mutual fund flows and performance in rational markets · mutual fund flows and performance in...

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[Journal of Political Economy, 2004, vol. 112, no. 6]� 2004 by The University of Chicago. All rights reserved. 0022-3808/2004/11206-0003$10.00

Mutual Fund Flows and Performance in RationalMarkets

Jonathan B. BerkUniversity of California, Berkeley and National Bureau of Economic Research

Richard C. GreenCarnegie Mellon University

We derive a parsimonious rational model of active portfolio manage-ment that reproduces many regularities widely regarded as anomalous.Fund flows rationally respond to past performance in the model eventhough performance is not persistent and investments with active man-agers do not outperform passive benchmarks on average. The lack ofpersistence in returns does not imply that differential ability acrossmanagers is nonexistent or unrewarded or that gathering informationabout performance is socially wasteful. The model can quantitativelyreproduce many salient features in the data. The flow-performancerelationship is consistent with high average levels of skills and consid-erable heterogeneity across managers.

One of the central mysteries facing financial economics is why financialintermediaries appear to be so highly rewarded, despite the apparentfierce competition between them and the uncertainty about whether

We would like to thank Avi Bick, Jennifer Carpenter, Hsiu-lang Chen, Gian Luca Cle-menti, John Cochrane (the editor), Jeff Coles, Josh Coval, Ravi Jagannathan, BurtonHollifield, Dwight Jaffee, Finn Kydland, Mike Lemmon, David Musto, Walter Novaes,George Pennacchi, Raj Singh, Peter Tufano, Uzi Yoeli, Lu Zheng, and two anonymousreferees for their helpful comments and suggestions. Seminar participants at several con-ferences and universities also provided useful feedback and suggestions. We also gratefullyacknowledge financial support from the Q Group, the National Science Foundation(Berk), and the International Center for Financial Asset Management and Engineeringthrough the 2003 FAME Research Prize.

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they add value through their activities. Research into mutual fund per-formance has provided evidence that deepens this puzzle. Since Jensen(1968), studies have shown little evidence that mutual fund managersoutperform passive benchmarks. Recent work has produced several ad-ditional findings. The relative performance of mutual fund managersappears to be largely unpredictable from past relative performance.1

Nevertheless, mutual fund investors chase performance. Flows into andout of mutual funds are strongly related to lagged measures of excessreturns (see Chevalier and Ellison 1997; Sirri and Tufano 1998).

The evidence that performance does not persist is widely regardedas implying that superior performance is attributable to luck rather thanto differential ability across managers.2 This implication, were it true,would be troubling from an economic point of view. If all performanceis due to luck, there should be no reason to reward it. Yet, in reality,managers appear to reap rich rewards from superior past performance.Together these findings have led researchers to raise questions aboutthe rationality of investors who place money with active managers despitetheir apparent inability to outperform passive strategies and who appearto devote considerable resources to evaluating past performance of man-agers (see, e.g., Gruber 1996; Daniel et al. 1997; Bollen and Busse, inpress). In the face of this evidence many researchers have concludedthat a consistent explanation of these regularities is impossible withoutappealing to behavioral arguments that depend on irrationality or toelaborate theories based on asymmetric information or moral hazard.One thing that has been missing from this debate is a clear delineationof what a rational model, with no moral hazard or asymmetric infor-mation, implies about flows and performance. Before appealing to theseadditional effects, we believe that it makes sense to first establish whichbehaviors in the data are qualitatively and quantitatively consistent withmore direct explanations.

Our simple model of active portfolio management and fund flowsprovides a natural benchmark against which to evaluate observed re-turns, flows, and performance outcomes in this important sector of thefinancial services industry. Using this model, we show that the effectsdiscussed above are generally consistent with a rational and competitivemarket for capital investment and with rational, self-interested choicesby fund managers who have differential ability to generate abnormalreturns.

The model combines three elements. There is competitive provision

1 While some controversial evidence of persistence does exist (see Gruber 1996; Carhart1997; Zheng 1999; Bollen and Busse, in press), it is concentrated in low-liquidity sectorsor at shorter horizons.

2 Indeed, various researchers have interpreted this fact as evidence for “market effi-ciency” (see, e.g., Malkiel 1995, p. 571; Ross, Westerfield, and Jaffe 2002, p. 353).

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of capital by investors to mutual funds. There is differential ability togenerate high average returns across managers but decreasing returnsto scale in deploying these abilities. Finally, there is learning aboutmanagerial ability from past returns. Some of these features have beenexamined in other models. Our contribution is to show that togetherthey can reproduce the salient features of the empirical evidence asequilibrium outcomes in a rational model.

In our model, investments with active managers do not outperformpassive benchmarks because investors competitively supply funds tomanagers and there are decreasing returns for managers in deployingtheir superior ability. Managers increase the size of their funds, andtheir own compensation, to the point at which expected returns toinvestors are competitive going forward. The failure of managers as agroup to outperform passive benchmarks does not imply that they lackskill. Furthermore, the lack of persistence does not imply that differ-ential ability across managers is unrewarded, that gathering informationabout performance is socially wasteful, or that chasing performance ispointless. It merely implies that the provision of capital by investors tothe mutual fund industry is competitive.

Performance is not persistent in the model precisely because investorschase performance and make full, rational, use of information aboutfunds’ histories in doing so. High performance is rationally interpretedby investors as evidence of the manager’s superior ability. New moneyflows to the fund to the point at which expected excess returns goingforward are competitive. This process necessarily implies that investorscannot expect to make positive excess returns, so superior performancecannot be predictable. The response of fund flows to performance issimply evidence that capital flows to investments in which it is mostproductive.

The logic underlying these results is similar to that of standard modelsof corporation finance. Capital is supplied with perfect elasticity to man-agers whose differential ability allows them to identify positive net pres-ent value opportunities. Since these opportunities, or the ability to iden-tify them, are the resource that is ultimately in scarce supply, theeconomic rents thus created flow through to the managers who createthem, not to the investors who invest in them. In corporate finance,new money is raised to fund investments on competitive terms. Investorssupplying these funds do not earn high expected returns, but this isnot evidence that managers are all the same or that firms lack valuableinvestment opportunities. Similarly, in our model, the competitive re-turns investors earn do not tell us that managers lack ability. What isunique about mutual funds is the equilibrating mechanism. In corporatefinance models the price of the firm’s securities adjusts to ensure thatreturns going forward are competitive. With open-ended mutual funds


this cannot happen. The adjustment must come through quantities, orfund flows, rather than price.

Our model is related to several recent papers, though none of themare directly aimed at reconciling the lack of return persistence with theresponsiveness of flows to past performance. Bernhardt, Davies, andWestbrook (2002) study short-term return persistence in a model inwhich privately informed managers trade to maximize funds under man-agement. The response of funds to performance in their model, how-ever, is exogenous. In our model, the flow-performance relationship isendogenous and consistent with no persistence in performance. Ippolito(1992) and Lynch and Musto (2003) endogenize the flow of funds. Asin our paper, investors and managers learn about managers’ abilitiesand the profitability of their strategies from past returns. However, inboth of these models, differences in ability lead to persistent differencesin performance. We show that rational learning and strong response offlows to performance can be consistent with no persistence in perfor-mance. Nanda, Narayanan, and Warther (2000) develop a three-datemodel with heterogeneous managerial ability. They use the model toderive endogenous heterogeneous fee structures involving loads as man-agers compete for clients with different expected liquidity needs. Intheir model, managerial ability is known, so there is no role for learningabout managerial ability and the response of the flow of funds to it.These are central concerns in our paper. Finally, in a corporate setting,Holmstrom (1999) considers a dynamic multiperiod moral hazard prob-lem in which learning about managerial ability is very similar to that inour model. He characterizes the resulting optimal contract. Our goalsare very different. There is no moral hazard or asymmetric informationin our model, so the simple compensation scheme we consider is op-timal, funds are allocated efficiently across managers, and all outcomesare first-best.

The paper is organized as follows. In Section I we lay out the elementsof the model, and in Section II we develop its implications for the flowof funds and fund life cycles. Section III derives the flow of funds andperformance relationship for a cross section of funds with differenthistories. In Section IV we parameterize the model and evaluate itsquantitative implications. Section V presents conclusions.

I. The Model

We begin by describing the simplest version of our model, which pro-duces the two central behaviors the literature has struggled to reconcile:flows that are responsive to performance and performance that is notpersistent. We shall then elaborate on the model to show that theseoutcomes are consistent not only with investor rationality but also with


efficient provision of portfolio management services. For convenience,we include a summary of the important notation in the Appendix (tableA1).

All participants in the model are symmetrically informed. Funds differin their managers’ ability to generate expected returns in excess of thoseprovided by a passive benchmark—an alternative investment opportu-nity available to all investors with the same risk as the manager’s port-folio. The model is partial equilibrium. Managers’ actions do not affectthe benchmark returns, and we do not model the source of successfulmanagers’ abilities. A manager’s ability to beat this benchmark is un-known to both the manager and investors, who learn about this abilityby observing the history of the managed portfolio’s returns. Let R pt

denote the return, in excess of the passive benchmark, on thea � et

actively managed funds, without costs and fees. This is not the returnactually earned and paid out by the fund, which is net of costs and fees(see below). The parameter a is the source of differential ability acrossmanagers. The error term, , is normally distributed with mean zeroet

and variance and is independently distributed through time. We2j

further assume that this uncertainty is idiosyncratic to the manager: byinvesting with a large number of these managers, investors can diversifyaway this risk. Denote the precision of this uncertainty as .2q p 1/jParticipants learn about a by observing the realized excess returns themanager produces. This learning is the source of the relationship be-tween performance and the flow of funds.

To achieve high returns, management must identify undervalued se-curities and trade to exploit this knowledge without moving the priceadversely. To do this, managers must expend resources and pay bid-askspreads that diminish the return available to pay out to investors. Weassume that these costs are independent of ability and are increasingand convex in the amount of funds under active management. Denotethe costs incurred when actively managing a fund of size as , andq C(q )t t

assume, for all , that , , and , with′ ′′q ≥ 0 C(q) ≥ 0 C (q) 1 0 C (q) 1 0and . These assumptions capture the notion′C(0) p 0 lim C (q) p �qr�

that with a sufficiently large fund, a manager will spread his information-gathering activities too thin or that large trades will be associated witha larger price impact and higher execution costs.

Assume that the fund managers are simply paid a fixed managementfee, f, expressed as a fraction of the assets under management, . (Weqt

shall relax this assumption shortly.) Managers accordingly accept andinvest all the funds investors are willing to allocate to them. The amountinvestors will invest in the fund depends on their assessment of themanagers’ ability and on the costs they know managers face in expand-ing the fund’s scale.


The excess total payout to investors over what would be earned onthe passive benchmark is

TP p q R � C(q ) � q f.t�1 t t�1 t t

Let denote the excess return over the benchmark that investors inrt

the fund receive in period t. Then

TP C(q )t�1 tr p p R � � ft�1 t�1q qt t

p R � c(q ), (1)t�1 t

where

C(q )tc(q ) { � f (2)t qt

denotes the unit cost associated with investing in the actively managedfund. The return corresponds to the return empirically observed.rt

At the birth of the fund the participants’ prior about the ability ofthe fund’s management is that a is normally distributed with mean

and variance . Since all participants are assumed to have rational2f h0

expectations, this is also the distribution of skills across new funds. Theprecision of the prior will be denoted . Investors (and the2g p 1/h

manager) update their posteriors on the basis of the history of observedreturns as Bayesians. The costs faced by the funds are common knowl-edge, and the resources each fund has under management are observ-able. Thus, by observing the history , investors can infer thet{r , q }s s sp0

history . Let the posterior mean of management ability at time tt{R }s sp0

be denoted

f { E(R FR , … , R ).t t�1 1 t

The timing convention is as follows. The fund enters period t withfunds under management and an estimate of managerial ability,qt�1

. Managers and investors observe (from which they can infer )f r Rt�1 t t

and update their estimate of the manager’s ability by calculating .ft

Then capital flows into or out of the fund to determine . In the nextqt

section we shall calculate this flow of funds explicitly.

II. The Flow of Funds and Performance

We assume that investors supply capital with infinite elasticity to fundsthat have positive excess expected returns. This can be justified as longas is idiosyncratic risk: by diversifying across funds with positive excesset

expected returns, investors can achieve the average excess expectedreturn with certainty. Similarly, they remove all funds from any fund


that has a negative excess expected return. At each point in time, then,funds flow to and from each fund so that the expected excess returnto investing in any surviving fund is zero:

E (r ) p 0. (3)t t�1

As in any equilibrium with perfect competition, the marginal return onthe last dollar invested must be zero. In this case, however, since allinvestors in open-ended mutual funds earn the same return, all investorsearn zero expected excess return in equilibrium. Condition (3) clearlyimplies that there is no predictability or persistence in funds’ excessreturns and that the average excess return of all managers will be zero,regardless of their overall level of skill. Finally, we shall now show thatit implies that flows into and out of the fund respond to performance.

Taking expectations of both sides of (1), and requiring expectedexcess returns of zero as in (3), gives

C(q )tf p c(q ) p � f. (4)t t qt

As changes, changes to ensure that this equation is satisfied at allf qt t

points in time. The following proposition derives the flow of fundsrelation.

Proposition 1. For any fund in operation at dates and t, thet � 1evolution of and the change in the amount of money under man-ft

agement at any date, as a function of the prior performance, are givenby the solutions to

qf p f � r (5)t t�1 t

g � tq

and

qc(q ) p c(q ) � r . (6)t t�1 t

g � tq

Proof. It is straightforward to show from DeGroot (1970, p. 167, the-orem 1) that the mean of investors’ posteriors will satisfy the followingrecursion:

g � (t � 1)q qf p f � R , (7)t t�1 t

g � tq g � tq

where is the mean of the initial prior. Next, use (4), evaluated atf0

both t and , to substitute for and in this expression. Finally,t � 1 f ft t�1

use the fact that the realized return is given by . Theser p R � c(q )t t t�1

substitutions yield equation (6). Appealing, again, to (4) gives (5).Q.E.D.


Equation (6) reproduces the observed relationship between perfor-mance and fund flows. Note that

1 C(q)′ ′c (q) p C (q) � 1 0,[ ]q q

so good performance (a positive realization of ) results in an inflowrt

of funds ( ).3 Similarly, bad performance results in an outflow ofq 1 qt t�1

funds. A smaller t increases the weight on , so flows to younger fundsrt

respond much more dramatically to performance than flows to moremature funds.

A. Efficient Provision of Portfolio Management Services

The assumption that fees are fixed may appear restrictive. A managercould, presumably, raise his compensation by choosing the fee optimallyin each period, and in this way always invest efficiently. We now showthat if managers can expand the fund by investing a portion of it inthe passive benchmark (i.e., “closet indexing”4), efficient outcomes canbe achieved with a proportional fee that does not change over time andis not contingent on performance.

We begin by first allowing managers to optimally choose the fee theycharge at each point in time, . Every manager is constrained by theft

investor participation constraint (4): investors in the fund must earn(at least) zero excess returns. Rewriting (4) now gives

q f � C(q ) p q f . (8)t t t t t

The manager’s objective is to maximize compensation—the right-handside of (8). He does this by first determining what level of maximizesqt

the left-hand side of (8). This choice will set the expected excess returnon the marginal dollar equal to the marginal cost of expansion:

′f p C (q ). (9)t t

3 We know by the mean value theorem that for any q there exists a point , ,¯ ¯q 0 ≤ q ≤ qsuch that . Strict convexity of and the assumption′ ¯[C(q) � C(0)]/q p C (q) C(7) C(0) p

then imply .′ ′ ¯0 C (q) 1 C (q) p C(q)/q4 While used in the industry and financial press, the expression “closet indexing” should

be interpreted with caution in this setting. There is no moral hazard in the model and,thus, no cost or inefficiency associated with the indexing. In the model, nothing is hiddenin the “closet.” Investing in the passive benchmark serves simply as a means of compen-sating managers for value they are creating.


Let be the solution of this equation.5 The manager then selectsq*(f)t t

a fee to ensure that investors choose to invest (by using [8]):q*(f)t t

C(q*(f))t tf * p f � . (10)t t q*(f)t t

This compensation scheme allows managerial compensation to de-pend on history, but it is not contingent on current performance. Thefee, , depends on a conditional expectation, so it is implicitly a functionft

of , , but it is not a function of . It is nevertheless efficient. Ther s ! t rs t

left-hand side of (8) is the value the manager adds. Since this contractmaximizes this value and then delivers it all to the manager, no othercontract exists that could make the manager better off. Investors areconstrained by competition to always make zero excess expected returnsin any case. The contract is also Pareto efficient: no other contract couldmake both investors and the manager better off.

If we allow managers to index part of their funds, we can use anyfixed fee, , to implement exactly the same compensation to thef 1 0manager and returns to investors. Assume that of the total amount ofmoney under management, , the manager always chooses to investqt

in active management, and let denote the remaining amountq*(f) qt t It

of money that the manager chooses to index. Investors still pay themanagement fee, f, on this money, but because it is not actively managed,it does not earn excess returns and does not contribute to the costs,

. Under these conditions the return investors earn on their moneyC(7)is

r p h(q )R � c(q ), (11)t�1 t t�1 t

where is the proportion of funds under active management,h(q )t, and the unit cost function is now given asq*(f)/qt t t

C(q*(f))t tc(q ) p � f. (12)t qt

Taking expectations of (11) and imposing the participation condition,(3), gives

h(q )f p c(q ). (13)t t t

Writing out and explicitly givesc(q ) h(q )t t

q*(f)f � C(q*(f)) p [q*(f) � q ]f. (14)t t t t t t t It

5 Existence of a solution is guaranteed under our assumptions on .C(q)


The left-hand side of (14) is identical to the left-hand side of (8), sothe right-hand sides must also be the same:

q*(f)f * p [q*(f) � q ]f. (15)t t t t t It

Because investors compete away the economic rents, managers earn thesame compensation under either contract. Since the contract underwhich the manager is allowed to set his fee is efficient, the fixed-feecontract must be efficient as well. Solving (15) for gives the amountqIt

of money the manager chooses to manage passively:

f * � ftq p q*(f) . (16)It t t f

If , then , implying that the manager borrows or short sellsf 1 f * q ! 0t It

the benchmark portfolio. As we shall discuss further in the next section,by simply choosing a fee , a manager can ensure that he will neverf ≤ f *tneed to short sell, and this involves no cost in terms of the manager’stotal compensation.

Adding the active and passive part of the manager’s portfolio together(using [14]) gives the total funds under management using the fixed-fee contract:

q*(f)f � C(q*(f))t t t t tq p q*(f) � q p . (17)t t t It f

In light of the revenue equivalence between the two contracts, wecould choose either in our subsequent analysis. We focus on the fixed-fee contract for several reasons. First, it comes closer to the institutionalsetting for retail mutual funds, which are the primary source of datafor empirical studies. Mutual funds show relatively little variation in fees,through time and across funds. As discussed in Christoffersen (2001),both historical practice and regulatory constraints limit the ability ofretail mutual funds to use performance-contingent fees. Our theorysuggests, instead, that rents can be collected through the flow of funds.Second, allowing for indexing in this way augments the responsivenessof fund flows to performance. Third, it leads to interesting and empir-ically realistic life cycle effects for funds. As funds survive, age, and grow,they will have an ever larger portion of their portfolio passively invested.They will exhibit less idiosyncratic volatility and lower attrition rates.These behaviors have often been attributed to an irrationally sluggishresponse by investors to mediocre performance and the opportunisticexploitation of it by fund managers. In our model, these behaviors ariseas a natural consequence of learning and of rents accruing to the talentsthat are their source.

From (11)–(13), application of the same steps that prove proposition


1 gives the following expressions for updating expectations about man-agerial ability when the fee is fixed and managers index part of theirfunds:

r qtf p f � (18)t t�1 ( )h(q ) g � tqt�1

and

c(q ) c(q ) r qt t�1 tp � . (19)( )h(q ) h(q ) h(q ) g � tqt t�1 t�1

These expressions are the basis for our analysis going forward. Notethat all the variables governing the flow of funds are observable toinvestors. There is no asymmetric information or moral hazard in themodel.

B. Entry and Exit of Funds

We now introduce assumptions that govern how funds enter and exit.This allows us to examine other behaviors that have been studied em-pirically for mutual funds, such as survival rates.

Suppose that managers incur known fixed costs of operation, denotedF, each period. These costs can be viewed as overhead, back-office ex-penses, and the opportunity cost of the manager’s time. Managers willchoose to shut down their funds when they cannot cover their fixedcosts. From (14), the manager’s compensation is , soq*(f)f � C(q*(f))t t t t t

the fund will be shut down whenever this expression is less than F. Letbe defined as the lowest expectation of management ability for whichf

the manager will still remain in business. It solves

¯ ¯ ¯q*(f)f � C(q*(f)) p F.t t

Funds shut down the first time . In each period a cohort of new¯f ! ft

managers enter, and their abilities are distributed according to the mar-ket’s initial prior. We shall also assume that when a manager goes intobusiness for the first time, he incurs additional costs he must recoverfrom his fees, so that when a cohort enters, .¯f 1 f0

To be consistent with the institutional setting for retail mutual funds,we shall assume that managers cannot borrow or short sell the bench-mark asset: . This rules out dynamic strategies in which managersq ≥ 0It

with low expected ability borrow in hopes of building or restoring areputation that would lead to high fees in the future. Managers canensure that the short-sale constraint will never bind by choosing a fee

. It is optimal from their perspective to make this choice. Sincef ≤ f *tmanagers extract all the surplus in the model, they bear the cost of any


inefficient outcomes, and a binding short-sale constraint would imposesuch costs. At any fee lower than , the manager’s compensation isf *tgiven by , the left-hand side of (14), which is inde-q*(f)f � C(q*(f))t t t t t

pendent of the fee. The particular fee chosen in the range is0 ! f ! f *ttherefore a matter of indifference to the manager. In our numericalimplementation, we set the fee equal to the highest level that will ensurethat the short-sale constraint will never bind over the range of outcomesin which the fund remains in business:

Ff p . (20)¯q*(f)t

At this fee, the fund goes out of business whenever . Define¯q ! q*(f)t t

as the lowest value of q for which the fund remains viable.¯q { q*(f)t

III. Cross-Sectional Distribution of Managerial Talent

So far we have analyzed the relation between performance and flowsin time series for a single manager. The empirical evidence concerningfund flows and survival rates, however, analyzes funds by age cohort onthe basis of the cross-sectional variation in returns across funds and thereaction of flows to these returns. In this section we derive the model’simplications for the cross-sectional distribution of funds by cohort. Thesections following this one use these results to predict cross-sectionalsurvival rates and the unconditional flow of funds.

All funds start with the same expectation of managerial ability, .f0

From this point, posteriors diverge depending on the manager’s trackrecord. As long as perceived ability in period t, , is greater than orft

equal to , we shall say that the fund survived through period t. For suchf

a fund, if perceived ability in period , , is less than , then the¯t � 1 f ft�1

fund goes out of business in period .t � 1For a fund that starts at time 0, let be the probability, conditionalG(f)t

on the fund’s survival though period t, that the perceived talent of themanager at t, , is less than f. The conditioning event means that thereft

is a zero probability assigned to values of . To derive this distri-¯f ! f

bution, we shall first consider the joint probability that a fund born atdate 0 both survives through period t and has . We denote thisf ≤ ft

joint probability as . Let and beˆ ˆˆG(f) g (f) { dG(f)/df g(f) { dG(f)/dft t t t t

the associated densities.We begin by conditioning on a, the actual talent of the manager. Let

be the probability, conditional on having a manager with talentaG (f)t

a, that the fund survives through period t and, at time t, that the per-ceived talent of the manager is f or less. Let be thea aˆg (f) { dG (f)/dft t

associated density.


Proposition 2. Suppose that a manager with true ability a beginsoperating at time 0 when the market’s prior on her ability is . Thenf0

is defined recursively as follows:ag (f)t

�g � tq g � tq g � (t � 1)q

a a aˆ ˆg (f) p g (v)n f � v dv, (21)t � t�1 ( )q q qf

with the boundary condition

q � g g � qa ag (f) p n (f � f ) � f1 0 0( )q q

and is a normal density function with mean a and precision q.an (7)Proof.

¯ ¯Pr [f ≥ f] p Pr [f ≥ fFf ≥ f] Pr [f ≥ f]t t t�1 t�1

g � (t � 1)q q ¯p Pr f � R ≥ fFf ≥ ft�1 t t�1[ ]g � tq g � tq

¯# Pr [f ≥ f]t�1

��

a aˆp n (u)du g (v)dv.� � t�1f [(g�tq)/q](f�v)�v

Thus

Pr [f ! f] p 1 � Pr [f ≥ f]t t

��

a aˆp 1 � n (u)du g (v)dv.� � t�1f [(g�tq)/q](f�v)�v

Differentiating this expression with respect to f, and thus eliminatingthe inner integral, gives the density in the proposition. Finally, we needto derive . By (7) we have thatg1

g qf p f � R ,1 0 1

g � q g � q

where is distributed normal . This implies that (the� ˆR [a, 1/q] g (f)1 1

density of ) isf1

�qg qnormal f � a,0[ ]g � q g � q g � q

over the range [ , ��) and zero for . Q.E.D.¯ ¯f f ! f

The expression for , the joint probability pooling all types, nowGt

follows immediately.


Proposition 3. Suppose that a fund begins operating at time 0 whenthe market’s prior on the fund manager’s ability is . Thenf0

�

u f0ˆ ˆg(f) p g (f)n (u)du (22)t � t��

andf

ˆ ˆG(f) p g(u)du, (23)t � tf

where is a normal density function with mean and precisionf0n (u) f0

g.Proof. Follows immediately from proposition 2 and the fact that the

prior distribution for a is normal with mean and precision g. Q.E.D.f0

The distribution of , conditional on survival through date t, can nowft

be computed directly using the unconditional probabilities:f ˆ¯ g(u)du∫f t

G(f) p , (24)t � ˆ¯ g(u)du∫f t

with associated density function

g(f)tg (f) p . (25)t � ˆ¯ g(u)du∫f t

Recall that the conditioning event, survival through period t, implies thatwhen , .¯f ! f G(f) p g(f) p 0t t t

Survivorship bias for mutual funds has been an important area ofempirical investigation. Clearly, empirical studies that ignore funds thatfailed will have biased estimates of the expected returns for the survivingfunds (see, e.g., Carhart et al. 2002). More important, survival ratesthemselves communicate information about the abilities of active man-agers. For example, one could examine the hypothesis that no skilledmanagers exist by comparing the fit of our model with to itsa p 0performance with parameters of the prior distribution freely estimated.Survival rates are natural moments to focus on in such a test. Our modelprovides explicit expressions for them.

Proposition 4. The unconditional probability that a fund that startsat time 0 survives through period t is

�

ˆˆP p g(f)df p G(�),t � tf

and the unconditional probability that a fund shuts down in period tis , where .P � P P p 1t�1 t 0


Proof. This result follows immediately from the definitions of andg(f)t

. Q.E.D.G(f)t

Now that we have derived the cross-sectional distribution of , we canft

use the results to analytically describe, within an age cohort of funds,the cross-sectional relation between past performance and fund flows.We can then quantitatively compare the theoretical flow-performancerelation to empirical findings in the literature. That is the goal of thenext section.

IV. Parameterization

We begin by first specifying a cost function and then using it to derivean explicit expression for the relation between performance and theflow of funds. We then parameterize this model and quantitatively com-pare the simulated results to the salient features of the data.

A. Parametric Cost Function

Assume , so the variable cost function is quadratic. Under2C(q) p aqour assumption that the manager sets his fees so that the constraint

does not bind,6 (9) impliesq ≥ 0It

ftq*(f) p . (26)t t 2a

The total amount of money under management will, by (18), be

2ftq p . (27)t 4af

Inverting this relationship and substituting the result into (26) gives thefraction of money under active management:

f�h(q ) p . (28)t aqt

Equation (27) together with (13) gives

c(q )t �p 2 aq f . (29)th(q )t

The percentage change in funds at time t as a function of themanager’s performance (as long as the fund survived to time t) can

6 The extension to the general case in which the constraint may bind is straightforwardbut tedious. It can be found in Berk and Green (2002).


now be obtained by substituting (28) and (29) into (19) and simplifying:

2 2q � q r q r qt t�1 t tp � . (30)( ) ( )2q f g � tq 4f g � tqt�1

Each of the parameters affecting the responsiveness of flows to perfor-mance has an intuitive role. As the noise in observed returns increases(q falls) relative to the precision of the priors, investors learn less fromreturns about ability, and a given return triggers less response in flows.As the age of the fund, t, increases, investors have more informationabout the fund’s performance, and flows respond less to the next return.As fees increase, the fund becomes less attractive relative to passivealternatives, and the manager earns his equilibrium compensation witha smaller amount of funds under management, making flows less sen-sitive to returns.

The flow of funds is zero at an excess return of zero. This suggeststhe empirically observed shape of the flow-performance relationship.Because of the quadratic term, flows respond more dramatically to ex-treme performance than to mediocre performance. While this quadraticbehavior slows down the response to extreme negative performance,this effect will be offset in the cross section by the closure of funds whenperformance is extremely bad. That is, for negative enough, the flowrt

of funds is simply .�qt�1

Either the flow of funds responds continuously to (given by [30])rt

or is so bad that the fund shuts down and investors withdraw all funds.rt

The latter will occur for any below a critical realization. Denote thisrt

critical realization as . It is determined by finding the such thatr*(f ) rt�1 t

the market’s posterior on the manager’s ability falls to . From equationf

(18), this value is given by

g � tq¯r*(f ) p (f � f )h(q ) .t�1 t�1 t�1q

Using (27) and (28) to simplify this expression gives

f � f g � tqt�1r*(f ) p 2 f. (31)( )t�1 ( )f qt�1


In summary, the overall change in the assets under management is givenby

q � qt t�1 pqt�1

f � f g � tqt�1�1 if r ! 2 f( )t ( )f qt�1(32)

2 2r q r qt t{ � otherwise.( ) ( )2f g � tq 4f g � tq

Empirical studies, however, commonly consider the flow of new funds,that is, the percentage change in new assets, which is typically definedto be

q � q (1 � r )t t�1 tn (r , q ) { . (33)t t t qt�1

Note that this measure uses in the denominator rather thanqt�1

. Unfortunately, this definition distorts the implications of veryq (1 � r )t�1 t

large negative returns that cause liquidation of the fund. For any r !t

, andr*(f) q p 0t t

q � q (1 � r )t t�1 t p �(1 � r ).tqt�1

That is, under this definition, for these returns, the measure becomesless responsive the worse the performance. In the limit when r pt

percent, the measure gives no response in the flow of funds. In�100an effort to address this issue while still maintaining consistency withthe empirical estimates for other returns, we set our measure of thepercentage flow of funds equal to �1 whenever liquidation occurs. Withthis caveat, (32) implies that

n (r , f ) pt t t�1

f � f g � tqt�1�1 if r ! 2 f( )t ( )f qt�1(34)

21 q 1 q 2{ � 1 r � r otherwise.t t( ) ( )2[ ]f g � tq 4f g � tq

When empiricists study the relation between past returns and the flowof funds, they often condition on age but do not condition on perceivedmanagerial ability. To derive this unconditional relation between the


flow of funds and past returns, we integrate over the cross-sectionaldistribution of surviving funds in a cohort:

�

N(r) p n (r, f)g (f)dft � t t�1f

r(r) �1 q

p � g (f)df � � 1 r� t�1 � ( ){[ ]f g � tqf r(r)

21 q 2� r g (f)dft�1( )2 }4f g � tq

1 qp � 1 {1 � G [r(r)]}rt�1( )[ ]f g � tq

21 q 2� {1 � G [r(r)]}r � G [r(r)], (35)t�1 t�1( )24f g � tq

where

f g � tqr 1 �2 f( )1 � (r/2f )[q/(g � tq)] q

r(r) {g � tq{� r ≤ �2 f.( )q

For a given realization of , gives the unconditional probabilityr G [r(r )]t t�1 t

that the fund will go out of business at time t. Clearly, when , nor ≥ 0t

fund goes out of business ( ); so and the flow of¯r(r ) ! f G [r(r)] p 0t t�1

funds is quadratic for any positive excess return. For a given negativeexcess return, the flow of funds is no longer quadratic because theunconditional probability that a fund will go out of business is positive.For very large negative returns (less than ), shutting�2[(g � tq)/q]fdown is certain.

It is clear that larger fees imply less sensitivity. Fund age also attenuatesflow sensitivity. Figure 1 plots the flow of funds relationship for fundsof different ages. The shapes of the relation between performance andflows are reminiscent of what researchers have found empirically (see,e.g., Chevalier and Ellison 1997, fig. 1).

With this cost function the conditional volatility of funds’ excess re-

Fig. 1.—Flow of new funds as a function of return. The curves plot the response in the flow of new funds to the previous period’s return (i.e., eq.[35]). The steepest curve shows the response for two-year-old funds (i.e., the return is from year 2 to year 3). The remaining curves show the responsefor five-, 10-, and 20-year-old funds, respectively. The parameter values used are , , , , and .f p 0.03 f p 0.065 f p 0.015 g p 277 q p 250


turns as a function of the assets under management (or the perceivedquality of the manager) is

2

q*(f) q*(f) 1t t t tVar (r ) p Var R pt t�1 t t�1[ ] [ ]q q qt t

2f 1 2fp p . (36)( )aq q q ft t

The conditional volatility is a decreasing function of the size of the fund.Since past positive returns increase the size of the fund, the volatilityof the fund will decrease when managers do well. The opposite occurswith negative returns. Thus the model delivers the relation between riskand past performance that has been attributed in past research to at-tempts by managers to mislead investors. In our model, funds withsuperior past performance invest a larger portion of their new capitalin passive strategies and thus lower their overall volatility, or “trackingerror.” Similarly, funds with poor performance increase their volatilitybecause as funds flow out, they preferentially liquidate capital that wasallocated to passive strategies.

B. Implementation

We begin by tying down the model parameters that can be inferreddirectly from existing evidence. The parameters that govern the distri-bution of skill level ( and g) are then inferred by matching the twof0

moments for which we have derived closed-form expressions—the sur-vival probabilities and the flow of funds.

The parameter f is reasonably straightforward to determine from pastempirical studies of mutual funds. We use percent. This is a bitf p 1.5higher than the averages reported in the literature for the expense ratio.For example, Chen and Pennacchi (2002) report average expense ratiosfor the funds in their study of 1.14 percent. Our use of a slightly highernumber is intended to account for the amortized loads that are notincluded in the expense ratio. We set j at 20 percent, which reflectshistorical levels of portfolio volatility. The volatility of returns in themodel is naturally interpreted as tracking error around a benchmarkreturn. Empirical estimates of tracking error for mutual funds are lowerthan this. The value of j in our model will be higher than the volatilityof observed returns, however, since indexing reduces the variance of afund’s return. When return histories for 5,000 funds are simulated, thislevel of j produces an average standard deviation of returns ( ) of 9.12rt

percent. This number is consistent with empirical estimates of average


TABLE 1Parameter Values

Variable Symbol Value

Percentage fee f 1.5%Prior precision g 277Prior standard deviation h 6%Return precision q 25Return standard deviation j 20%Mean of prior f0 6.5%Exit mean f 3%

levels of tracking error and idiosyncratic volatility.7 From (20), the quad-ratic cost function and (26) give percent. These parameterf p 2f p 3values are summarized in table 1. It is straightforward to see (from [22]and [35]) that the only remaining parameters that affect the survivalprobabilities and flow of funds relationship are and g.f0

Our approach is to match the two parameters governing the distri-bution of skill level using an “eyeball metric” that matches the empiricalsurvival rates and relation between the flow of funds and performance.Table 1 contains the values of and g that resulted from this process.f0

The lighter bars in figure 2 represent quantity-weighted averages ofsurvival rates for all funds in the Center for Research in Security Prices(CRSP) mutual fund database from 1969 to 1999. Each bar representsthe fraction of funds that survived for one through 19 years over thisinterval, that is, the ratio of the number of surviving funds over the totalnumber of funds that could have survived.8 The dark bars are thematched survival rates computed using the probabilities from propo-sition 3. In our model, survival rates drop off geometrically with age,as they will in any model based on learning, whereas in the data theyfall in a more linear fashion. We chose to match the shorter-term survivalprobabilities. For obvious reasons many more data exist for youngerfunds (see fig. 2), which means that the early data are estimated muchmore precisely. The behavior of the longer-term survival rates is puzzlingand may be due to managerial turnover within the mutual fund. Goodmanagers might be promoted or defect to other firms. Thus the low

7 Koski and Pontiff (1999, table 3) report mean levels of idiosyncratic risk for funds withdifferent investment styles. When annualized, these levels vary from 9.97 percent foraggressive growth funds to 3.67 percent for equity-income funds. Chen and Pennacchi(2002) provide estimates of unconditional tracking error ( in the model estimatedd � d0 1

in their table 3). The median value for all funds when annualized is 9.66 percent.8 We are very grateful to Hsiu-lang Chen and George Pennacchi for providing us with

the raw survival rates.

Fig. 2.—Percentage of surviving funds. The bars show the fraction of funds as a function of the number of years in business. Light bars are the actualsurvival rates computed from the CRSP mutual fund database. The dark bars are what the model predicts the survival rates should be. The scale ismarked on the left-hand axis. The line marks the total number of funds that could have survived at each age. The scale for this line is marked on theright-hand axis.


survival rates over longer periods may reflect a renewal in the learningprocess our model does not capture.9

Matching the relatively high survival rates for the first few years re-quires that funds begin operating at a scale that is considerably higherthan the size at which they liquidate, with correspondingly high expec-tations for managerial ability. The priors are that managers are expectedto earn an annual excess return of 6.5 percent when they beginoperating.

High precision in priors relative to the variability in returns is requiredto reproduce the fund flow and performance relationship. The slopeof this relationship is determined by the fees, f, and the ratio of theprecision in priors and returns. Figure 3 provides the explicit compar-ison between the model’s results and behaviors documented in empir-ical studies. In it the flow of funds relationship for two-year-old fundsis superimposed over the plot of the nonparametric estimates and 90percent confidence intervals in Chevalier and Ellison (1997). The curvefrom the model seems to pick up the general curvature in the rela-tionship. A notable discrepancy is that the empirical curve does not gothrough the origin, as it does in our model. A natural explanation forthis is the general growth in the mutual fund sector during their sampleperiod, something that is missing in our model.

An important question in financial economics is whether active port-folio managers have skill. Alternatively, are other market participantsenforcing such efficiency in financial markets that there is no oppor-tunity for active managers to add value or create rents for themselves?Figure 4 plots the prior distribution over management ability using ourparameter values. It also shows the level of the management fee (1.5percent). If level of skill in the economy is defined as the fraction ofmanagers who can generate an a in excess of the fees they charge, thenthis fraction is the area of the curve to the right of the line. About 80percent of managers satisfy this criterion—they generate value in thatthey can beat their fees on at least the first dollar they manage—andthe average manager has an a of 6.5 percent.

These estimates might appear high, given the skepticism in the aca-demic literature about whether active managers add value at all. It is adirect consequence, however, of the very high survival rates and em-pirically observed flow of funds relationship. Furthermore, the otherparameters implied by these estimates seem reasonable. For example,the implied value of the ratio (the size of a new fund over the¯q /q0

minimum fund size) is 4.7. So, for example, if the minimum fund size

9 In contrast, for hedge funds in which there is little distinction between the managerand the fund company, Getmansky (2003) finds that survival rates do drop offgeometrically.

Fig. 3.—Flow of funds. The dashed line shows the flow of funds for two-year-old funds produced by the model (using the parameters reported intable 1) superimposed over the actual flow of funds plot (solid lines) for these funds as reported in Chevalier and Ellison (1997, fig. 1). Chevalier andEllison report the estimated curve (middle line) as well as the 90 percent confidence intervals, the outer lines.


Fig. 4.—Distribution of management skill. The vertical line marks the level of themanagement fee—1.5 percent. Approximately 80 percent of the area below the curve liesto the right of this line. The parameter values (mean and precision) are andf p 0.0650

.g p 277

is $5 million, the model implies that new funds would start around $25million. At , the fees equal the fixed costs, , so million¯ ¯ ¯q F pqf q p $5implies periodic fixed costs F of $75,000. Far from implying that mostmanagers are unskilled, therefore, the data seem consistent with a highlevel of skill for active managers.

V. Conclusion

In this paper we derive a number of empirical predictions of a rationalmodel for active portfolio management when managerial talent is ascarce resource and is dissipated as the scale of operations increases.Many of these predictions reproduce empirical regularities that oftenhave been taken as evidence of investor irrationality or agency costsbetween managers and investors. Not only is the rational model con-sistent with much of the empirical evidence, but it is also consistent witha high level of skill among active managers.


Appendix

TABLE A1Important Notation

Rt Excess return earned on the first dollar actively managed by the fundin period t

a Expected excess return earned on the first dollar actively managed bythe fund in period t: our measure of the manager’s ability

j2 Variance of Rt

q Precision of Rt: q p 1/j2

h2 Variance of the market’s prior over managerial ability, ag Precision of the market’s prior over managerial ability, a: g p 1/h2

ft The market’s expectation of managerial ability, conditional on the re-turn history up to date t

f0 Mean of the market’s initial prior over managerial abilityf The minimal expectation of managerial ability at which the fund can

recover its fixed costs and continue to operateqt Funds under management at date tq*(ft) Optimal amount of funds under active management, given the mar-

ket’s expectations of managerial ability¯q { q*(f) The lowest value of q for which the fund remains viable

qIt Funds the manager raises but passively invests in the benchmark port-folio: qt p q*(ft) � qIt

h(qt) Proportion of funds under active management: h(qt) p q*(ft)/qt

ft Management fees, where the optimal level of fees is denoted ; whenf *tfees are fixed through time, ft p f

F Fixed costs of operating the fund each periodC(q) The total variable costs of actively managing a fund of size qc(q) The unit costs and fees borne by investors in a fund of size q: c(q) p

[C(q)/q] � frt Return to investors per dollar invested in the fund: rt p Rt � c(qt)Pt Survival rate: the probability that a fund born at date 0 survives until

date tr*(ft�1) The minimal return realization such that a fund with expected mana-

gerial ability ft�1 survives through the next period

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