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Journal of Financial Economics

Journal of Financial Economics 108 (2013) 506–528

0304-40

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Mutual fund risk and market share-adjusted fund flows$

Matthew Spiegel a,n, Hong Zhang b,c,1

a Yale School of Management, P.O. Box 208200, New Haven CT 06520, USAb CKGSB, 20/F Tower E2 Oriental Plaza, 1 East Chang An Avenue, Beijing 100738, Chinac INSEAD, 1 Ayer Rajah Avenue, Singapore 138676, Singapore

a r t i c l e i n f o

Article history:

Received 28 September 2010

Received in revised form

19 April 2012

Accepted 18 May 2012Available online 20 December 2012

JEL classification:

G11

G1

Keywords:

Mutual funds

Fund flows

Investment flows

Convexity

Market share

5X/$ - see front matter & 2012 Elsevier B.V

x.doi.org/10.1016/j.jfineco.2012.05.018

would like to thank seminar participants at

versity of Texas at Austin, University of W

Singapore International Conference on Fina

ity for helpful comments. Hong Zhang thank

r financial support.

esponding author. Tel.: þ1 203 981 1184.

ail addresses: [email protected] (M

[email protected] (H. Zhang).

l.: þ65 97879206.

a b s t r a c t

Several papers use a fractional specification (net inflow/ assets under management) to

infer a convex relation between flow and past performance. However, heterogeneous

linear response functions combined with the pooled analysis commonly used in these

studies can yield false convexity estimates. We show that such heterogeneity obtains in

practice. Along these same lines, the paper also finds that several previously unexamined

implications of a convex flow-performance relation fail to hold. Moreover, convexity

with fractional flows (which we confirm) largely disappears in a conditional analysis that

controls for heterogeneity. Market shares offer an alternative specification for flow that

is more resilient to heterogeneity. Using this alternative specification, we again find no

evidence of convexity in the flow-performance relation. We conclude that the widely

held belief that the flow response function is convex is due solely to misspecification of

the empirical model. The flow-return relation is linear.

& 2012 Elsevier B.V. All rights reserved.

1. Introduction

Numerous studies show a convex relation betweenmutual fund flows and past returns, including those byChevalier and Ellison (1997), Sirri and Tufano (1998), Fantand O’Neal (2000), and Huang, Wei, and Yan (2007). Thesefindings have since been used as a foundation for theoreticalmodels of inter-fund competition (Carpenter, 2000; Lynchand Musto, 2003; Taylor, 2003; and Basak and Makarov,2012). Convexity plus Jensen’s inequality naturally leads

. All rights reserved.

Columbia University,

estern Ontario, the

nce, and Vanderbilt

s the INSEAD Alumni

. Spiegel),

these articles to conclude that increasing a fund’s riskincreases its expected capital inflows. While these papersdiffer in many ways, they share a common element: Returnsare assumed to map directly into flows divided by assetsunder management (AUM). While this fractional flow speci-fication may appear innocuous, it is not. If these models areproperly specified, then aggregate flows should be linked tothe cross-sectional distribution of fund returns. Our tests,however, indicate otherwise, suggesting that the standardfractional flow model is misspecified. Market shares offer analternative, more resilient specification and show no evi-dence of convexity in the fund growth-performance rela-tion. As the paper demonstrates, misspecification can fullyaccount for the previously empirically documented convexflow-return relation. In reality, the relation appears to belinear.

A simple example shows how the fractional flow modellinks the distribution of individual fund returns to aggre-gate flows. Consider an economy with two funds: one has$100 under management, the other, $10. The fractional flow

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M. Spiegel, H. Zhang / Journal of Financial Economics 108 (2013) 506–528 507

model states that the better performing fund will see aninflow of 10%, the other, 0%. If the large fund does better,the aggregate flow equals $10, and if the small one doesbetter, the aggregate flow is $1. This relation between fundreturns across different size groups and aggregate flows is,in fact, a general implication of the standard fractionalflow model: When large funds do relatively well, aggregateflows should be larger. However, our tests yield littleevidence that this is the case. Aggregate flows are see-mingly determined by economy-wide events, such as theoverall market return, not by whether large or small fundshave recently done relatively well. A model based onmarket share changes provides a simple way around thisproblem. By construction, market share changes add tozero. Thus, in the absence of additional specification, amarket share model does not link aggregate flows to thedistribution of individual fund returns.

Any empirical specification is necessarily misspecified tosome degree. What is important is its robustness to sucherrors—in the case of mutual funds, an ability to handleunaccounted for heterogeneity in the cross section andacross time. In practice, this can be difficult to accomplishwith a fractional flow-return model as the (nonexhaustive)example depicted in Fig. 1 illustrates.

In Fig. 1, the dashed blue lines represent the relationbetween period t flows (ft) and period t�1 returns (rt�1)within a period for two different fund types. They arelabeled hot and cold money funds. The picture has threeimportant elements. First, heterogeneity exists in thecross section of functions mapping flows to returns, whichthe empirical model has not fully controlled for. (In thisexample, the researcher does not estimate separate func-tions for the hot and cold money funds.) Second, when afund’s inflows are poised to be relatively high (meaning arelatively high cross on the y-axis), the flow-return rela-tion steepens. Third, hot money funds have more volatilereturns than cold money funds. As a result, the cold fundshave relatively more observations close to the y-axis andthe hot ones relatively more further away, as the red ovalsindicate. Importantly, in all cases the relation betweenflows and returns is linear, implying volatility will not

Fig. 1. Spuriously estimated convexity example. Fractional flows are

graphed against relative returns.

affect a fund’s expected asset accumulation. Nevertheless,a regression with fractional flows as the dependentvariable and returns as the explanatory variable willproduce the convex green line. The convexity is entirelydue to misspecification error. In this case, the curve triesto minimize the sum of squared errors by moving closerto the relatively densely populated areas, i.e., closer to thered ovals.

To see how Fig. 1 might come about in actual data,consider a scenario in which one fund is growing (hot) whileanother has essentially stabilized at its current asset level(cold). To fix the general idea, the stable fund might be old,large, and housed within a well-established mutual fundfamily. While the cold fund’s many long-term investorsmight not look to it as the place to put their money whenseeking out the latest and greatest, neither are they likely topull out if recent returns are subpar. This could arise fromsimple inertia (Choi, Laibson, Madrian, and Metrick, 2002,2004a, 2004b; Duflo and Saez, 2002; and Choi, Laibsonand Madrian, 2009) or if the fund’s assets derive largelyfrom essentially automated 401(k) retirement flows (Choi,Laibson, Madrian, and Metrick, 2002, 2004b; and Mitchell,Mottola, Utkus, and Yamaguchi, 2006). These and otherpossible factors produce for the older (cold money) fundthe flow-return relation described by the lower blue shortdashed line. The high long dashed blue line in Fig. 1 can bethought of as potentially representing young (hot money)funds with relatively few assets under management. Due totheir size, these hot money funds can easily hold portfoliosthat, compared with large cold money ones, are relativelyundiversified. Consequently, the hot money funds havea higher return volatility than the cold money funds.The important point is that the red ovals, indicating wheredata for the cold money funds are relatively dense, lie closerto the origin than those for the hot money funds.

Finally, consider the impact of the age of the hot andcold money funds. If a hot money fund’s investors havebeen with it for only a short period of time, its flow willlikely vary dramatically in response to its recent returns,at least relative to the cold money fund. Combining theimpact of heterogeneity in the unconditional flow, returnvolatility, and investor responsiveness to past returnsyields the flow-return relation displayed in Fig. 1. Atypical flow-return model will now produce the convexsolid green line. Can controls fix the problem? Yes, if oneknows how to properly divide up the funds.

In this paper, we show that separating out funds thatare both young and small from the rest yields a pair oflinear relations. These young-small funds have all therelative properties depicted in Fig. 1: volatile returns, highrelative unconditional growth, and a particularly steepflow-return relation. Thus, even accounting for just thisone source of cross-sectional heterogeneity, the convexityfound by running a pooled regression largely disappears.

While we show that the empirical problems that leadto false convexity can be mitigated, it requires knowingexante what cross-sectional controls are needed. But anynumber of scenarios also yield the pattern depicted inFig. 1, so there is no guarantee that any single set willbe sufficient. Consider that the relation depicted in Fig. 1could arise dynamically. High aggregate market returns

M. Spiegel, H. Zhang / Journal of Financial Economics 108 (2013) 506–528508

induce high aggregate flows (Warther, 1995; Edelen andWarner, 2001; Goetzmann and Massa, 2003; Boyer andZheng, 2009; and, using Israeli data, Ben-Rephael, Kandel,and Whol, 2011)). During these high aggregate flowperiods, one might expect hot money funds to experienceparticularly high unconditional flows and an increase inthe slope of the flow-return relation. The relative numberof hot and cold fund types may also be impacted. Highaggregate inflows encourage entry leading to an abundanceof hot funds in such periods. (We test for and find theseintertemporal patterns within the data and show that theyexplain a sizable fraction of the estimated convex fractionalflow-return relation in standard models.) The importantpoint, however, is that whatever the underlying causality,a nonlinear panel data regression of fractional flows givenreturns will generate the convex green curve in the absenceof a full set of controls.

What happens if one uses market share changesinstead of flows as the dependent variable? While marketshares are not impervious to misspecification, they aremore robust. This is illustrated via the simulation resultsin Fig. 2.

Fig. 2 derives from four thousand simulations of asingle period economy. In each simulation, there are one

0.04

0.02

0

0.02

0.04

0.06

0.08

0.1

-0.3 -0.2 -0.1 0 0.1 0.2 0.3

Cold Δ Hot Δ Avg. Δ

Hot money fund

Cold money fund

Estimated shape of the flow return relation

Areas of relatively high observation density

m m mCold /n Hot /n Avg. /n

Fig. 2. Spurious convexity simulation, fractional flow and market share

models. This figure simulates an economy over four thousand independent

periods (t), each of which has one thousand identically sized funds. In each

period, aggregate fractional flows (Ft) are normally distributed with mean

4.73% and standard deviation 0.1%. The overall market return is normally

distributed with mean 0.8% and standard deviation 2.2%. All funds have

normally distributed returns (ri,t) with means of zero net of the market.

Relative to the market’s standard deviation, hot fund returns are 257%

greater and cold funds are 230% greater. Let rt�1 represent the value

and response weighted average return across mutual funds. In the

simulations it equalsP500

i ¼ 1 0:198ri,t�1þP500

i ¼ 1 0:002ri,t�1

� �=1000 and

is defined to ensure that the individual fund flows add to the aggregate

flow irrespective of the realized returns. For hot fund i in period t

fractional flows, dollar flows (f) over assets under management (n), equal

f i,t=ni,t�1 ¼ Ft=2000þ3:3 ri,t�1�r t�1

� �exp Ft=3101

� �and for a cold fund it

equalsf i,t=ni,t�1 ¼ 0:11 ri,t�1�r i,t�1

� �exp Ft=3101

� �. In each period, a fund’s

fractional flow, market share change Dm, return, and return rank for that

period are recorded. The simulated periods are then pooled together.

Finally, market shares are scaled so that at rt�1¼0 the pooled line (Avg.

Dm) has the same slope as the pooled fractional flow line (Avg. f/n).

thousand funds split evenly between hot and cold money.The parameter values have been selected to match thesimulated data with some of the moments in the actualdata for monthly fund flows (after trimming the top andbottom 5%) and returns. In both the actual data and thesimulation, the flows have a mean value of 4.7%, with astandard deviation of 1.35%, and fund returns net of themarket have a standard deviation of 5.4%. The three upperlines display the results using fractional flows and essen-tially reproduce the idealized lines depicted in Fig. 1. Thethree lower ones represent the results using market sharechanges. The upper orange long dashed line displays thehot fund flows for a given return; the lower short dashedone, the cold fund flows. The solid green line in themiddle is the estimated impact of relative returns onmarket share changes using the pooled data.

To make the comparison between fractional flow andmarket share change estimates easier to visualize, Fig. 2scales the two measures so that at the origin the estimatedresponse functions (the two solid lines) have the same slope.Even though the pooled regressions have identical slopesaround the origin, the lines representing the flows to hot andcold funds are closer to each other when market shares,instead of fractional flows, are used. This greatly attenuatesthe distorting effect of uncontrolled for heterogeneity. Theresult holds whenever flows are an affine function of returns;market shares produce a smaller angle between the responselines, which reduces the impact of misspecification error.At the same time, if the data exhibit true flow convexity,our power tests (based on the estimated flow-performancerelation) indicate that with very high probability the marketshare model will yield a statistically significant convexrelation as well. One can adjust the flow-return estimatorto account for cross-sectional and intertemporal heterogeni-ety. However, market shares provide a simple alterna-tive. Relative to flows, market shares mitigate many of theproblems associated with intertemporal variation in thefund-by-fund cross section of investor responses to pastreturns without needing to know what they are exante.

This paper concludes that the widely held belief thatincreasing a fund’s risk will help it grow is due tomisspecification error and is not reflected in the dataeither in the short or long term. Nevertheless, this is aseparate issue from whether funds vary their risk overtime. It is true that the risk-leads-to-growth idea hasoften motivated the empirical time-varying risk literature.However, it is not needed to accept the empirical conclu-sion that such variation is occurring. Managers may besystematically varying their fund’s risk (Brown, Harlow,and Starks, 1996; Chevalier and Ellison, 1997; Taylor,2003; Busse, 2001; Qiu, 2003; and Goriaev, Nijman, andWerker, 2005) but this does not appear to be motivatedby some sort of fund flow tournament, as is typicallyhypothesized.

If the underlying flow tournament which motivatedmany studies of mutual funds, does not exist, what then isresponsible for the observed results? A number of otherpotential forces can provide an explanation. For example,mutual fund managers may vary their fund’s risk inresponse to tournament-like incentives from forces otherthan fund flows. Career issues can play a role, as in Qiu


(2003), who attributes the time-varying fund volatility inhis data set to the risk of termination. Undoubtedly, othercompensation factors such as bonus payments and thelike could provide similar incentives.

The paper is organized as follows. Section 2 looks atsome long-term patterns that seem difficult to reconcilewith a convex flow-performance relation. Section 3 dis-cusses the mathematical implications of various empiricalspecifications of the fund return-flow relation. Section 4introduces a market share based model as an alternativeempirical specification. Section 5 discusses the data used.Section 6 presents a statistical summary and compares thelevel of observed convexity in the raw data across thefractional flow and market share-based models. Section 7tests whether or not aggregate flows have patterns con-sistent with the embedded implications from empiricalmodels that regress fractional fund flows on performance.Section 8 shows how the observed convex fractional flow-return relation can potentially be reconciled with the linearmarket share change-return relation found here. Section 9concludes.

2. Some long-term patterns

Table 1 presents some long-term trends regardingmarket shares across various volatilty groups within themutual fund industry. The data in the table combine thestandard Center for Research in Security Prices (CRSP)mutual fund database from 1990 to 2006 with matchingdata from Cremers and Petajisto (2009) on fund activeshares and tracking errors. As in the rest of the paper, onlydomestic equity funds are included. (See Section 5 for afuller description of the data.)

If the flow-performance relation is convex, Jensen’sinequality implies that volatile funds should grow relativeto their peers. The long-term trends displayed in Table 1do not indicate that this has been happening. In Panel A,each column displays the aggregated year-end marketshares of funds grouped according to various measures ofrisk.2 The second column shows that index funds ended1991 with 4.3% of the market. By 2006 they had garnered17.5%. Arguably, these are the funds with the least risk.Yet their market share growth has been substantial.Perhaps index funds have grown despite their low riskbecause they offer lower expenses and, thus, somewhathigher returns. Under this hypothesis, the reduced growthfrom low risk is offset by higher growth from the higherreturns they produce. But, even restricting attention toactively managed funds, the table offers no evidence thathigh-volatility funds have seen their market share growrelative to other managed funds or that low-volatilityfunds have lost out. The columns displaying the marketshares of funds with very low and low active shares (thoseunable to attract flows via volatility) show some com-bined growth until about 1998 and then level off withabout 10% of the market.3 Over this same time period,

2 See Cremers and Petajisto (2009) for definitions and www.sfsrfs.

org for the underlying data.3 Very low active share is defined as an active share measure below

1%. Low is defined as an active share measure between 1% and 10%.

high active share funds, which can presumably usevolatility to garner flows, have seen essentially no growth.

The final eight columns in Table 1 list the marketshares of funds classified by both return standard devia-tion net of the market volatility (VOL), and tracking error(TE, the volatility of return net of the closest index). Aswith the other measures, no evidence exists that highlyvolatile funds have seen any long-term market sharegrowth. The columns VOL D1 and VOL D10 display themarket shares of funds in the lowest and highest volatilitydeciles, respectively. While each category has seen somevariation in market shares over time, no apparent long-term trend emerges one way or the other. Also, the lowestvolatility decile has had a market share of around 15%,while the high volatility decile has never gone above 8%and is typically under 5%. Tracking error deciles producesimilar conclusions. Little evidence exists of long-termgrowth within the highest decile. Similarly, the lowtracking error decile commands a much greater marketshare (around 20%) than the highest decile (around 5%).

The final four columns of Table 1, Panel A list thevolatility and tracking error decile cutoffs over time. Toget into the first decile in a particular year, a fund needs aVOL or TE standard deviation below the value in VOLCOD1 or TECO D1, depending on the column. Similarly, afund is included in the tenth decile if its VOL or TE lieabove VOLCO D10 or TECO D10. If volatile funds aregrowing relative to their peers, one expects the low andhigh decile group’s cutoffs to increase over time as theaverage fund’s tracking error increased. At the very least,the high decile group’s cutoff should increase as firmscompete to produce the highest volatility portfolios. Noneof this appears to be happening. If fund flows are convexin returns, it is not apparently showing up in either thelong-term relative market shares or as an inducement forfunds to increase their portfolio’s volatility on either anabsolute basis (VOL) or relative to their benchmark (TE).

Table 1, Panel A includes the impact of both changesin market shares due to each fund’s own growth and entryby new funds. Arguments can be made both for and againstconsidering entry. If investments tend to flow towardvolatile funds, then one would expect entrants to cater tothat. In this case, including entrants reveals whethervolatility is being rewarded with greater levels of assetsto manage. The counterargument is that an existing fundcannot control entry. Thus, for an individual fund manager,given that he already has an existing fund, the questioncould be whether or not increasing its return volatility willhelp it to grow faster than its current peers as opposed tothe overall market. In this case, new entrants should beexcluded from the analysis. To gain insight into this versionof the problem, Panel B does just that.

In Table 1, Panel A, actual market shares are calculated. InPanel B, year-to-year changes in market share are displayedwith new entrants excluded. This is done by calculating eachfund’s market share in year t, then taking this same set offunds and calculating their market shares relative to eachother in year tþ1. For example, suppose Funds A, B, and Ceach manage $1,$2, and $3 respectively, in year t. In yeartþ1, they manage $3, $4, and $6 respectively, and in thesame year Fund D enters with $1 under management. In this

www.sfsrfs.org

www.sfsrfs.org

Table 1Market shares and market share changes by category over time.

The index fund, enhanced index fund, active share values (AS), and tracking error values (TE) are based on Cremers and Petajisto (2009). The data come from the paper’s addenda page on www.sfsrfs.org/.

Definitions: very low AS if ASr1%, low AS if 1%oASr10% and high AS if ASZ90%. For the tracking error columns (TE D1 and TE D10), funds are ranked by tracking error and placed into deciles. The aggregate

market share of those in the bottom decile is given in the TE D1 column. Similarly, the TE D10 column displays the aggregate market share of funds in the top tracking error deciles. The cutoff values used to

create the D1 and D10 tracking error deciles are displayed in the TECO D1 and TECO D10 columns, respectively. Panel A reports market shares by group on a year-by-year basis. Panel B reports year-over-year

market share changes holding the set of funds constant from one year to the next. For example, only funds that existed in 1990 are used to calculate the market share changes from 1990 to 1991. The results are

listed in the row labeled 1990. The t-statistics are for the hypothesis that the average change is different from zero. Asterisks indicate significance levels: n¼10%, nn

¼5%, and nnn¼1%. The final two rows labeled

‘‘þ ’’ and ‘‘� ‘‘ indicate the total number of years with positive and negative changes, respectively. Panel C displays the average return (plus 1) by category. Blue cells indicate a category that came in first and red a

category that came in last relative to those in its comparison group. The total number of first, second or third place entries by comparison group are displayed in the last row. For example, the final row in the

very low AS column contains the entry [4/6/7]). This indicates that relative to low AS and high AS funds those in the very low AS category had the highest return in 4 years, the second highest in 6 years and the

lowest in 7 years.

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5 In some models, ranks are replaced by relative returns in g. This

paper explores both specifications.6 This dependence on aggregate flows and the performance of funds

by size can be broken by dividing c(ki,t�1) by ni,t�1. However, this

independence relies on what is essentially a knife-edge specification.

Any other variation in the relation between c and n will destroy the

independence of F on the ranks. Nevertheless, this knife-edge was tested

against the data via statistical tests in which additional terms involving

c and n were included in the estimation of Eq. (1). Not only were many

of these additional terms statistically significant but the parameter


case, Fund A’s change in market share would be calculated as3/13�1/6. Because Fund D enters only in date tþ1, its AUMis excluded when calculating Fund A’s tþ1 market share forthe purposes of this statistic.

Table 1, Panel B shows little evidence that fund volatilityleads to systematic growth relative to the existing peer group.At the 1% level, only index and low active share funds displayconsistent year-over-year gains in market share. Increasingthe critical value to 10% adds the very low active share fundsto those exhibiting statistically significant relative growth. Forthe other groups, every indication exists that market sharesessentially change randomly from year to year. If risk leads toadditional AUM, it has not worked out that way over the longrun for funds employing high volatility strategies.

Panel B also addresses, to some degree, the possibilitythat the Panel A results are driven by larger funds necessa-rily having smaller active shares and tracking errors thansmaller funds. Because of this, Panel A may simply reflectthe fact that as funds grow they depart the high activeshare and tracking error categories and transition into thelow ones.4 However, on an annual basis at least, Panel Bcontrols for this by using constant cohorts. Thus, suppose afund grows and transitions out of the high active sharecategory in 1995. It nevertheless remains in that categorywhen looking at the year-over-year 1994 to 1995 change inthe category’s market share.

The costs incurred by the more active funds could resultin systematically lower returns. Thus, these funds retainmarket share based on their relatively larger volatility andlose out due to their lower returns. The general hypothesisproffered in the flow-return literature is that investors chasereturns, funneling their money into the winners. Assumingthat is correct, then the right comparison is year-to-yearrank across groups. Table 1, Panel C examines this idea bycomparing the annual average returns for each category. Redcells indicate a category that came in last relative to thedisplayed comparison group, and blue ones are the winners.The final row indicates how often each group won, lost, or, inthe case of the active share groups, came out in the middle.The categories win or lose with nearly equal probability. Nosystematic evidence exists that, however measured, lowervolatility funds outperform high volatility funds, at least tothe extent that it might be expected to dramatically impactoverall retail flow allocations.

Finally, what could be happening is that volatile fundsin general attract fewer inflows for any given performancedue to the standard arguments regarding risk aversion. Inthis case, the gain from convexity is offset by the loss frominvestors generally avoiding risker funds. However, if thisis the case, then the conclusion based on the flow-returnliterature that managers’ gain from increasing risk isproblematic from the start.

3. Fractional flow model econometrics and implications

The standard flow-performance model has a numberof implications for how aggregate flows should relateto the distribution of fund returns. A simple example in

4 This is the finding in Brown, Harlow, and Starks (1996).

which there exists a convex flow-return relation illus-trates one of the potential issues. Suppose there are threefunds and aggregate flows are zero. The highest rankedfund receives $20 and the two lower ranked ones lose $10each. While this creates a convex relation, if the funds areof different sizes this brings with it an empirical predic-tion: A fund’s flows as a percentage of its AUM willdecline with AUM. Because this prediction seems highlyunlikely to hold, most flow-performance models definethe relation in terms of flows relative to AUM. A typicalspecification starts by assuming that for firm i on date t itspercentage flow is governed by

f i,t

ni,t�1¼cðki,t�1Þ, ð1Þ

where f is the dollar flow in the period, n is the fund’s totalAUM, and c is a function that maps the firm’s rank k intoits flow.5 Going back to the example, the two losing fundsmight see a percentage outflow of 1%, and the winner, aninflow of 2%.

It is true that mapping a fund’s fractional flows to itsreturn rank [as in Eq. (1)] avoids the implicit hypothesisthat larger funds will see smaller percentage flows forany given performance level. However, aggregate flows(Ft ¼

PIi ¼ 1 f i,t) are now a function of the industry’s

performance distribution. Consider what happens if, asone might expect, better performing funds see largerflows (qc/qki,t�140). Then aggregate flows must berelated to how large funds perform relative to small ones.Take two funds labeled 1 and 2. Consider the total flowsgoing to these two funds. Next, suppose they switchranks. Using Eq. (1), the sum of their dollar flows (f1,tþ f2,t)will be invariant to this swap in rankings only if the twofunds are of identical size (n1,t�1¼n2,t�1). Otherwise, ifthe large fund moves up in rank, then the total flows willincrease, whereas if the large fund moves down, they willdecrease. One can also equivalently define c in terms ofrelative returns and produce identical conclusions. Inmany places, the paper will make this substitution.

Based on the above analysis, if Eq. (1) holds, then onehas the following testable empirical implication:

Proposition 1. If Eq. (1) holds and flows are increasing in

performance, then aggregate fund flows increase as the rank

(or relative returns) of the industry’s larger funds increases.6

Proposition 1. requires that fractional flows increase onlyin returns. However, if they are also convex, then themodel has another implication: Aggregate flows should be

estimate for c/n was not. A copy of the exact set of models that were

estimated along with the results is available upon request from the

authors.


larger when large funds are located in the tails of thereturn distribution. This is nothing more than a variationof the idea that a mean-preserving spread in the value of arisk-loving individual’s future consumption increaseshis utility. In this case, a mean-preserving spread impliesthat the change in the relative performance of funds bysize leaves the sum of the size-weighted ranks (Kw,t ¼PI

i ¼ 1 ni,tki) and the total AUM (TAUM, Nt ¼PI

i ¼ 1 ni,t)unchanged. Holding Kw,t constant highlights how size andperformance impact aggregate flows. A mean-preservingspread in size– weighted performance, along with alinear C function, leaves aggregate flows unchanged. Incontrast, a convex C results in an increased aggregateflow. To simplify the exposition, Proposition 2 is statedunder the assumption that the set of ranks and funds arecontinuous with n(k) mapping a rank k into a fund size n.

Proposition 2. Suppose fractional flows are convex in perfor-

mance: C0 and C00 are both positive. Let the dummy parameter

f alter the size of funds assigned to particular performance

ranks. Assume f leaves both Ft and Kw,t unchanged while

inducing a mean-preserving spread over the size of funds so that@@f

R k0

Rn k,fð Þdk

� �dk40 for all k. Then @F/@f40.

Because the proof follows that for the impact of a mean-preserving spread on a utility function, it is omitted here tosave space.7 However, the point is easy to illustrate.Suppose there are four funds: two with AUM of $100 andtwo with AUM of $10. Further assume that fractional flowsin rank order are �5%, 0%, 10%, and 25%. If the small fundsoccupy ranks 1 and 4, then aggregate flows are 12. How-ever, if the large funds are ranked 1 and 4, then aggregateflows are 21.

The implications derived in Proposition 1 and 2 can betested in numerous ways.8 Ultimately, what is needed is ameasure relating aggregate flows to the size of the fundswith the highest rankings. The empirical section of thispaper employs two such measures. The first uses

Ft

Nt�1¼ a0þ

a1

Nt�1I

XI

i ¼ 1

ni,t�1ki,t�1þa2

Nt�1I

XI

i ¼ 1

ni,t�1k2i,t�1

þcontrols: ð2Þ

Because the first summation in Eq. (2) increases withthe rank of the larger funds, it should yield a positive valuefor a1 under Proposition 1. Similarly, shifting the largestfunds toward the tail of the distribution will increase thesecond summation, and from Proposition 2 the estimate of

7 To apply the standard proof that a risk-loving agent prefers gamble

1 over 2 if the payoff distribution of 1 second order dominates that of 2,

one can treat the set (n1,y, nI) as the distribution of the payoff function,

g as the utility function, and Ft as the expected utility.8 Another specification that is sometimes used relates percentage flows

to a fund’s excess returns (e.g., Chevalier and Ellison (1997). However, this

specification creates implicit relations similar to Proposition 1, and others

that are even stronger. For example, assume that the return-flow relation is

convex. Then Jensen’s inequality implies that an increase in overall realized

dispersion in fund returns should lead to greater aggregate flows. However,

because most studies now use ranks, we do not explore this or other

implications that are unique to the excess return approach.

a2 will be positive if the flow-performance relation isconvex.9

The second test conducted here simply replaces thesummation in Eq. (2) with the covariance between fundsize and fund rank:

Ft

Nt�1¼ a0þ

a1I

Nt�1covðni,t�1,ki,t�1Þþ

a2I

Nt�1covðni,t�1,k2

i,t�1Þ

þcontrols: ð3Þ

Again, under the flow-performance model, a1 shouldbe positive if flows are increasing in returns, and a2

should be positive if they are convex.10

4. Market share model

Producing a fractional flow model based on Eq. (1) thatdoes not link aggregate flows to the relative performance offunds by size is not easy. Ultimately, the model needs toadjust each fund’s flow not only for its own return but alsofor the returns produced by all other funds, as well as theaggregate flow in that period. Market shares provide analternative solution to this specification problem becausethey are easily applied without forcing a link between thedistribution of fund performance and aggregate flows.

To operationalize the market share model, let Dmi,t

represent fund i’s change in market share in period t. ThenDm i,t can be written as

Dmi,t ¼ni,t_Nt

�ni,t�1

Nt�1: ð4Þ

The left-hand side of Eq. (4) captures the change in afund’s market share due to both the flows it receives andthe returns generated to garner them.

_Nt and Nt�1 are

based on the funds that are in existence only in periodt�1. The arc over the N term indicates that it is theaggregate AUM of the funds that were present at timet�1. This is in contrast to the N term without the arc,which includes the AUM of all funds that exist in thecorresponding period. Thus, in Eq. (4) and in all of thetests that follow, period t market share growth is definedas growth relative to the peer group in existence as ofperiod t�1.11

Any error term based on Eq. (4) is unlikely to behomoskedastic across funds. Larger funds naturally seea greater variation in their market share changes. Onemight consider mitigating this by dividing the right-handside of Eq. (4) by the fund’s initial market share. However,doing so destroys the adding-up condition that the errorssum to zero under all circumstances. This is not trivial

9 One can use the demeaned ranks for a somewhat more intuitive

measure with regard to the squared ranks. But the results are mathe-

matically identical except for the estimated constant.10 While both Warther (1995) and Fant (1999) examine aggregate

flows, neither examines how they vary in response to the rank perfor-

mance of funds of various sizes. Warther does, however, look at what

might be viewed as the inverse question: the impact of fund flows on

stock returns across stock size deciles.11 Industry can always be defined as the set of funds currently being

ranked against each other. Thus, market share is relative to the market

for which data are being drawn. This is similar to when someone

compares an automotive company’s luxury car market share rather

than its overall automobile market share.


because it is likely to imply that the sum of the marketshare changes depends upon whether larger or smallerfirms do particularly well, which is clearly impossible.Another route, the one used here, is to examine the resultsboth on a fund-by-fund basis and in funds-of-fundstests that have equal initial market shares.12 Becausethe synthetic funds in the fund-of-funds approach haveequal initial market shares, this should eliminate anyeffects due to size-related heteroskedasticity.13 In anycase, the fund-by-fund and fund-of-funds tests can alwaysbe viewed as robustness checks for each other.14

Based on the above discussion, the paper employs

Dmi,t ¼ a0þg ki,t�1

� �þcontrolsi,tþui,t , ð5Þ

where ui,t is an error term. Often g is assumed to follow astep function of some sort. In this case, funds are placed inone of a smaller number of groups (for instance, deciles)based on their returns, and g is then estimated via a set ofdummy variables. Eq. (5) now becomes

Dmi,t ¼XD

j ¼ 1

ajd ki,t�1

� �þcontrolsi,tþui,t , ð6Þ

where aj is an estimated dummy parameter and d amapping from rankings into zero-one dummies for eachof the D-cile groups. This leads to the Proposition 3 whichspells out the empirical questions that are typically ofinterest.

Proposition 3. Under the alternative that fund flows are

convex in rank, the difference between aj and aj�1 should

be increasing in j. In that case, increasing a fund’s return

volatility will yield a positive expected change in market

share. Equivalently, E[Dmi,t] is increasing in a fund’s return

volatility.

While most of the flow-return literature has focusedon the alternative hypothesis of whether or not flowsare convex in ranks, the economically important issue iswhether or not fund flows react to returns in a way thatencourages managers to increase the risk of their portfolio.This paper compares the standard percentage flow specifi-cation to using changes in market share. Other possibleempirical models exist that one might consider in the hopeof retaining the fractional flow model while simultaneouslyconstraining the sum of the in-sample individual fund flowestimates, no matter the return ordering, to equal thesample aggregate value. This is simply an adding-up con-straint that ensures the in-sample flow estimates have amean zero error, a condition similar to that yielded by aproperly specified ordinary least squares model. For

12 An analog to this procedure can be found in studies that compare

results when firms are equally weighted and value weighted. The former

corresponds to this paper’s tests on a firm-by-firm basis and the latter to

tests using equal market share groupings.13 Heteroskedasticity does not lead to biased coefficients. At worst,

it should reduce the efficiency with which parameters are estimated.14 This does not rule out alternative hypotheses. For example, one

might suspect that market share changes are serially correlated. This

might occur if a particular fund goes from being excluded to be included

in a large 401(k) plan. Naturally, the equation can be adapted to this and

any number of other scenarios.

example, instead of using changes in market share, onecan use deviations from the fund’s expected flow given itssize and the period’s aggregate flow. In this case, thedependent variable becomes fi,t/Ft�ni,t�1/Nt. This specifica-tion satisfies the adding-up constraint because its sum overall funds always equals zero. However, it places aggregateflows (Ft) in the denominator, and in many periods Ft isnear zero. Even ignoring that, a specification such as thisforces the regression line to pay particular attention to lowaggregate flow periods as that is when the dependentvariable takes on its largest values.15 Using fi,t�Ftni,t�1/Nt

solves the problem of having aggregate flows in thedenominator but now leads to a regression equation thatis no longer unit free, in this case dollars. Another optionwould be to substitute percentage flow deviations from themean fi,t/ni,t�1�Ft/Nt�1 instead of the absolute deviation inEq. (1). This avoids having aggregate flows in the denomi-nator, but the adding-up constraint is now violated, withthe error again systematically related to how well smallfunds do relative to large ones. For example, suppose thereare two funds of size 10 and 90 and aggregate flows of 30.If they all go to the small fund, the aggregate error is30/10þ0/90�2�30/100 or 2.4. If they all go to the largefund, it becomes 0/10þ30/90�2�30/100 or �0.67.

5. Data

The data come from the CRSP survivorship bias-freemutual fund data set and the addenda page for Cremersand Petajisto (2009) at www.sfsrfs.org. From the CRSP data,only nonspecialty domestic equity funds are included in thefinal sample (Lipper Objectives EI, EIEI, ELCC, EMN, G, GI, I,LCCE, LCGE, LCVE, LSE, MC, MCCE, MCGE, CMVE, MLCE,MLGE, MLVE, MR, SCCE, SCGE, SCVE, and SG). The analysisthat follows uses category-adjusted returns. These arecreated by taking a fund’s return and subtracting theaverage return for funds that have the same Lipper Objec-tive code in that period.

To be included in a particular period, a fund’s datahave to include its AUM and returns. Our data containquarterly AUM since 1970 and monthly AUM since 1991.Both data sets end in 2006. Funds are also dropped fromperiods in which their flows appear to be due to dataentry errors.16 If flows in periods t and tþ1 have oppositesigns and are both at least ten times larger than flows inboth periods t�1 and tþ2, then that fund is removedfrom that period’s data.

6. Summary statistics

This section shows that aggregate patterns in the dataexhibit substantial variation over time. It then goes on to

15 We ran this specification with aggregate flows in the denomi-

nator on the historical data. The results were plagued by econometric

problems due to small values of Ft.16 Similarly to Huang, Wei, and Yan (2007), we filter out the top and

bottom 2.5% tails of the net flow data, which could be problematic due

to misreported fund mergers and splits. Our main results are robust to

this filtering.

www.sfsrfs.org


show that even simple graphs and regressions indicate thatflows and market shares yield different conclusions regard-ing how investors react to a fund’s past performance.

6.1. Variation in the independent variables

Table 2, presents summary statistics for traditionalflow measures fi,t and fi,t/ni,t�1. While statistics werecalculated for each year, in the interest of space the tablepresents data only from every five years for the quarterlydata and every other year for the monthly data. Theimportant item to note is that aggregate fund flowsrelative to AUM have been fairly volatile over time. Theaverage across funds ranges from a low of �4.8% in 1976to a high of 17.8% in 1996 and is typically positive. Thisvariation can have a large impact on a fractional flowmodel that does not properly account for it. In contrast,market share changes handle this variation without anyadditional controls because they automatically adjust foraggregate flows.

The latter columns of Table 2 under the Quarterly andMonthly Data headers display the time series distributionof market shares in the sample. Over time, the mutualfund industry has become less and less concentrated. Inpart, this implies the data include a constant inflow ofnew funds that are likely to produce data similar to thehot funds displayed in Figs. 1 and 2.

Table 3 tabulates the distribution of the two measures[Eqs. (2) and (3)] proposed for testing whether or notaggregate fund flows are positively correlated with how

Table 2Flow data summary statistics.

Summary statistics for the quarterly (1971–2006) and monthly flow data (199

the number of funds with sufficient data [non-missing period flow, assets und

year. The next two columns report the average and t-statistics (in parenthesis) fo

the average dollar inflows, in millions of dollars, across funds. The f t=nt�1 colum

across funds. The median market share column represents the median market

Asterisks indicate significance levels: n¼ 10%, nn

¼ 5%, and nnn¼ 1%.

Quarterly data

Year Number of

fundsf t f t=nt�1

Market share basis

points

Yea

(millions) (percentage) 10% Median 90%

1971 208 �1.92 1.52 0.92 11.66 117.01 199

(3.65)nnn (2.90)nnn

1976 219 �4.76 �2.50 1.96 11.81 126.57 199

(6.91)nnn (11.55)nnn

1981 241 �2.61 �0.77 2.58 15.28 97.14 199

(3.00)nnn (2.44)nn

1986 396 15.47 8.00 0.70 8.48 64.57 199

(3.69)nnn (9.14)nnn

1991 837 8.69 5.69 0.21 2.53 28.94 200

(5.01)nnn (11.32)nnn

1996 2513 17.78 21.41 0.01 0.40 7.04 200

(8.03)nnn (21.77)nnn

2001 5467 7.52 8.99 0.01 0.16 2.83 200

(8.48)nnn (29.62)nnn

2006 4274 �2.77 �1.29 0.02 0.28 3.98 200

(1.70)n (11.65)nnn

well larger funds perform. These statistics indicate thata great deal of intertemporal variation exists in howwell large funds do relative to small ones. Furthermore,because ranks are calculated net of the market return,large fund performance is not highly correlated with theoverall market.

6.2. The classic flow-performance relation

Fig. 3 displays the classic fractional-flow lagged returnconvexity found in the literature, starting with Sirri andTufano (1998). To create the graph, a fund is placed ina vigintile based upon its period t�1 performance asmeasured by its market or category-adjusted return. Themarket-adjusted return equals the fund’s return minusthat of the CRSP market portfolio’s return for that period.Similarly, a fund’s category-adjusted return is defined asits return minus its Lipper category’s average return forthat period. The lowest performing funds are placed intoVigintile 1; the best, Vigintile 20. Each vigintile includes5% of all assets under management as of period t�1. Next,the average period t change in market share or percentageflow is calculated for each vigintile. The resulting valueis then averaged over periods to generate the displayedvalues.

While fractional flows have the standard convex shape,market share changes yield a much straighter line. Asusual, fractional flows seem to indicate a clear incentivefor funds to increase their risk to attract assets. However,the evidence for this tactic’s value is considerably lessobvious when market share changes are used.

1–2006) used in the paper. The first column reports the year followed by

er management, and return information] in the middle of each selected

r two flow measures found in the prior literature. The f t measure reports

n presents the period’s average flow to lagged funds under management

share across funds in a particular period in basis points, i.e. 10000ni,t/Nt.

Monthly data

r Number of

fundsf t f t=nt�1

Market share basis

points

(millions) (percentage) 10% Median 90%

2 978 4.688 2.862 0.149 2.031 23.01

(7.04)nnn (13.07)nnn

4 1671 3.695 3.101 0.043 0.894 11.57

(7.44)nnn (17.20)nnn

6 2568 3.906 3.548 0.014 0.388 7.08

(5.99)nnn (24.02)nnn

8 3977 3.162 3.882 0.008 0.239 4.05

(6.55)nnn (28.37)nnn

0 5173 1.881 2.381 0.005 0.154 2.94

(3.70)nnn (22.81)nnn

2 5574 �1.148 0.238 0.0061 0.182 3.00

(3.23)nnn (4.73)nnn

4 4890 1.545 0.056 0.013 0.235 3.45

(3.32)nnn (8.48)nnn

6 4263 �2.044 �0.785 0.016 0.267 3.93

(4.21)nnn (22.74)nnn

Table 3Summary statistics for the correlation between fund size and performance rank.

Summary statistics for the distribution of the n�Rank/N and the I� cov(n, Rank)/N variables. The variable n equals the assets under management for an

individual fund and Rank is its normalized rank from zero to one based on the performance measure being used. The variable I is the total number of

funds and N their aggregate assets under management. The last two columns report the correlations between the variables and market excess return and

the corresponding p-values.

Category-adjusted return-based ranks Correlation to market excess return

Correlation Measure Mean Standard deviation 10% 25% 50% 75% 90% Correlation p-value

Quarterly dataSum(n�Rank)/N 54.35 6.50 48.23 51.02 54.49 57.66 60.75 0.20 0.013

Sum(n�Rank2)/N 36.67 6.05 29.91 33.00 36.41 39.99 44.23 0.19 0.018

I� cov(n, Rank)/N 2.32 4.95 �4.42 �1.55 2.10 5.44 8.68 0.20 0.013

I� cov(n, Rank2)/N 1.10 5.51 �5.75 �2.90 0.56 4.33 8.79 0.19 0.022

Monthly dataSum(n�Rank)/N 53.66 5.97 47.63 50.45 53.87 56.97 60.09 �0.09 0.231

Sum(n�Rank2)/N 35.94 5.57 29.60 32.17 35.81 39.42 42.71 �0.08 0.271

I� cov(n, Rank)/N 1.51 4.77 �5.10 �1.95 1.44 4.70 7.99 �0.06 0.382

I� cov(n, Rank2)/N 0.26 5.18 �6.60 �3.72 0.00 3.73 7.19 �0.06 0.421

-0.5

0

0.5

1

1.5

2

2.5

-4

-3

-2

-1

0

1

2

3

4

5

0 2 4 6 8 10 12 14 16 18 20

∆m in

bas

is p

oint

s

Vigintile

Δm (in basis points, left axis) of market adjusted return vigintilesΔm (in basis points, left axis) of category adjusted return vigintilesf/n (in percent, right scale) of market adjusted return vigintilesf/n (in percent, right scale) of category adjusted return vigintiles

/n in

per

cent

Fig. 3. Fractional flows and market share changes graphed against a fund’s return performance rank. Performance ranks are calculated monthly for each

month t from the start of 1991 through the end of 2006. Fractional flows and market share changes are for month tþ1.


The differences displayed in Fig. 3 could arise from alack of controls. Tables 4–6 examine whether this is truein a standard Fama and MacBeth regression framework.Table 4 begins the analysis by examining a series of piece-wise linear regressions of the form {100Dmi,t or fi,t/ni,t�1}¼aþb1� Lowi,t�1þb2�Midi,t�1þb3�Highi,t�1þcontrolsþei,t,where Low, Mid and High refer to break points in the linearspecification. Break points are estimated by using the R2

maximizing values. The control variables used in the Famaand MacBeth-style regressions are Cat_flow, the monthlydollar flow divided by the one-month lagged AUM for allfunds in the category; Fee, the fund’s annual expense ratioplus actual 12b-1 fees in the prior calendar year; LogAge, thelog of the fund’s age since inception in years; LogSize, the logof the fund’s AUM in millions; Vol, the standard deviation ofthe fund’s return over the prior calendar year; Turnover, thefund’s turnover ratio from the prior calendar year; and Lag_u,the value of the dependent variable in the prior period. Whenapplicable, control variables are averaged over funds in eachvigintile.

Table 4, Panel A and Panel C report the parameterestimates using market shares and fractional flows as thedependent variables, respectively. Panel B and Panel Dtest whether the slope parameters Mid�Low and High-

�Mid are statistically different from each other. Usingmarket shares, some evidence exists of convexity (Models1, 2, 6, and 7), so long as the model does not include manyof the standard controls. However, once these are addedthe convexity disappears. In Model 10, the difference inthe Mid-Low slope parameter is statistically significantbut has the wrong sign. By contrast, in the fractional flowspecifications, every single High�Mid estimate is statisti-cally significant and has the right sign.

The flow performance literature contains innumerablespecifications, many of them continuous in nature. Table 5examines the degree to which this influences the results.Models 1 through 3 regress subsequent market share changeson a fund’s return rank (Rank). Models 4 through 6 repeat theanalysis using a fund’s actual return (Perf). Panel A measuresreturns using a fund’s market-adjusted return. Panel B

Table 4Piecewise linear models of market share change and flow-performance relation.

This table measures both monthly market share changes and flows via Dmi,t ¼ ni,t=Nt�ni,t�1=Nt�1 and f i,t=ni,t�1 respectively, in percentage terms. Assets under management for fund i in period t equal ni,t and

the sum across funds Nt. In Panel A, market share changes are regressed on ranks using the piecewise linear regression: 100Dmi,t ¼ aþb1 � Lowi,t�1þb2 �Midi,t�1þb3 � Highi,t�1þcontrolsþei,t , where

Lowi,t�1 ¼ ranki,t�1 � Ifi 2 LowBing are the ranks (normalized to be 0.05, 0.1,y, 1) of the low performance bin and zero otherwise, (I{} is an indicator function), Midi,t�1 ¼ ranki,t�1�MaxðLow� �

Þ � Ifi 2MidBing, and

Highi,t�1 ¼ ranki,t�1�MaxðLow� �

�MaxðMidÞÞ � Ifi 2 HighBing. To generate the three bins begin by considering the set of all possible three-group splits of the 20 rank vigintiles. The algorithm then selects the

division with the largest average R2 in the cross-sectional regressions. Panel A reports the time series average of regression parameters as well as the Newey and West adjusted with three lags t-statistics. The

last two rows report the break points for the three bins. Panel B displays the difference between the break points with the t-statistics in parenthesis. Panels C and D repeat the analysis with fractional flows as the

dependent variable. Asterisks indicate significance levels: n¼ 10%, nn

¼ 5%, and nnn¼ 1%.

Panel A: Piecewise regressions on market share changes

Performance measure ¼ Market-adjusted Return Performance measure ¼ Category-adjusted return

Regression parameters Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 Model 10

Constant �0.0375 �0.0345 �0.0140 0.0085 0.0153 �0.0230 �0.0592 0.0781 0.0738 0.0697

(3.90)nnn (2.34)nn (0.40) (0.23) (0.38) (4.64)nnn (4.08)nnn (2.63)nnn (2.17)nn (1.14)

Low 0.0697 0.0632 �0.0030 �0.0088 �0.0021 0.0390 0.0120 0.0152 0.0085 0.1445

(3.80)nnn (3.52)nnn (0.32) (0.97) (0.10) (3.85)nnn (0.18) (1.24) (0.69) (2.88)nnn

Mid 1.4395 1.1642 0.0549 �0.0356 0.0728 0.8585 0.0565 0.2014 0.1579 �0.1130

(4.04)nnn (3.54)nnn (0.90) (0.32) (0.88) (4.28)nnn (3.34)nnn (1.69)n (1.17) (1.30)

High 1.6565 1.2858 0.1439 0.0399 0.0684 0.5081 1.1372 0.2215 0.2156 0.0862

(3.96)nnn (3.54)nnn (2.18)nn (0.47) (0.23) (4.52)nnn (3.32)nnn (2.06)nn (1.71)n (1.28)

Cat_flow 0.0201 0.0158 0.0132 0.0659 0.0509 �0.0010

(1.44) (0.72) (0.54) (3.57)nnn (2.00)nn (0.02)

Fee 0.1431 �1.1030 �2.2822 �2.0455 �3.3651 �2.5750

(0.07) (0.55) (0.76) (0.94) (1.19) (0.47)

LogAge �0.1184 �0.1161 �0.0973 �0.0807 �0.0780 �0.0897

(4.12)nnn (4.55)nnn (3.32)nnn (2.81)nnn (2.59)nnn (2.96)nnn

LogSize 0.0347 0.0327 0.0252 �0.0026 �0.0063 �0.0161

(3.34)nnn (2.98)nnn (2.04)nn (0.24) (0.53) (1.20)

VOL 0.4465 0.2425 0.5653 0.7281 0.7078 1.9310

(0.62) (0.28) (0.48) (0.82) (0.76) (1.40)

Turnover �2.0866 �2.1152 �2.4706 �2.7283 �2.2182 �2.6984

(1.11) (1.14) (1.10) (1.11) (0.93) (1.07)

Lag_u 0.0188 0.0029 0.0157 0.0140 0.0231 0.0087

(0.50) (0.08) (0.34) (0.37) (0.54) (0.18)

Dummy_Mid �0.0039 0.0844

(0.25) (3.95)nnn

Dummy_High 0.0072 0.0614

(0.23) (3.70)nnn

Piecewise break points

Mid-Low 18 18 13 14 14 16 2 15 15 9

High-Mid 19 19 16 16 18 17 19 17 17 14

Panel B: Difference between piecewise regression parameters in Panel A

High-minus-Mid 0.2170 0.1216 0.0890 0.0756 �0.0043 �0.3504 1.0807 0.0202 0.0577 0.1992

(1.25) (0.75) (1.61)n (0.88) (0.01) (2.77)nnn (3.30)nnn (0.20) (0.46) (1.74)n

Mid-minus-Low 1.3698 1.1010 0.0579 �0.0268 0.0749 0.8194 0.0445 0.1861 0.1494 �0.2575

(4.03)nnn (3.52)nnn (1.00) (0.25) (0.97) (4.22)nnn (0.73) (1.67)n (1.17) (2.20)nn

M.

Spieg

el,H

.Z

ha

ng

/Jo

urn

al

of

Fina

ncia

lE

con

om

ics1

08

(20

13

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06

–5

28

51

7

Table 4 (continued )

Panel A: Piecewise regressions on market share changes

Performance measure ¼ Market-adjusted Return Performance measure ¼ Category-adjusted return

Panel C: Piecewise Regressions on f/n

Constant 0.0438 �0.1567 0.7766 0.7270 0.9031 0.0807 �0.4138 0.7101 0.5168 0.0931

(0.38) (1.18) (2.61)nnn (2.37)nn (1.66)n (0.69) (2.71)nnn (2.56)nn (1.41) (0.16)

Low �1.8601 �4.1297 �1.5602 0.3980 0.4652 �0.3626 �2.7737 �2.2685 �4.6619 1.6281

(0.70) (1.69)n (1.10) (0.12) (0.26) (0.12) (0.96) (0.76) (1.44) (0.48)

Mid 1.1585 0.9778 0.6408 0.5174 0.2527 1.0985 1.0496 0.9677 0.5482 0.7294

(6.58)nnn (5.37)nnn (2.95)nnn (3.46)nnn (0.96) (7.56)nnn (6.65)nnn (6.32)nnn (3.27)nnn (3.32)nnn

High 41.9407 36.8623 29.2007 15.5057 8.0538 39.0263 36.4472 29.3096 11.9701 8.9318

(8.49)nnn (8.11)nnn (6.17)nnn (6.56)nnn (4.12)nnn (8.95)nnn (8.49)nnn (7.08)nnn (5.59)nnn (3.01)nnn

Cat_flow 0.5748 0.5131 �0.0223 0.7832 0.5408 0.6198

(4.09)nnn (2.56)nn (0.07) (4.28)nnn (1.83)n (1.57)

Fee 3.8626 2.3523 �9.5667 77.3117 49.3525 30.9978

(0.17) (0.10) (0.38) (3.54)nnn (1.96)nn (1.14)

LogAge �1.7015 �1.4316 �1.4364 �1.6432 �1.5637 �1.3224


LogSize 0.3801 0.3590 0.3278 0.1151 0.1305 0.0204

(3.86)nnn (3.63)nnn (2.56)nn (1.06) (1.12) (0.17)

VOL 14.3121 1.9452 1.7310 3.9698 3.2266 7.5177

(1.80)n (0.24) (0.18) (0.51) (0.32) (0.71)

Turnover �13.1872 �3.0487 4.6388 �20.8002 �19.3581 3.5106

(1.16) (0.41) (0.64) (0.88) (0.86) (0.64)

Lag_u 0.2073 0.5053 0.5734 �0.3638 �0.1479 �0.2297

(0.56) (1.30) (1.39) (0.97) (0.36) (0.54)

Dummy_Mid 0.2144 0.3005

(0.80) (0.87)

Dummy_High 0.0901 0.7002

(0.24) (1.80)n

Piecewise break points

Mid-low 1 1 2 1 3 1 1 1 1 2

High-mid 19 19 19 18 17 19 19 19 18 18

Panel D: Difference between piecewise regression parameters in C

High-minus-Mid 40.7822 35.8846 28.5599 14.9882 7.8012 37.9278 35.3976 28.3419 11.4219 8.2025


Mid-minus-Low 3.0186 5.1075 2.2010 0.1195 �0.2125 1.4611 3.8233 3.2362 5.2101 �0.8988

(1.15) (2.11)nn (1.52) (0.04) (0.12) (0.50) (1.33) (1.10) (1.60) (0.26)

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Table 5Market share change-performance relation continuous models.

This table measures monthly changes in market share via Dmi,t ¼ ni,t=Nt�ni,t�1=Nt�1 in percentage terms (100Dmi,t). Here, ni,t�1, Nt�1, ni,t , and Nt are

lagged and concurrent assets under management of the fund and of all funds in the sample. Panel A reports on a regression of market share changes using

the following regression: 100Dmi,t ¼ aþg Ranki,t�1

� �þg Perf i,t�1

� �þcontrolsþei,t , where the independent variables Ranki,t�1 and Perf i,t�1 are the ranks

(normalized to be 0.05, 0.1, y, 1) and real performances (in percentage terms) of the 20 rank vigintiles, respectively, and g(.) is a function of ranks and

performances. Panel A reports the time series average of regression parameters as well as the Newey and West adjusted with three lags t-statistics. Panel

A uses a fund’s return net of the market to calculate its performance rank or return. Panel B uses a fund’s return net of the average return in its Lipper

category. Asterisks indicate significance levels: n¼10%, nn

¼5%, and nnn¼1%.

Panel A: Market-share changes to market adjusted return

Coefficient Model 1 Model 2 Model 3 Model 4 Model 5 Model 6

Constant �0.0023 �0.0224 �0.0191 0.0205 �0.0042 0.0005

(0.07) (0.65) (0.48) (0.62) (0.13) (0.01)

Rank 0.0469 0.0412 0.0260

(2.42)nn (1.72)n (0.98)

Rank2 0.0432 0.0645

(0.91) (1.26)

Rank3 0.0721

(0.81)

Perf 0.0023 0.0032 0.0073

(1.09) (1.39) (1.67)n

Perf2�0.0005 0.0002

(0.87) (0.16)

Perf3�0.0002

(0.62)

Cat_flow 0.0141 0.0140 �0.0033 0.0151 0.0192 0.0012

(0.73) (0.63) (0.13) (0.82) (1.00) (0.06)

Fee 1.4780 0.4219 0.2629 1.4231 0.7251 0.5000

(0.82) (0.21) (0.16) (0.95) (0.45) (0.29)

LogAge �0.1055 �0.0941 �0.0925 �0.1088 �0.1034 �0.0940


LogSize 0.0272 0.0302 0.0288 0.0277 0.0292 0.0324


VOL 0.6827 0.3505 0.2735 �0.0391 0.5330 �0.1328

(1.01) (0.42) (0.27) (0.06) (0.71) (0.14)

Turnover �2.1989 �2.1988 �2.3386 �1.8873 �2.0572 �2.1723

(1.14) (1.14) (1.15) (1.12) (1.12) (1.11)

Lag_u �0.0123 �0.0138 �0.0065 �0.0008 0.0311 0.0093

(0.35) (0.36) (0.16) (0.02) (0.91) (0.24)

Panel B: Market-share change to category-adjusted return

Constant 0.0651 0.0670 0.0863 0.0582 0.0581 0.0779

(2.30)nn (2.35)nn (2.69)nnn (1.87)n (2.02)nn (2.63)nnn

Rank 0.0617 0.0598 0.0443

(3.35)nnn (3.18)nnn (1.57)

Rank2�0.0460 �0.0720

(0.89) (1.34)

Rank3 0.1225

(0.82)

Perf 0.0083 0.0080 0.0119

(4.39)nnn (4.12)nnn (3.77)nnn

Perf2�0.0002 0.0001

(0.31) (0.07)

Perf3�0.0003

(1.15)

Cat_flow 0.0409 0.0663 0.0371 0.0568 0.0470 0.0244

(1.46) (3.34)nnn (1.54) (2.45)nn (1.92)n (1.31)

Fee �0.7938 �0.0193 1.4354 �0.5745 �0.7958 0.4799

(0.37) (0.01) (0.53) (0.24) (0.27) (0.19)

LogAge �0.0543 �0.0658 �0.0739 �0.0802 �0.0976 �0.0867

(2.35)nn (2.66)nnn (2.66)nnn (3.20)nnn (3.71)nnn (3.32)nnn

LogSize �0.0119 �0.0179 �0.0184 �0.0081 �0.0035 �0.0075

(1.20) (1.58) (1.42) (0.76) (0.34) (0.64)

VOL 0.6054 1.2972 0.8699 0.8740 0.7823 0.8181

(0.73) (1.56) (0.97) (1.01) (0.84) (1.01)

Turnover �3.0710 �2.6257 �3.5006 �2.6550 �2.8301 �3.2889

(1.25) (1.06) (1.34) (1.10) (1.14) (1.28)

Lag_u �0.0098 �0.0125 0.0025 0.0066 0.0186 0.0111

(0.34) (0.37) (0.07) (0.21) (0.60) (0.32)


Table 6Fractional flow-performance relation continuous models.

This table measures monthly flows via f i,t=ni,t�1 in percentage terms. Here, f i,t and ni,t�1 are concurrent flows and lagged assets under management. The

independent variables Ranki,t�1 and Perf i,t�1 are the ranks (normalized to be 0.05, 0.1, y, 1) and real performances (in percentage terms) of the 20 rank

vigintiles, respectively. Panel A reports the time series average of regression parameters as well as the Newey and West adjusted with three lags

t-statistics. Panel A uses a fund’s return net of the market to calculate its performance rank or return. Panel B uses a fund’s return net of the average

return in its Lipper category. Asterisks indicate significance levels: n¼10%, nn

¼5%, and nnn¼1%.

Panel A: f/n to market-adjusted returns

Coefficient Model 1 Model 2 Model 3 Model 4 Model 5 Model 6

Constant 0.7343 0.9577 1.0062 1.3030 1.3401 0.9504

(1.99)nn (2.41)nn (2.32)nn (3.85)nnn (4.09)nnn (2.63)nnn

Rank 0.8818 0.9444 0.1010

(3.88)nnn (3.90)nnn (0.35)

Rank2 1.2176 1.7271

(2.37)nn (3.03)nnn

Rank3 6.9135

(3.53)nnn

Perf 0.1777 0.1649 0.1169

(5.41)nnn (5.56)nnn (2.42)nn

Perf2 0.0248 0.0363

(3.10)nnn (2.93)nnn

Perf3 0.0024

(0.72)

Cat_flow 0.5075 0.3597 0.3439 0.4204 0.4806 0.3223

(2.71)nnn (1.78)n (1.51) (2.17)nn (2.47)nn (1.55)

Fee 45.2438 34.3499 6.2017 28.8977 19.4399 27.1525

(1.72)n (1.31) (0.25) (1.19) (0.78) (1.09)

LogAge �2.0626 �2.0520 �1.7507 �1.5653 �1.2968 �1.3428


LogSize 0.4050 0.4544 0.3672 0.2742 0.2946 0.4273


VOL 11.2776 6.4077 4.5569 �3.7539 �8.9780 �6.1746

(1.36) (0.77) (0.49) (0.45) (1.03) (0.61)

Turnover �6.5899 �9.0258 �5.4088 �1.6767 �7.9016 �8.8822

(0.91) (0.95) (0.65) (0.31) (0.97) (0.90)

Lag_u 0.0369 0.1004 0.2220 0.2868 0.1161 0.1367

(0.10) (0.29) (0.61) (0.78) (0.30) (0.36)

Panel B: f/n to category-adjusted return

Constant 0.1280 0.1793 0.3005 0.6088 0.8488 0.8634

(0.32) (0.44) (0.73) (1.67)n (2.37)nn (2.25)nn

Rank 1.0239 1.0661 0.4711

(5.66)nnn (5.24)nnn (1.90)n

Rank2�0.4831 �0.5584

(0.68) (0.91)

Rank3 5.3279

(3.44)nnn

Perf 0.2116 0.1894 0.1804

(8.19)nnn (7.71)nnn (4.15)nnn

Perf2 0.0174 0.0198

(2.31)nn (2.05)nn

Perf3 0.0011

(0.20)

Cat_flow 0.6092 0.5141 0.2877 0.5535 0.3978 0.2688

(2.76)nnn (2.27)nn (1.05) (2.63)nnn (1.97)nn (1.03)

Fee 77.0647 90.5576 97.7693 75.3953 68.2900 74.2156


LogAge �1.6829 �1.6857 �1.5228 �1.5789 �1.5426 �1.5453


LogSize 0.1430 0.1317 0.0766 0.0883 0.0801 0.1074

(1.25) (1.10) (0.54) (0.76) (0.73) (0.92)

VOL 17.8114 19.4200 17.7367 12.5532 7.2065 8.0909

(1.86)n (1.92)n (1.51) (1.20) (0.77) (0.86)

Turnover �27.4627 �24.2120 �23.0203 �25.2063 �22.7954 �23.2045

(1.15) (0.99) (0.97) (1.06) (1.00) (1.08)

Lag_u �0.0912 �0.0704 �0.4250 �0.0257 �0.0350 �0.4232

(0.23) (0.18) (1.18) (0.07) (0.10) (1.11)



conducts the same analysis but this time uses a fund’scategory-adjusted return. If market shares are convex inperformance, then this relation should result in positive Rank2

and Perf2 parameter estimates.The evidence for convexity in Table 5, Panel A is very

weak. While three out of four estimates using either Rank2

or Perf2 are positive, none is statistically significant. Incomparison, the linear Rank and Perf parameters are bothpositive in the six specifications in which they appear, aswell as significant at the 10% level in two cases and at the5% level in one case. Panel B offers even less evidence ofconvexity. In this case, three of the four estimated para-meters for the quadratic terms are negative. Again, noneis statistically significant. Furthermore, this time not onlyare all the parameter estimates for the linear rank andperformance terms positive, but five out of six aresignificant at the 1% level as well. The contrasting resultsboth in terms of consistency and statistical significancebetween the linear and quadratic terms indicate thatpower is not the issue. If it were, one would expect thelinear terms to come out with inconsistent signs and to beinsignificant, yet that is not what happens.

Table 6 repeats the analysis with the traditional fractionalflow measure as the dependent variable. With respect toconvexity, they differ considerably from those in Table 5. Inline with prior studies, the convexity parameter is positive insix out of the eight models in which it appears. In the six runswhere it is positive, it is statistically significant at the 5% levelin three cases and at the 1% level in three others. The linearterm is again positive in all 12 runs and significant at the 1%level in nine of those. Overall, there appears to be enoughdata to detect convexity where it exists. The fact that there isminimal evidence for it in Table 5 would thus seem to implythat if a fund’s change in market share is convex in perfor-mance, then these tests cannot detect it.

While a market share model can detect a positive relationbetween relative fund growth and past performance, itsinability to detect a convex relation could be due to a lackof power with respect to that particular parameter. To checkthis possibility, a bootstrapped power test was conducted.In each period the funds that actually existed are randomlyassigned a performance rank. Their fractional flow is thengiven by the estimates displayed in Fig. 3. To this is added anormally distributed mean zero error term with varianceequal to the fund-by-fund residual variance based on theFig. 3 estimates. This is repeated for each year in the data toform a panel data set. The Table 5 regression Model 2 is thenrun and the resulting estimates and t-statistics are recorded.This is repeated two thousand times. Overall, these testsproduce a squared rank and squared performance parameterthat is statistically significant at the 5% level over 85% of thetime. It, thus, appears that the model has sufficient power todetect convexity in the flow performance relation if it exists.

6.3. Robustness: alternative performance flow periods

All of the tests to this point have involved monthly data.Tests that use other periodicities or allow for seasonalitycould yield results consistent with prior studies, indicating aconvex flow-performance relation. To that end, tests usingsix-month return periods followed by six month flow periods

were also run. The results are qualitatively unchanged.Another set of robustness tests were conducted using theperiod lengths found in Brown, Harlow, and Starks (1996).They hypothesize that consumers pay particular attention tocalendar year returns. To test this they suggest looking atfunds that do poorly in January through June in year t andthen seeing if they increase their risk from July throughDecember of year t. If flows are convex in the prior calendaryear returns, then this strategy increases the expected flowsto such funds in the full calendar year tþ1. We repeat theirtests and find results that are again qualitatively identical tothose presented so far. Funds that do poorly in the first sixmonths of year t and then increase their risk over the next sixmonths do not, on average, grow any faster than other fundsin the subsequent calendar year in terms of market share.

7. Aggregate flows

Is the convexity displayed by the fractional flow curve inFig. 3, as well as the results found in Table 4 and Table 6, trulyindicative of the flows a fund manager can expect, or doesthe market share curve more accurately represent the actualincentives? Aggregate flows can provide some insight intothis question. From Proposition 1, if the fractional flow modelis properly specified, and if fractional flows are increasingin returns, then aggregate flows should increase followingperiods when large funds perform relatively well comparedwith small ones. From Proposition 2, if fractional flows areconvex in returns, then aggregate flows should also be largerfollowing periods when large funds are in the tails of theperformance distribution than when small funds are in thetails.

Aggregate flows are measured using the procedure foundin Warther (1995). First, the raw aggregate flow measure isconstructed as dollar flows into and out of existing mutualfunds. Next, this value is divided by the total value of the USstock market at the prior month’s end as reported by CRSP.The purpose of this procedure is to correct for changes in thevalue of the investment pool over time. Thus, we now havethe fraction of the market’s total value moving into and out ofmutual funds in each period.

If fund flows are increasing in return ranks, then, asshown in Proposition 1, this implies that aggregate flowsshould be positively correlated with how well the largerfunds perform. Table 7 presents several tests of this hypoth-esis using the measures proposed in Eqs. (2) and (3). Fundsare ranked using the same procedure described at thebeginning of Subsection 6.2, assigning each a rank from 1to 20. A fund’s rank and AUM are then interacted and usedas independent variables to explain aggregate flows.

Overall, the results in Table 7 offer little support for theidea that aggregate fund flows are impacted to a statisti-cally significant degree by the performance of the largestfunds in the industry. Using quarterly data, only four ofthe eight relevant t-statistics are even significant at the10% level, and those all have the wrong sign. Withmonthly data, none of the estimates is significant. Theseresults do not appear to arise from a lack of power. Thestatistical model has little trouble detecting the impactof past market returns on aggregate fund flows. Also,from Table 2, Panel B, the variable Sum(n�Rank)/N has

Table 7Aggregate flows regressed on total assets under management based measures.

In each period funds are sorted into 20 groups using lagged performance. Aggregate flows are then divided by aggregate assets under management and regressed on the following model:

Ft=Nt�1 %ð Þ ¼ g1

Pini,t�1=Nt�1 %ð Þ � ki,t�1þg2

Pini,t�1=Nt�1 %ð Þ � k2

i,t�1þg1 � I � covðni,t�1=Nt�1ð%Þ,ki,t�1Þþg2 � I � covðni,t�1=Nt�1ð%Þ,k2i,t�1Þ,

where Ft is the aggregate flow, ni,t�1 the lagged assets, and ki,t�1 the lagged ranks by group. The value, Nt�1 ¼P

ini,t�1is aggregate total assets and cov ni,t�1 ,ki,t�1

� �is the cross-sectional covariance between ni,t�1

and ki,t�1 : The label Sum(n�Rank)/N equalsP

ini,t�1=Nt�1 %ð Þ � ki,t�1 and Sum(n�Rank2)/N equalsP

ini,t�1=Nt�1 %ð Þ � k2i,t�1. Similarly, I� cov(n, Rank)/N equals I � cov ni,t�1=Nt�1 %ð Þ,ki,t�1Þ

�and I� cov(n, Rank2)/N

equals I � covðni,t�1=Nt�1 %ð Þ,k2i,t�1Þ. The market return MKT is in fractional values. The t-statistics are in parentheses. Asterisks indicate significance levels: n

¼ 10%, nn¼ 5%, and nnn

¼ 1%.

Panel A: Quarterly data

Market-adjusted return-based ranks Category-adjusted return-based ranks

Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 Model 10

Constant 0.0022 0.0052 0.0446 0.0013 0.003 0.0022 0.0091 0.0501 0.0015 0.004

(2.49)nn (0.80) (1.72)n (1.55) (2.08)nn (2.49)nn (1.23) (1.86)n (1.69)n (2.27)nn

Sum(n�Rank)/N �0.0001 �0.002 �0.0001 �0.0021

(0.64) (1.70)n (1.10) (1.84)n

Sum(n�Rank2)/ N 0.0017 0.0018

(1.69)n (1.83)n

I� cov(n, Rank)/N �0.0001 �0.0019 �0.0001 �0.002

(0.64) (1.70)n (1.10) (1.84)n

I� cov(n, Rank2)/N 0.0016 0.0017

(1.69)n (1.83)n

MKT 0.045 0.0465 0.045 0.0465 0.0453 0.047 0.0453 0.047

(2.88)nnn 2.87)nnn (2.88)nnn (2.87)nnn (2.95)nnn (2.96)nnn (2.95)nnn (2.96)nnn

Lag_MKT 0.0081 0.0092 0.0081 0.0092 0.0101 0.0112 0.0101 0.0112

(0.70) (0.81) (0.70) (0.81) (0.93) (1.02) (0.93) (1.02)

Lag_F/N 0.6725 0.6819 0.6816 0.6819 0.6816 0.6726 0.6769 0.6663 0.6769 0.6663


R2 0.4438 0.5204 0.5329 0.5204 0.5329 0.4438 0.5217 0.533 0.5217 0.533

Panel B: Monthly data

Constant 0.0009 0.0008 �0.0023 0.0007 0.0006 0.0009 0.0015 0.0035 0.0007 0.0009

(3.64)nnn (0.59) (0.74) (2.71)nnn (1.89)n (3.64)nnn (0.89) (0.75) (2.74)nnn (2.20)nn

Sum(n�Rank)/N 0.0000 0.0002 0.0000 �0.0001

(0.09) (1.10) (0.49) (0.54)

Sum(n�Rank2)/N �0.0001 0.0001

(1.13) (0.48)

I� cov(n, Rank)/N 0.0000 0.0001 0.0000 �0.0001

(0.09) (1.10) (0.49) (0.54)

I� cov(n, Rank2)/N �0.0001 0.0001

(1.13) (0.48)

MKT 0.0207 0.0206 0.0207 0.0206 0.0206 0.0205 0.0206 0.0205

(4.21)nnn (4.19)nnn (4.21)nnn (4.19)nnn (4.20)nnn (4.19)nnn (4.20)nnn (4.19)nnn

Lag_MKT 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001 0.0001

(2.09)nn (2.02)nn (2.09)nn (2.02)nn (2.10)nn (2.13)nn (2.10)nn (2.13)nn

Lag_F/N 0.5205 0.52 0.524 0.52 0.524 0.5205 0.5158 0.5132 0.5158 0.5132


R2 0.2636 0.3914 0.3939 0.3914 0.3939 0.2636 0.3921 0.3927 0.3921 0.3927

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a relatively wide distribution, as does the covariancebetween size and rank, which should let the model detecttheir impact, if any, on aggregate flows.

From Proposition 2, if fund fractional flows are convex,then aggregate flows should be larger in periods whenthe largest funds are in the performance tails. In Table 7,Models 3 and 8 test for this by including as a regressor sizetimes a fund’s squared demeaned rank. Overall, the resultsoffer little support for this implication of the hypothesis thatfractional flows are convex in performance. With quarterlydata, the terms using squared ranks have the right sign andare statistically significant at the 10% level. However, at thesame time the linear Rank terms turn negative and are alsosignificant at the 10% level. For these estimates to reflectreality, one has to conclude that fractional flows are decliningand convex in performance. Because the former seemsundeniably wrong, it is more likely that something is awrywith the fractional flow model. Monthly data also offer littlesupport for the convex fractional flow model’s implications.In Panel B, neither the linear Sum(n�Rank)/N parameter norits squared rank counterpart are ever statistically significant,although they generally have the right signs.

The evidence in Table 7 is inconsistent with the idea thatinvestors react to the relative performance of large and smallfunds when determining how much to invest in the mutualfund industry. As shown in Propositions 1 and 2, this iscontrary to what the typically estimated performance-flowmodel derived from Eq. (1) implies.17

While Table 7 fails to find a link between aggregateflows and relative performance of large and small funds,perhaps the tests lack the power to do so. To check forthis possibility bootstrapped power tests were conducted.These tests use the same data generation algorithmdescribed in Subsection 6.2 to produce simulated datapanels. As in Subsection 6.2, two independent panel datasets are created. One set is created by taking the actualper period returns, randomly reallocating them acrossthe existing funds, and then using the parameter valuesdisplayed in Fig. 3 to produce flow data. A second set wascreated by simulating normally distributed return datawith the same mean and variance found in the historicaldata and then applying the parameter values in Fig. 3 togenerate the fund flows. Using either data generationmethod, the results are dramatic. In every single case, theregression model produces a positive parameter valueand t-statistic greater than 1.96 on the Sum(n�Rank)/Nand I� cov(n,Rank)/N variables.

One critique of the above power tests might be thatthe Fig. 3 parameter values are derived without controls.To ensure this is not driving the results, the Table 7 powertests are also run with the regression model in Table 6as the basis for calculating fund flows. The results areidentical: One hundred percent of the time both of theparameters relating the distribution of size and returns toaggregate flows are positive and significant. It seems clearthat if fund managers face a flow-performance relation as

17 In general, when aggregate flows are partially affected by shocks

exogenous to the mutual fund industry, the equation could be mis-

specified because its right side only contains endogenous variables. This

mismatch can have a nontrivial impact.

strong as typically estimated, our tests should pick up itsimpact on aggregate flows.

Contrary to what is implied by the fractional flowmodel, it does not appear that investors look at the relativeperformance of funds when determining how much toinvest in the stock market. If they did, the results inTable 7 would include a number of patterns that appearto be absent. The alternative explanation is that investorsfirst decide how much to invest (based on, for example, themarket’s return) and then determine how to allocate theamount across funds. While this behavior is fully compa-tible with a market share-based model, in the absence ofsufficient controls and a well-specified functional form, it islikely to be problematic for a fractional flow model.

8. Reconciling the flow-performance results

So far, the paper has focused on how past fund returnsimpact market share changes and flows. Section 7 showsthat the connection between aggregate flows and thedistribution of fund returns is inconsistent with the percen-tage flow-return relation typically estimated. Subsection 6.2provides evidence that while percentage flows could beconvex in a fund’s relative return, its change in market shareis not. How can these two conflicting results be reconciled?That issue is explored below.

8.1. Regression tests

Returning to the hypothesis proposed in the Introduc-tion—that variations in cross-sectional flow sensitivity leadto the observed empirical patterns —this subsection of thepaper explores the degree to which this might be leadingpercentage flow and market share change regressions todraw different conclusions regarding the impact of risk onfuture expected fund size. The patterns displayed in Fig. 1need to appear only in the cross section to induce specifica-tion problems. However, intertemporal forces can generateproblems as well. For example, the relative position of thedashed lines describing flows to hot and cold funds couldvary with aggregate flows. If high aggregate flows induce thepublic to seek out new investments, then hot money fundswill see particularly large unconditional inflows relative tocold money funds. Also, high aggregate inflows tend to followgood market returns. That, in turn, tends to increase theamount of press the mutual fund industry receives. Addi-tional press can raise and steepen the hot money fund line,relative to the cold money one, as investors not only add totheir portfolio but also reevaluate their existing fund hold-ings.18 For these and perhaps many other reasons, one mightexpect particularly high unconditional flows to both raise theintercept and increase in the slope of the hot money flow-return relation relative to that for cold funds.

The results in Table 8 provide evidence on whether timeseries issues influence flow-return regressions. The testsbegin with period-by-period convexity parameter estimates.Three measures are used: squared performance, squared

18 At the fund level, Sirri and Tufano (1998) find that media attention

is increasing in performance, as do Kaniel, Starks, and Vasudevan (2007).

Table 8Relation between flow convexity and aggregate flow.

This table examines the time-series correlation between flow convexity and the aggregate flow in the mutual fund industry. Convexity values are estimated period by period in the cross section either as the

flow sensitivity to squared performance or ranks from the quadratic regressions in Table 6 or as the difference between High and Mid coefficients of the piecewise regressions in Table 4. The time series of the

convexity measure are then regressed on the aggregate flow or the summation of absolute flows of all funds in the cross section, standardized by lagged total assets. The table reports the regression parameters

and the Newey and West adjusted t-statistics. The last two lines report the R2 of the time series regression, as well as the hypothetical impact of a one standard deviation change in the aggregate flows on

convexity (including both F/N and Sum(9f9)/N, when applicable). Asterisks indicate significance levels:n ¼ 10%, nn¼ 5%, and nnn

¼ 1%.

Panel A: f/n convexity estimated from quadratic regressions (Table 6) regressed on aggregate flows

Convexity of Rank2 from Model 2 of Panel A Convexity of Perf2 from Model 5 of Panel A (multiplied by 100)

Parameter Model 1 Model 2 Model 3 Model 4 Model 5 Model 6 Model 7 Model 8 Model 9 Model 10 Model 11 Model 12

Constant 1.2176 �0.3263 �0.8033 1.3974 1.3671 1.1941 2.4846 0.8495 �2.2156 �0.4024 2.7426 �0.8846

(2.37)nn (0.51) (0.24) (0.39) (2.63)nnn (0.33) (3.10)nnn (0.84) (0.67) (0.12) (3.28)nnn (0.26)

F/N 2.4884 2.8658 3.4213 2.6354 2.3613 3.2514

(2.26)nn (2.58)nn (3.07)nnn (1.95)n (1.53) (1.92)n

Sum(9f9)/N 1.0115 �0.98 �0.9173 2.3526 0.7117 0.8769

(0.55) (0.49) (0.45) (1.29) (0.35) (0.44)

MKT �1.8554 �13.1162 �18.5665 �32.6287

(0.15) (1.17) (0.99) (1.53)

Lagged MKT �22.0009 �29.3113 �22.4515 �30.9436

(2.21)nn (2.84)nnn (1.72)n (2.32)nn

R2 0.0442 0.0041 0.047 0.0165 0.0799 0.0194 0.0086 0.02 0.0114 0.0447

One standard deviation

impact of aggregate flow

1.4595 0.4417 2.1088 2.4072 1.5457 1.0273 1.6957 2.2899

Panel B: f/n convexity estimated from piecewise regressions (Table 4) regressed on aggregate flows

Constant 14.9882 9.1135 �12.9582 �8.625 14.9399 �8.3016 11.4219 7.8428 �20.0357 �19.9404 10.2413 �18.5507

(6.65)nnn (2.74)nnn (1.27) (0.84) (6.62)nnn (0.82) (5.67)nnn (2.25)nn (1.98)nn (1.69)n (5.14)nnn (1.71)n

F/N 9.5269 5.6429 6.1496 5.76 0.1595 �1.7373

(2.17)nn (1.18) (1.21) (1.44) (0.03) (0.32)

Sum(9f9)/N 14.0059 10.0847 9.9309 13.161 13.0796 12.5802

(2.61)nnn (1.72)n (1.74)n (2.92)nnn (2.08)nn (2.23)nn

MKT 68.2979 25.8039 91.6227 69.474

(1.33) (0.45) (1.59) (1.18)

Lagged MKT �55.4974 �78.8662 96.0488 86.8146

(1.02) (1.40) (1.98)nn (1.70)n

R2 0.0352 0.0422 0.0513 0.0138 0.0636 0.0159 0.0696 0.0696 0.0425 0.0972

One standard deviation

impact of aggregate flow

5.5877 6.1157 7.7131 7.9432 3.3058 6.921 6.9697 7.6126

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19 Fant and O’Neal (2000) also find that the convexity in the flow-

return relation depends upon aggregate flows. They split their data into

1978-1987 and 1988-1997 periods and find that convexity was stronger

in the latter period. They attribute this change, in part, to the larger

aggregate flows during the second half of their sample period.20 One can find a somewhat analogous result in Sawicki (2001).

Using Australian data she finds that smaller and younger funds exhibit

greater fund flow-return convexity than do other funds.


ranks, and the difference between the High and Mid coeffi-cients from a piecewise regression. The former two specifi-cations come from the quadratic regressions in Table 6; thelatter, from the piecewise regressions in Table 4. Thisprocedure produces a time series of the convexity measurein question. Using the time series regression parameters asthe dependent variable, Table 8 displays the degree to whichthey can be explained by aggregate flow measures.

The hypothesis tested in Table 8 follows from thediscussion around Fig. 1. That is, individual fund flowsare linear in returns, but the functions mapping one to theother vary in the cross section. Now add to this thepossibility that this variation is itself sensitive to aggre-gate flows. Assuming that is right and of significantexplanatory power, then much of the observed time seriesconvexity should be explained directly by changes inaggregate flows. The constant term in Table 8 providesthe relevant test statistic. To the degree it remainspositive and significant; the control variables fail toaccount for the observed level of convexity in the flow-return regressions.

Based on Table 8, it appears that estimates of fundflow-return convexity arise at least in part from timeseries variation in aggregate flows. Panel A presentsresults using the quadratic parameter estimates fromearlier specifications with either ranks or actual returnsas the dependent variable. In every case, the aggregateflow parameter enters as a significant explanatory vari-able at the 5% level or higher when ranks are the flow-return model’s basis. Using returns instead of ranksweakens the statistical significance. In two cases, theaggregate flow term is significant at the 10% level and inone, not at all. However, in every case the sign remains ashypothesized.

Another item of note in Table 8 is the constant termand how it changes across models. When aggregate flowsare excluded, the constant is both positive and statisti-cally very significant. This implies that, during the periodgenerating the data, on average one finds a convexperformance-flow relation. Given the many studies show-ing fractional flow convexity, one would expect this.However, once aggregate flows are added not only doesthe constant become statistically insignificant in everyspecification, but also often turns negative. The implica-tion is that if aggregate flows were symmetrically dis-tributed around zero, studies would not have found aconvex flow-performance relation in the overall data.In each panel, the last row displays how a 1 standarddeviation change in aggregate flows changes the esti-mated degree of convexity in the time series flow-returnrelation. It is generally economically large. A 1 standarddeviation change in aggregate flows in the joint modelspecifications (Models 6 and 12 in Panel A) increases theestimated convexity by 2.41 and 2.29. These values are ofthe same magnitude as the original Fama and MacBethconvexity estimates of 1.2 and 2.48 (Models 1 and 7 ofPanel A).

Table 8, Panel B repeats the above analysis, but thistime with the time series difference between the linearregression terms from a piecewise regression as thedependent variable. The results are not as strong. In many

cases, the aggregate flow parameter is not statisticallysignificant, and in one case has the wrong sign. However,now the aggregate absolute flow is often significantand positive. Again, the constant becomes very unstable,taking on positive and then negative values whenever theregression’s independent variables are altered. Similar tothe results in Panel A, the estimated convexity in thepercentage flow-return relation is very sensitive to theaggregate flow. Using the High-minus-Mid flow sensitivity,the full sample estimated Fama and MacBeth flow-returnconvexity is 15 and 11 using market- and category-adjusted returns, respectively. At the same time, the finalrow in Panel B shows that 1 standard deviation changein aggregate flows changes the estimated value by 7.9and 7.6.19

Table 8 indicates that over time the estimated convexitylevel is very sensitive to aggregate flows. This, however,leaves open the question of whether or not the types ofcontrols suggested by the flow sensitivity aggregate flowhypothesis proposed in the introduction can significantlydampen estimated flow-return convexity levels. Table 9takes on this task.

During the discussion of Fig. 1, it was suggested that thehigh blue line might equate to the flow-performance relationfacing small young funds; the lower blue line, to the onefacing the remaining funds. If that is true, then reformulatingthe fractional flow model to account for this should reducethe estimated convexity. (It would not eliminate it unless thisaccounted for every driver of the relation shown in Fig. 1.)Table 9 tests for this by breaking out funds that are small andyoung from the rest of the industry and then repeating thetests from the equation run as Model 6 in Panels A and Bfrom Table 6. A fund is classified as small and young if it isunder five years of age and smaller than the median fundthat year. The complementary set is labeled large or old.Thus, every fund falls into one group or the other.

For ease of exposition the results from Table 6 areredisplayed in the first set of columns. The second setdisplays the estimates on the small and young funds only;the final columns, those on the funds outside the small andyoung category. As shown in Table 6, running fractionalflows against returns yields the typical convexity result: ahighly significant quadratic term. However, when thesmall young funds are estimated separately the relationdisappears.20 In the four models, only one is positive andsignificant and then only at the 10% level. In the otherthree specifications it is insignificant, and in one case, it isnegative. For the funds outside the small and youngcategory the results are also mixed. If returns are onlymarket-adjusted, a convex relation remains, suggestingthat there are additional factors that lead to relationssimilar to those depicted in Fig. 1. However, when returns

Table 9Augmented flow convexity estimation with hot money in the cross section.

This table revisits the quadratic flow convexity estimation as reported by Model 6 in Panels A and B of Table 6. The original convexity values are estimated period by period in the cross section over the whole

sample of available domestic equity funds. This table extends the convexity estimations to two subgroups of funds that can be classified as ‘‘hot’’ and ‘‘cold.’’ Hot funds are funds that are both young (five or

fewer years since inception) and have relatively few assets under management (below the period’s cross section’s median value). Cold funds contain the complement. Thus, every fund is classified as either hot

or cold. In each period, hot and cold funds are sorted into vigintiles based on their group’s returns. Vigintile fund flows are then regressed against the vigintile’s prior period performance and a list of control

variables. Each model is run both with and without category flows to highlight the relation between fund flow and category flow. Panel A reports the Fama and MacBeth regression parameters and the Newey

and West adjusted t-statistics. To save space, constants and other control variables are not tabulated. Panel B reports the standard deviation of the returns by group.

Panel A: Subsample f/n regression, modified from Table 6

Original models (full sample, Table 6) Small and young funds Large or old funds

Parameter f/n to Market-adjusted return f/n to Category-adjusted

return

f/n to Market-adjusted return f/n to Category-adjusted

return

f/n to Market-adjusted return f/n to Category-adjusted

return

Perf 0.1169 0.1130 0.1804 0.1991 0.5045 0.7555 0.5941 0.6669 0.1492 0.1454 0.2332 0.2261

(2.42)nn (2.98)nnn (4.15)nnn (5.23)nnn (3.80)nnn (2.60)nnn (3.34)nnn (2.16)nn (5.77)nnn (5.82)nnn (6.48)nnn (6.73)nnn

Perf2 0.0363 0.0365 0.0198 0.0194 0.0880 �0.0557 0.0186 0.0216 0.0234 0.0222 0.0148 0.0143

(2.93)nnn (2.92)nnn (2.05)nn (2.06)nn (1.62)n (0.45) (0.28) (0.19) (2.24)nn (2.12)nn (1.16) (1.30)

Perf3 0.0024 0.0025 0.0011 �0.0005 �0.0130 �0.1000 �0.0264 �0.0055 0.0005 0.0009 �0.0027 �0.0011

(0.72) (0.89) (0.20) (0.10) (0.63) (1.57) (0.85) (0.12) (0.19) (0.33) (0.44) (0.20)

Cat_flow 0.3223 0.2688 2.1883 2.0166 �0.0002 �0.0003

(1.55) (1.03) (6.71)nnn (5.96)nnn (2.40)nn (2.95)nnn

Constant and controls Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes Yes

Panel B: Return standard deviation by category

Measure Small and young funds Large or old funds

Standard deviation of pooled fund return 3.1115 2.9295

Standard deviation of pooled category-adjusted return 2.4409 2.1199

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are category-adjusted, then even for the funds that falloutside the small and young category the convexity para-meter becomes insignificant. This could be occurring for anumber of reasons. For example, when investors allocatetheir money, they could allow for whether or not acategory has recently performed well. It could also be thecase that capital flows more heavily into hot categories.Whatever the underlying reason, unless one controls forthis dichotomy (and likely others), a regression analysiscould generate the convex green line depicted in Fig. 1even if every single fund faces a linear flow-return relation.In any event, Table 8 and Table 9 indicate that the flow-return convexity result is not particularly robust to theempirical specification employed.21

So far, the analysis has shown that splitting the data upinto young and small funds and their complement eliminatesmost of the observed convexity in the estimated fractionalflow—return relation. However, if the hypothesis depicted inFig. 1 is the underlying cause, then other conditions must besatisfied as well. One is that, between the two groups, theyoung and small funds should exhibit greater flow-returnsensitivity. This does appear to be the case. In all of thespecifications, the young and small funds have a far higherestimated linear term relative to other funds (Table 9,Panel A, Perf estimates). A second condition is that the youngand small funds should be more volatile than their peers.Table 9, Panel B checks this condition and finds that it issatisfied. Small and young funds have more volatile returnsthan their complement, the large or old funds. Thus, asthe hypothesized model requires, hot money funds have asteeper flow-return relation and more volatile returns thancold money funds. Uncontrolled for, these patterns cangenerate convexity in standard flow-return regressions evenwhen every individual fund faces a linear relation.

8.2. Misspecification robustness

This subsection formalizes why an empirical modelthat uses market shares as the dependent variable willgenerally be more robust to misspecification error thanone that uses fractional flows. Assume fund flows aredetermined by the affine function

f i,t

ni,t�1¼ aiþbigi,t ri,t�1�rw

m,t�1

� �þ

Ft

Nt�1ð7Þ

where ai represents flows that the fund will earn irre-spective of its return, bi is a fund-specific parameter thatalters only the slope of the flow-return relation, gi,t is afunction that depends on fund and market characteristics(e.g. market share and aggregate flows), and rw

m,t�1 is thebigi,t weighted average fund return.22 Assuming flows areincreasing in returns, then bigi,t�1 is strictly positive. Tofacilitate the exposition, and without loss of generality,

21 An analysis identical to that in Table 8 Panel A was also

conducted using market share changes as the dependent variable. Given

the results with pooled data in Table 6, not too surprisingly, the

convexity parameter is insignificant in every run.22 One can build b into the function g. However, it is easier to

present the proofs with an explicit separate parameter for altering the

slope of the flow return relation.

both are assumed to be positive. Thus, the hot fund inFig. 1 has a higher bi than the cold fund.

Adding up implies that, no matter what the set of fundreturns may be, the sum of the individual fund flowsdefined in Eq. (7) must be the aggregate flow Ft. This, inturn, implies that the ai have to add to zero (

Pini,t�1ai ¼ 0).

Similarly, rwm,t�1 must satisfyX

i

ni,t�1bigi,t ri,t�1�rwm,t�1

� �¼ 0 ð8Þ

for all possible ri,t�1. Rearranging Eq. (8) then implies

rwm,t�1 ¼ ½

Xi

ni,t�1bigi,tri,t�1�=½X

i

ni,t�1bigi,t�: ð9Þ

The relation in Eq. (9) posits that there exists a singlesufficient statistic for the set of fund returns (rw

m,t�1) thatallows for mapping between individual fund flows andindividual fund returns.23 Finally, from Eq. (7) a fund’schange in market share equals

Dmi,t ¼ni,t�1 1þri,t�1

� �þ f i,t

Nt�

ni,t�1

Nt�1

¼ni,t�1

Nt½ri,t�1�rm,t�1þaiþbigi,tðri,t�1�rw

m,t�1Þ�, ð10Þ

where rm,t�1 equals the standard value-weighted marketreturn, but in the case of assets collectively held bythe sample mutual funds. The last equality comes fromrearranging the prior one and using the relation Nt¼Nt�1

(1þrm,t�1)þFt.Based on Eqs. (7) and (10), the first derivatives with

respect to ri,t�1 equal

@ f i,t=ni,t�1

� �@ri,t�1

¼ bigi,t and@Dmi,t

@ri,t�1¼

ni,t�1

Nt1þbigi,t

� �ð11Þ

and the cross-derivatives with respect to bi are

@2 f i,t=ni,t�1

� �@ri,t�1@bi

¼ gi,t and@2Dmi,t

@ri,t�1@bi

¼ni,t�1gi,t

Nt: ð12Þ

The ultimate goal is to assess the impact of uncon-trolled heterogeneity on each measure, that is, whichmeasure’s first derivative with respect to ri,t�1 is moresensitive to uncontrolled for differences in bi. As seenin Fig. 1, the greater the sensitivity, the easier it is togenerate spurious convexity in a misspecified regression.

While it is tempting to simply compare the cross deriva-tives in (12), the two measures are in different units. Asimple way around this problem is to instead take the ratioof the terms in (11) (@Dmi,t=@ri,t�1 over @ f i,t=ni,t�1

� �=@ri,t�1)

and then see how it changes with bi.

@

@bi

(@Dmi,t

@ri,t�1

@ f i,t=ni,t�1

� �@ri,t�1

)¼�

ni,t�1gi,t

Nt bigi,t

� �2o0:

,ð13Þ

From Eqs. (11) and (12), both measures have positivefirst and cross-derivatives. The fact that Eq. (13) isnegative, therefore, implies that an increase in bi inducesa larger change in the flow-performance model’s

23 One can imagine more complex relations between fund returns

and flows such that a single summary statistic such as rwm,t�1 is

insufficient to map one into the other. However, as a practical matter,

far more complex structures are likely to present significant empirical

hurdles.


sensitivity to returns than it does in the market sharemodel’s sensitivity.24 Proposition 4 states this formally.

Proposition 4. Suppose flows are an affine function of returns

of the form given by Eq. (7). Then changes in bi will have a

relatively larger impact on the fractional flow measure Eq. (7)

than the market share measure Eq. (10).

9. Conclusions

An extensive literature indicates that current fund flowsare convex in past performance. This linkage has beenestablished within an empirical model that regresses indi-vidual fund flows divided by AUM on past performancerankings and controls. However, the fractional flow modelspecification is not without its own economic and empiricalimplications. If fractional flows increase in performance,then aggregate flows should be larger when large fundsdo well relative to small ones. If fractional flows are alsoconvex in performance, then aggregate flows should belarger when large funds populate the performance distribu-tion tails. The empirical tests conducted here do not lendsupport to either hypothesis. The relation between aggre-gate flows and relative fund performance by size implied bythe fractional flow model appears to be absent from thedata. A power analysis shows that a lack of data is notbehind this non-result, implying that the standard frac-tional flow model is misspecified in some way.

Part of the problem with a fractional flow model is that itis sensitive to uncontrolled-for heterogeneity in theway investors respond to past returns either in the crosssection of funds or over time. While in principle it is possibleto correct for this, it requires knowing exante the full list ofnecessary controls. Absent that, simulations show that afractional flow model can easily yield a spuriously estimatedflow-performance relation, like that seen in the mutual funddata. Our analysis shows that allowing for various types oftypically unaccounted-for heterogeneity greatly reduces theestimated convexity from a fractional flow model.

An alternative to the fractional flow model is one basedupon market shares. As our study shows both theoreticallyand empirically, it is considerably more robust to unaccoun-ted-for heterogeneity within the data. Furthermore, our testsindicate that investors first decide how much to invest andthen determine how to split it up—a sequence more in linewith a market share than a fractional flow model. In contrast,a fractional flow model assumes investors first decide howmuch to invest based on each fund’s individual performancein isolation and then generate the aggregate flow necessary toyield the required amount.

Using market share changes instead of fractional flows asthe dependent variable produces no evidence that increasinga fund’s volatility will help it grow. This is in line with the

24 Similar conclusions are drawn if one instead uses a measure

analogous to Arrow-Pratt relative risk aversion. In this case, the units

can be removed by dividing the cross derivatives in Eq. (12) by their

respective first derivatives in Eq. (11). These ratios can then be

compared directly and also show that the fractional flow-performance

model exhibits greater sensitivity to changes in bi than does the market

share model. The advantage is that it eliminates the Nt�1 term, but the

ratio itself seems somewhat less intuitive than the one in Eq. (13).

long-term data showing that investors are not, in theaggregate, moving their investments into more volatilemutual funds. If anything, the opposite seems to be occurring.

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