prior performance and risk chen and pennacchi

JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS Vol. 44, No. 4, Aug. 2009, pp. 745-775COPYRIGHT 2009, MICHAEL G. FOSTER SCHOOL OF BUSINESS, UNIVEHSITY OF WASHINGTON, SEAHLE. WA 98195doi:1Q.1017/S002210900999010X

Does Prior Performance Affect a Mutual Fund'sChoice of Risk? Theory and Further EmpiricalEvidence

Hsiu-lang Chen and George G. Pennacchi*

Abstract

Recent empirical studies of mutual fund competition examine the relation between a fund'sperformance, the fund manager's compensation, and ihe finid manager's choice of portfo-lio risk. This paper models a manager's portfolio choice for compensation rules that canbe either a concave, linear, or convex function of the fund's performance relative to thatof a benchmark. For particular compensation structures, a manager increases the fund's"tracking error" volatility as its relative perfomiance declines. However, declining perfor-mance does not necessarily lead the manager to raise the volatility of the fund's return. Thepaper presents nonparameiric and parametric tests of the relation between mulua! fund per-formance and risk taking for more than 6.üt)ü equity mutual funds over the l%2 to 2006period. There is a tendency for mutual funds to increase the standard deviation of trackingerrors, but not the standard deviation of returns, as their perfomiance declines. This risk-shifting behavior appears more common for funds whose managers have longer tenures.

I. Introduction

As mtitual fund investing has grown, the management of mutual funds hascome under closer scrutiny by financial ecotwmists. One strand of research ex-amines potential agency problems between a mutual fund's shareholders and itsportfolio manager. Several studies investigate whether a manager might unneces-sarily shift the fund's risk in response to changes in its performance relative toother funds. This behavior is linked to the way the manager is compensated andto the actions of mutual fund investors. The manager's compensation depends onher success in generating flows of new investments into the fund, while mutualfund investors "chase returns" by channeling investments into funds with better

'Chen, [email protected]. College of Business Administration. University of Illinois at Chicago.601 S. Morgan St.. Chicago. IL 60607; Pennacchi, gpennaccCa'illinois.edu, College of Business. Uni-versity of Illinois at Urbana-Champaign. 515 E. Gregory Dr.. Champaign, IL 61820. We are gratefulfor valuable comments pmvided by Wayne Fersnn (associate editor and referee), Zoran Ivkovich.Jason Karceski, Maiii Keloharju, Paul Malafesta (Ihe editor), and participants al the 2000 .WAMeetings and at seminars al llie Federal Reserve Bank of Cleveland, the University of Illinois, theUniversity of North Carolina at Chapel Hill, and the University of Notre Dame.

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746 Journal of Financial and Quantitative Analysis

relative performance. This creates a situation described as a mutual fund "tour-nament," where portfolio managers compete for better performance, greater fundinflows, and, ultimately, higher compensation.

Inflows rise nonlinearly with a fund's relative performance. Numerous stud-ies document that mutual funds with the best recent performance experience thelion's share of new inflows, but poorly pertbrming funds are not penalized withsharply higher outflows.' If the fund manager's compensation rises in propor-tion to the fund's inflows, this convex pertbrmance-fund flow relation producesa convex performance-compensation structure.^ Research such as Sirri andTufano (1998) notes that such compensation is similar to a call option, creat-ing an incentive for a manager to raise the risk of the fund's relative returns.Chevalier and Ellison (1997), Brown, Harlow, and Starks (BHS) (1996), Basse(2001 ), and Goriaev, Nijman, and Werker (GNW) (2005) have empirically exam-ined the behavior of a cross-section of mutual funds for which this risk-takingincentive is predicted tti differ.

The current paper adds to this mutual fund tournament literature by pro-viding new theoretical and empirical insights into risk taking by mutual funds. Itmodels the optimal intertemporal portfolio strategy of a mutual fund manager thatfaces the competitive tournament environment assumed by recent empirical work.Explicit solutions for this manager's portfolio allocation are derived when her util-ity displays constant relative risk aversion and compensation is either a concave,linear, or convex function of the fund's relative calendar-year performance.

The model shows that the deviation of a fund manager's optimal portfo-lio from the benchmark portfolio is a function of the fund's performance. If thepenalty for poor performance is limited so that the manager's total compensationcan never fall to zero, then the fund manager chooses to deviate more from thebenchmark portfolio as the fund's relative performance declines. In other words,when a fund is performing poorly it displays more "tracking error" than when itperforms relatively well. However, it is not necessarily true that an underperiorm-ing fund chooses to raise the volatility (standard deviation) of its returns.

Almost all empirical studies that have tested for tournament risk shiftinghave analyzed fund risk measures other than tracking error. Most commonly, em-pirical research has tested for whether underpertbrming funds increase the stan-dard deviation of their total returns, rather than the standard deviation of theirtracking errors. The most comprehensive of these studies conclude that there isno evidence for tournament behavior. However, based on our model's insights.

'Siudies examining the fund flow-performance relalionship include IppoliUi (1992). Gruber(1996), Chevalier and Ellison (1997), Sirri and Tufano (1998). Gœtzmann and Peles (1997), andDel Guercio and Tkac (2002).

-The literature on mutual fund tournaments distinguishes between a fund's investment advisor,the entity responsible for ponfoho management, and the portfolio manager hired by the advisor. Thetypical investment advisor is paid a fixed fraction of the fund's assets, assets that depend on both netfund inflows(externali;rowthof assets) and the fund's return (internal growthof assets). However, theptirtfoUu manager's compensation is assumed to depend only on his ability lo generate extraordinarygrowth in fund assets, growth that depends on the fund's return relative to (the average of) other funds'rettims. Common or systematic shocks to all funds' returns (affecting internal asset growth) are notdue to the individual manager's ponfolio selection ability and would not alïect compensation. Hence,compensation is assumed to depend on relative, not absolute, performance.

Chen and Pennacchi 747

we argue that this conclusion may be unwarranted because it is based on tests thathave employed risk measures that are inappropriate for examining the tournamenthypothesis.

This paper reexamines the empirical evidence for tournament behavior inlight of our model's predictions. We construct two different types of tests. One isa nonparametric test that modifies the standard deviation ratio (SDR) tests usedin prior studies. It analyzes risk shifting for a cross-section of mutual funds basedon the SDR of their tracking errors, rather than the SDR of their returns. Anotherparametric test exatiiines individual mutual funds' time-series behavior based onan empirical model that nests our paper's theoretical one. Unlike the nonparamet-ric tests that allow a fund's risk to change only once per year, this parametric testpermits each fund's risk to vary at every (monthly) observation date.

Both tests are performed using data on tiiore than 6.00() mutual funds thatoperated during the 1962 to 2006 period. As predicted by our model, the empiricalevidence suggests that an underpertorming mutual fund manager increases thestandard deviation of tracking error, but not the standard deviation of returns.Evidence is strongest for managers that have longer tenures at their funds, a resultthat also is consistent with our model.

The plan of the paper is as follows. Section II briefly discusses related the-oretical and empirical work on the risk-taking incentives of mutual fund man-agers. In Section III we present our model. Section IV discusses nonparametricand parametric empirical methods for testing the risk-taking behavior implied byour model. Section V describes our data, and Section VI presents the empiricalresults. Concluding comments are in Section VII.

II. Related Literature

This section begins by discussing sotne of the theoretical research thatrelates to our paper's model. It then reviews empirical studies of mutual fundtournaments.

A. Models of Portfolio Management

A growing literature examines links between a fund manager's compensa-tion contract and his portfolio choice. Grinblatt and Titman (1989) show howcompensation contracts that include a bonus for good performance can producemoral hazard incentives. Mutual fund managers can maximize the present valueof their option-like bonus by choosing a fund portfolio with excessive risk. More-over, the fund manager can risklessly capture the increased value of this bonus ifshe can hedge using her personal wealth.

Starks (1987) considers the moral hazard incentives of a bonus contract, fo-cusing on situations of asymmetric information between investors and fund man-agers. When investors cannot observe a manager's choice of portfolio risk or themanager's effort level, compensation contracts with symmetric payoffs dominatecontracts that include a bonus. However, Das and Sundaram (2002) show that therelative advantages of symmetric and bonus contracts can be reversed if investors'choice of funds is made endogenous to the funds' risk levels and compensation

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748 Journal of Financial anä Quantitative Analysis

contracts. In their model, bonus contracts provide better risk sharing between in-vestors and fund managers when investors take account of a fund's risk and con-tract choice. Dybvig. Farnsworth. and Carpenter (2009) also find that contractsshould include a bonus proportional to the fund's return in excess of a benchmarkreturn when a fund manager's effort determines the quality of her information.-'

Other research, such as Huberman and Kandel ( 1993), Heinkel and Stoughton(1994). and Huddart (1999), considers environments where fund managers pos-sess different abilities that are unknown to investors. In these screening mod-els, there is typically an initial period when investors learn of managers* abilitiesbased on their relative performances, followed by a second period when investorscan switch their savings to those managers perceived to have the highest abilities.Hence, these "investor learning" models can explain the link between fund flowsand prior performance. In addition, if managerial ability displays decreasing re-turns to scale. Berk and Green (2004) show that fund flows determine the relativesizes of mutual funds such that, in equilibrium, investors expect no future superiorreturns net of fund fees and expenses.

The model in the current paper differs from this previous work by focusingon how prior performance affects a fund manager's intertemporal choice of port-folio risk. We take the structure of compensation as given and study a manager'sdynamic portfolio choice during an annual mutual fund tournament. Models byCarpenter (200Í)), Cuoco and Kaniel (2001), and Basak, Pavlova, and Shapiro(2007) are related to ours. These papers assume that the performance-related coni-f>onents of a manager's compensation are piece-wise linear functions of perfor-mance. Carpenter (2000) specifies compensation equal to a fixed fee plus a calloption written on the value of the managed portfolio with an exercise price equalto a benchmark asset. Cuoco and Kaniel (2001) permit compensation to containa penalty for poor performance in the form of the manager's writing a put optionon the managed portfolio. Basak, Pavlova, and Shapiro (2007) assume a man-ager's compensation is linear in portfolio returns relative to a benchmark but sub-ject to fixed minimum and maximum payoffs, equivalent to a bull spread optionstrategy.

While we also assume that a manager's compensation depends on the port-folio's performance relative to a benchmark, the contract is not strictly in the formof standard call or put options. Rather than being an option-like, piece-wise lin-ear function of performance, our compensation contract is a smooth function thatcan be concave, linear, or convex in relative performance. An advantage of oursmooth compensation schedule is that it leads to simple and intuitive closed-formsolufions for a fund manager's optimal portfolio choice. This simplicity allowsthe model to guide our later empirical tests of mutual fund behavior.

Arguably, a smooth performance-compensation function better captures theenvironment of a mutual fund tournament where compensation is proportional toa fund's assets under management that, in tum, depend on how investor inflows

^Becker, Ferson. Myers, and Schill (1999) also analyze the consequences for portfolio choicewhen a manager has compensation based on the fund's return in excess of a benchmark and also hasthe ability to time the market.


respond to the fund's relative performance.'' Chevalier and Ellison (( 1997). p. 1181 )contend that assuming a smooth relationship between miitu;il fund performanceand fund flows is preferable because it avoids imposing the strong restrictions onrisk incentives that occur with a piece-wise linear contract. Under a piece-wiselinear contract, a manager's risk incentives are always maximized or minimizedat the contract's kink points, yet for mutual funds, identifying the location of thesekink points is subject to error since they must be estimated from past fund flows.

B. Empirical Research on Mutual Fund Tournaments

Several empirical studies have analyzed the relationship between a mutualfund's prior pertbrmance and its choice of risk. Chevalier and Ellison ( 1997) esti-mate the shape of mutual funds" performance-fund How relation and use it to inferdifferent funds' risk-taking incentives. They, like other researchers, assume thata fund's inflows respond primarily to its relative performance calculated over theprevious calendar year. Thus fund managers compete in annual tournaments thatbegin in January and end in December.- A fund's risk-taking incentive over thefinal quarter of the year is assumed to be proportional to the estimated convexityof fund inflows measured locally around the fund's September performance rank-ing. Using 1983 to 1993 data on the equity holdings of mutual funds at the endsof September and December. Chevalier and Ellison (1997) find that a fund tendsto change the standard deviation of its return relative to a benchmark return as theperformance-fund flow relation predicts. For example, young mutual funds thatperform relatively poorly from January to September tend to raise the standarddeviation of their return in excess of a benchmark return (standard deviation oftracking error) during October to December/'

Another study by BHS (1996) performs SDR tests of whether a fund thatperforms relatively poorly at midyeai- tends to raise the standard deviation of itsreturn over the latter half of the year more than does a fund that performs relativelywell at midyear. They use monthly returns data for a cross-section of mutual fundsduring 1980 to 1991 and ñnd support for the tournament hypothesis that midyear"losers" gamble to improve their relative end-of-year performance by raising tbeirfunds' standard deviation of returns more than do midyear "winners." Koski andPontiff (1999) also use monthly returns data over the 1992 to 1994 period to

"•The ciinlract in Carpenter {2()(K)} may better represeiU ihe irompensalion of a nonfiniinciiil timimanager who receives stock options. Cuoco ¡md Kaniel's {2(K)I ) compensation contract miyht be moslappropriate for other poritblio managers, such as managers of pension funds. Their analysis focuseson tlie equilihriuni asset pricing consequences of portfolio management. Basak el ai. (2007) assume acomplete markets environment where portfolio managers lack asset selection ability and choose onlysystematic risk.

^This assumption is justified because sources of mutual fund information, such as Momingstar,Inc., typically compute relative fund performances using this calendar-year period. Hence, flows ofinvestor funds and. in turn, managerial compensation should be most sensitive to a fund's calendar-year performance- Empirical evidence in Koski and Pontiff ( 1999) indicates that changes in a fund'srisk are most strongly related to performance calculated over calendar years.

''Chevalier and Ellison (1997) also test their estimated performance-flow relation using monthlydata on fund returns. They measure a fund's risk as ihe standard deviation of its return in excess ofthe return on a value-weighted index of NYSE. AMEX. and NASDAQ stocks and lind thai it movesin the predicted direclion during ihe last quarter of the year.


calculate various measures of a fund's risk, including the standard deviation, beta,and idiosyncratic risk of a fund's returns. Their results are similar to those of BHS(1996) in that a mutual fund's performance in the first half of the calendar year isnegatively related to its change in risk during the second half.

However, more recent research finds that some of these results are not ro-bust to other testing methods and sample periods. Busse {2001) uses a differentdatabase of daily, rather than monthly, mutual fund returns from 1985 to 1995to calculate more accurate estimates of a fund's SDRs. He duplicates the SDRtests in BHS (1996) and finds no evidence that midyear poor-performing fundsincrease their standard deviation of return more than midyear better-performingfunds. He also shows that if standard deviations are calculated using monthly re-turns measured from the middle of each month, rather than from the beginning ofeach month as in BHS ( 19%), the evidence disappears for raising return standarddeviations as relative performance declines.

Similariy. GNW (2005) replicate BHS's (19%) SDR tests using an expanded1976 to 2001 sample of monthly fund returns and correct their test significancelevels for cross-correlation in fund returns. They also find no evidence that un-derperfonning funds raise their standard deviations of returns. Hence, the morecomprehensive studies of Busse (2001 ) and GNW (2005) conclude that the BHS(1996) finding of tournament behavior is fragile. It does not hold up to more pre-cise tests and larger samples of mutual fund returns.

As our model of mutual fund tournaments in the next section demonstrates,under plausible conditions, an optimizing manager chooses to raise the standarddeviation of her fund's tracking error (return in excess of a benchmark) as thefund's relative performance declines. Such behavior does not necessarily implya rise in the fund's standard deviation of returns, beta, or residual risk. Hence,with the exception of Chevalier and Ellison (1997). prior tests of tournament be-havior are based on arguably inappropriate risk measures. In particular. Busse's(2001) and GNW's (2005) rejection of tournament behavior may be unjustifiedsince their SDR tests, like those of BHS (1996). employ standard deviations oftotal returns, rather than tracking error. Reexamining the evidence for tournamentbehavior with a focus on mutual funds' tracking errors, rather than their totalreturns, motivates our paper's empirical work.

III. Modeling a Mutual Fund Manager's Portfolio Decisions

We now describe our model's specific assumptions. A fund manager's com-pensation is assumed to depend on tbe fund's performance relative to a benchmarkindex. The fund's portfolio can be invested partly in this benchmark index andpartly in a set of "alternative" securities cbosen by its fund manager. These alter-native securities are defined as the portion of the fund's total assets that accountsfor the difference between the fund's portfolio and one that is invested solely inthe benchmiu-k portfolio. Tbe Appendix shows that when securities' excess re-turns and return covariances are conslant, the fund manager's optimal choice ofindividual alternative securities is one where their relative portfolio proportionsdo not vary over time. This implies that tbe manager's intertemporal portfoliochoice problem can be transformed to one of allocating a portion of the fund's


portfoiio to tiie i^enchmarii index and the remaining portion to a single alternativecomposite security.' Hence, we simplify the presentation by assuming at the startthat the portfoiio allocation prohiem involves only two types of securities: thebenchmark index and a single alternative security.

Defme 5, as the value of the relevant benchmark index at date / and A, as thedate / value of the alternative securities. Then 5, and A, are assumed to follow theprocesses

àS . . .( I ) — = asdt + asdz and

(2) — = aAdt + aAdq>

where tT rf-fT.Aí/ —tTAsííí-For analytical convenience, CT^. 0-5. and (Tasare assumedto be constants. Here, a^ and as may be time varying, as might be the case ifmarket interest rates are stochastic.** However, we require that the spread betweentheir expected rates of return, o^ - as. be constant.

If the fund manager allocates a portfolio proportion of 1 -u; to the benchmarkindex and a proportion ui to the alternative securities, then the portfolio's value,V. follows the process

dV ,, JS dA(3) y - (l-.^)y.^^

= [( 1 — u;) Qs + U;Û;^] dt + (] — uj) fT

Note that whenever CJ ^ 0, the fund's return in equation (3) deviates from thebenchmark return. We can also calculate the process followed by the fund's rela-tive performance. Define G, = V¡/S, to be the date t ratio of the value of a share ofthe fund's portfolio to that of the benchmark. A simple application of Itô's lemmashows that

(4) —r = uj(aA- as + o-¡ - ffAs) ät + uj [aAdq - asdz).Lf

The fund manager is assumed to compete in a tournament for inflows intothe fund. At the start of the tournament's assessment period. G = 1 by definition,but it then changes stochastically according to equation (4). Thus, G, measuresthe date t ratio of the fund's return to that of the benchmark since the start of thetournament, and hence D, ^ ln(G,) is the difference between the fund's contin-uously compounded return and that of the benchmark index since the beginningof the tournament.^ The tournament ends at date 7", which, for example, could be

'This "two-fund separaiion" resull is similar to Merton's ( 1971 ) case of lognormal asset prices.^The Appendix derives the values of a,\ and a^ in terms of the p:irameters o!" processes for n

individual altemalive securities.^We refer to relative performance as the ratio of returns. G,. rather than the difference in returns.

Dr. but this distinction is nonessential. The manager's compensation function could be rewritten interms of In (Gf). rather than G,. As will be shown, a manager's optimal portfolio choice is independentof prior performance when compensation is proportional to a power of G, rather than D,, so the ratiois a natural variable to use.


the last trading day of the calendar year. The manager's compensation is a func-tion of the fund's relative perfonnance at the end of the tournament, so that hiscompensation or "pay" can be written as /'[G7].'"'"

The fund manager maximizes his expected utility of compensation (wealth)at the end of the tournament by choosing the fund's asset allocation at each pointin time during the assessment period.'^ This maximization problem can be writtenas

(5) Max E.{U{PIGT])],uj{.,) V sç[,,T]

subject to the process followed by G given in equation (4). If we define J{G, t)as the derived utility of wealth (or performance) function, then assuming thatU{P[GT]) is concave in Gj, the first-order condition of the Bellman equationwith respect to u; implies that the portfolio proportion invested in the alternativesecurities is

(0) ÜJ —

where «c = QA - as + CT| - (TAS and (J¿- ^ ffj - 2(T^5 + «rj. Substituting this backinto the Belltnan equation, one obtains an equilibtium partial differential equationfor J that must satisfy the boundary condition J{GT, T) = U(P[GT]). A solutionrequires that the manager's utility function and compensation schedule be speci-fied. We make the standard assumption that utility displays constant relative riskaversion, U{P[GT]) — {P[GT\)'' /-y, where 7 < 1. For the manager's compensationschedule, we choose a flexible specification that can be either a concave, linear,or convex function of fund performance:

(7) P[GT] -

where b > 0, c > 0, a > -b/c, and c-y < 1. When 0 < c < I. the manager'scompensation is a concave function of performance. Gr- Compensation is linear

'"In practice, compensation may depend also on the performance of the ovenill equily (mutualfund) market. Kareeski (:îOO2) shows thai a fund's inflows are highest when the fund performs rela-tively well and. simultaneously, the overall stock markei performs well. He studies the implicationsof ihis pheTH)menon ¡br fund managers' selection of high versus low beta stocks and the equilibriumetïecls on as.sei prices. Our analysis omils (his market effect.

' The assumption that compensation depends only on ihe fund's performance relallve to a singleindex is a simplificaiion for another reasim. In general, a fund's nei inflows, and hence ils manager'sportfolio choices, might depend on the iinal pertonnances of each of the mutual fund's compeiitors.Oursimplitied siructure can be justified i]i an environment where there arc a large (intiniie) number ofmutual funds that choose dilferenl "altemaiive" securities. Their relative performances over the yearwmild be a smooth, approximately normally distributed function around a mean performance. Thiswould justify (as we do in our empirical work) using the "average" performance of all mutual fundsas a sufficieni statistic for comparing any given mutual fund's performance.

'•'Our model selling is very similar lo thai of Diitfie and Richardson (1991), who study ihe tradingstrategy of a risk-averse hedger. We assume that the manager does noi hedge his compensation riskvia his personal ponfolio. This is a standard assumption, though Grinblatt and Titman (1989) is anexceplion.


in performance when c = 1, whereas when c > 1 the funcfion is convex.'-* Ifa is set equal to 1 and the limit of P is taken as c goes to infinity, the functionbecomes exponential, P[Gr] - QxpihGr). While most prior empirical .studies oftiiutual funds emphasize the convexity of fund flows and compensation to perfor-mance, the allowance in equation (7) for a nonconvex function may be useful formodeling money managers in other industries.'*

As will be shown, the sign of the parameter a is critical to a manager's in-cenfive to shift a fund's risk. In the linear case ofc=\,a can be interpreted as thefixed component of a manager's net compensation or end-of-period wealth. Plau-sible arguments can be made for a to be positive or negative, depending on theparticular fund manager. For example, if a manager incurs fixed expenses (over-head) that are not explicitly reimbursed by the fund, then a could be negative.On the other hand, if P[GT\ is interpreted as a fund manager's total wealth thatincludes personal wealth as well as net compensation, then a may be positive ifpersonal wealth is substantial. Personal wealth may tend to be greater for moreexperienced managers for at least two reasons. First, experienced managers aremore likely to have savings from past compensation. Second, Chevalier andEllison (1999) find that tnutual fund managers who are older or have longertenures at their funds are less likely to be terminated for underperformance; thatis, they have more job stability. Hence, due to saved past compensation and thepresent value of future compensation, one might expect the parameter a to begreater for more experienced managers.

In general, when c ^ \, the parameter a does not translate directly to afixed component of wealth or total compensation, but its sign continues to de-termine whether total compensation has the potential to be nonpositive. Loweringa (possibly below zero) decreases total compensation, but sensible solutions tothe manager's portfolio choice problem require that a > —h/c. This restrictionprovides the manager with a feasible portfolio strategy that guarantees positivewealth at the end of the tournament. The manager can avoid zero wealth (and infi-nite marginal utility) by investing solely in the benchmark portfolio for the entireassessment period, since then compensation equals {a+G-ih/cY = {a+b/cY > 0.

The restriction cf < I ensures that the boundary condition U{P[GT]) —{P[GT])''/"/ is a concave function of Gj and that an interior solution to the man-ager's portfolio choice problem exists. This is always the case when 7 < 0, thatis, the manager's risk aversion exceeds that of logarithmic utility. However, if0 < 7 < 1 and c is sufficiently greater than 1 so that 7c > I, then U(P[GT]) isconvex, and the manager chooses u; to maximize the expected rate of return onG. From equation (4), this implies setting oj — +00 if ac > 0, and uj — —00 ifdo < 0.

'-'The function could be generalized to P{GT] = íi{a + {b¡c)GrY. where d >0. bul this extensionhas no effect on portfolio choice, if performance is defined as the difference in, rather than the ratioof. returns, then compensation is convex (concave) in Dj = ln(G7 ) whenever a + hG-¡ is positive(negative). Since it will be shown that a+{b/c)GT is always positive in equilibriutii. compensation canbe a convex function of the difference in returns even when 0 < c < 1. Thus, denning performanceas the difference in returns expands the range for which compensation is convex in performance.

'''Empirical evidence in Del Guercio and Tkac (2002) finds a performance-fund flow relation thatappears to be linear for pension fund managers.


Assuming 17 < 1, the solution to the Bellman equation is

f8) 7(G,0 ^ i j ^^ + ^

whereö = -c-ya};/ [2(1 - f7) (T¿] .Ifequation (8) is substituted into equation (6),then the manager's optimal proportion invested in the alternative securities is

( I - n l

Note from the restriction a > -b/c that the term ( 1 + (ac/hG) ) is always nonneg-ative in equilibrium, even when a < 0. To see this, suppose that a is negative andthat G declines sufficiently from its initial value of unity, so that (I -1- {cic/hG))approaches zero. Then from equation (9). the manager's optimal strategy is to in-vest fully in the benchmark portfolio. But at this point, with UJ* ^ 0, equation (4)implies that dG — 0. Wiih no further changes in the relative performance of thefund, the manager optimally prevents compensation from falling to zero.

Equation (9) shows that the manager chooses a long (short) position in thealternative securities whenever »G is positive (negative), and the magnitude ofthe position is decreasing in risk aversion, - 7 , but increasing in the fixed com-ponent of compensation. aP Also, the manager's position is independent of thetournament's time horizon. T - i.'^ For the special case of a = 0, so that cotn-pensation is proportional to a power of relative performance, P\GT\ = {bCrlcY,the alternative securities' portfolio weight is constant and invariant to changes inthe fund's performance. However, for the general case of û ^ 0, w* varies withchanges in G. When a > 0. the manager moves closer to the benchmark portfoliowith improvements in fund performance. The reverse occurs when a < 0. For thespecial case of a = 1 and c -^ 00, that is. compensation is the exponential formP[GT] = exp{hGi). equation (9) becomes

(10) u;* - "'^

so that the alternative securities' portfolio weight responds inversely to relativeperformance.

We summarize the manager's portfolio behavior with the followingproposition:

' tf, as discussed earlier, the parameter a tends to be greater for more experienced fund managers.then equation (9) predicts that more experienced fund managers lend to deviate íTnire from the bench-mark portfolio, ail else being equal. Chevalier and Ellison's (199*J) empirical resulls confirm thisprediction. They find that older mutual fund managers tend to deviate more from the average portfoliochosen by other managers having the same investment style.

""That portfolio choice is independent of the investment hori/on is a common feature of standardponlolio choice problems such as Merton ( 19711. The solution in equation (9) is analogous to thai ofa standard portfolio choice problem where the alternative securities ponfolio plays the role of a riskyasset portfolio and the benchmark portfolio is the risk-free asset. From this perspective. Qf; and ac,are the risky asset's excess return and its standard deviation of return, respectively, and ri.sk aversionis (I - c~))bG I {ac A- hG). Hence, the manager acts as if risk aversion varies wilh performance.


Proposition I. If a fund manager's utility displays constant relative risk aversionand has compensation given by equation (7), then when 07 < 1 an interior solu-tion to the portfolio choice problem exists. Moreover,

clad— —ÜT^dG

whose sign is opposite to that of the compensation parameter«. When « is positive(negative), a decline in the fund's relative performance leads the fund manager todeviate more (less) from the benchmark index.

The case a > 0 provides theoretical justification for Lakonisbok, Shleifer,and Vishny's (1992) argument that successful managers attempt to lock in gains.Furthermore, such behavior is likely to increase with the convexity of compen-sation, since when 7 < 0, one can show that a larger value of c makes portfoliochoice more sensitive to prior perfonnance. In contrast, the a < 0 case leads tomanagerial risk-shifting behavior that is opposite to that assumed in recent empir-ical studies of tournaments. For this case, total compensation is not automaticallybounded at zero, and a manager more closely matches the benchmark as perfor-mance declines to prevent zero compensation and infinite marginal utility.'^

Proposition 1 has implications for the risk measures chosen by BHS (1996),Busse (2001), and GNW (2005). These studies test the SDR relation:

(II) -^ >

where íTy denotes the standard deviation of the rate of return on mutual fundy's portfolio during the ith half of the year. Mutual fund 7 = Í, is a "loser" thatdisplayed relatively poor performance in the first half of tbe year, while mutualfund 7 — W in a "winner" that had relatively good performance during the firsthalf. The implication is that a mutual fund that is a midyear loser should increasethe standard deviation of its return more than that of a fund that was a midyearwinner.

Does our model imply tbe inequality in equation (11)',' Note that tbe pro-portion invested in the alternative securities that would minimize the standarddeviation of the mutual fund's rate of return given by equation (3) is

'^Proposition I is consistent with Carpenter (2000) and Cuoco and Kaniel (2001). When theyassume that compensation equals a positive componenl plus a cali option written on the portfolio'srelative performance, they lind that a manager increases tracking error as performance declines. Thiscompares to our ca.se of a > 0. since in both instances the compensation rule is always positive andportfolio managers need not fear obtaining zero wealth as tracking error risk increa.ses. In contrast.when Cuoco and Kaniel (2001) assume that compensation also includes [he manager writing a putoption on his performance, the manager reduces his tracking enor as performance declines. Here.compensation includes a penalty for poor performance and compares to our case with a < 0, since inhoth instances compensation may be nonpositive.

756 Journal oí Financial and Quantitative Analysis

Written in terms of equation ( 12), the optimal portfolio allocation in equation (9)becomes

(13) to' = CJîn-r^ ^ ^ ( ' " ^ n ) '

This allows us to state the following proposition:

Pmposuion 2. Suppose C7 < I and a > 0. so that an interior solution existsin which the manager deviates more from the benchmark index as performancedeclines. If

-x > 0 and —^ 1 + -p^ < 1,^s ~ ^'^^ i'^s ~ Âs)\^ ~ o ) hG/

so that ÜJ* and ujmia are of the same sign but u;* is smaller in magnitude than uJmin.then a decline in G that moves w* farther from zero and closer to Umin reduces thestandard deviation of the mutual fund's return. Otherwise, a decline in G raisesthe standard deviation of the fund's return.

Therefore, our model does not necessarily imply the SDR relation equation(II), and empirical evidence against equation (II) found by Busse (2(X)I) andGNW (2005) catinot rule out tournament behavior. Propositions 1 and 2 clarifythat worsening performance may, indeed, cause a mutual fund manager to deviatemore from the benchmark portfolio, but this could move the portfolio closer toone that minimizes standard deviation. Hence, the correct empirical indicator ofrisk shifting should be the standard deviation of a fund portfolio's return relativeto the benchtnark's return (tracking error), not the portfolio's total return stan-dard deviation.'** Indeed, perhaps the most publicized "gamble" by a mutual fundtiianager was one that increased tracking errors, but reduced total return standarddeviation. In late 1995, Jeffrey Vinik shifted the portfolio of Fidelity's MagellanFund out of technology stocks and into allocations of 19% bonds and 10% cash.With the subsequent stock market rally, the bet turned sour, leading to RobertStansky's replacing Vinik as the fund's portfolio manager.

We can characterize the model's prediction for the time .series of a mutualfund's tracking error, which will be a basis for our empirical tests. Let R,+ [ ~In(V,+i/V,) be an individual mutual fund's rate of return from the beginning ofmonth t to the start of month / + 1, and let Rs.!+¡ ^ ln(5,+ i /5,) be the benchmarkportfolio's rate of return from the beginning of month / to the start of montht + I. Since G, - V,/S„ note that R,î - Rs.,^\ = ln(V;+,/V,) - ln(S,+i/5,) =

"*Koski and Pontiff's (199')) tests calculate alleniative rhV. nioa.'iures equal to the fund return'stotal standard deviation, ils beia. and its residual (idiosyncratic) risk. None of these measures areequal to the standard deviaiion of a fund's return in excess of a benchmark return (tracking error),even if the benchmark is assumed to be the market portfolio from which a fund's beta is calculated.For example, if a > 0 so thai a fund manager increases tracking error risk as perfonnance declines,then it can be .shown that the fund's beta deviates more from unity as performance declines. That is,d\ßv - II/OG = \ßfl - I \d\iu\/dG < 0 for a > 0, where ßv and ß^ are the betas of the fund'sportfolio and the alternative asset portfolio, respectively. However, the fund's beta or residual riskneed not necessarily increase or decrease following poor performance. A derivation of this result isavailable from the authors.


ln(G,+i/G,) A; d\n{Gi) = dD,.'^ The Appendix shows that tracking error, R,+i —^5 ,4.], satisfies

( C \ 1

^ 1 A AG,

where the standard deviation of tracking error equals

(15) y/h, =

and rit^i ~ N{0, 1), ßa ^ \OIG\/<^G^ ào = ac\aa\/[b{\ -n)aG]. and i/ | =I"GI / [(' ~ -7) < c]- The intuition in equation (15) is that the standard deviationof tracking error, ^/JT,, is inversely related to prior performance when do > 0,whieh, as shown in Proposition 1, occurs when cy < 1 and a > 0.

IV. Empirical Methodology

This section outlines two empirical methods that we use to examine a mutualfund's performance and its choice of risk. The first is a nonparametric test thatmodifies the SDR tests used in prior studies. The second is a parametric test thatnests our theoretical model.

A. Standard Deviation Ratio Tests

We first replicate prior studies' test of the SDR given in equation (11). Asemphasized in the preceding section, this hypothesis is not one that is predictedby our model. However, we then test a different SDR hypothesis that is closer inspirit to our model because it is based on the standard deviation of tracking errorsrather than the standard deviation of total returns:

(16) ^ ^ > ^ ^ ,

where CTC.JJ denotes the standard deviation of the rate of return on mutual fund/ s portfolio in excess of its benchmark rate of return (tracking error) during the/th half of the year. Mutual fund / — L is a "loser" that displayed relatively poorperformance in the first half of the year, while mutual fund j — W \s a "winner"that had relatively good perfonnance during the first half.

B. Parametric Tests

We propose a new time-series estimation method that permits a fund's risk torespond to calendar-year performance at each (monthly) observation date, ratherthan just once per year. Allowing frequent changes in the fund's risk is arguably

'^Notc that since tournaments are assumed to ocetir each calendar year. G, is reset to ! at thebeginning of January each year, even though we use multiple years of fund returns to estimate thesepr(x:es!ves. Thus. G, always equals the return on n share of the fund relative to that of ihe benchmarkportfoliu since ihe start of the calendar year.


more logical, since our model predicts that an optimizing manager continuouslyadjusts the fund's risk as its relative perfonnance changes. Another benefit of ourapproach is that it allows the estimation of risk-shifting behavior for each indi-vidual mutual fund, enabling us to examine whether risk shifting appears strongerlor particular types of funds.

As with the SDR tests, our parametric te.sts examine the hypothesis of priorstudies that a fund's standard deviation of total returns should vary inversely withits relative prior performance.^" More importantly, we also test our model's pre-diction that a fund's standard deviation of tracking error varies inversely with itsrelative prior performance. In both cases, the parametric forms that we estimatepermit a mutual fund's returns to display generalized autoregressive conditionalheteroskedasticity (GARCH). Prior research has shown that the returns on indi-vidual stocks and stock indices reflect GARCH-like behavior, so that a time seriesof equity mutual fund returns is likely to exhibit this property.^'

We estimate our model's predicted process in equation (14) but modify equa-tion (15) to allow tracking error variance, /i,, to change stochastically due to fac-tors in addition to prior performance. This is done by generalizing h¡ to follow anexponential GARCH (EGARCH) process first introduced by Nelson ( 1991):

(17)

This log variance process allows for persistence by including the lagged variableln(/i,_ i). The variable in(/i,) is also influenced by the prior period's absolute inno-vation, |7/,|. The.se two variables are standard in EGARCH models. What is uniquein equation (17) and critical for examining the effect of prior performance on amutual fund manager's choice of risk is the variable G^ ' / \/hi~\. As with a pos-itive value of ¿yo in equation (15), a positive value of «i in equation (17) supportsthe hypothesis that a fund manager increases tracking error volatility as the mu-tual fund's performance declines. A key advantage of this EGARCH specificationversus equation (15) or a standard GARCH form is that the parameter, aj, canbe of either sign without creating the possibility that /i, could become negative.The rationale for the functional form G,~ ' / yjh,- \ is that it leads to an effect of

• °A test of this hypothesis may be of independent interest. While mosl research assumes ihatchanges in a fund's risk are due to managerial incenlives, Ferson and Wanher (1996) offer anotherexplanation of why a fund's standard deviation could be inversely related to its prior performance.If better-performing funds receive greater ca.sh inflows, then a funü"s reium standard deviation willdecrease until its new (riskless) cash is fully invested in equities or until the fund increases its ex-posure by purchasing equity derivatives. Because transactions costs are mitigated by graduai, ratherthan immediate, purchase of stocks, and some mutual funds are restricted from holding derivatives,the standard deviaiion of a fund's returns may decline temporarily following a cash inflow. KoskJ andPontiff ( 1999) find evidence consistent with this explanation, since funds that hedge with derivativesdisplay less risk shifting.

-'For example, see French. Schwert, and Stambaugh (1987), who fit different GARCH models tothe Standard & Poor's 500 stock index.


performance on tracking error volatility that locally approximates equation (15)where í32 w 2do}^

The alternative hypothesis assumed by prior studies, that a fund's prior per-formance predicts its future standard deviation of total returns, is tested in a simi-lar manner. It is assumed that a mutual fund's total returns, rather than its returnsin excess of a benchmark return, satisfy

(18) /?f+i = ßy/ii, - -It, + v^T/,4.1,

where A, is assumed to follow an EG ARCH process as in equation (17). However,here we see that h¡ is the mutual fund's total return variance, not its tracking errorvariance. A positive value of «2 from joint estimation of equations (17) and (18)would indicate that a mutual fund manager increases the standard deviation of thefund's total return when its performance declines.

V. Data Description

Information on monthly mutual fund returns and fund characteristics comesfrom the Center for Reseiu-ch in Security Prices (CRSP) Survivor-Bias-Free U.S.Mutual Fund Database. The sample covers mutual funds that operated during theperiod from January 1962 to March 2006. We selected domestic equity fundswhose investment style could he broadly classified as either "growth" or "growthand income."^^ To have sufficient observations for estimating the parameters ofeach mutual fund's time-series processes in equations (14), (17). and (18). werequired that a mutual fund report at least 36 consecutive monthly returns. Thefinal .sample consists of 4,188 growth (G) funds, 861 growth and income (GI)funds, and 1.129 "style-mixed" (SM) funds, this last category being funds whosereported investment style was either growth or growth and income during onlypart of their life. Each of these three fund groups was given a different benchmarkreturn, Rs,i+\ ^ ]n{S,+\/Si), equal to the equally weighted average return on allfunds within the group that operated during month ir'* By benchmarking a fundrelative to others in its broad style classification, we avoid attributing style-related

that our theory of

+e¡\ implies —f = -2 Í /ÜC,OU/

Taking the derivative of equation ( 15) with respect to G, leads to

dh, G~ ~ h,

dG, VV^Therefore, the parameter oj ^ Id» and will be positive when c7 < I and « > 0.

-^Mutual funds with a Wiesenberger or Investment Company Data. Inc. (ICDI) objective of "Ag-gressive Growth." "Growth." "Maximum Capital Gains." "Small Capitalization Growth," or "LongTerm Growth." are classitied as growth funds. Mutual funds with an objective of "Growth and Income"or "Growth wiih Current Income" are classified as growth and income funds. Index funds are excluded.

^This assumption implies thai ftmd managers within a group can identify their benchmark asthe "average" of the security holdings of the funds in their same group. Given that there are a largenumber of mutual funds in each group, this average of security holdings is likely to be close to iheholdings of a style-class index. For example, managers of G funds may identity their benchmarkas approximately the Russell 3000 Growth Index while managers of GI ftinds may identify theirbenchmark as approximately the S&P 500 index.


differences to performance differences as would occur if the same bencbmarkwere used for all funds.-"

Figure I sbows the sample's number of G, Gl, and SM mutual funds in oper-ation during each month of our sample period. G funds operating during the pastdecade account for a majority of our sample observations. The number of funds

FIGURE 1

Mutual Fund Sample Population

Graph A. Growth Funds

3600

3200

o ) i T ) c > o > 0 > o i O ï o > o a ) O ) < T ] d ï m C î C f i A a > o s o ï O )

Year

Graph B Qrowth & Income and Slyle-Mixed Funds

900

700 - ->-

600-

500-

200-

100 ,-,-,-,=n>

03 Oï 05

YearGrowth & Income Style-Mixed

-''However, in our notiparamelric tesis, we do examitie ihe case of funds' performances relative tothe universe of all firms, so thai [he benchmark is effectively the same for all funds.


grew rapidly over the 1990s and peaked approximately three years prior to thesample period's end, since no new funds were added during the last 36 months.This reflects the parameter estimation constraint that a fund report al least threeyears of returns. The proportions of all G, GI, and SM sample funds that survived(were in operation) as of March 2006 are 67%. 65%. and 62%. respectively.

Table 1 presents summary statistics of our mutual fund sample, broken downby fund style. Age is defined as the number of years from the fund's inceptiondate until the date it expired or. for surviving funds, the end-of-sample date ofMarch 2006. Row 1 shows that G and GI funds have, on average, similar ages ofapproximately 9.9 years, whereas SM funds, whose definition is conditioned on astyle change having occurred, are much older. On average, G funds have greaterfront loads, expense ratios, and turnover ratios than do GI funds. Manager tenureis the average number of years that the same individual or group manages a fund'sportfolio. Average manager tenure is similar for G and GI funds, but somewhatgreater for SM funds.

Table 1 also reports statistics on funds' asset size scores, which measuretheir relative sizes by assigning a rank score from 0 (smallest) to 1 (largest) foreach fund according to its total net asset value at the end of each year. This score isthen averaged over each year of the fund's life. The average SM fund is larger thantypical GI and G funds, likely a reflection of the greater average age of SM funds.However, on average SM funds display slower but less volatile asset growth, whileGI funds grow slightly faster than G funds.

The final information in Table 1 relates to a fund's number of share clas.sesand the number of funds in its fund family. These numbers are computed eacbyear and then averaged over all years of a fund's life. SM funds tend to havefewer share classes and belong to smaller fund families compared to G and GIfunds.

VI, Results

A. Standard Deviation Ratio Test Results

Following prior studies. SDR tests are performed using an annual 2 x 2 clas-sification of whether a fund's return performance over the first six months (RTN)was above (winner) or below (loser) the median and whether its SDR computedover the second versus first halves of the year was above or below the median,where these medians are for all funds in its style class (G. GI. or SM). First, wereplicate the tests performed by past research by calculating SDR as in equation(11), equal to the ratio of second to first half-year standard deviations of a fund'stotal returns. A finding that the frequency of funds in the category (low RTN, highSDR) significantly exceeds 25% would be evidence in favor of the hypothesis thatunderperforming funds increase their total return standard deviation. Second, wecalculate SDR as in equation (16). equal to the ratio of .second to first half-yearstandard deviations of a fund's returns in excess of benchmark returns (trackingerror).

The statistical significance of our SDR tests is determined by the methodof Busse ((2001). pp. 58, 62-63). His method controls for the auto- and


TABLE 1

Summary Statistics of Mutual Fund Sample

statistics are based on a sample taken ttom thG CRSP Survivor-Bias-Free U.S. Mutual Fund Database covering Ihe periodJanuary 1962 lo March 2006 Mutual funds witfi a Wiesentierger or Inveslment Company Data, Inc. (ICDI) objective o("AggiessivB Growth." "Growth," "Maximum Capiial Gains." "Small Capdaljzation Growth,' or "Long Term Growth," areclassilied as growth funds. Mutual tunds with an ob|eclive of "Growth and Income" Of "Growth with Curren! Income" areClassified as growth and income funds. Style-mixed funds are mulual funds with a style of growth or growth and incomefor only part of their lives Index funds as well as funds with less than 36 consecutive monthly returns are excluded. Thesample contains 4.188 growth funds. 861 growth and income funds, and 1,129 style-mixed funds. Age is defined as thenumber ot years from the fund's inception date until the date it expired or. lor surviving funds, the end-of-sample dateof March 2006. A fund's fees and turnover ratios are calculated as the average annual numbers. Managef tenure is theaverage number of years that an individual manages a fund's portfolio A fund's asset size score is computed by assigninga rank score from 0 (smallest) to 1 (largest) for each fund according to its lotal net asset value at the end of each year. Thisscore IS then averaged over each year ol the tund s life A fund's asset growth equals the average annual log cfiange intotal net assets, and (T of a fund's asset growth rates is the time-series standard deviation of annual log changes In totalnet assets. The number of share classes records for each year the number of funds that have the same fund name butdifferent share classes. Fund family size equals the number of mutual funds that have an identical management oompanyname in each year Both fund family size and numbers ol fund share classes are then averaged over each year of a fund'slife.

FundChaiacietistic

Panel A. Growth Mulual Funds

Age (years)Front loadBack toadExpense ratioTurnover ratioManager tenure (years)Asset size scoreAsset growth (%)(T of asset growth (%)Number of share classesFund family size

PaneJ S Growth and Income Mulual Ft/nds

Age (years)Front loadBackktadExpense ratioTurnover ratioManager tenure (years)Asset size scoreAsset growth (%)(T of asset growth (%)Number of share classesFund family size

Panel C. Style-Mixed Mutual Funds

Age (years)Front loadBack loadExpense ratioTurnover ratioManager tenure (years)Asset size scoreAsset gfowlh (%)<T of asset growth (%)Number of share olassesFund family size

Panel D. All Mulual Funds

Age (yeafs)Front loadBack badExpense ratioTurnover ratioManager tenure (years)Asset size scoreAsset growth {%)a of asset grovrth (%)Number ol share olassesFund family size

No, OfFunds

4,1884,1864,1884,1884,1454,0114,1854,1764,1384.1884,022

861861861860857847861858850861848

1.1291.1291.1291,1291,1221,0361,1291,1281,1261,1291,045

6,1786,1786,1786,1776,1245,8946,1756,1626,1146,1785,915

Meen

9.920.0140.0110.0161.1546.390.41

32.5071.882.78

146.74

9.940.0110.0110.0140,703

6.310.42

34.3569 95

2.94143 11

16.290.0210.0070.0140.905

7.850 49

21.9959.39

1,93133.34

11.450.0150.0100.0151.0456.630.43

30 8369.31265

143.85

Std.Dev.

6.890.0230.0170.0072.3963.300.24

47 0056.16

139128.90

9.030.0210.0170.0060.5083 100.26

45.6858.89

1.51113.96

14.500.0260.0140.0082.0304.820.24

30.9843.06

1.24123.20

9.610.0230.0170.0072 169

3.630.24

44.5054.59

1.42125.94

Min.

300000.0000.0000.000

1.000.00

-277.800.22

11

30.0000.0000.0000.018

2.000.00

-139.950.58

11

40.0000.0000.0000.0002000 02

-174.475,32

11

30.0000.0000.0000.000

1.000.00

-277.800.22

t1

25%

60.0000.0000.0110.5154.000.217.39

35.581

42

60.00000000.0090.387

4.330.216.19

31.932

47

900000.0000.0090.4134,500.305.42

31.821

32

6O.OOO0.0000.0100,469

4.330.227.02

34.351

41

50%

80.0000,0030.0150.858

6.000.40

26.2556.09

3117

e0.0000.OO20.0130.611

6.000.41

27.1353.06

3120

130.0000.0010.0130.674

6.500.50

16,9848.97

1105

90.0000,0020.0140.771

6.000,42

23 8754.14

3115

75%

110.0200,0100.0201.3637 750.59

50.9889.69

4224

11O.0O70.0100.0190.880

7.500.62

50.6285.51

4212

210.0450.00600171.0349.000.68

34.1070.52

3199

120.0300.0100.0191.23B8.000.62

47.4665,99

4

216

Max.

740.0870.0600.167

130.10448.000.97

452.18565.11

9667

820.0850.0600.0355.95641 000.99

283.49513.00

9626

830.0850.0500.119

60.99641.00

0.98183.41338.83

6626

B30.0870.0600-167

130 10448.000.99

452 18565,11

9667


cross-correlation of fund returns that he and GNW (2005) document.^^ It involvessimulating fund returns from a Fama-French-Carhart four-factor model for eachyear and style class to obtain an empirical distribution for the 2 x 2 classifica-tions. Specifically, for each of the 44 years in our sample, we take the Ny funds ofa particular style class that operated in year y and regress each of their 12 monthlyreturns on market, size, book-to-market, and momentum factors. The four factorsand regression residuals are arranged into two matrices: a 12 x 4 matrix of thefour factors for each month; and a 12 x Ay matrix of the monthly residuals foreach fund. To simulate factors, we randomly select a row from the factor matrixand use the following 11 rows in order, continuing with row one of the factor ma-trix after row 12. To simulate residuals, we resample randomly with replacement12 rows from the residual matrix. We then create simulated monthly returns forA'y artificial funds using the betas and intercepts from the Ny regressions with thesimulated factors and simulated residuals.' ^

We compute RTN and SDR (based on either total returns or tracking error)for each artificial fund and allot funds to cells in 2 x 2 contingency tables based onthe median fund RTN and the median fund SDR. This procedure is repeated foreach year during a particular test sample period (e.g., the entire 1962-2005 sampleor the 1995-2005 subsample) to obtain a single simulation. The entire procedureis then repeated 10,000 times to generate an empirical distribution of monthly 2x2contingency table allotments under the null hypothesis of no tournament behavior.The 10%. 5%, and 1% tails of this empirical distribution provide the significancelevels for our SDR tests. Entries in Table 2 are marked by asterisks *. **. and ***when results exceed these respective significance levels in the predicted direction(low RTN, high SDR significantly greater than 25%) and by daggers ^ ^\ ^ ^when results are significantly opposite (low RTN. high SDR significantly lessthan 25%).

Panel A of Table 2 reports SDR test results when SDR is based on the stan-dard deviation of total returns as in equation (11), Tests are performed by styleclass for the entire sample period, 1962-2005, and for four 11-year subsampleperiods. In addition to these separate tests by fund style, we performed tests byaggregating funds across the style classes. This aggregation was done in two ways:The first "style ranked" (SR) method aggregates funds based their separate styleclass categorizations. For example, if a fund was categorized as (low RTN, highSDR) based on its particular style class ranking, it remained a (low RTN. highSDR) observation when funds were aggregated across styles to compute aggregate2 x 2 frequencies. The second "universe ranked" (UR) method aggregates fundsof all styles prior to calculating RTN and SDR rankings. Hence, this method es-sentially assumes a tournament where each G, GI. and SM fund competes againstall others irrespective of stated style. This aggregation method is consistent with

-*'The chi-square tesi of significance used by BHS (1996) is valid only under the assumption thatmutual fund returns are serially and cross-sectionally independent.

method preserves cross-correlation in the tactor returns and the fund return residuals, aswell as most of the autocorrelation in the factors. However, using constant factor loadings throughoutthe year and resampling Ihe factor retums and fund return residuals removes any relationship betweena fund's prior perlbrmance and risk.


the tests performed in Busse (2001 ) and GNW (2005), which assume that G, GI,and SM funds are a single style.

It is clear from Table 2 that there is no evidenee of underperforming fundsraising the standard deviation of their total returns in the second half of the year.The only statistically significant results occur for the 1995-2005 period and forthe entire 1962-2005 sample period- But these results are exactly opposite tothe findings of BHS (1996): Funds that underpertbrmed in the first half of theyear tended to reduce the standard deviations of their fund returns in the second

TABLE 2

Standard Deviation Ratio Tests of Risk-Taking Behavior

Cell frequencies are reported tor a 2 x 2 classification based on a fund's, i) standard deviation ratio (SDR), and ii| relumperformance for the lirst six months of each year (RTN). Funds are divided annually into four groups based on whetheri) RTN is below (low of "loser") or above (high or "winner") the median, and ii) SDR is above (high) or below (low) the median.In Panel A, risk is defined by the total return standard deviation In Panel B, risk is defined by the standard deviation oftracking error, using as a benchmark the equaiiy weighted return of funds with the same style. Results aggregated acrossstyles are reported m two ways. "All SR" aggregates funds using the previous styie-ranked categonzations ot RTN andSDR: and "All UR" aggregates funds using a uni verse-ranked categorizaiion of RTN and SDR. ', " , and * " indicate 10%,5%, and 1% two-tailed p-values, respectively, when the frequency of iosers with high SDR is significant in the predicteddirection. '', ' '*, and " ^ indicate similar signilioance levels for the nonpredicted direction. Significance is calculated forBusses ((2001), pp 62-63) cross-and au to-corr el at ron adjusted test (symbols following low RTN and fiigh SDR entries).

Panel A SDR Defined by Ihe Standard Deviation of Tolal Returns

Type of Funds

Samp/e Period. 1962-1972GrowthGrowth-IncomeSlyle-MiïedAllSFIAilUR

Sample Petioa. W73-I983GrowthGrowth-IncomeStyte-MixedAilSRAHUR

Sample Period: !984-l994GrovithGrowth-IncomeStyle-MixedAIISRAMUR

Sample Period: 199&-2005GrowthGrowth-IncomeStyle-MiKebAIISHAIIUR

Sample Period: 1962-2005GrowthGrowth-IncomeStyie-MixedAIISRAHUR

Obs

1,062221

1,2262.5092,609

1,800282

1,8933,9753,975

4,659812

5,31710,78810,78B

27,3415,6338,556

41,53041,530

34,8626,948

16,99258,80258.602

LowSDR

22.8820.3622.6822.5622.76

26.1127.3026.0426.1627.22

24.1023.8925.1824.6224.34

27.9327 .«27.5727 7628.33

27.1726.7826.3026.8727.29

Sample Frequency(% of obs.)

Low RTN("iosers*)

HighSDR

26.9328 5127.0027,1027.10

23.7221.6323.7723.6022.74

25.8025 7424.7725 2925 63

2 2 . 0 6 " '22.51 ^22.39''^22.19tt '21.66ttt

22 8 0 ^ ' '23.0Í23.62 f23 0 6 ^ "2 2 . 6 9 ^

LowSDR

26.9328.5127.0027.1027.10

23.7221.6323.7723.6022.74

25.8025.7424,7725.2925.63

22.0622.5122.3922.1921.66

22.B023.0423.6223.0622.69

High RTN("winners")

HighSDR

23.2622.6223.3323.2423.04

26.4429.4326.4126.6427.30

24.3024.6325.2824 8124.40

27.9527.5527.6427.8328.35

27.2427.1326.4527.0027.32

(continued on next page)


TABLE 2 (continued)

Standard Deviation Ratio Tests of Risk-Taking Behavior

Panel B. SDR Defined by the Standard Deviation of Returns in Excess of Benchmark (trackinq error)

Type of Funds

Sarriple Perioa: 1962-1972GrowthGrowth-IncomeSiyie-MixedAIISRAHUR

Sample Period: 1973-1983Grow! hGrowth-IncomeSlyle-MiJtedAIISRAMUR

Sample Period: 1984-1994GrowthGrowth-IncomeStyle-MixedAIISRAHUR

Sample Penotí- 1995-2005GrowthGrowth-IncomeSiyle-MixedAIISRAIIUR

Sample Period: I962'2œ5GrowthG towlh-IncomeStyle-MixedAIISRAHUR

1

Obs.

1,062221

1,2262,5092,509

1,800292

1.8933,9753,975

4.659812

5,31710,78610.788

27,3415,633a,556

41,53041,530

34,8626.948

16.99258.80258.802

LowSOR

24.2020.8124.2323.9124.11

23.7225.8923.4523.7523 60

23.7623.0324.5824 1124 07

24.1524.6924.0824.2124.22

24,0724,4224,1824 152d,15

Sample Frequency( •

Low RTN("losers")

HighSDR

25.6128.05"25.4525.7525.75

26.11"23.0526.36-26.01 —26.36'"

26.14*26,6025.3725.8025.90

25.84—25.2425.8925.77"25.77-

25.89-"25.4025.7525.79-"25.83—

% o1 obs.)

High RTN("winners")

LowSDR

25.6128,0525.4525.7525.75

26.1123.0526,3626 0126,36

26 1426,6025,3725,8025,90

25.8425.2425.8925.7725,77

25.B925.4025.7525.7925.63

HighSDR

24.5823.0824.8824.5924,39

24.0628.0123.8224.2323.67

23.9523.7724.6824.3024.13

2A 1724.Ô224.1524.2524.24

24.1524.7724.3324.2724.18

half.-** Our results confirm those of Busse (2001 ) and GNW (2005). who also findevidence contrary to BHS (1996).

Panel B of Table 2 reports SDR test results when the SDR is based on thestandard deviation of tracking errors as in equation (16). Recall that according toour theory, this type of SDR is the appropriate statistic for testing tournament be-havior. Indeed, we now see that there is substantial evidence that underperformingfunds raise the standard deviations of their tracking errors. This behavior is sta-tistically significant over the entire 1962-2005 sample period for G funds and forthe aggregation of funds of all styles using either SR or UR rankings. Evidenceof our theory's tournament behavior appears for at least some fund styles during

'**NoteUial BHS's (1996) SDR lesls used Morningstar dala on the returns of G mutual funds from1980 lo 1991. When we perfurm SDR lests using our CRSP Jala on the returns of G funds overthe exact .siime period. 1980 lo 1991. we match their Undings. Namely, underperforming G funds raisetheir standard deviations of returns, and this result is slatistically signilicani at the 5'íí' level, even whenadjusted for correlation in fund returns. As .shown in Panel A of Table 2. the fact that this significancedisappears when different periods of I973-I9S3 or 1984-1994 are tised underscores the fragility ofthe BHS (1996) findings.


each of the four subsamples. There is tio contrary evidence of the sort found inPanel A. Measuring risk appropriately as tracking error volatility, rather than totalreturn volatility, appears to make a big difference for tests of tournament behavior.

While the results are not tabulated, we followed BHS {1996) by performingSDR tests with samples split by fund age and fund size.^^ Similar to Panel A ofTable 2, underperforming funds, both new and old. in addition to small and large,did not raise the standard deviations of their total returns. The only statisticallysignificant results were always in the opposite direction of finding that losingfunds decrease their total return standard deviations. However, like Panel B ofTable 2. there was evidence that underperforming funds, both new and old, inaddition to small and large, raised the standard deviations of their tracking errorsas our theory predicts. Evidence for this behavior was strongest for older andlarger funds.

In summary, these SDR tests provide no evidence for the traditional hypoth-esis that underperformance leads to an increase in the standard deviation of fundreturns. In contrast, there is substantially more evidence from SDR tests of aninverse relationship between pertbrmance and the standard deviation of trackingerrors.

B. Parametric Test Results

The previous section's SDR tests are crude in the sense that they allow afund's risk to change only once per year and that they require a cross-sectionalgrouping of funds that implicitly assumes these funds engage in similar behavior.We now perform parametric tests for each individual mutual fund that exploitsthe time-series properties of its returns. These tests permit a fund's risk to changeat each observation date and allow risk-taking behavior to differ across funds.Maximum likelihood estimation of the EGARCH equation ( 17), along with eitherthe total returns equation ( 18) or the tracking error equation ( 14), was carried outfor the 4,t88 G funds, 861 G! funds, and 1,129 SM funds that had at least 36monthly observations over the period from January 1962 to March 2006. *''- '

Table 3 reports summary statistics of the estimates of funds' total return pro-cesses, equations (17) and (18). Recall that this process is not implied by ourtheoretical model because here, \/h¡ represents the standard deviation of a fund's

(old} funds were classified as having been in exisleiice for less (greater) than seven years.Small (large) funds were classified as being below (ahove) the median in asset size.

-"'We obtained convergence of the likelihood function for greater ihar 99.R% (99.9%) of the fund.when estimating the total returns (tracking error) processes. For the few cases where we could nolobtain convergence, the processes were estimated with the GARCH mean reversion coefficient a\constrained to equal the median estimate obtained from the other funds for which it was possible tofind unconstrained e.stimates. These median values are reported in Tables 3 and 4.

"As a robustness check, we also estimated each fund's total return process (18) and tracking errorprocess ( 14) under the assumption that their means were constants. Specihcally. the equations

were estimated, where co and IT, are constant intercept terms. This parametric change had ver>' littleeffect on the estimates of the other parameters and did not alter the qualitative results that we reportbelow. Results of this alternative specification are available from the authors.


total returns and is permitted to vary with performance, G,. The first five columnsof Table 3 give the minimum, first quartile, median, third quartile, and maxiniumfor each parameter's point estimates for the sample of individual funds. Column 6reports the proportion of estimates that is strictly greater than zero. Columns 7and 8 give the proportions of estimates that are significantly positive and nega-tive, respectively, at the 5% confidence level.

TABLE 3

Parameter Estimates for Mutual Funds' Return Processes

fulaximum likelihood estimates are for the monthly return processes of 4,188 growth, 861 growth-income, and 1,129 style-mined mutual funds hawing at least 36 mcnthly return observations during the January 1962 to fvlarch 2006 sample period.The benchmark returns are the equally weighted returns of all (unds. including funds not having at least 36 observations,within each investment style category, Likefihood function convergence allowed us to obtain estimates of ail parameters for4,168 growth, 838 growth-income, and 1,129 style-mixed mutual funds. The value of a^ for the remaining growth-incomefunds was fixed tc the median estimate of the converged growth-income funds, and constrained maximum likelihoodestimates of these remaining tunds' other parameters were obtained.

r- ^ r-

Panel A

1'äQ3\32

33

Panel B

Mao^1«203

Panel C

faoa i

azasPane; D

aoa i

3333

Min.

= ao •r 3] ln(/7[_ 1 )

Distribution of Parameler Estimates

25%

, Growth Mutual Funds

-31,6042^0.5662-5,0833-1,2514

-287.5825

0,0026-11,9096-1,4663-0 1477-0,1093

Growth-Income Mutual Fundí

-4,6868-344.1056-379.9965-80,9025

-157,3799

0.0697-12.5432-1,6006-0.1239-0,0149

Style-MixecJ Mutual Funds

-1.7164-36,0890-5,5808-0,9953-7,6088

0 0717-10,8326-0,8069-0,07370,0071

, All Mutual Funds

-31.6042-344.1056-379.9965-80.9025

-287,5825

0.0189-11.5747

-1.2882-0,1294-0 0743

Quartiies

50%

0.1313-3.67640.0349

-O.05450.1068

0.2089-2,75080.5618

-0.01830,1808

0.2214-2.30140.6350

-0,01190,1763

0,1598-3,30730,2227

-O.03500 1371

75%

0.2515-0.29160.96420.02440.3474

0.32210.47941,33360.09480.3521

0.3059-0,3614

1,04420,03510,3364

0 2740-0.2627

1.00860,03190,3471

Max,

26.573566.6288

5,16316,35559.1991

33.204618.97026.00080.60124.5886

30,683215.71075.06440 50673.5623

33,204666.62886.00086,35559.1991

Est > 0

0.7550,2250,5090.3210.631

0.8490.2710,5700.3830,727

0,8170,2140,6200.3920,758

0.7790.2290.5380,3430668

Summary Statistics

Proportion of

Í > 1.96 f

0,2880,0740.2610,1250.230

0.4690,1020.3500,1890.305

0,5630.0990.4250.1730.395

0,3640 0830,3040.U20.271

< -1 .96

0.0430.3200,1990.2450,093

0,0200.3080.1890.1850,102

0.0430.3360.1540.2020.081

0,0400,3210.1890.2290.092

A positive value for the parameter ui would support the hypothesis that afund increases its standard deviation of returns as its performance declines. How-ever, the second-to-last row of Table 3 shows that only 34.3% of all mutual fundshave a positive estimate for aj and that the median estimate. -0.0350, is negative.Some 14.2% of all funds have a significantly positive estimate of ÍÍ2, but 22.9%have a significantly negative estimate. Hence, more funds appear to lower, ratherthan raise, the standard deviations of their returns as their performance declines.

768 Journal of Financia! and Quantitative Analysis

GI funds are the only .style class that have marginally more estimates of Ü2 thatare significantly positive than are significantly negative (18,9% vs. 18,5%). How-ever, the general results of this estimation exercise are consistent with the previousSDR tests in finding relatively few funds that raise the volatility of their returnswhen their performance is poor.

Table 4 presents summary statistics of the estimates of funds' tracking errorprocesses, equations (14) and (17), In this case, sfh, equals the standard devia-tion of a fund's return in excess of its style benchmark return (tracking error), Afund having a positive ai increases its standard deviation of tracking error as itsperformance declines, which is the type of tournament behavior that is consistentwith our theory. As shown in the second-to-last row of Table 4, there are relativelymore funds with a significantly positive value of «2 than a significantly negativeone (16,8% vs. 12,6%), The result that relatively more funds significantly raisetracking error volatility with poor performance holds for each style category. Butclearly this tournament behavior is not widespread. Indeed, the majority of fundshave coefficient estimates of «2 that are insignificantly different from zero, and themedian point estimate is just slightly below zero. Still, the effect of tournamentbehavior could be sizeable for many funds. At the third quartile point estimateof «2 = 0,03. a fund that was underperforming by one standard deviation (G, =0.70) would have an annual standard deviation of tracking error that was 64 basispoints (bps) greater than if its performance equaled the benchmark (G, = I).''^However, a similarly underperforming fund having the first quartile point esti-mate of «2 ~ -0,048 would lower tracking error volatility by 102 bps. Thus,underperformance can lead to economically significant rises or falls in trackingerror volatility, depending on the fund.

To investigate whether funds that significantly raised tracking error withunderperformance (Ö^ > 0) difïer from those that significantly lowered track-ing error with underperformance («2 < 0)., we compared the characteristics ofthese two groups of funds. Table 5 reports the results of a univariate comparison(Panel A) and a multivariate regression analysis (Panel B), In Panel A. column Igives the total number of funds for which a particular characteristic is rept)rted.and columns 2 and 3 show the numbers of these funds for which we obtainedestimates of «2 that were significantly positive and negative, respectively. Then.,columns 4 and 5 report the average values of fund characteristics for these twogroups of funds whose estimates of 02 were significantly positive and significantlynegative, respectively. Column 6 calculates the difference in the two groups"averages, while column 7 reports the /-statistic for the test of whether the groupmeans are statistically different.

-''The monthly standard deviation of G, across ail funds and years is Ü.O87 and 0.30, respectivety,on an annual basis. Since from equation (17)

dG, 2

the total change in standard deviation at C, = 1 - 0.30 = 0,70 versus G, = I equals

/

0.7(l 1 1 / I \

-ujG-'^dt = - a , I 1 I = 0.214ii2 = 0.0064.2 ^ 2 - \ 0 . 7 0 )


TABLE 4

Parameter Estimates for Mutual Funds' Excess Return (Tracking Error) Processes

Maximum likelihood estimates are for the monthly rettjrn processes o( 4,188 growth. 851 groiMh-income, and 1,129 style-mixed mulual funds having al least 36 montttly return observalions durihg the January 1962 to March 2006 sample period.The benchmark returns are the equally weighted returns ol all lunds, including tunds nol having al leasl 36 oDservations,within each investment style category. Likelihood function convergence allowed us to obtain estimates of ali parameters tor4,188 growth, 861 growth-income, and!,128 siyie-mixed mutual tunds. The value ol a\ for the remaining single slyie-mixedlund was tixed lo the median estimate of the converged style-mixed funds, and constrained maximum likelihood estimatestor this remaining ftjnd's other parameters were obtained.

= In (vN(0,

Distribution of Parameter Estimates

•32

Quartiles

Panel A. Growiti Mutual Funds

-11.3536-44.2961-13.3081

-1.5855-991.2410

-0.0980-11.1542

-0.8859-0.05760.0119

0 0079-1.654S0.7511

-0.00120,2866

Panel B. Growth-Income Mutual Funds

t'Gao

33

-10.7731-53.8398

-5.2996-0.7395

-66.5036

-0.1200-9.4249-0.5546-0.03770.0177

Panel C. Slyle-Mixed Mutual Funds

l^a -2.1955-36.6229-4.1114-0,3530-5.0661

-0.0958-10.2616

-0.4509-0.01740.0460

Panel D. All Mutual Funds

-11.3536-53.8398-13.3081-1.5855

-991.2410

-•0 0993-11.1430-0.7861-0.04750.0147

-0.0171-2.20380.7961

-0.00050.3287

-0.0059-1.51740 84880.00090.2402

G.OGIO-1.65560.7705

-0.00010.2856

0 11400 45661.17680.02880.5372

0 1G320.65811.18410.02230 5412

0.07620.56451 22680.03620.4075

0 10590.49951.18470.02980.5088

31.2935227.9097

7.56790.78147.8778

16.073333 87935.86280.4279

17 6268

3.071330.1132

5.62450.39933.7967

31 2935227.9097

7 5679O.7B14

17.6268

Summary Statisties

Proportion of

Est. > 0

0.5210 2960.6440 4360.766

0.4430 2940 6850 4530.774

0.4770 2910.6930.5210.801

0.502G.2950.659G.4540.773

I > 1.96

0.0950 0820.4070.1520.377

0.1010.1000.4190.1630.417

0.0900.1300.5070 229G.5G0

0 0950.0930 4270.1680 405

f < -1.96

0.0800.2340 11!G.I 350064

0.10602060.0950 1230.079

0.1070.2770.1210.0970.078

O.OBB0238O l l i0 1260.068

Four of the 11 chiiracteristies are signitieantly different between the twogroups. Relative to a fund displaying a positive relationship between performanceand tracking error, a fund displaying an inverse relationship (tournament behav-ior) tends to be older, larger, have fewer share classes, and have a portfolio man-ager with a longer tenure at the fund. That older and larger funds appear moreprone to tournament behavior is consistent with our (untabulated) SDR test find-ings. However, because many fund characteristics are correlated and may proxyfor one another, more insight regarding the determinants of tournament behaviorcan be obtained through multivariate regression. -^

-'-'For example, ihe correlation between fund age and manager tenure is 0.4 i across all funds and0.42 across funds whose estimates of ^2 are statistically significani.

770 Journai of Financial and Quantitative Analysis

TABLE 5

Characteristics of Risk-Shifting Mutual Funds

Foj Ihe regression analysis, the dependent variable ap was winsorized to remove Ihe oHects ot extreme outliers This wasctone by ranking the estimates of 3^ for all funds and setting those above the 99th perceniNe to the value ot ^2 ai the 99tnpercentile. Similarly, all of the values ofa-j Below the fifst percentiie were set to ttie valuB of a^ at the first percenliie. ", " ,and • " Indicate 10%, 5%, and 1% significance levéis, respectively.

Panel A. Univanate Analysis

FundCharacterislic

Age (years)Front loadBack loadExpense ratioTufnover ratioManager tenure (years)Asset size scoreAsset growth (%)a ol asset growth ¡%)No. of share classesFund family size

- R s ,

VI

^, = In

.1 — N

ln(fi,) = a

No. ofFunds

6,1786,1786,1786,1776,1245,9946,1756,1626,1146,1785,915

No.

a? > 0

1,0361,0361,0361,0361,027

9811,0361,0341,0251,036

988

Panel B Mullivariaie Reoression Analysis

Independent Variables

Age (years)Frcnt loadBack loadExpense ratioTurnover ratioManager lenure (years)Asset size scoreAsset growtn (%)a of asset growth (%)No. of share classesFund family sizeAdjusted R^No. ol oQs.

Coefficieni

4 44E-04"2.B4E-021.80E-023.34E-012.2aE-048.51E-04"

-1.D9E-034.93E-06

-2.21 E-05-3.63E-03*-

1.S9E-050.655,829

\ G( /

'(0,1)

of Funds

a * <

780780780780770732779776766780734

Aii Funds

•

Gf'- 1 ) + a? ^

y''f-i

0 32 > 0

12.5710.0160.0100,0151.1926.9480.428

31,33870.1512.545

14^.45

1 /—-hi * •Jhirtt-^t2 ^

Characterislic Average

a^ < 0 Diff

10 503 2 069—0.015 0.0010.010 0 0000.015 0.0001.096 00966.022 0.926—0.402 0 026"

32.237 -0.90072.779 -2.6282.723 -O.179*"

150.65 -6.19

Dependent Variable a2

r-Vaiue

2 750.470.201.470.402.21

-0.180.15

-0.83-3.26

1.38

Funds withSfgniticant a2

Coefficient

1.55E-041.55E-O15.88E-037.29E-013.26E-042.17E-03"8.49E-O3

-5.05E-051.10E-05

-4.54E-03'1,71 E-050.591,688

i-Vaiue

4.191.11

-0.28-0 820.565,092.27

-0.41-0.97-2.61-0,97

f-ValuB

0.441,120,031,360.402.470,62

-O.eo0.17

-1,750.65

Panel B of Table 5 reports the results of regressions where ihe dependentvariable is a fund's estimate of 02 und the explanatory variables are the fund'scharacteristics. However, before performing these regressions, the dependent vari-able 02 wa.s winsorized at tbe first and 99th percentiles to eliminate ihe effects ofextreme outliers in the estimates of this variable.-*^ Columns 1 and 2 report regres-sion coefficients and their /-statistics based on the sample of all 5,829 funds thatreported each of the 1 ! characteristics. These results indicate that only a fund's

''*The existence of exlreme outliers is clear from the minimum and maximum values of ni estimatesin Table 4. This variable wa.s winsorized by ranking Lhe estimates oí ai Ibr all funds and then settingthose estimates below (above) the first (99th) percentJIe to that of the first (99th) percentile estimate.


age. its number of share classes, and its manager's tenure are significantly relatedto its estimate of «2. However, recall that a majority of funds in this regression'ssample have estimates of «2 that are themselves statistically insignificant. Thus,in columns 3 and 4 of Table 5. Panel B. we perform the same regression but usethe subsample of 1,688 firms whose estimates of «2 i'" statistically significant.Arguably, this sample includes only those funds that appear to actually shift theirtracking error risk as their performance changes.

What we find with Ihis sharper sample is that only a fund manager's tenure isa statistically significant determinant of the fund's tendency to increase trackingerror with underpeiformance." This result is consistent with our model's predic-tions. Assuming that a manager with longer tenure is more likely to accumulatepersonal wealth from past compensation, her fixed component of total compen-sation (wealth) is more likely to be positive: that is, the parameter a in equation(7) is positive. Furthermore, Chevalier and Ellison ( 1999J find that managers withlonger tenures tend to have more job sectarity, which also increases the likelihoodthat the manager's wealth remains positive even if the current year's performanceis poor. Hence, Proposition 1 predicts that a manager with this characteristic(positive á) would tend to raise her fund's tracking error as performance declines.

In contrast, a short-tenured manager is tnore likely to have less personalsavings and to suffer termination with poor performance. He would act as if totalcompensation were characterized by a negative value for parameter a. Our modelpredicts that this manager would decrease tracking error as performance declines,precisely what our empirical results confirm.

VII. Conclusion

To help guide empirical tests of mutual fund tournaments, we have modeledthe optimal portfolio choice of a fund manager whose compensation depends onthe fund's relative pertbrmance. Perhaps the most important implication of ourmodel is that a mutual fund's tracking error volatility., not necessarily its total re-turn volatility, is what a portfolio manager changes as the fund's performance de-clines. Hence, prior studies that have found no link between a fund's performanceand the standard deviation of its returns cannot be taken as conclusive evidenceagainst tournament behavior.

Based on our model's insights, this study also developed new nonparametricand parametric methods tor estimating the relation between a mutual fund's priorperformance and its tracking error risk. In particular, our parametric estimationmethod permits a fund's risk to react to its relative performance at each observa-tion date, not just once per calendar year. It also is able to estimate risk-shiftingbehavior for individual mutual funds, thereby allowing a fund's behavior to varywith its particular characteristics. We illustrated this technique by applying it to

is a iiegiitlve relaiionship between a fund's number of share classes and its tendency forlournament behavior, but it is statistically significan! only at a 10% level. A potential explanation forthis relationship is that more share clas.ses increase the fixed administrative expense (overhead) ofmanaging a fund, thereby making the fixed component, a, of the compensation function in equation(7} lend to be negative.

772 Journal of Finanoial and Quantitative Analysis

the monthly returns of more than 6,(X)0 growth and growth and income mutualfunds over the period 1962 to 2006.

As our theory predicts, the paper's empirical results show that most fundsdo not increase their standard deviations of returns as their relative performancedeclines. Rather, the evidence is more supportive of the theory's prediction thatunderperforming funds raise their standard deviations of tracking errors. Alsoconsistent with our model is the empirical finding that such tournament behav-ior is more common when fund managers have longer tenures.

Appendix

The Appendix shows that equation (2) is the equilibrium process for a portfolio ofindividual alternative securities that are chosen optimally by the fund manager. The Ap-pendix then derives the equilibrium process for a mutual fund's relative perfonnance whenthe manager's compensation takes the general form given in equation (7) in the text.

As before, assume that the benchmark index follows equation (1) in the text, butlet there be n different alternative securities. The date / value of the ith security, A¡{t), isassumed to follow the process

(A-1) —— — a i d t + fTidqi, i = l , . . . , / i ,

where cTsdza¡dq¡ = as¡dt and Uidq¡ajdq- = a^dt for all /.y = 1 . . . . , n. It is assumed that as,ai, asi. and (T,y are constants. Here, as and ai may be time varying, but the spread betweentheir expected rates of return, a; — as. is assumed to be constant.

If the fund manager :illt>cate.s a portfolio proportion of w, to alternative security /' andI -X!^|L| VI'; to the benchmark index, then the fund's portfolio value, V, follows the process

/JA

I - ^ n; j as + dt + I'-P'hLet w = [w\ H'2... H',,]' denote the/i x 1 vector of portfolio weights. A simple applicationof Itô's lemma shows that the fund's relative perfonnance. G = V/5, follow.s the process

{A-3) = V ^ VI',- I ft; — as + iTÏ — (Ts/) di +G ''-^ V ' )c

where a« i,s an n x I vector whose ith element equals a^; = <*; — Q.V + al - asi andiagdZ) is an n X 1 vector whose (th element equals Gg.dzi. where (7 , = a} + (T| - lastand dzi = {cTidq¿ — asd:.)/cr^i. The Bellman equation for the manager's optimal portfoliochoice problem is then

(A-4) 0 ^ Max J, + Jr.w'a^G + -JGGW'ÜWG^,


where Í2 is an n x n matrix whose ijth element equals i?,y = a¡j + (TS ~ <^.si — ^sj- Thefirst-order conditions with respect to the w,'s are

n

(A-5) JGCtgi + JGGG ^ Wjßy = 0, / = 1 , , . , , AI.

The n linear equations in (A-5) can be solved

(A-6) Vf,- =

where i/i/ is the ijth element of J7 '. If we define 5¡ as the portfolio proportion invested inalternative asset í relative to the portfolio proportion invested in all n alternative securities,then

(A-7)

E

which is independent of the fund's relative performance, G. Since the optimal amountsinvested in ihc individual alternative securities are constant shares of the total amountinvested in alternative securities, the fund manager's problem can be simptilied to oneof choosing between two different assets at each poinl in lime. One asset is the bench-mark portfolio following the process in equation {!), and the other is the alternative se-curity portfolio following the pr(.>cess given in equation (2), where Q^ = ELi*^'*^''(^A = E"=i E"=i aiójaij. and dq = X ILi (5ÍCT,/ÍT^} Jy,.

We next derive the equilibrium process for a mutual fund's relative performance. Fora given asset allocation, the fund's relative performance follows the process of equation (4).This can be rewritten as

(A-8) dG = ùj'acGdl-t-Lü'Gac.äx,

where dv = {(T,\dq — (Tsdz.) ¡CTU is a standard Brownian motion process and LJ' is the port-folio manager's optimal proportion invested in the alternative asset. Using equation (9) tosubstitute for u;' gives

(A-9, äG (

which can be rewritten as

(A-10) dG =

, \aa\ , ac \aG\ ,where fic, = -, dt) = —7- r— andb {\ - C'y) au (1 - c-y)

•'''Equation (A-6) holds even when one of the alternative securities is assumed to earn a risk-freereturn of r. For example. If ihe ilh security is risk-free, ihen a^i = r — as + <T^ and Í2jj = cr^. Ifthere are no redundant alternative securities, then Ü is a positive-definite, nonsingular matrix, and itsinverse exists.


With no loss in generality {dx can be redefined as -dx), the standard deviation in equation(A-9) can be assumed positive, implying thatí/(i + í/iG > Oand also/ia = |af;| /(TC. > 0."Thus, one can conclude itiat sgn (i/o) - sgn {a) and d] > 0. Since ii > 0 implies trackingerror increases with underperfbrmance. a positive value oí do implies a negative trackingerror-pertbrmance relation.

Tests of equation {A-10) are equivalent to tests of the fund's relative (excess) returnprocess. i/ln(G). Using ltó's lemma and equation (A-10) implies

(A-ll)

Note that the volatility of the fund's relative return equals {{do/G) + d]). Hence, underpropotiional compensation, du = 0 and Ihe volatility of the fund's relative return i.s inde-pendent of perfonnance. If compensation were exponential, do > 0 and i/| = 0 so that thefund's relative return volatility would be inversely proportional to performance, G. "

The process in equation (A-11) is the continuous-time analogue of equation (14),since iyin(G) = d\niV) - d\n{S] ^ In (V,,i/V,) - In (S„i/5,) = Rn\ - Rsj*\. where R,and Rsi are defined as the returns on the fund's portfolio and on the benchmark portfolio,respectively. If we define the relative return volatility as \/hi = {{do/G) + di), then indiscrete time equation (A-11 ) can be approximated as

( G \ I

- ^ I ^ ß<i\/% - i.h, + V^T/,+1,u. J 2

where r},^\ ^ N(0, 1),

References

Basak. S.; A. Pavlova; and A. Shapiro. "Optimal Asset Allocation and Risk Shifting in Money Man-agement."/ít'i';>H'o/f/ífíiíiíiíí/.Cfiíí/(f.(, 20(2007). 1583-1621.

Becker, C ; W, Ferson; D. Myers; and M. Schill, "Conditional Market Timing with BenchmarkInvestors." Jounuil of Financial Economics. 52 (1999). 119-148,

Berk, J., and R. Green. "Mutual Fund Flows and Performance in Rational Maikelfi." Journal of Poiiti-cal Economy. II2(2(KM), 1269-1295.

Brown, K.; W. V. Harlow; and L. Starks, "Of Tournaments and Temptations: An Analysis of Manage-rial Incentives in the Mutual Fund Industry." .loiimai of Finance. 51 (1996), 85-110.

Busse, J. "Another Look at Mutual Fund Tournaments." Journul of Financial und Quantitative A/iti/-vi-ij, 36 (200!), 53-73.

Carpenter. J. "Dties Option Compensation Increase Managerial Risk Appetite?" Journai of Finance.55 (2(K)0), 2.111-2331.

Chevalier, J., and G. Ellison. "Risk Taking by Mutual Funds as a Response to Incentives." Journal ofPoUticai Economy. IO5il'J97), 1Í67-12(X).

Chevalier, J., and G. Ellison. "Career Concerns of Mutual Fund Managers." Quaneri\ Journal ofEconomics. 114 ( 1999), 389-432.

Cuoco, D., and R. Kaniel. "Equilibrium Prices in the Presence of Delegated Portfolio Management,"Working Paper. University of Pennsylvania (2(X)I).

Das, S., and R. Sundaram. "On the Regulation of Fee Structures in Mutual Funds." In QuantitativeAnalysis in Financial Markets, Vbl, 3, M. Avellaneda, ed, Singapore: World Scientific (2002),1-36,

Del Guercio, D., and P. Tkac. "The Determinants of the Row of Funds of Managed Portfolios: MutualFunds versus Pension Funds." Journai ofFinamiai and Quantitative Analysis. 37 12(K)2). 523-557,

Duffie. D.. and H- Richardson. "Mean-Variance Hedging in Continuous Time." Annals of AppiiedProhabiiily. I (1991). 1-15,

•"Recall that ((ac/6) + C) is always positive in equilibriuin and an interior solution requiresC7 < 1.

-"'The limiting case of exponential compensation requires a = I and c -*• oo implying that dn > 0and d\ = 0.


Dybvig, R; H, Famsworth; and J, Carpenter, "Portfolio Performance and Agency,"Sntdiex. fonhcoming (2009).

Ferson. W., and V. Wanher. "Evaluatitig Fund Performance in a Dynamic Market." Financial Analysts7()Hm«/, 52 (1996), 20-28,

French. K.; W. Schwert; and R. Stambaugh. "Expected Siock Returns and Volatility," Journal ofFinancial Economics. 19 (1987), 3-29,

Goetzmann, W., and N. Peles. "Cognitive Dissonatice and Mutital Fund Investors," Journal of Finan-íiul Research. 1(M [mi). 145-158.

Goriaev. A,: T, Nijnian; and B, Werker. "Yet Another Look at Mutual FtJtid Tournaments." Journal ofEmpirical Finance, 12 (2005). 127-137.

Grinblatt. M,, and S. Titmati, "Adverse Risk Incentives and the Design of Performance-Based Con-tracts," Management Science. 35 (1989), 8()7-íi22.

Gruber. M. "Anolher Puzzle: The Growth in Actively Managed Mutual Funds," Journal of Finance,51 (1996). 783-810.

Heinke!. R., and N. Stoughton. "The Dynamics of Portfolio Management Contracts." Review ofFinancialSnulies.l (\994). :S5\-3H7.

Hubcrman. G,, and S. Kandcl. "On the Incentives for Money Managers: A Signalling Approach."European Economic Review. 37 f i 993), 1065-1081.

Huddart. S. "Reputation and Performance Fee Effects on Portfolio Choice by Investment Advisers."Journal of Financial Markets., 2 (1999). 227-271.

Ippolitn. R, "Consumer Reaction to Mea,sures of Poor Quality: Evidence from the Mutual FundIndustry." Journal of Law and Economics. 3.1 ( 1992), 45-70.

Karceski, J. "Relums-Chasing Behavior. Mutual Funds and Beta's Death." Journal of Financial andQuantilative Analysis, 37 (2002). 559-594,

Küski. J.. and J. Pontiff. 'How Are Derivatives Used? Evidence from the Mutual Fund Industry."Journal of Finance. .54 (1999), 791-816.

Lakonishok. J.; A. Shieifer; and R. Vishny. 'The Structure and Petformance of the Money Manage-ment Industry." Bmokings Papers on Economic Activity. Microeconomics. (1992). 339-391.

Merton, R, "Optimal Cotisumption and Portfolio Rules in a Continuous-Time Model." Journal ofEconomic Theory. 3 ( 1971), 373^13.

Nelson, D. "Conditional Heteroskedasticity in Assel Returns: A New Approach." Economeirica. 59(1991). 347-370.

Sirri. E., and P. Tufano. "Costly Search and Mutual Fund Rows." Journal of Finance, 53 (1998),1589-1622.

Starks. L. "Pertbrmatice Incentive Fees: An Agency Theoretic Approach." Journal of Financial andQuantitative Analysis, 22 ( 1987), 17-32,

prior performance and risk chen and pennacchi

Economy & Finance

funds risk

funds performance relative

funds inflows

fund managers compensation

mutual funds shareholders

crosssection of mutual

funds relative returns

pertbrming funds