular, the optimal fees are αi = 1/(1+h) = λ/(λ+(1−λ)m) in our symmetric model. The optimalfixed fee can be obtained by equating the expected utilities of the households with and without fund i, and isreadily obtainable from our expressions in the appendix.

21

5 Concluding remarks

We study the fund formation decision of rational informed investors in order to offer an ex-planation for why we observe so many mutual funds in practice. Our findings indicate thatthe creation of mutual funds in equilibrium is more the rule than the exception. As opposedto trading solely for their own accounts, informed investors in our model are always better offby establishing mutual funds and marketing their investment strategies to the public. House-holds are also better off when mutual funds are established. Since the fund managers haveprivate information, a stock market bet that is achieved by purchasing a mutual fund is verydifferent than a direct stock market bet. Essentially, mutual funds offer a new asset class tothe investing public. Our model makes this statement precise.

We endogenize the information acquisition decision of the informed agents in order to derivethe equilibrium size of the mutual fund industry. This also allows us to compare our mutualfund model to the case in which a mutual fund sector does not exist (see, e.g., Verrecchia,1982; Diamond, 1985), and to make precise statements about the equilibrium relationshipbetween the level of information acquisition and the model’s economic primitives. It alsofacilitates our discussion of information revelation – namely, we outline conditions under whichour mutual fund model produces a more informative risky asset price. These conditions aredirectly related to the trading aggressiveness of the fund managers (i.e., the informed agents’effective risk aversion is lower if they establish mutual funds). Lastly, we use our model tostudy the equilibrium relationship between information precision, contingent management fees,total fees, and mutual fund size (e.g., assets under management).

There are several natural areas for future research. For example, given the multi-assetnature of actual financial markets, generalizing the model to the case of multiple risky assetsappears to be a worthy topic. In this case, agents might purchase signals on only a subset ofthe available risky assets and therefore establish specialized mutual funds. This type of modelwould facilitate an examination of the relationship between fund specialization (i.e., mutualfund styles) and contingent fees. While this topic has received considerable attention in theempirical literature, there are few theoretical results in this area. A second natural area is tostudy the existence and formation of mutual fund families. Essentially, one could examine thecase in which managers share information and possibly collude in the setting of their fees. Thispotentially could explain the structure of the mutual fund industry, i.e., in practice we tendto observe large fund companies (such as Fidelity and Vanguard) that each manage an arrayof mutual funds. This type of analysis could potentially shed some light onto the equilibriumrelationship between the size of a mutual fund family and their contingent fees, among otheritems.

22

Appendix

Proof of Proposition 1: Using the standard properties of Gaussian random variables, itis straightforward to evaluate the conditional expectation in expression (3). Solving the ithmanager’s problem at date 1 produces the familiar mean-variance expression in (10).

To evaluate the jth household’s problem, we let V denote the variance-covariance matrixof (X, ε1, . . . , εm) conditional on price. Further define γi ≡ riYi + qiPs, where ri and qi arereadily obtainable from (10). The household’s problem in expression (4) can be expressed as

maxθ,φ

τM(θ, φ) +12

log(|B(φ)|)− τ2

2g(θ, φ)>B(φ)−1V g(θ, φ).

where

M(θ, φ) = θ(E [X|Px]− Px)−m∑

i=1

αiφiPi + (E [X|Px]− Px)

(m∑

i=1

(1− αi)φi [qiPx + riE [Yi|Px]]

)B(φ) = I + 2τV A(φ);

g(θ, φ) =

θ + (E [X|Px]− Px)

∑mi=1 φi(1− αi)ri +

∑mi=1(1− αi)φi[riE [Yi|Px] + qiPx]

(E [X|Px]− Px)r1(1− α1)φ1

...(E [X|Px]− Px)rm(1− αm)φm

Aij(φ) ≡

∑m

i=1 ri(1− αi)φi : i = j = 1;ri(1− αi)φi/2 : i = 1, j 6= 1 or j = 1, i 6= 1;

0 : otherwise

where I ∈ R(m+1)×(m+1) denotes the identity matrix, g(θ, φ) ∈ Rm+1, and B(φ), A(φ) ∈R(m+1)×(m+1). Let C ≡ B(φ)−1V . Then the first-order conditions yield

E [X|Px]− Px

τ=

m+1∑j=1

C1j(φ)gj(φ); (27)

Mφi+ trace(C(φ)Aφi

) = τ(g>φi

− τg>CAφi

)Cg, i = 1, . . . ,m; (28)

where Hφidenotes the derivative of the function H with respect to φi. Some basic manipula-

tions of (27) and (28) yield

Mφi+ trace(C(φ)Aφi

) = Pxqi(1− αi) + ri(1− αi)E [Yi|Px] ; i = 1, . . . ,m;

so thatαiPi = trace(C(φ)Aφi

) = (1− αi)ri (C11 + C1,i+1) .

23

The standard matrix inversion lemma and some simple algebra yields (14).

Given the symmetry of the model, market clearing in the mutual fund sector implies themutual fund demands are φij = 1

h . Substituting these equilibrium demands into (27) yields,after some simple manipulations, the optimal stock demands in (11).

We extract the price coefficients a, b and d from the market clearing condition for therisky asset (6). Using the previous expressions for the optimal demands of both informedand uninformed agents in (6) yields three equilibrium conditions for a, b and d. Note that thisverifies the functional form for Px conjectured in (8). Solving this system yields the expressionsin (12)-(13). This completes the proof.

Proof of Proposition 2: Taking the fees of the other m−1 managers as given and noting thatthe price coefficients b and d are independent of any individual manager’s fees, the first-ordercondition for (17) gives the equation

1− 2ρi

h−(

2hσ2

ε

)ρi

(1− ρi

h

)2(var(X|Px)−1 + 2

h

∑mk=1 ρk − 1

h2σ2ε

∑mk=1 ρ2

k

) = 0 (29)

where ρi ≡ (1−αi)/αi. Substituting for Pi and bd , letting α1 = α2 = . . . = αm ≡ α, and noting

that α = α, the above expression reduces to

2(

1−αασ2

ε

) (1− 1

h

(1−α

α

))2[1σ2

x+ 1

σ2u

(λ

τασ2ε

)2+ 2m

hσ2ε

(1−α

α

)− m

h2σ2ε

(1−α

α

)2] = 1− 2h

(1− α

α

)(30)

Some simple algebraic manipulation produces the cubic equation that is presented in Proposi-tion 2 with coefficients

k0 =2η3(m− 1)

m3σ2ε

− 2ηλ2

τ2mσ2uσ4

ε

k1 =λ2

τ2σ2uσ4

ε

+2ηλ2

τ2mσ2uσ4

ε

− (5m− 4)η2

m2σ2ε

− 6η3(m− 1)m3σ2

ε

k2 =2η(m− 1)

mσ2ε

− 2η

mσ2x

+2(5m− 4)η2

m2σ2ε

+6η3(m− 1)

m3σ2ε

k3 =1σ2

x

− 2η(m− 1)mσ2

ε

+2η

mσ2x

− (5m− 4)η2

m2σ2ε

− 2η3(m− 1)m3σ2

ε

where η ≡ λ1−λ .

The second-order condition for a maximum is easily verified by differentiating the left-hand side of (29) with respect to αi. It is straightforward to check that the sign of the secondderivative of αiPi is the same as the sign of (1/τ − αiPi/h). Since the first-order condition in

24

(30) implies that ταiPi < h, the second-order condition is verified.

Next, recalling the definitions of k0, k1, k2, and k3, let us define the function G(α) as

G(α) = k3α3 + k2α

2 + k1α + k0

It is straightforward to check that G(1) < 0 and that G(1/(1 + h/2)) > 0. Since G(α) isa continuous function of α, this shows that a solution in the interval (1/(1 + h/2), 1) exists.Moreover, it can be shown that dG

dα < 0 for every point that lies in the interval (1/(1+h/2), 1).This immediately leads to uniqueness.

Lastly, note that in general G(α) has three roots. For some parameter values G(α) hastwo complex roots and one real root, while for other parameter values G(α) has three realroots. Obviously, in the former case, we take the sole real root as the solution to G(α) andthis real root lies in the interval (1/(1 + h/2), 1). In the case of three real roots, one of theroots is negative and we therefore discard this root. While the remaining two roots may liein [0, 1], only one of these two roots produces a non-negative total fee. Hence, by a process ofelimination we conclude that the unique real root that lies in (1/(1+h/2), 1) is the economicallymeaningful root.

Proof of Proposition 3: We let µx = µu = 0 in the expressions below. The general casefollows with minor modifications. The ith manager’s indirect utility function at date 1 is

E[−e−ταi(Pi+γi(X−Px))+τc

∣∣∣Px, Yi

]= −e−τ(αiPi−c) e

− (E[X|Px,Yi]−Px)2

2var(X|Px,Yi) (31)

where E [X|Px, Yi] and var(X|Px, Yi) are given in (15)-(16). Taking the expectation of bothsides of (31) produces the manager’s date 0 indirect utility function, i.e.,

E[−e−ταi(Pi+γi(X−Px))+τc

]= −e−τ(αiPi−c)

√1

FΛ(32)

whereΛ = d2σ2

u + (b− 1)2 σ2x. (33)

Some tedious algebra reveals that the date 1 indirect utility of household j is given by

E[−e−τWj

∣∣∣Px

]= −

√S

De− (E[X|Px]−Px)2

2var(X|Px)+ τ

h

Pmk=1 αkPk (34)

where Wj denotes the household’s optimal wealth, S ≡ var(X|Px)−1. Taking the expectation

25

of both sides of (34) produces the household’s date 0 indirect utility function, i.e.,

E[−e−τWj

]= −

√1

DΛe

τh

Pmk=1 αkPk (35)

where Λ is given in (33). Recalling that the manager’s date 0 indirect utility function is givenby (32), we can now equate (32) and (35) in order to identify λ∗. This completes the proof.

Proof of Proposition 4: The demand functions for the informed traders are given by theusual mean-variance form (10). The uninformed agents’ optimization problem can be solvedby the same approach as in Proposition 1. In particular, it is straightforward to check thatthe first-order conditions to household j optimization problem yield

αiPi =ρi

(1− φijσ

2i ρi

)var(X|Px)−1 + 2

∑mk=1 φkjρk −

∑mk=1 φ2

kjσ2kρ

2k

; (36)

where ρi ≡ (1 − αi)/(αiσ2i τi). An immediate calculation shows that the demands φij both

satisfy the market clearing condition (7) and the above first-order conditions. Using the optimaldemands for the informed agents and the market clearing condition for the stock one can easilyobtain (24). Finally, by the same arguments as in Proposition 3 we have that each fund managersolves maxαi αiPi, which yields the first-order conditions (25). This completes the proof.

26

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