Stale prices and performance qian


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Stale prices and performance qian

  1. 1. JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS Vol. 46, No. 2, Apr. 2011, pp. 369–394COPYRIGHT 2011, MICHAEL G. FOSTER SCHOOL OF BUSINESS, UNIVERSITY OF WASHINGTON, SEATTLE, WA 98195doi:10.1017/S0022109010000773Stale Prices and the Performance Evaluation ofMutual FundsMeijun Qian∗AbstractStaleness in measured prices imparts a positive statistical bias and a negative dilution effecton mutual fund performance. First, evaluating performance with nonsynchronous data gen-erates a spurious component of alpha. Second, stale prices create arbitrage opportunitiesfor high-frequency traders whose trades dilute the portfolio returns and hence fund perfor-mance. This paper introduces a model that evaluates fund performance while controllingdirectly for these biases. Empirical tests of the model show that alpha net of these biasesis on average positive although not significant and about 40 basis points higher than alphameasured without controlling for the impacts of stale pricing. The difference between thenet alpha and the measured alpha consists of 3 components: a statistical bias, the dilu-tion effect of long-term fund flows, and the dilution effect of arbitrage flows. Whereas theformer 2 components are small, the latter is large and widespread in the fund industry.I. Introduction Mutual fund performance evaluation has long been an important and inter-esting question to finance researchers as well as practitioners. The evidence ofnegative alpha (Jensen (1968)) for actively managed U.S. equity funds on aver-age is particularly puzzling, given that these funds receive billions of dollars innew money flows each year and control trillions of dollars in assets. Further stud-ies show that measures of alpha are sensitive to sample periods (Ippolito (1989)),benchmark indices (Lehmann and Modest (1987), Elton, Gruber, Das, and Hlavka(1993)), and trading dynamics (Ferson and Schadt (1996), etc.). This paper ex-plores how measures of alpha are impacted by fund return properties. Studies of mutual fund performance typically use monthly return data with-out controlling for the issue of stale pricing, an inequality between the current ∗ Qian,, National University of Singapore, Business School and Risk Manage-ment Institute, 15 Kent Ridge Dr., MRB 07-66, Singapore, 119245. I thank Paul Malatesta (the editor),an anonymous referee, Wayne Ferson (my dissertation chair), and committee members Jeff Pontiff,David Chapman, Edith Hotchkiss, and Hassan Tehranian for valuable discussions and comments. Anyremaining errors are mine. 369
  2. 2. 370 Journal of Financial and Quantitative Analysisprice of fund shares and the current value of the underlying assets. This discrep-ancy occurs when the share price set by funds at the end of each day fails toreflect the most current market information on the assets, because the underlyingsecurities are thinly traded or traded in a different time zone. In funds with stalepricing, observed returns differ from true returns. Scholes and Williams (1977)and Dimson (1979) show that when data are nonsynchronous, estimates of betaand alpha are biased and inconsistent. In addition, stale prices in mutual fundshares create arbitrage opportunities for short-term traders who purchase shareswhen the fund’s net asset value (NAV) is lower than the value of the underly-ing securities and sell shares after the true value is incorporated into the NAV.The round-trip transactions can be as quick as overnight. Chalmers, Edelen, andKadlec (2001) call this opportunity a “wild card” option. Greene and Hodges(2002) show that arbitragers take advantage of such opportunities, and their tradescan dilute fund returns up to 0.5% annually.1 This paper proposes a performance evaluation model that estimates alphabased on true returns of underlying assets picked by fund managers rather thanobserved returns that capture price changes of fund shares and that are staleand diluted. The key element of the model is the incorporation of stale pric-ing and return dilution relative to the true return process. Because the evalua-tion model directly controls for statistical bias and dilution effects, the estimatedmanagerial picking ability depends only on the true return process of the portfo-lio and is independent of these biases. The model gives 2 alpha measures: truealpha that is based on true returns of portfolios and reflects the actual pickingability of managers, and observed alpha that is based on observed returns offund shares. Empirical tests show that true alpha is about 40 basis points (bp) higher thanobserved alpha and on average positive although not significant. Moreover, truealpha does not correlate with stale pricing of fund shares, while observed alpha,just like performance evaluated using various traditional models, is negativelyinfluenced by stale pricing. As this paper illustrates, for each 1-standard-deviationincrease in stale pricing, traditionally evaluated performance decreases by about10 bp to 70 bp. Given an average expense ratio of 0.78% for the sample funds,this magnitude is economically important. These empirical findings have important implications for the existence ofpicking abilities among mutual fund managers. Positive alpha in the after-expensereturns implies that fund managers have superior information/abilities to pickinvestments and that these abilities are higher than the fees they have charged.In contrast, observed alpha in previous studies measures values remaining forlong-term shareholders after covering arbitragers’ profits, management fees, liq-uidity costs, and any other costs not fully compensated through other charges.Therefore, observed alpha underestimates the actual picking ability of fundmanagers. The model also generates an interesting industry application: performancedecomposition. The difference between observed alpha and true alpha consists 1 However, Goetzmann, Ivkovi´ , and Rouwenhorst (2001) report a much lower dilution and pro- cpose an alternative “fair pricing” mechanism to alleviate the stale pricing problem.
  3. 3. Qian 371of 3 components: statistical bias, dilution by long-term flows, and dilution byarbitrage flows. The model derives both alphas and bias components and esti-mates them simultaneously. According to the empirical evidence, the statisticalbias from stale pricing is positive but has little economic significance, while thedilution effect, mainly due to arbitrage flows, is negative and significant at the 1%level for most fund style groups. The dilution effect increases with the stalenessin pricing, with funds in the stalest quintile having an average dilution of 98 bpannually and those in the lowest quintile having a dilution of only 18 bp. This ev-idence provides a measure of potential gains to mutual fund investors from fundmanagement actions to reduce stale pricing. The model fits into the literature on nonsynchronous data issue.2 Chen,Ferson, and Peters (2010) and this paper are the first to address this issue in mu-tual fund evaluation models. Whereas the former considers the impact of stalepricing on fund managers’ timing ability, this paper instead examines their pick-ing ability. Correspondingly, the 2 studies model stale pricing differently. Theformer assumes a systematic and time-varying component of stale pricing thatcauses a bias in estimating fund managers’ timing ability. This paper assumesthat stale pricing is constant but analyzes the arbitrage timer’s response to stalepricing, thereby controlling for the arbitrage dilution effect on fund performance. This paper contributes to the literature on fund return dilution due to stalepricing arbitrage3 in the following 3 dimensions: First, it takes a model-basedapproach to directly estimate dilution. The model is applicable to both individ-ual funds and group portfolios. Whereas previous studies treat flows as exoge-nous, by modeling the arbitrage timer’s decision, this approach also allows flowsto be endogenous to stale pricing. Second, it uses a larger and more convenientdata sample than those in previous studies. Whereas previous investigations coverabout 20% of U.S. open-end mutual funds from 1998 to 2001, the present analysiscovers all U.S. domestic equity open-end mutual funds from 1973 to 2007. It alsouses monthly rather than daily observations. Although stale pricing arbitrage hap-pens on a daily basis and is based on daily returns, the dilution effect is cumulativeand the approach here allows for a monthly aggregation. Using monthly data notonly avoids the reporting problems with daily flow data but also makes alpha es-timation comparable to those in the performance evaluation literature. Last andmost important, empirical analysis in this paper reveals that the arbitrage dilutionis indeed a problem for domestic funds just as it is for international funds. Thisfinding is important to investors and fund managers. It also reconciles Chalmerset al.’s (2001) finding that arbitrage opportunity allows 10%–20% profits in both 2 In addition to Scholes and Williams (1977) and Dimson (1979), who are mentioned earlier,Brown and Warner (1985) show when and how the nonsynchronous trading data can affect the es-timation of abnormal performance in the case of event studies. Lo and MacKinlay (1990) show thatnonsynchronous trading could be more evident in portfolios than in individual stocks. A more recentstudy by Getmansky, Lo, and Makarov (2004) reveals that hedge fund managers can purposely smoothreturns to achieve higher measured performance. 3 Chalmers et al. (2001), Bhargava, Bose, and Dubofsky (1998), Goetzmann et al. (2001), andBoudoukh, Richardson, Subrahmanyam, and Whitelaw (2002) show that stale pricing provides a prof-itable arbitrage opportunity. Greene and Hodges (2002) and Zitzewitz (2003) empirically documentreturn dilutions resulting from the arbitrage behavior.
  4. 4. 372 Journal of Financial and Quantitative AnalysisU.S. equity funds and international equity funds with Greene and Hodges’s (2002)documentation of dilution in international funds only. The rest of the paper proceeds as follows. Section II introduces the model,Section III describes the data, and Section IV presents the empirical results.Section V then compares the alpha from the proposed model with those from var-ious traditional models, and Section VI presents robustness checks and discussespossible extensions of the model. Section VII concludes the paper.II. The Model There are two meanings of “market timing” in the literature on mutual funds.In one definition, fund managers increase fund beta by changing portfolio hold-ings when they expect the market to rise (hereafter, market timing); in the other,daily timers trade in and out of funds frequently (hereafter, arbitrage timing). Thispaper focuses only on arbitrage timing and refers to the arbitragers as arbitragetimers.A. Stale Pricing Nonsynchronous trading (Scholes and Williams (1977), Lo and MacKinlay(1990)) and na¨ve methods for determining the fair market value or “marks” for ıunderlying assets (Chalmers et al. (2001), Getmansky et al. (2004)) yield seriallycorrelated observed returns. Thus, assuming that information generated in time tis not fully incorporated into prices until 1 period later,4 the observed fund returnbecomes a weighted average of true returns in the current and last periods:(1a) rt = α + βrmt + εt ,(1b) rt∗ = η rt−1 + (1 − η)rt ,where rt denotes the true excess return of the portfolio with mean μ, and varianceσ 2 , rmt denotes the excess market return with mean μm , and variance σ 2 . Both rt mand rmt are independent and identically distributed (i.i.d.), and the error term εt isindependent of rmt . Here, rt∗ is the observed excess return of the portfolio with 0flows, while η is the weight on the lagged true return. That is, the higher the η, thestaler the prices. Assumedly, arbitrage traders can earn the return rt∗ by trading atthe fund’s reported NAVs. This form can be extended to a variety of more complicated models; forexample, to estimate market timing ability by assuming Cov(rt , rmt ) is nonzero 2or to incorporate time-variant stale pricing by specifying the form taken by theparameter η (Chen et al. (2010)). In this paper, for simplicity, η is constant overtime. 4 This assumption can be verified empirically. I apply Getmansky et al.’s (2004) measure to mutualfund data and show that, on average, fund returns smooth over the current and 1 previous period (PanelB of Table 4).
  5. 5. Qian 373B. Analysis of Mutual Fund Flows The new money flows into the mutual funds consist of 2 components. The ar-bitrage timers’ actions imply a short-term flow component. Nonetheless, the timerfaces a choice between fund shares and risk-free assets. By choosing the opti-mal weights in fund shares, the timer maximizes a utility function that dependson the expected excess return conditional on observations of past fund returns.The change in weights from one period to the next forms flows in and out ofthe funds. Following Admati, Bhattacharya, Pfleiderer, and Ross (1986), Becker,Ferson, Myers, and Schill (1999), and Stein’s (1973) lemma, I find that the opti-mal weights equal the conditional expected fund excess return over its conditionalvariance and Rubinstein (1973) risk aversion.5 Therefore, the change in weightsfrom period t − 1 to period t equals η(rt−1 – rt−2 )/γ(1 – η)2 σ 2 , where γ is riskaversion, η measures stale pricing, rt−1 is the true portfolio excess return in periodt − 1, and σ 2 is its volatility. In addition, the model assumes a long-term flow component Ct−1 , measuredas a fraction of the fund total net assets (TNA) and occurring at the end of periodt − 1. This long-term flow component responds to past long-run returns and isuncorrelated with stale pricing or the short-term returns, rt−1 . For simplicity, itfollows a normal distribution:(2a) Ct−1 ∼ N(c, σc ), i.i.d.,where c is the mean and σ c is the standard deviation of the long-term flows. The total fund flow at the end of period t − 1, measured as a percentage ofthe fund’s TNA, is the sum of the long-term flows and the short-term flows fromarbitrage timers:(2b) ft−1 = Ct−1 + η(rt−1 − rt−2 )/λ(1 − η)2 σ 2 ,where λ equals γTNAFUND,t−1 /ASSETSTIMER,t−1 and ft−1 represents flow as apercentage of the fund’s TNA.C. The Impact of Fund Flows As Greene and Hodges (2002) show, daily flows by active mutual fundtraders cause a dilution of returns, which results from the lag between the timethat money comes in and the time that fund managers purchase risky assets. Ede-len and Warner (2001) note that fund managers typically do not receive a reportof the day’s fund flow until the morning of the next trading day, by which timethe prices of underlying risky assets have changed. For simplicity, this model as-sumes that the fund manager reacts immediately on seeing the flows so that theresponse lags by only 1 day. 5 The optimal weights equal γ −1 E(rt∗ |I o )/Var(rt∗ |I o ), where It−1 includes all the information t−1 t−1 oin past returns {rt−1 t−2 o , r o , . . . , r o }, r o 1 t−1 is the observed return in period t − 1. Given equations (1a)and (1b), the conditional mean and variance are E(rt∗ | I o ) = ηrt−1 + (1 – η)μ and Var(rt∗ | I o ) = t−1 t−1(1 − η)2 σ 2 , respectively.
  6. 6. 374 Journal of Financial and Quantitative Analysis The flow occurring at the end of the previous trading day is ft−1 = CFt−1 /(Nt−1 Pt−1 ), where CFt−1 is the dollar amount of flows, Nt−1 is the number ofshares, and Pt−1 is the observed share price, Pt−1 = TNAt−1 /Nt−1 . By the endof the current trading day t, the observed share price is Pt = TNAt /Nt . Because ofthe lag in investment, the new money flow overnight earns only a risk-free rate ofreturn, TNAt = TNAt−1 (1 + rt∗ + Rf ) + CFt−1 (1 + Rf ). As noted earlier, rt∗ is themeasured TNA excess returns with 0 flows. The number of shares also increaseswhen money flows in, Nt = Nt−1 + CFt−1 /Pt−1 . The observed excess fund return with flows, rto , is the change in the shareprices from trading day t − 1 to t, in excess of the risk-free rate: rto = (Pt −Pt−1 )/Pt−1 − Rf . Given the relations specified above, simple algebra shows that6(3) rto = rt∗ /(1 + ft−1 ).Equation (3) indicates that rto differs from rt∗ and the dilution, rto − rt∗ = rto ft−1 ,depends on the covariance between fund flows and observed returns on the sub-sequent trading day.D. The System The model is now fully defined by equations (1a), (1b), (2a), (2b), and (3).The following are the moment conditions:7(4a) E[rto (1 + ft−1 )] = μ,(4b) Cov[rto (1 + ft−1 ), rmt ] = (1 − η)Cov(r, rm ),(4c) Cov[rto (1 + ft−1 ), rmt−1 ] = η Cov(r, rm ),(4d) Var[rto (1 + ft−1 )] = [η 2 + (1 − η)2 ]σ 2 ,(4e) E( ft ) = c,(4f) Cov( ft−1 , ft ) = −η 2 /λ2 (1 − η)4 σ 2 . Combining these equations with(4g) E(rm ) = μm ,(4h) Var(rm ) = σm ,produces a system with 8 parameters [μm , σ 2 , μ, Cov(r, rm ), σ 2 , η, c, λ] by mwhich fund performance can be estimated. The notations follow definitions fromthe model: rto is the observed fund excess return in period t, rmt is the excessmarket return in period t, ft is fund total flows in period t, and r is the true excessfund return. Parameters μm and σ 2 are the mean and variance of excess market mreturns, μ and σ 2 are the mean and variance of fund excess returns, Cov(r, rm )is the covariance between fund excess returns and market returns, η measures theextent of stale pricing, c is the mean of long-term fund flows, and λ is the multipleof risk aversion and relative asset size between the timer and the fund. 6 r o =(P −P ∗ t t t−1 )/Pt−1 −Rf =(TNAt /Nt )/Pt−1 −1−Rf = [Nt−1 Pt−1 (1+rt +Rf ) + CFt−1 (1+Rf )]/[Nt−1 Pt−1 + CFt−1 ] − 1 − Rf = [1 + rt∗ + Rf + ft−1 (1 + Rf )]/(1 + ft−1 ) − 1 − Rf = rt∗ /(1 + ft−1 ). 7 Formulating moment conditions involves simple mean and variance-covariance transformation.Details are available from the author.
  7. 7. Qian 375E. Components of Performance and Biases I define the “true alpha” or true performance as the alpha estimated with thetrue but unobserved returns. It measures fund managers’ picking ability withoutthe influence of stale pricing or arbitrage timing:(5a) α = μ − Cov(rt , rmt )μm /σm .In contrast, I define the observed alpha or observed performance as the tradition-ally measured alpha, which treats the observed returns as if they were the truereturns: αo = E(rto ) − Cov(rto , rmt ) μm /σ 2 . Thus, m(5b) αo = [μ − Cov( ft−1 , rto )]/(1 + c) − {[(1 − η)Cov(r, rm ) − Cov(rto rmt , ft−1 ) + Cov(ft−1 , rto )μm ]/(1 + c)}μm /σm . 2In an ideal situation, in which there is neither stale pricing nor flows (i.e., η = 0,c = 0, and ft−1 = 0), the observed performance is the true performance. The difference between the observed alpha and the true alpha has the follow-ing 3 components:(6a) B1 = ηCov(r, rm )μm /σm , 2(6b) B2 = [−c/(1 + c)][μ − (1 − η)Cov(r, rm )μm /σm ], 2(6c) B3 = [−1/(1 + c)]{Cov(ft−1 , rto ) − [Cov(rto rmt , ft−1 ) − Cov(ft−1 , rto )μm ]μm /σm }. 2Specifically, B1 is the statistical bias in the performance measurement resultingfrom the nonsynchronicity in the returns data; that is, there is stale pricing, η ≠0, but no influence of flows, c = 0 and ft−1 = 0. If η = 0, then B1 = 0. B2 isthe dilution bias from long-term flows; that is, c ≠ 0 but Cov(ft−1 , rto ) = 0 andCov(rto rmt , ft−1 ) = 0. If c = 0, then B2 = 0. B2 is nonzero as long as c is nonzero,disregarding whether η is 0 or not. As Edelen (1999) shows, the unexpected flowsdilute fund performance because of liquidity-motivated trading. If c is completelypredictable, fund managers can avoid this dilution through certain cash budget ar-rangements. Finally, B3 is the dilution effect from arbitrage flows; that is, ft−1 ≠ 0,and it is correlated with subsequent fund returns. The higher the correlation be-tween arbitrage flows and fund returns, the larger the bias (downward). In addi-tion, as Ferson and Warther (1996) argue, flows also reduce performance througha decrease in the portfolio beta when Cov(rto rmt , ft−1 ) < 0. The relation among the observed alpha, true alpha, and these biases is asfollows:(6d) α o = α + B 1 + B 2 + B3 ,where αo is the observed alpha; α, the true alpha; B1 , the statistical bias; B2 , thedilution of long-term flows; and B3 , the dilution of arbitrage flows, all defined asabove. Appendix A presents the details of performance and its decomposition.
  8. 8. 376 Journal of Financial and Quantitative AnalysisIII. Data Both Lipper and Trimtab currently have data sets of daily fund flows. How-ever, it is widely recognized that mutual funds have no accurate day-end TNA fig-ures because they do not know how much money has been received on the currentday. Therefore, some funds report TNA on day t including the current day’s flows,some report it excluding day t flows, and some report a mixture that includes partbut not all of day t flows. In addition, funds may report one way on one day andanother way on the next day. In other words, as Edelen and Warner (2001) andGreene and Hodges (2002) argue, the reported daily flows may sometimes lag thetrue flows by 1 day, without it being clear when or for which funds. Furthermore,the data cannot be checked against another source because all daily fund flowdatabases suffer from this problem. The Securities and Exchange Commission’s(SEC’s) Form N-SAR includes the current day’s flow in TNA by requirement, butbecause they aggregate all share classes, these data cannot be merged with otherdata. As a result, it is currently infeasible to calculate the true day-by-day flows. Practically, a monthly return dilution is the aggregation of daily return dilu-tions within the month. In addition, it is favorable to use monthly returns for es-timation in order to make comparisons with traditionally measured performance.Therefore, based on equation (3), I aggregate the daily return dilution by monthto give a measure of the monthly return dilution. Appendix B presents the de-tails of the aggregation. It turns out that modifying the flow measure to be a dailyaverage flow of the contemporary month is not only an intuitive but also a safeapproximation that has a downward rather than upward bias.8 Therefore, I use monthly data on U.S. equity funds in the Center for Re-search in Security Prices (CRSP) Mutual Fund Database from January 1973 toDecember 2007, excluding observations during incubation periods and funds withless than 1 year of observations of monthly returns or TNAs smaller than $5 mil-lion. The final sample has 3,545 fund portfolios and 7,423 in term of fund shareclasses. I sort these funds into 8 style groups based on the Wiesenberger objectivecodes, Strategic Insight codes, and Lipper objective codes. They are small com-pany growth, other aggressive growth, maximum capital gain, growth, income,growth and income, sector, and timing funds.9 Panel A of Table 1 summarizes fund returns by style group during the sam-ple period. The small company growth funds show the highest mean in returns(1.15% per month), while the income funds show the lowest mean in returns 8 The sum of daily flow dilutions, if documentable, is larger than the monthly dilution documentedhere with monthly flows. At the monthly frequency, the relation between the observed returns with ∗and without flows becomes rp ≈ rp /(1 + fp /D). Intuitively, the monthly return is diluted by the daily oaverage flows of that month. 9 Denoting the objective codes from Wiesenberger as OBJ, those from Strategic Insight as SI, andthose from Lipper as LP, I classify the styles as following: small company growth funds = OBJ SCG,SI SCG, or LP SCGE or SG; other aggressive growth funds = OBJ AGG, SI AGG, or LP LCGE orMCGE; growth funds = OBJ G, G-S, S-G, GRO or LTG or LP G; income funds = OBJ I, I-S, IEQ, orING or LP EI, EIEI or I; growth and income funds = OBJ GCI, G-I, G-I-S, G-S-I, I-G, I-G-S, I-S-G,S-G-I, S-I-G, or GRI or LP GI; maximum capital gains funds = OBJ MCG or LP CA; sector funds =OBJ ENR, FIN, HLT, TCH, or UTL or LP UI, FS, TK or TL; and timing funds = OBJ BAL, SI BAL,or LP B.
  9. 9. Qian 377(0.72% per month). Panel B of Table 1 summarizes fund flows computed as thepercentage change of TNA after fund returns are controlled for:(7) FLOWit = (TNAit − TNAit−1 (1 + Rit )) / TNAit−1 ,where Rit is the return of fund i in period t. The income funds exhibit moreflows than any other group (1.20% of the TNA monthly), while the other aggres-sive growth funds exhibit the least flows (−0.02% of the TNA monthly). Exceptfor the growth funds, fund flows are persistent, with 1st-order autocorrelationsof about 0.46 ∼ 0.75. TABLE 1 Summary Statistics for Monthly Returns and Flows by Fund StyleTable 1 summarizes the fund returns and flows. All funds in each fund style are grouped month by month to form an equal-weighted portfolio whose time series of returns and flows are summarized here. Flows are computed as in equation (7):FLOWit = (TNAit – TNAit−1 (1 + Rit ))/TNAit−1 . Panel A presents the returns; Panel B gives the flow summary. Returns arereported in percentage rate per month; fund flows are calculated as a percentage of the fund TNA, and ρ1 is the 1st-orderautocorrelation. Fund Style Begin End Mean Min. Max. Std. Dev. ρ1Panel A. Summary of Fund ReturnsSmall company growth 1986 2007 1.15 –27.42 15.79 5.33 0.11Other aggressive growth 1987 2007 1.07 –33.18 19.35 5.75 0.05Growth 1973 2007 0.95 –23.56 15.61 4.40 0.07Income 1973 2007 0.72 –5.68 8.50 1.89 0.16Growth and income 1973 2007 0.92 –16.82 12.62 3.58 0.05Sector 1973 2007 1.01 –25.01 15.73 4.84 0.10Maximum capital gains 1986 2007 0.94 –20.99 12.64 4.93 0.08Timing 1987 2007 0.77 –8.84 16.62 2.70 –0.06Panel B. Summary of Fund FlowsSmall company growth 1991 2007 0.84 –2.18 6.52 1.31 0.69Other aggressive growth 1991 2007 –0.02 –26.98 18.99 3.30 0.48Growth 1973 2007 0.42 –3.65 28.99 2.06 0.07Income 1974 2007 1.20 –4.36 64.07 4.22 0.75Growth and income 1973 2007 0.11 –4.35 4.90 0.97 0.58Sector 1991 2007 0.41 –2.77 3.31 0.88 0.60Maximum capital gains 1990 2007 0.19 –6.18 8.25 1.60 0.46Timing 1990 2007 0.71 –2.09 3.99 0.94 0.75 Section V compares the proposed performance evaluation model with tra-ditional models. The benchmarks used there include market returns, style index(Sharpe (1988), (1992)), Carhart (1997) 4 factors, and Pastor and Stambaugh’s(2003) liquidity factor. Style index returns are constructed with returns of the fol-lowing 8 asset classes:10 90-day T-bills, 1-year T-bonds, 10-year T-bonds, BAAcorporate bonds, a broad equity index, value stocks, growth stocks, and small-cap stocks. The conditional model (Ferson and Schadt (1996)) uses 11 economyinstruments as conditional variables. They are short-term interest rate, term struc-ture slope, term structure concavity, interest rate volatility, stock market volatility, 10 The style-matched benchmark portfolio has weighted average returns of the 8 asset classes. Theweights are obtained by minimizing the tracking error between the fund returns (for all funds of thesame style) and the style benchmark.
  10. 10. 378 Journal of Financial and Quantitative Analysiscredit spread, dividend yield, inflation, industrial output growth, short-term corpo-rate illiquidity, and stock market liquidity.11 Data on these variables are accessiblethrough the Wharton Research Data Services (WRDS).IV. Empirical Results Empirical specifications for the model require transformations of the mo-ment conditions in system (4) into pricing errors.12 In response to the modifica-tion in using monthly data, the flows are contemporary and take the daily averageof each month. Estimation employs the generalized method of moments (GMM)(Hansen (1982), Hansen and Singleton (1982)) that searches for parameter val-ues to minimize the weighted average of pricing errors using the inverse of theircovariance matrix as a weighting matrix. Once the parameters are estimated, Ican compute the true alpha (equation (5a)), the observed alpha (equation (5b)),the statistical bias (equation (6a)), the dilution effect of long-term flows (equa-tion (6b)), and the dilution effect of arbitrage flows (equation (6c)). The followingdiscussion presents the estimation results at the style group and the individualfund level with particular attention to the relation between stale pricing and theperformance components. Table 2 gives the estimation results by style group for the style group port-folios. Returns of each style portfolio are equally weighted, month-by-month,average returns of funds within that investment style. As the table indicates, theestimated true alpha is on average positive but not significant. The spurious biasis small for the style portfolios, but the dilution effects of flows are significant.For example, the dilution effect for short-term flows is 38 bp for maximum capi-tal gain funds, 31 bp for small company growth funds, 22 bp for other aggressivegrowth funds, and 14 bp for sector funds. These results are striking, since stalepricing should be quickly diversified away when aggregating individual fund re-turns to style group average. I also estimate the model for each individual fund. Figure 1 plots the empiri-cal distributions of the t-statistics of the true alpha and the bias components for in-dividual funds by style group. The distributions are based on estimations that usebefore-expense returns at the fund portfolio level. The returns for the fund portfo-lios are generated by adding back expenses and value-weighting them by TNA ofshare classes. Distributions based on after-expense returns at the fund share class 11 These variables take the same definitions as in Ferson and Qian (2004). The short-term interestrate = the bid yield to maturity on a 90-day T-bill; term structure slope = the difference between and5-year and a 3-month discount Treasury yield; term structure concavity = y3 − ( y1 + y5 )/2, whereyj is the j-year fixed maturity yield from the Federal Reserve; interest rate volatility = the monthlystandard deviation of 3-month Treasury rates, computed from the days within the month. Stock marketvolatility = the monthly standard deviation of daily returns for the Standard & Poor’s (S&P) 500 indexwithin the month; dividend yield = the annual dividend yield of the CRSP value-weighted stockindex; inflation = the percentage change in the consumer price index, CPI-U; industrial productiongrowth = the monthly growth rate of the seasonally adjusted industrial production index; short-termcorporate illiquidity = the percentage spread of 3-month high-grade commercial paper rates over 3-month Treasury rates; and stock market liquidity = the liquidity measure from Pastor and Stambaugh(2003) based on price reversals. 12 For example, the moment condition (4a), transformed into a GMM-convenient pricing error,becomes gt = rto (1 + ft ) − μ.
  11. 11. Qian 379 TABLE 2 Alpha and Bias Components at Style Group LevelTable 2 presents the estimation results of equation (6d). The GMM system (based on moment conditions in equations(4a)–(4h) at the style portfolio level is estimated for the 1973–2007 sample period. The style portfolio is equal-weightedmonth by month with funds within that style group. The estimated performance and biases are in annual percentage. Here,α is the true alpha; αo , the measured alpha; B1 , the statistical bias; B2 , the dilution of long-term flows; and B3 , the dilutionof arbitrage flows. *, **, and *** represent significance at the 10%, 5%, and 1% levels, respectively. Style Groups α αo B1 B2 B3Small company growth 4.36 4.01 0.00 –0.03 –0.31*** (0.03) (0.03) (0.00) (–1.45) (–18.18)Other aggressive growth 4.27 4.04 0.00 –0.01 –0.22*** (0.03) (0.03) (0.00) (–0.79) (–8.51)Growth 0.86 0.73 0.00 0.00 –0.13*** (0.01) (0.01) (0.00) (–0.81) (–12.30)Income 2.23 2.20 0.00 –0.01*** –0.03*** (0.04) (0.04) (0.00) (–2.50) (–18.28)Growth and income 1.26 1.20 0.00 –0.01*** –0.06*** (0.01) (0.01) (0.00) (–1.85) (–6.62)Sector 0.65 0.51 0.00 0.00 –0.14*** (0.00) (0.00) (0.00) (–0.39) (–15.78)Maximum capital gains 1.37 0.98 0.00 0.00 –0.38*** (0.01) (0.01) (0.00) (–0.23) (–20.06)Timing 1.21 1.17 0.00 –0.01*** –0.03*** (0.01) (0.01) (0.00) (–2.52) (–5.53)level are similar. These distributions suggest that fund managers generally haveno significant picking ability (although the distribution of the t-statistics of alphais centered above 0). In addition, there is little spurious bias or dilution effect oflong-term flows; however, the dilution effect due to short-term flows is significantand widespread. Not surprisingly, the dilutions, while mostly negative, are posi-tive for some funds, which may result, at least partly, from a net outflow causingthe cash balance to shift downward or, in a downturn market, from a higher cashbalance making the return look better. I next examine the relation between stale pricing and performance compo-nents. Table 3 summarizes the fund-level estimates of true alpha and biases by theextent of stale pricing. Funds are sorted into 5 quintiles according to stale pricing.Panel A presents the means of the true alpha estimated with after-expense returnsat the fund share class level, while Panel B presents the means of the true alphaestimated with before-expense returns at the fund portfolio level. In both panels,neither the estimated true alpha nor the dilution effect of long-term flows differssignificantly across groups; however, the statistical bias is significantly larger inthe top quintile than in the bottom quintile. The dilution effect of arbitrage flows isalso larger (in a negative direction) in the top quintile than in other quintiles. Sincethe magnitude of the dilution effect is larger than that of the spurious bias, the ob-served alpha is significantly smaller in the top quintile. Specifically, in Panel B,the true alpha is about 40 bp higher than the observed alpha in the sample average;the dilution effect of arbitrage flows is −0.98 in the top quintile and −0.18 in thelowest quintile for fund portfolios. These results are consistent with the theoret-ical derivation that the statistical bias is linearly and positively related with stalepricing, whereas the dilution effect of arbitrage flows is negatively related with
  12. 12. 380 Journal of Financial and Quantitative Analysis FIGURE 1 Distribution of t-Statistics of True Alpha and Biases in the Observed AlphaThe empirical transformation of system (4) is estimated with the GMM approach fund by fund (portfolio level) with before-expense returns, and their t-statistics are grouped and then plotted by style group. The 4 plots are for true alpha (GraphA), statistical bias (Graph B), dilution of long-term flows (Graph C), and dilution of arbitrage flows (Graph D). The x-axispresents the fund style group. The y-axis presents the range of the t-values: t > 2.36, 1.96 < t < 2.36, 0 < t < 1.96,0 < t < –1.96, –1.96 < t < –2.36, t < –2.36. The z-axis presents the percentage of funds that fall into the range of thet-values for each style group.Graph A. Distribution of t-Statistics for the True AlphaGraph B. Distribution of t-Statistics for the Statistical Bias (continued on next page)
  13. 13. Qian 381 FIGURE 1 (continued) Distribution of t-Statistics of True Alpha and Biases in the Observed AlphaGraph C. Distribution of t-Statistics for the Dilution of Long-Term FlowsGraph D. Distribution of t-Statistics for the Dilution of Arbitrage Flowsarbitrage flows, hence stale pricing. Most important, the true alpha is independentof stale pricing and flows. One interesting question is how the magnitude and significance of stale pric-ing and arbitrage dilution have changed over time. To address this issue, for eachyear, I pool the monthly return and flow observations of all funds existing in thatyear and apply GMM procedure. This approach forces the parameters to be iden-tical across all funds in the same year, and searches for values that minimize the
  14. 14. 382 Journal of Financial and Quantitative Analysis TABLE 3 Performance Evaluation Considering Stale Pricing and Endogenous Flows by Individual FundTable 3 summarizes the estimated results of the GMM system (based on moment conditions in equations (4a)–(4h) at theindividual fund level for 1973–2007. The system is first estimated for each fund, then the estimated true alpha, observedalpha, and biases are summarized in annualized percentage. In the summary, the funds are sorted into 5 quintiles ac-cording to their stale pricing, and the mean of each quintile and the statistical differences between the highest and lowestquintiles are presented. Estimates in Panel A are based on after-expense returns at the fund share class level. Estimatesin Panel B are based on before-expense returns at the fund portfolio level. The before-expense returns are obtained byfirst adding back expenses then averaging to the portfolio level with TNA as weights. *, **, and ** represent significanceat the 10%, 5%, and 1% levels, respectively. α: True αo : Observed B1 : B2 : Dilution of B3 : Dilution of Quintiles Alpha Alpha Statistical Bias Long-Term Flows Arbitrage FlowsPanel A. Sorted by Stale Pricing: Share Class Level after ExpensesFull sample 2.24 1.91 <0.001 –0.03 –0.311 (lowest) 2.33 2.13 <0.001 –0.02 –0.172 2.59 2.36 <0.001 –0.03 –0.203 2.42 2.12 <0.001 –0.03 –0.274 2.22 1.86 <0.001 –0.03 –0.345 (highest) 2.35 1.12 0.004 –0.02 –0.51Highest – Lowest 0.02 –1.01*** 0.004** 0.00 –0.34***t-stat. of the difference (0.13) (–5.18) (2.05) (1.63) (–5.32)Panel B. Sorted by Stale Pricing: Portfolio Level before ExpensesFull sample 3.60 3.19 0.001 –0.03 –0.381 (lowest) 3.76 3.56 <0.001 –0.02 –0.182 3.94 3.59 <0.001 –0.03 –0.323 3.10 2.61 <0.001 –0.03 –0.454 3.09 2.82 <0.001 –0.01 –0.265 (highest) 3.75 1.75 0.011 –0.01 –0.98Highest – Lowest –0.01 –1.81*** 0.011** 0.01 –0.80***t-stat. of the difference (–0.04) (–5.14) (2.23) (–0.36) (–4.19)moments’ estimation errors both in time series and cross section. The yearly esti-mation of stale pricing and dilution effects are then plotted in Figure 2, in whichthe x-axis represents year, the left-hand y-axis represents stale pricing and the es-timated values are designated by columns, and the right-hand y-axis denotes thearbitrage dilution measured in annualized percentage and the estimated values aredesignated by line-connected dots. As the graph illustrates, stale pricing fluctuates only a little, no more thana 0.1 difference from year to year without any trend. Arbitrage dilution, how-ever, exhibits a clear pattern: From a small negative of around −0.20 in 1991,it increases in magnitude until reaching the largest negative of −0.65 in 1998and then reverses to the smallest negative of −0.10 in 2006. This pattern of dif-ferences is not surprising given that the magnitude of dilution depends on theamount/frequency of both arbitrage flows and degree of stale pricing. Dilutionpeaks in the years 1998–2001 with a magnitude of more than −0.50 persistently.This finding is consistent with the SEC litigations claiming that during this timeperiod, mutual funds allowed hedge funds to time and late trade mutual fundshares widely and frequently. In all years, estimations are significant at the 1%level for arbitrage dilution but weak for stale pricing. The overall results indicate that the true alpha estimated with the proposedmodel is positive although not significant and free of stale pricing biases, whilethe observed alpha is negatively impacted by stale pricing. The negative impact
  15. 15. Qian 383 FIGURE 2 Time-Series Trend of Stale Pricing and Arbitrage DilutionFigure 2 plots the estimated stale pricing and arbitrage dilution with fund-level observations by year. The x-axis denotesyear. The left-hand y-axis denotes the stale pricing level and the corresponding values are designated by the columns in thefigure. The right-hand y-axis denotes dilution in the annualized percentage, and the corresponding values are designatedby the line-connected dots.results primarily from the arbitrage flow dilution effect, while the statistical biasand the dilution effect of long-term flow are small. Moreover, the arbitrage activ-ities vary over time in spite of the relatively stable stale pricing level.V. Comparison: Traditional Performance Measures and Stale Pricing The previous section shows that alpha estimated with the proposed modelis uncorrelated with stale pricing. As a comparison, this section examines therelation between alpha estimated with traditional models and stale pricing.A. Measures of Performance and Stale Pricing I compute fund performance using 6 conventional alpha estimates from thefollowing equations:(8a) rit = αi + βi rmt + εit ,(8b) rit = αi + βi rst + εit ,(8c) rit = αi + βi rmt + ci rm Zt−1 + εit ,(8d) rit = αi + βi rst + ci rst Zt−1 + εit ,(8e) rit = αi + βi rmt + ci SMBt + di HMLt + ei MOMt + εit ,(8f) rit = αi + βi rst + ci PREMLIQt + εit ,where rit is fund i’s return in period t, rmt is the market return, rst is the stylebenchmark return, and Zt−1 are lagged economic instruments.13 The first 4 alphas 13 I use average alpha alone in the conditional model, because it is well specified in large samplesand unaffected by problems of data mining and persistent lagged regressors. The time-variant alpha,however, has a spurious bias (Ferson, Sarkissian, and Simin (2003), (2008)).
  16. 16. 384 Journal of Financial and Quantitative Analysisare Jensen’s (1968) alpha and the conditional alpha of Ferson and Schadt (1996)estimated with a market benchmark or style benchmarks.14 The 5th alpha controlsfor the Carhart (1997) 4 factors, and the last alpha controls for the Pastor andStambaugh (2003) liquidity factor. I use 4 measures of stale pricing. The 1st measure, introduced by Lo andMacKinlay (1990), is designed for a nontrading scenario:(9a) π = −Cov(rto , rt+1 )/μ2 , o if Cov(rto , rt+1 ) < 0, o and 0 otherwise,where rto is the observed returns, and μ is the mean of the true return. The truereturn follows a 1-factor linear model. In each period, the security has π, theprobability of nontrading; the larger the π, the staler the price. The 2nd measure, the smooth index (Getmansky et al. (2004)), is designedfor thin trading scenarios in which trades are not deep enough to absorb allinformation:(9b) ξ = Σθj2 , where rto = Σθj rt−j , Σθj = 1, θj ∈ [0, 1], and j = 1, 2, . . . , k, owhere rto is again the observed returns, θj is the autoregressive coefficient of ob-served returns up to j periods, and the true return follows a 1-factor linear model.The value of smooth index ξ is confined: 1/(1 + k) ≤ ξ ≤ 1. The wider thedistribution of θj , the smaller the ξ; the more concentrated the θj , the larger the ξ.Indeed, Getmansky et al. show that smoothing increases the Sharpe (1966) ratioof the observed returns of hedge funds. In addition to π and ξ, stale pricing can also be measured by the covariancebeta of fund returns and lagged market returns or the autocovariance beta of fundreturns:(9c) BETA(rt , rmt−1 ) = Cov(rt , rmt−1 )/Var(rm ),(9d) BETA(rt , rt−1 ) = Cov(rt , rt−1 )/Var(r),where r represents fund returns, rt is the fund return in period t, rm representsmarket returns, and rmt−1 is the market return in period t − 1. Table 4 summarizes the estimated stale pricing by fund style with each mea-sure separated out by panel. Less than half the funds display nontrading prop-erties. Among the funds with π > 0, the probability of nontrading is as high as0.52. The minimum smoothing index ξ is 0.17 for all style groups, implying thatthe return can smooth up to 5 periods, k = 5. The average smoothing index ξis about 0.5, implying that prices on average smooth back only 1 period, k = 1.Both BETA(rt , rmt−1 ) and BETA(rt , rt−1 ) are on average highest in the growthfunds and lowest in the timing funds. All the stale pricing measures display widevariations in cross section. Table 5 presents the joint probability of a fund’s stale pricing being high orlow in period t and t + τ , τ = 1, 2, . . . , 5, with each block in the table summingup to 1. As is apparent, except for BETA(rt , rt−1 ) and τ = 2 or 3, whose diagonalsums only to 0.45, the sum of the diagonal in most blocks is larger than the sum 14 Tables on the style benchmark weights and returns are available from the author.
  17. 17. Qian 385 TABLE 4 Summary of Estimated Stale Pricing of Individual Funds by Style GroupsTable 4 summarizes the stale pricing of funds. For each fund, stale pricing is measured by estimating equations (9a)–(9d) with 36-window returns rolling by year. Panel A summarizes the estimated stale pricing, π, by fund styles. Here, π =–Cov(ro , ro )/μ2 , if Cov(ro , ro ) < 0, and 0 otherwise (equation (9a)). Only the means of the uncensored observations t t+1 t t+1are presented, and the percentages of uncensored observations are in parentheses. Panel B summarizes the estimatedstale pricing, ξ, by fund styles. Here, ξ = Σθ 2 , where Ro = Σθ j Rt−j , and Σθ j = 1, θ j ∈ [0,1], j = 1, 2, . . . , k (equation (9b)). j tPanels C and D summarize the estimated stale pricing BETA( rt , rmt−1 ) and BETA(rt , rt−1 ), respectively, from equations(9c) and (9d). Style Groups Mean (π > 0) Std. Dev. Min. Max.Panel A. Stale Pricing π (Lo and MacKinlay (1990))Growth funds 0.46 (9.6%) 0.29 0.000 1.00Maximum capital gain 0.48 (21.0%) 0.31 0.016 0.98Other aggressive growth 0.50 (20.4%) 0.28 0.000 1.00Income funds 0.43 (14.5%) 0.29 0.002 1.00Growth and income 0.52 (22.0%) 0.29 0.003 1.00Sector funds 0.48 (15.7%) 0.29 0.005 0.99Small company growth 0.44 (18.1%) 0.29 0.003 1.00Timing 0.51 (21.3%) 0.28 0.002 1.00Panel B. Stale Pricing ξ (Getmansky et al. (2004))Growth funds 0.47 0.22 0.17 1.00Maximum capital gain 0.44 0.23 0.17 1.00Other aggressive growth 0.44 0.22 0.17 1.00Income funds 0.46 0.22 0.17 1.00Growth and income 0.44 0.22 0.17 1.00Sector funds 0.45 0.22 0.17 1.00Small company growth 0.47 0.23 0.17 1.00Timing 0.41 0.20 0.17 1.00Panel C. Stale Pricing BETA(rt , rmt−1 )Growth funds 0.03 0.24 –0.84 1.11Maximum capital gain 0.02 0.22 –0.44 0.69Other aggressive growth –0.01 0.16 –0.60 0.78Income funds –0.03 0.10 –0.55 0.42Growth and income –0.02 0.13 –0.60 0.66Sector funds 0.00 0.19 –1.89 0.83Small company growth –0.04 0.22 –0.95 0.71Timing –0.05 0.10 –0.46 0.30Panel D. Stale Pricing BETA(rt , rt−1 )Growth funds 0.09 0.14 –0.44 1.66Maximum capital gain 0.03 0.14 –0.46 0.43Other aggressive growth 0.03 0.14 –0.65 0.52Income funds 0.10 0.19 –0.61 0.99Growth and income 0.02 0.15 –0.63 0.58Sector funds 0.05 0.13 –0.50 0.49Small company growth 0.04 0.13 –0.46 0.50Timing 0.01 0.15 –0.58 0.48of the off-diagonal, even when τ > 2. BETA(rt , rmt−1 ) particularly never switchesbetween high and low subsamples from one period to another. These results implythat stale pricing is widespread and persistent in the mutual fund industry.B. Explanatory Relation between Measured Performance and Stale Pricing I examine the relation between performance and stale pricing in a 2-step pro-cess. First, I estimate each fund’s performance α and stale pricing π, ξ, BETA(rt ,rmt−1 ), and BETA(rt , rt−1 ) with a rolling window of 3 years (36 months), in whicheach rolling moves observations forward 1 year (12 months). In the 2nd step, I ap-ply the Fama-MacBeth (1973) method to examine their cross-sectional relation.
  18. 18. 386 Journal of Financial and Quantitative Analysis TABLE 5 Persistence in Stale PricingFor each year, funds are classified into high (H) or low (L) according to their stale pricing. Table 5 presents the jointprobability of a fund falling into the H or L category in years t and t + τ . Staleness, π, ξ, BETA(rt , rmt−1 ), and BETA(rt , rt−1 )are from equations (9a), (9b), (9c), and (9d). For π, H means π > 0 and L means π = 0. For ξ, H means ξ ≥ 0.5 andL means ξ < 0.5. For BETA(rt , rmt−1 ) and BETA(rt , rt−1 ), H means higher than the sample average and L means lowerthan the sample average. t (τ = 0) τ =1 τ =2 τ =3 τ =4 τ =5Categories H L H L H L H L H LPanel A. Stale Pricing π (Lo and MacKinlay (1990))H 0.05 0.08 0.03 0.11 0.04 0.14 0.03 0.16 0.02 0.17L 0.11 0.76 0.13 0.72 0.14 0.68 0.10 0.71 0.10 0.71Panel B. Stale Pricing ξ (Getmansky et al. (2004))H 0.11 0.21 0.10 0.20 0.11 0.22 0.11 0.24 0.11 0.24L 0.20 0.47 0.21 0.50 0.20 0.47 0.20 0.45 0.20 0.46Panel C. Stale Pricing BETA(rt , rmt−1 )H 0.46 0.00 0.48 0.00 0.50 0.00 0.51 0.00 0.50 0.00L 0.00 0.54 0.00 0.52 0.00 0.50 0.00 0.49 0.00 0.50Panel D. Stale Pricing BETA(rt , rt−1 )H 0.36 0.17 0.22 0.28 0.20 0.24 0.21 0.17 0.24 0.19L 0.14 0.33 0.28 0.23 0.30 0.25 0.29 0.32 0.26 0.31Because the rolling window approach causes the estimated coefficient to followan MA(2) process, I adjust the standard errors of the coefficients on stale pricingwith the Newey and West (1987) approach.15 The controlling variables in the 2nd-stage regression include log(TNA), expense ratio, log(age of the fund), portfolioturnover, income distributed, capital gains distributed (CAP GNS), net flows, to-tal loads, and fund style. All explanatory variables are studentized, and fund stylesare controlled for with dummy variables. Table 6 presents the Fama-MacBeth (1973) mean and t-statistics of the co-efficients on stale pricing and other fund characteristics, with each panel address-ing 1 measure of stale pricing. Overall, the results indicate that the traditionallyestimated alpha is negatively associated with stale pricing. The results are con-sistent and significant for all 6 measures of alpha when stale pricing is mea-sured with smooth index ξ and BETA(rt , rmt−1 ). For each 1-standard-deviationincrease in ξ, α decreases by 11 bp to 42 bp, with significance at the 1% level.For each 1-standard-deviation increase in BETA(rt , rt−1 ), α decreases by 18 bp to73 bp, with significance at the 5% or 1% level. The coefficients on π are negativeon average but significant only when performance is evaluated with the Carhart(1997) 4-factor model. The results for BETA(rt , rt−1 ), however, are puzzling inthat the coefficients switch signs and are sometimes significant on the positiveside. Nonetheless, this outcome is not intuitively surprising. Traditional perfor-mance evaluation suffers from a missing variable problem when the returns arepartially explained by a lagged pricing factor. The mean of the missing compo-nent goes into the estimated alpha and is positively associated with the importance 15 Denoting γ as the coefficient of stale pricing on alpha, I compute σ(γ ) as follows: σ(γ ) = 1 1 1{(1/T) × [Σt=1:T gt gt + (4/3)Σt=2:T gt gt−1 + (2/3)Σt=3:T gt gt−2 ]}1/2 , where gt = γ 1t − Mean(γ 1 ).
  19. 19. Qian 387of the lagged pricing factor, which can be proxied by BETA(rt , rt−1 ).16 Overall,these results suggest that the cross-sectional differences in conventional alphasare largely explained by stale pricing even after various fund characteristics havebeen controlled for.VI. RobustnessA. Two-Stage Approach with the Proposed Measure of Performance Section IV shows that the true performance derived from the proposed modelis free of biases from stale pricing. In contrast, Section V shows that performancemeasures from traditional models are affected by stale pricing. However, per-formance and stale pricing are estimated separately in Section V but simultane-ously in Section IV using the proposed model. Therefore, it is necessary to checkwhether or not the true alpha from the proposed model is correlated with stalepricing estimated separately as in Section V. As is apparent from Table 7, the truealpha estimated using the proposed model is also free of the stale pricing biaswhen the 2-stage approach is used.B. Errors in Variables The procedure for estimating the relation between traditional performancemeasures and stale pricing consists of 2 stages: estimation of both performanceand stale pricing for each fund with time-series data and regression of the esti-mates from the 1st stage on each other in cross section. As a result, the 2nd-stageregression is likely to suffer from an errors-in-variables problem. This sectionaddresses the issue by calculating the bias caused by this problem. Two biases are involved here. The first is the attenuation bias, A = Var(a2 )/[Var(a2 ) + (x2 x2 )−1 Var(ε2 )], where a2 is the explanatory variable in the 2ndstage, and x2 and ε2 are, respectively, the explanatory variable and the observationerror in the 1st stage, where a2 is estimated. This is the bias studied in a traditionalerrors-in-variables problem, in which only the explanatory variable, a2 , is ob-served with an error. The 2nd bias is caused by the correlated errors in the depen-dent and explanatory variables, Φ=(x1 x1 )−1 (x1 x2 )(x2 x2 )−1 Cov(ε1 , ε2)/[Var(a2)+ (x2 x2 )−1 Var(ε2 )], where a1 is the dependent variable in the 2nd stage, and x1and ε1 , respectively, are the explanatory variable and the observation error in the1st stage in which a1 is estimated. The problem now becomes the calculation ofA and Φ, whose values are determined by Cov(ε1 , ε2 ) and Var(ε2 ).17 To estimate the cross-sectional variance and covariance of ε1 and ε2 , theequations in the 1st stage must be regressed simultaneously for all funds. How-ever, this procedure is plausible only if the number of funds in the cross sec-tion is much smaller than the number of periods in the time series. Assuming 16 I also examine the role of loads in impeding opportunistic trading by imposing high transactioncosts and find that high loads reduce the dilution of returns in funds with stale pricing. In addition, Iexamine the predictive relation between stale pricing and future performance and find that stalenesshas a weak predictive power. Tables of these results are available from the author. 17 Details of the errors-in-variables issue are available from the author.
  20. 20. 388 Journal of Financial and Quantitative Analysis TABLE 6 Traditional Performance Measures and Stale Pricing (Fama and MacBeth Approach)Table 6 displays the relation between fund stale pricing and traditionally evaluated performance. The dependent variablesare alphas in annual percentages before expenses. The explanatory variables are stale pricing, lagged fund characteris-tics, and fund styles. The discretionary turnover is the turnover component that is orthogonal to fund flows. The explanatoryvariables are studentized. Both alphas and stale pricing are estimated with 36 months of returns and rolled over by year.The Fama-MacBeth (1973) coefficients and t-statistics are presented. The standard errors of the coefficients on stalenessare adjusted by a moving average (MA(2)) process. *, **, and *** represent significance at the 10%, 5%, and 1% levels,respectively. Alpha from Alpha from Unconditional Conditional Market & Unconditional Conditional and Style and Style Carhart’s Liquidity Variables CAPM CAPM Benchmarks Benchmarks 4 Factors PremiumPanel A. Stale Pricing π (Lo and MacKinlay (1990))Intercept 3.66 5.58 –0.70 –0.40 0.22 5.06 (12.07) (15.19) (–3.14) (–0.99) (1.00) (13.49)Staleness –0.12 –0.15 0.09 –0.09 –0.54*** 0.07 (–1.08) (–1.12) (0.87) (–0.71) (–2.94) (0.76)Flow 1.22*** 1.27 1.10* 1.27*** 1.11*** 0.96*** (3.50) (1.59) (1.82) (6.97) (5.75) (6.60)log(Age) 0.00 –0.09** 0.12*** –0.01 –0.05 0.04 (–0.09) (–2.11) (4.49) (–0.28) (–1.51) (1.21)log(TNA) –0.04 0.04 –0.01 –0.09 –0.11 –0.01 (–0.60) (0.54) (–0.25) (–1.34) (–1.56) (–0.13)Income 0.77*** 0.86*** 1.30*** 1.66*** 0.40*** 0.57*** (11.09) (9.15) (12.74) (12.06) (5.37) (8.72)CAP GNS 0.28*** 0.36*** –0.08 0.14 0.08** 0.29*** (3.35) (2.80) (–1.16) (1.14) (2.12) (3.30)Discretionary –0.18*** –0.15*** –0.31*** –0.33*** 0.08 –0.36*** turnover (–5.95) (–4.25) (–8.69) (–10.74) (1.38) (–12.47)Total load 0.29*** 0.12* 0.13*** 0.05 0.44*** 0.35*** (4.74) (1.86) (3.33) (1.13) (5.58) (4.91)Expense 0.07 0.44*** 0.20*** 0.44*** –0.18 0.39*** (0.82) (5.87) (2.69) (4.18) (–1.37) (5.85)Styles Controlled with dummy variablesPanel B. Stale Pricing ξ (Getmansky et al. (2004))Intercept 1.40 2.22 –0.14 –0.38 0.62 1.70 (2.52) (4.67) (–0.27) (–0.71) (2.12) (2.76)Staleness –0.29*** –0.33*** –0.42*** –0.42*** –0.11*** –0.25*** (–3.53) (–3.83) (–4.73) (–4.45) (–2.78) (–3.19)Flow 0.78*** 1.16*** 1.07*** 1.71*** –0.24 0.61*** (4.22) (7.54) (5.30) (8.11) (–1.07) (3.12)log(Age) 0.25** 0.19* 0.28** 0.27* –0.04 0.29*** (2.07) (1.68) (2.04) (1.78) (–0.67) (2.41)log(TNA) –0.21 –0.17 –0.33 –0.55** 0.30*** –0.16 (–1.21) (–0.91) (–1.58) (–2.19) (6.11) (–0.88)Income 0.78*** 0.84*** 0.99*** 1.20*** 0.18** 0.83*** (5.09) (4.95) (5.15) (5.28) (2.13) (5.38)CAP GNS 1.46*** 1.25*** 1.40*** 1.38*** 0.58*** 1.49*** (5.44) (4.65) (4.30) (3.58) (6.92) (5.62)Discretionary –1.64*** –1.00** –1.53*** –0.95* –1.84*** –1.64*** turnover (–4.05) (–2.22) (–3.78) (–1.73) (–5.80) (–3.57)Total load 0.26*** 0.16*** 0.22*** 0.20*** 0.37*** 0.36*** (10.37) (4.23) (9.70) (5.65) (8.29) (9.71)Expense –0.60*** –0.40** –0.57*** –0.60*** –0.14** –0.47*** (–3.34) (–2.13) (–3.07) (–2.76) (–2.18) (–2.55)Styles Controlled with dummy variables (continued on next page)
  21. 21. Qian 389 TABLE 6 (continued) Traditional Performance Measures and Stale Pricing (Fama and MacBeth Approach) Alpha from Alpha from Unconditional Conditional Market & Unconditional Conditional and Style and Style Carhart’s Liquidity Variables CAPM CAPM Benchmarks Benchmarks 4 Factors PremiumPanel C. Stale Pricing BETA(rt , rmt−1 )Intercept 2.08 3.45 –2.00 –1.28 –0.06 3.39 (6.96) (11.29) (–8.52) (–3.06) (–0.19) (8.47)Staleness –0.26** –0.61*** –0.54*** –0.73*** –0.18** –0.50*** (–2.07) (–5.12) (–3.00) (–3.56) (–2.25) (–2.87)Flow 1.13*** 1.21*** 1.31*** 1.49*** 0.24 1.21*** (7.71) (5.15) (6.81) (4.36) (1.48) (7.73)log(Age) –0.12*** –0.20*** –0.01 0.00 –0.01 –0.04 (–3.24) (–5.22) (–0.11) (0.07) (–0.13) (–1.01)log(TNA) 0.19*** 0.46*** 0.17*** 0.20*** 0.18*** 0.25*** (3.35) (6.58) (2.82) (2.63) (4.33) (3.89)Income 0.10 0.04 0.41*** 0.62*** 0.08 –0.14 (1.23) (0.50) (3.69) (5.49) (0.92) (–1.46)CAP GNS 0.46*** 0.01 0.08 –0.19*** 0.42*** 0.41*** (5.85) (0.28) (1.02) (–2.69) (7.26) (4.81)Discretionary 0.08 0.88*** –0.14 0.75** 0.73*** 0.22 turnover (0.59) (3.26) (–1.34) (2.26) (2.59) (1.13)Total load 0.10 –0.05 0.20*** 0.12* 0.18** 0.17*** (1.57) (–0.72) (3.31) (1.82) (2.24) (2.63)Expense 0.30*** 0.44*** 0.23*** 0.39*** 0.36*** 0.40*** (3.40) (5.50) (3.13) (4.80) (3.79) (4.73)Styles Controlled with dummy variablesPanel D. Stale Pricing BETA(rt , rt−1 )Intercept 0.53 2.41 –0.38 0.87 –1.00 0.42 (1.03) (4.73) (–1.13) (1.61) (–2.53) (0.73)Staleness 0.20** 0.01 –0.01 –0.14 0.46*** 0.62*** (2.00) (0.08) (–0.08) (–0.72) (5.74) (9.68)Flow 0.51*** 0.92*** 0.72*** 1.32*** –0.18 0.33*** (4.22) (9.17) (6.27) (8.73) (–0.86) (2.59)log(Age) 0.06 –0.01 0.06 –0.01 –0.01 0.12 (0.81) (–0.10) (0.75) (–0.12) (–0.23) (1.43)log(TNA) 0.26*** 0.26*** 0.20*** 0.03 0.29*** 0.36*** (4.16) (2.88) (3.49) (0.32) (6.02) (4.77)Income 0.41*** 0.54*** 0.60*** 0.76*** 0.12** 0.41*** (5.40) (5.45) (5.56) (5.92) (2.25) (4.66)CAP GNS 0.57*** 0.36*** 0.41*** 0.27*** 0.47*** 0.53*** (7.03) (6.71) (6.61) (3.33) (7.39) (6.11)Discretionary –1.08*** –0.44* –0.83*** –0.15 –2.06*** –1.05*** turnover (–4.69) (–1.66) (–4.53) (–0.49) (–5.72) (–3.93)Total load 0.20 0.09*** 0.21*** 0.19*** 0.24*** 0.28*** (4.15) (4.27) (4.10) (7.98) (9.74) (5.32)Expense –0.22 –0.08 –0.15 –0.18 –0.10 –0.02 (0.20) (0.01) (–0.01) (–0.14) (0.46) (0.62)Styles Controlled with dummy variablesCov(ε1 , ε2 ) and Var(ε2 ) are the same for all funds within the same style groupenables a random selection of 1 fund from each style to form a 16-equation sys-tem and compute A and Φ. This procedure is repeated 10 times, each time with8 funds randomly redrawn. Table 8 presents the 10 pairs of estimated A and Φ. It indicates that the at-tenuation bias A ranges from 0.888 to 0.997, while the bias caused by correlatederrors is smaller than 0.005. The right-hand column illustrates the implied rela-tion between performance and stale pricing, that is, the true coefficient g, if the
  22. 22. 390 Journal of Financial and Quantitative Analysis TABLE 7 Performance Estimated with the Proposed Model and Stale Pricing Using a 2-Stage ApproachTable 7 displays the relation between the proposed true performance and various measures of stale pricing of funds. Thedependent variable is the true alpha estimated from the proposed model with after-expense returns in annualized percent-age. The explanatory variables are 4 different measures of stale pricing (equations (9a)–(9d)), fund characteristics, andfund styles. The stale pricing measures and the fund characteristics are studentized. *, **, and *** represent significanceat the 10%, 5%, and 1% levels, respectively. Explanatory Variables (1) (2) (3) (4)Intercept 2.66 2.18 2.67 2.68 (6.68) (4.63) (6.78) (6.81)Stale pricing π (Lo and MacKinlay (1990)) –0.31 (–0.47)Stale pricing ξ (Getmansky et al. (2004)) 0.47 (0.80)Stale pricing BETA(rt , rmt−1 ) –0.01 (–0.01)Stale pricing BETA(rt , rt−1 ) 0.56 (0.40)Flow 0.62 0.66 0.63 0.62 (1.25) (1.27) (1.25) (1.24)log(Age) –0.44*** –0.40*** –0.44*** –0.44*** (–4.74) (–4.21) (–4.71) (–4.74)log(TNA) 0.78*** 0.82*** 0.79*** 0.79*** (3.44) (3.42) (3.47) (3.49)Income –0.12** –0.13*** –0.12** –0.12** (–2.21) (–2.35) (–2.26) (–2.24)CAP GNS 0.23 0.19 0.23 0.23 (1.15) (0.95) (1.14) (1.14)Discretionary turnover –0.11 –0.14 –0.13 –0.13 (–0.51) (–0.65) (–0.62) (–0.61)Total load 0.70** 0.69** 0.70** 0.70** (2.24) (2.21) (2.24) (2.24)Expense –0.52*** –0.44** –0.51*** –0.52*** (–2.68) (–2.19) (–2.68) (–2.69)Styles Controlled with dummy variablesNo. of obs. 2,541 2,318 2,541 2,541F-test 27.99 24.39 27.8 27.88R2 0.15 0.15 0.15 0.15estimated coefficient g is 40 bp in the 2nd stage, where g = gA + Φ. According to ˆ ˆthe estimation, the overall bias due to the errors in variable is minimal in 9 out of10 cases. Therefore, it is safe to conclude that the relation between traditionallyestimated performance and stale pricing is robust.C. Staleness in Indices The estimation of the proposed model uses the S&P 500 index returns asbenchmark returns, a favorable choice given that other benchmark returns, evenstyle indices, can be stale. I empirically test this assumption with a simplified ver-sion of the model that involves 0 flows. Applying the simplified model to 12 port-folios (NYSE, AMEX, NASDAQ equal-weighted indices, value, growth, smallstock indices, and the 6 Fama-French (1992) portfolios) shows that most of themare stale relative to the S&P 500 index.18 18 Results are available from the author.
  23. 23. Qian 391 TABLE 8 The Attenuation Bias and the Bias Caused by Correlated ErrorsTable 8 presents the estimated attenuation bias and the bias caused by correlated errors in the 2-stage regressions. Thesebiases are estimated using 8 funds, each randomly drawn from a style group. Each row in the table presents a randomdrawn sample. Cov(ε1 , ε2 ), Var(ε2 ), A, Φ, and implied g are given in the table. Here, ˆ g = Cov(a1 , a2 )/Var(a2 ) ˆ ˆ ˆ = gA + Φ, ˆwhere g is the true slope, g is the estimated slope, −1 A = Var(a2 )/[Var(a2 ) + (x2 x2 ) Var(ε2 )], −1 −1 −1 Φ = (x1 x1 ) (x1 x2 )(x2 x2 ) Cov(ε1 , ε2 )/[Var(a2 ) + (x2 x2 ) Var(ε2 )];x1 and x2 are explanatory variables in the 1st-stage regressions, in which a1 and a2 are estimated; a1 is the dependentvariable in the 2nd stage, and a2 is the explanatory variable in the 2nd-stage regression; ε1 and ε2 are observation errors inthe 1st-stage regressions; Cov(e1 , e2 ) is computed to proxy for Cov(ε1 , ε2 ) and Var(e2 ) for Var(ε2 ); and Var(a2 ) equalsVar(a2 ) – (x2 x2 )−1 Var(ε2 ). The attenuation bias A, and the bias caused by correlated errors Φ, are also computed. ˆThis procedure is repeated 10 times with funds redrawn randomly for each iteration. Subsample Cov(ε1 , ε2 ) Var(ε2 ) A Φ ˆ Implied g If g = 40 bpJan. 1992–Dec. 2000 1 0.15 0.31 0.90 0.001 33 2 0.21 0.38 0.91 < 0.001 40 3 0.21 0.38 0.91 < 0.001 40 4 0.20 0.38 0.89 < 0.001 43 5 0.20 0.38 0.89 < 0.001 43Jan. 1981–Dec. 1989 6 0.26 0.28 0.93 0.005 –8 7 0.19 0.21 1.00 < 0.001 37 8 0.21 0.22 1.00 < 0.001 38 9 0.25 0.27 1.00 < 0.001 38 10 0.22 0.23 0.99 < 0.001 36D. Endogenously Chosen Stale Pricing Getmansky et al. (2004) document that some hedge fund managers smoothprices on purpose to boost the measured Sharpe ratio. This opportunistic behavioris beneficial to hedge fund managers because flows in and out of hedge funds arefar less frequent than those in and out of open-end mutual funds. As a result, theyare less concerned with the negative dilution effect. The present analysis showsthat the negative dilution effect on the measured mutual fund performance is largerand more significant than the positive statistical effect. Therefore, as regards fundperformance, the argument that stale pricing is endogenously chosen by mutualfund managers is unconvincing. To explore the possibility of endogenously cho-sen stale pricing, the analysis needs to be extended into other areas, such as fundgovernance (Qian (2011)).VII. Conclusion Stale pricing is prevalent in the mutual fund industry and impacts fund per-formance through 2 channels: a statistical bias and an arbitrage dilution. This pa-per introduces a performance evaluation model that accounts for stale prices andendogenous fund flows. Specifically, the model differentiates true performancefrom observed performance—that ignores stale pricing and treats the observedreturns as true returns—and attributes the difference to these effects. Comparedto the true alpha, the observed alpha has a positive spurious bias that is small,but a negative dilution bias that is large and significant. Such decomposition isparticularly interesting for performance attribution studies and applications.
  24. 24. 392 Journal of Financial and Quantitative Analysis The true picking ability of fund managers should not be explained by stalepricing of fund shares. Alpha estimates from the proposed model meet this stan-dard. An empirical test of the model also shows that the true alpha is on averagepositive, although not significant, and higher than the observed alpha. Compara-tive analysis shows that fund performance evaluated with conventional methodsis negatively associated with stale pricing. Overall, these results suggest that re-ducing stale pricing through fair-value pricing to impede arbitrage activities maynot only protect long-term investors but also benefit portfolio managers.Appendix A. Derivation of the Observed Alpha andComponents of the Biases The main elements of the observed alpha are E(rto ) and Cov(rto , rmt ). Since(A-1) E(rto (1 + ft−1 )) = Cov(rto , ft−1 ) + E(rto )(1 + c),combining with equation (4a) gives(A-2) E(rto ) = [μ − Cov(ft−1 , rto )]/(1 + c).Since(A-3) Cov(rto (1 + ft−1 ), rmt ) = E(rto (1 + ft−1 )rmt ) − E(rto (1 + ft−1 ))E(rmt ) = Cov(rto rmt , ft−1 ) + E(rto rmt )E(1 + ft−1 ) − E(rto (1 + ft−1 ))E(rmt ) = Cov(rto rmt , ft−1 ) + (1 + c)[Cov(rto , rmt ) + E(rto )E(rmt )] − E(rto (1 + ft−1 ))E(rmt ) = Cov(rto rmt , ft−1 ) + (1 + c)Cov(rto , rmt ) − E(rmt )Cov(ft−1 , rto ),combining with equation (4c) gives(A-4) Cov(rto , rmt ) = [(1 − η)Cov(r, rm ) − Cov(rto rmt , ft−1 ) + Cov(ft−1 , rto )um ]/(1 + c).With E(rto ) and Cov(rto , rmt ) derived, αo can be derived, which gives equation (5b). When η is nonzero, c = 0 and ft−1 = 0, the observed alpha becomes α = μ − (1 −η) Cov(r, rm )μm /σ 2 . The difference between α and the true performance is purely due to mstale pricing. Denote this bias as B1 , B1 = α − α. It gives equation (6a). When c is nonzero but Cov(ft−1 , rto ) = 0 and Cov(rto rmt , ft−1 ) = 0, the observed alphabecomes α = [1/(1+c)] [μ − (1 − η)Cov(r, rm )μm /σ 2 ]. The difference between α and mα is the dilution effect due to the long-term flows. Denote this effect B2 , B2 = α − α . Itgives equation (6b). When there are arbitrage flows responding to the expected short-term returns andcorrelated with fund returns (i.e., Cov(ft−1 , rto ) and Cov(rto rmt , ft−1 ) are nonzero), the ob-served alpha now becomes αo as in equation (5b). The difference between αo and α isthe dilution effect due to the arbitrage flows. Denote the dilution effect of arbitrage flowsas B3 . B3 = αo − α . This gives equation (6c). In algebra, αo = (αo − α ) + (α − α ) + (α − α) + α = α + B1 + B2 + B3 .
  25. 25. Qian 393Appendix B. The Aggregation of Daily Dilution to MonthlyDilution Supposing that there are D days in each month and for each day d within the month,the following is derived: ∗(3a) o rd = rd /(1 + fd−1 ), o ∗where rd is the observed excess return of the fund, rd is the would-be excess return of thefund with 0 flows, and fd−1 is the net flow as a percentage of the TNA. The return dilutionon day d is ∗(B-1) rd − rd o = rd fd−1 . o o Denoting the observed excess return for the month as rp , the excess return without ∗flows as rp , and the flow of the month as fp , the monthly dilution is then ∗ ∗(B-2) rp − rp o = rd − o rd = o rd fd−1 = (1/D) o rd fd−1 + DCov(rd , fd−1 ) o(B-3) ≈ rp fp /D. o From expressions (B-2) to (B-3), there are 2 biases, both infeasible to address with-out observations of daily flows. First, the covariance between daily flows and subsequentreturns is ignored. This omission biases the measured dilution downwards. Second, a dif-ference between the flows on the last day of this month and that of last month is ignored.This bias may be positive or negative but of much smaller magnitude compared to thefirst one. Therefore, at the monthly frequency, the observed return is approximated by ∗rp = rp /(1 + fp /D). That is, when the proposed model is estimated with monthly flows, oflows dilute contemporary returns. This approximation is quite intuitive. Overall, the sumof the daily dilutions, if documentable, should be stronger than those found here usingmonthly returns.ReferencesAdmati, A. R.; S. Bhattacharya; P. Pfleiderer; and S. A. Ross. “On Timing and Selectivity.” Journal of Finance, 41 (1986), 715–730.Becker, C.; W. E. Ferson; D. H. Myers; and M. J. Schill. “Conditional Market Timing with Benchmark Investors.” Journal of Financial Economics, 52 (1999), 119–148.Bhargava, R.; A. Bose; and D. A. Dubofsky. “Exploiting International Stock Market Correlation with Open-End International Mutual Funds.” Journal of Business Finance and Accounting, 25 (1998), 765–773.Boudoukh, J.; M. P. Richardson; M. Subrahmanyam; and R. F. Whitelaw. “Stale Prices and Strategies for Trading Mutual Funds.” Financial Analysts Journal, 58 (2002), 53–71.Brown, S. J., and J. B. Warner. “Using Daily Stock Returns: The Case of Event Studies.” Journal of Financial Economics, 14 (1985), 3–31.Carhart, M. M. “On Persistence in Mutual Fund Performance.” Journal of Finance, 52 (1997), 57–82.Chalmers, J. M. R.; R. M. Edelen; and G. B. Kadlec. “On the Perils of Financial Intermediaries Setting Security Prices: The Mutual Fund Wild Card Option.” Journal of Finance, 56 (2001), 2209–2236.Chen, Y.; W. E. Ferson; and H. Peters. “Measuring the Timing Ability and Performance of Bond Mutual Funds.” Journal of Financial Economics, 98 (2010), 72–89.Dimson, E. “Risk Measurement When Shares Are Subject to Infrequent Trading.” Journal of Financial Economics, 7 (1979), 197–226.Edelen, R. M. “Investor Flows and the Assessed Performance of Open-End Mutual Funds.” Journal of Financial Economics, 53 (1999), 439–466.Edelen, R. M., and J. B. Warner. “Aggregate Price Effects of Institutional Trading: A Study of Mutual Fund Flow and Market Returns.” Journal of Financial Economics, 59 (2001), 195–220.