Model Uncertainty and Mutual Fund Investing

Transcription

1 Georga State Unversty Georga State Unversty Fnance Dssertatons Department of Fnance Model Uncertanty and Mutual Fund Investng Yee Cheng Loon Follow ths and addtonal works at: Part of the Fnance and Fnancal Management Commons Recommended Ctaton Loon, Yee Cheng, "Model Uncertanty and Mutual Fund Investng." Dssertaton, Georga State Unversty, Ths Dssertaton s brought to you for free and open access by the Department of Fnance at Georga State Unversty. It has been accepted for ncluson n Fnance Dssertatons by an authorzed admnstrator of Georga State Unversty. For more nformaton, please contact scholarworks@gsu.edu.

2 PERMISSION TO BORROW In presentng ths dssertaton as a partal fulfllment of the requrements for an advanced degree from Georga State Unversty, I agree that the Lbrary of the Unversty shall make t avalable for nspecton and crculaton n accordance wth ts regulatons governng materals of ths type. I agree that permsson to quote from, to copy from, or publsh ths dssertaton may be granted by the author or, n hs/her absence, the professor under whose drecton t was wrtten or, n hs absence, by the Dean of the Robnson College of Busness. Such quotng, copyng, or publshng must be solely for the scholarly purposes and does not nvolve potental fnancal gan. It s understood that any copyng from or publcaton of ths dssertaton whch nvolves potental gan wll not be allowed wthout wrtten permsson of the author. Yee Cheng Loon

3 NOTICE TO BORROWERS All dssertatons deposted n the Georga State Unversty Lbrary must be used only n accordance wth the stpulatons prescrbed by the author n the precedng statement. The author of ths dssertaton s: Yee Cheng Loon 3117 Burrs Road, Apt 71, Vestal, NY The drector of ths dssertaton s: Dr. Vkas Agarwal Assstant Professor of Fnance Department of Fnance J. Mack Robnson College of Busness Georga State Unversty 35 Broad Street, Sute 107 Atlanta, GA Tel: Fax:

4 Model Uncertanty and Mutual Fund Investng BY Yee Cheng Loon A Dssertaton Submtted n Partal Fulfllment of the Requrements for the Degree Of Doctor of Phlosophy In the Robnson College of Busness Of Georga State Unversty GEORGIA STATE UNIVERSITY ROBINSON COLLEGE OF BUSINESS 007 3

5 Copyrght by Yee Cheng Loon 007 4

6 ACCEPTANCE Ths dssertaton was prepared under the drecton of the Yee Cheng Loon s Dssertaton Commttee. It has been approved and accepted by all members of that commttee, and t has been accepted n partal fulfllment of the requrements for the degree of Doctoral of Phlosophy n Busness Admnstraton n the Robnson College of Busness of Georga State Unversty. H. Fenwck Huss, Dean DISSERTATION COMMITTEE Dr. Vkas Agarwal (Char) Dr. Jason Greene Dr. Jayant Kale Dr. Ajay Subramanam 5

7 ABSTRACT Model Uncertanty and Mutual Fund Investng BY Yee Cheng Loon August 6, 007 Commttee Char: Major Academc Unt: Dr. Vkas Agarwal Department of Fnance Model uncertanty exsts n the mutual fund lterature. Researchers employ a varety of models to estmate rskadjusted return, suggestng a lack of consensus as to whch model s correct. Model uncertanty makes t dffcult to draw clear nference about mutual fund performance persstence. We explctly account for model uncertanty by usng Bayesan model averagng technques to estmate a fund s rsk-adjusted return. Our approach produces the Bayesan model averaged (BMA) alpha, whch s a weghted combnaton of alphas from ndvdual models. Usng BMA alphas, we fnd evdence of performance persstence n a large sample of US equty, bond and balanced mutual funds. Funds wth hgh BMA alphas subsequently generate hgher rsk-adjusted returns than funds wth low BMA alphas, and the magntude of outperformance s economcally and statstcally sgnfcant. We also fnd that mutual fund nvestors respond to the nformaton content of BMA alphas. Hgh BMA alpha funds receve subsequent cash nflows whle low BMA alpha funds experence subsequent cash outflows. 6

8 Model Uncertanty and Mutual Fund Investng 1. Introducton Model uncertanty exsts when there are many plausble models and a decson maker s not sure whch model s correct. Model uncertanty s mportant n fnancal economcs. Investors concerns about model uncertanty result n an addtonal rsk premum n securty prces (Hansen, Sargent and Tallarn (1999); Hansen, Sargent and Wang (00); Anderson, Hansen and Sargent (003); Hansen and Sargent (006)). In asset allocaton, gnorng model uncertanty leads to perceved utlty loss as hgh as 4.8% per year (Avramov, 00). 1 Model uncertanty exsts n the mutual fund lterature. Researchers employ a varety of mutual fund return generatng models, suggestng a lack of consensus as to whch model s correct. 3 Mutual fund return generatng models are used to address a number of research questons. One queston of nterest to both fund nvestors and researchers s whether fund performance perssts. Performance persstence s the noton that past performance contnues nto the future. Funds that performed better (worse) than other funds contnue to do so n the future. If markets are effcent, then mutual fund returns should not be predctable usng past nformaton (Fama, 1991). On the other hand, snce a mutual fund sells ts shares at net asset value, superor fund management skll, the source of performance, may not be prced. Thus, fund returns may be predctable (Gruber, 1996). Fund nvestors care about performance persstence. If performance 1 Avramov (00) nvestgates return predctablty by explctly accountng for model uncertanty. He does not consder estmaton error of the explanatory varables n the predctve regressons. Accordng to the Investment Company Insttute, nsttutons and ndvduals nvested approxmately $6. trllon n U.S. equty, bond and balanced open-end mutual funds at the end of 004 (Mutual Fund Fact Book 006, Table 41). Clearly, mutual funds have become popular nvestment vehcles for both nsttutonal and retal nvestors. Gven the large sums nvolved, the behavor of mutual funds naturally attracts the attenton of researchers. 3 We provde a quck overvew of mutual fund return generatng models n secton. Appendx A contans detals of each model.

9 perssts, then nvestors should nvest n consstently good performers and take money out of consstently poor performers. Pror research n performance persstence nvestgates persstence condtonal on a partcular model of fund return. The general research methodology can be summarzed by the followng steps. The researcher specfes a mutual fund return generatng model and uses the chosen model to compute a fund s rsk-adjusted return, or alpha. The researcher then checks f funds wth hgh (low) alphas n the past have hgh (low) alphas n the future. Consequently, nference regardng performance persstence s potentally senstve to the choce of mutual fund return generatng model. Ths approach does not account for model uncertanty. The extant lterature on performance persstence has produced mxed fndngs. Grnblatt and Ttman (199), Hendrcks, Patel, and Zeckhauser (1993), Goetzmann and Ibbotson (1994), Brown and Goetzmann (1995), Elton, Gruber and Blake (1996a), Bollen and Busse (005), among others, demonstrate some degree of predctablty n fund returns. However, the momentum effect n stock returns and survvorshp bas seem to account for return predctablty (Carhart, 1997; Brown et al, 199). Model uncertanty s one possble factor contrbutng to the mxed fndngs. For nstance, usng a condtonal verson of the CAPM, Brown and Goetzmann (1995) fnd evdence consstent wth performance persstence. On the other hand, usng a 4-factor model that attrbutes fund performance to the market, sze, growth and momentum, Carhart (1997) concludes that there s lttle evdence of persstence n manageral ablty. In ths paper, we nvestgate mutual fund performance persstence by explctly accountng for model uncertanty. Specfcally, we measure a fund s alpha as a weghted combnaton of alphas from a wde varety of models employed n the mutual fund lterature. Ths technque places hgher (lower) weghts on the alphas of models wth hgher (lower) posteror

10 model probabltes. Roughly speakng, models that ft fund data better have hgher posteror model probabltes and ther alphas receve bgger weghts. Ths makes sense because f a model fts the data better than other models, ts alpha estmate should contan more nformaton about future returns. By weghtng ndvdual model alphas, our approach pools nformaton from a range of plausble return generatng models. Ths represents a departure from past studes whch mplctly rely on complete certanty n specfc models. We employ Bayesan econometrc technques to compute the posteror model probabltes and so our alpha measure s a Bayesan model averaged (BMA) alpha. Usng our BMA alpha, we test for fund return predctablty n a large sample of US actvely managed mutual funds comprsng of equty, balanced, and bond funds. To nclude bond funds n our study, our BMA alpha pools nformaton from models of equty and bond fund returns. 4 Snce balanced funds nvest n equtes and bonds, our BMA alpha seems hghly suted to predctng balanced fund returns. Hence, we nclude balanced funds n our sample. We sort funds nto decles based on ther BMA alphas, placng the hghest (lowest) BMA alpha funds nto the top (bottom) decle. We then track the subsequent monthly decle returns. We fnd that BMA alphas are able to predct fund rsk-adjusted return (as measured by BMA alphas) for all three categores of actvely managed mutual funds. 5 Hgh BMA alpha funds outperform low BMA alpha funds n the post-rankng perod. In our equty fund sample, the dfference n rsk-adjusted return between the top and bottom decles ranges from 4.56% to 5.5% per year. BMA alphas also demonstrate an ablty to forecast balanced fund returns. The dfference n rsk-adjusted return between the top and bottom decles ranges from.64% to 5.04% per year. In the bond fund 4 Between , US equty funds nvested an average of.6% of ther portfolos n long-term US government bonds and corporate bonds (Investment Company Insttute, 006 Table 9). Thus, the factors used to model bond fund returns may also be useful for equty funds. 3

11 sample, the dfference n rsk-adjusted return between the top and bottom decles ranges from.5% to 3.1% per year. We obtan these results when we employ a one-month post-rankng perod and reform the decle portfolos every month. When we extend the post-rankng perod to sx months and re-form the decle portfolos every sx months, BMA alphas contnue to exhbt predctve ablty. Smlarly, reformng decle portfolos every twelve months does not change our fndngs qualtatvely. When we extend the post-rankng perod up to 60 months, we contnue to fnd evdence of predctablty. We do observe that the performance dfferental between the top and bottom decles narrows wth the length of the post-rankng perod. For example, n our balanced fund sample, as the post-rankng perod ncreases from one month to sxty months, the dfference n rsk-adjusted return between the top and bottom decles declnes from.64% per year to 0.60% per year. 6 Extant research shows that mutual fund nvestors respond to past performance, whch s typcally measured as raw returns, market-adjusted returns or alphas defned by ndvdual models (see, e.g., Srr and Tufano (1998) and Chevaler and Ellson (1997)). Ths approach s restrctve because t assumes that nvestors behave as f they use a sngle mutual fund return generatng model to measure past performance. A more plausble assumpton s that nvestors behave as f they employ a varety of models to measure past performance. In the aggregate, we would expect fund flows to respond to a performance measure that combnes nformaton contaned n a varety of mutual fund return generatng models. The BMA alpha s such a performance measure because t s a weghted combnaton of alphas from a range of models. To nvestgate whether aggregate flow behavor s consstent wth fund nvestors usng a range of 5 Avramov (00), Cremers (00) and Tang (003) show that Bayesan model averagng technques mprove the forecastng of stock ndexes and passve portfolos. 4

12 models to evaluate fund performance, we relate past BMA alphas to subsequent fund flows. We fnd that nvestors respond strongly to the nformaton contaned n BMA alphas by adjustng ther fund allocatons. Funds wth hgh BMA alphas receve cash nflows whle funds wth low BMA alphas experence cash outflows n the post-rankng perod. 7 For example, n our equty fund sample, the dfference n monthly flow between the top and bottom decles ranges from 3.11% to 4.11%. In addton, Spearman rank correlatons exceed 0.9, ndcatng a close correlaton between BMA alphas and future fund flows. Furthermore, nvestors respond to BMA alphas up to sxty months after decle formaton. Results for balanced and bond funds are smlar, ndcatng that nvestors respond to the BMA alphas of a wde range of mutual funds. Our fndng of sgnfcant cash outflows from poorly performng funds contrasts wth the low senstvty of flows to poor past performance documented n Srr and Tufano (1998, Table 1). 8 Across all three fund types, we fnd that flows nto good past performers exceed flows out of poor past performers n magntude. Ths s consstent wth the asymmetrc relaton between flows and past performance as documented by Srr and Tufano (1998), Chevaler and Ellson (1997), Huang, We and Yan (007), among others. Our study relates to recent artcles that examne varous ways of ncorporatng addtonal nformaton for predctng mutual fund returns. Cohen, Coval and Pastor (005) show that stock holdngs and trades of mutual funds provde addtonal nformaton that helps to predct future returns. Busse and Irvne (006) demonstrate that seemngly unrelated passve assets also provde 6 These results are based on BMA alphas estmated wth a skeptcal pror belef n manageral skll and a 36-month estmaton wndow. Usng an alternatve pror belef or a longer estmaton wndow does not change our conclusons qualtatvely. 7 Flow s defned as new cash flow dvded by lagged total net assets. 8 Gruber (1996) also reports outflows from poorly performng funds. To sort funds nto decles, he uses a model that attrbutes fund return to the equty market, the bond market, a sze factor and a growth factor. 5

13 useful nformaton for forecastng. 9 Avramov and Wermers (006) fnd that condtonng on macroeconomc ndcators also help to predct future returns. Cremers and Petajsto (006) show that the actvely managed porton of equty fund portfolos also predct fund performance. In contrast, we consder the poolng of nformaton from dfferent return generatng models for predctng fund returns. Furthermore, we examne not just equty mutual funds, but also balanced and bond funds. The rest of the artcle proceeds as follows. In secton, we brefly dscuss the mutual fund return models that contrbute nformaton to the BMA alpha. We defer the detals of these models to Appendx A. In secton 3, we descrbe the econometrc framework and the computaton of BMA alphas. We provde detaled dervatons n Appendx B. In secton 4, we descrbe the constructon of our data set. In secton 5, we present return forecastng results usng BMA alphas and n secton 6, we provde evdence of nvestors cash flow response to BMA alphas. We conclude the paper n secton 7.. Mutual fund return generatng models We consder 6 separate mutual fund return generatng models that have been employed n the mutual fund lterature. In ths secton, we provde the reader wth a quck overvew of the models and defer detals to Appendx A. Jensen (1968) s probably one of the earlest to use a lnear return model to explan equty mutual fund returns. Specfcally, he uses the Captal Asset Prcng Model (CAPM) of Sharpe (1964) and Lntner (1965) to evaluate mutual fund performance. 10 More recently, Elton, Gruber, Das and Hlavka (1993) and Elton, Gruber and Blake (1996b) propose a 3-factor model that 9 Pastor and Stambaugh (00a) use seemngly unrelated passve assets to evaluate equty mutual fund performance. Kosowsk, Nak and Teo (007) fnd that the methodology of Pastor and Stambaugh (00a) also helps to predct 6

14 captures the rsk from holdng S&P 500 stocks (.e., large captalzaton stocks), non-s&p 500 stocks (.e., small captalzaton stocks) and bonds. Carhart (1997) employs the Fama and French (1993) 3-factor model to evaluate mutual fund performance. He also proposes a 4-factor model, whch s a combnaton of the Fama and French (1993) 3-factor model and an addtonal factor that captures the momentum effect documented by Jegadeesh and Ttman (1993). Jones and Shanken (005) augment the Carhart model wth three factors desgned to capture ndustry effects n mutual fund returns. Elton, Gruber and Blake (1996a) and Gruber (1996) ntroduce a 4- factor model whch attrbutes fund return varatons to the overall market, the return dfferental between large and small stocks, the return dfferental between growth and value stocks and the returns from corporate and government bonds. These models are uncondtonal models n the sense that the regresson coeffcents do not depend on observable quanttes. In contrast, Ferson and Schadt (1996), Brown and Goetzmann (1995), and Kosk and Pontff (1999) employ condtonal models n whch the regresson coeffcents are modeled as functons of macroeconomc ndcators and fund characterstcs. Besdes securty selecton, actvely managed mutual funds can add value by market tmng,.e., shftng allocatons between cash and rsky assets at opportune moments. Thus, we also consder the market tmng models of Treynor and Mazuy (1996), Henrksson and Merton (1981) and Goetzmann et al (000). We also nclude the condtonal versons of the Treynor-Mazuy and Henrksson-Merton models as mplemented by Ferson and Schadt (1996). We complete our collecton of models by addng bond mutual fund models employed by Blake et al (1993), Elton et al (1995) and Khorana et al (001). 3. Econometrc framework 3.1 Pror and Lkelhood hedge fund returns. 7

15 Our study pools nformaton from 6 separate models for predctng mutual fund returns. We use the subscrpt j to ndex mutual fund models, so that j = 1,,, 6. We specfy equal pror probablty for each of the 6 models. An alternatve approach s to dentfy the set of possble factors that affect mutual fund return. If there are K factors, then there are K possble models of mutual fund return. 11 Each of these models receves equal pror probablty equal to 1/ K (see, e.g., Avramov, 00; Cremers, 00). Gven our large sample and the number of potental factors, such an approach s too computatonally ntensve to be feasble. We llustrate our econometrc framework for the jth model, M j. The same econometrc framework apples to all other models under consderaton. Specfcally, for each model, we have the lnear regresson model, K t, = α + β k= 1 k, k + t, (1) r x u where r t, s fund s month t net return n excess of the rsk free rate, α ( alpha ) s the ntercept, x k s the kth explanatory varable, β k, s fund s regresson coeffcent wth respect to the kth factor and u,t s the dsturbance term, whch s assumed to be normally, ndependently and dentcally dstrbuted,.e., ( σ ) ut, N 0, u t. Thus, the lkelhood functon of rt, s normal. To smplfy the econometrc analyss, we make the followng assumptons. We assume the dsturbance terms ( u t, s) are uncorrelated across funds, whch mples that the lkelhood functons are ndependent across funds. In addton, we assume that pror belefs on the regresson coeffcents n (1) are ndependent across funds. Pror and lkelhood ndependence mply that we can conduct our analyss on a fund by fund bass. Jones and Shanken (005) and 10 Ippolto (1989) also uses the CAPM n evaluatng mutual fund performance. 11 All such models contan an ntercept term. 8

16 Fresen (004) relax the pror ndependence assumpton by specfyng a herarchcal pror for fund alphas. A complete relaxaton of the pror ndependence assumpton requres the specfcaton of herarchcal prors for both the fund alpha and regresson coeffcents. Such prors result n analytcally ntractable posteror dstrbutons and requre the use of Markov Chan Monte Carlo smulaton technques (see e.g., Koop, 003). The denttes of the explanatory varables depend on the specfc model under consderaton. If the model s the CAPM, then α s the measure of abnormal performance proposed by Jensen (1968), K = 1 and x 1 s the market rsk premum. Alternatvely, f the model s the Fama and French (1993) three factor model, then α s the abnormal return wth respect to that model, K = 3 and the three explanatory varables are the market rsk premum, MKT, the sze factor, SMB and the book-to-market factor, HML. Equaton (1) can be wrtten more compactly as: r = Zφ + u () where r s the S 1 vector contanng the S observatons of r t, (we assume the fund has monthly returns for S months); Z = ( ls, X) s the S ( K + 1) matrx contanng l S, the S 1 unt vector n the leftmost column and X, the S K matrx contanng the explanatory varables specfc to the jth model; φ = ( α, β,1,, β, ), the ( K + 1) 1 coeffcent vector and K u s the S 1 vector contanng the dsturbance terms. To facltate subsequent exposton, defne the ( K 1) sub-vector, b = ( β,1,, β, ). Followng Pastor and Stambaugh (00a), we K employ the natural conjugate normal-nverted gamma pror for for σ u follows an nverted gamma dstrbuton (Zellner, 1971), σ u and φ. Specfcally, the pror 9

17 σu IG(, ν s ) (3) where IG stands for nverted gamma and ν and s are parameters of the nverted gamma dstrbuton. Condtonal on σ u, α and b are normally dstrbuted σ u α σu N α, σ α E ( σu ) (4) σ u b σu N b, V b E ( σu ) (5) where α s the pror mean of α, b s the pror mean vector of b, σ α s the margnal pror varance of α ( pror varance of alpha ) and V b s the margnal pror covarance matrx of b. We assume α and b are ndependent of each other. Gven ths assumpton, φ s multvarate normal ( V φ ) φ σ N φ, σ (6) u u where φ = ( α, b ) and V φ s defned as V φ 1 σ 0 α = E ( σ u ) 0 V b (7) The dagonal structure of V b stems from the assumed ndependence of α and b. To mplement Bayesan estmaton, we need to specfy values for the pror hyperparameters α, s, ν, b, and V b. We follow Pastor and Stambaugh (00b) and set α to σ α, 10

18 1 α = expense (8) 1 where expense s fund s average annual expense rato. Followng Pastor and Stambaugh (00b), we specfy two values for σ α to reflect dfferent pror belefs about a fund manager s skll. Thus, we can nvestgate the senstvty of our results to dfferent belefs about manageral skll. 1 Specfcally, we set σ α to 0.01 to represent a skeptcal pror belef n skll and we set σ α to 0.03 to represent a less skeptcal pror belef n skll. A value of 0.01 mples a tghter dstrbuton of α centered around the fund s monthly expense and s consstent wth the vew that t s hard for a fund s net return to exceed ts expense. In contrast, a value of 0.03 mples a less tght dstrbuton around the fund s monthly expense. Such a specfcaton admts a stronger possblty that a fund s net return can exceed ts expenses. In short, a larger pror varance of alpha represents a greater wllngness to entertan the possblty of skll. We employ an emprcal Bayes approach n specfyng values for s, ν, b, V b. In general, the emprcal Bayes approach means that researchers use the data to obtan values for the pror hyperparameters. Ths s an attractve and practcal soluton to researchers who do not wsh to use non-nformatve (dffuse) prors but have dffculty n elctng subjectve nformatve prors. 13 Each fund s vewed as a draw from the cross secton of funds wth the same nvestment objectve. Thus, pror uncertanty about a fund s parameter s drven by the cross sectonal varaton n that parameter. For each nvestment objectve, we select all funds havng at least 60 1 It would be nterestng to conduct further senstvty analyss by consderng a wder range of pror belefs about fund manageral skll. 13 See Carln and Lous (000) for a detaled dscusson of the emprcal Bayes approach. Studes that employ ths method nclude Fama and French (1997), Frost and Savarno (1986), Pastor and Stambaugh (1999, 00a, 00b). To apply ths approach, we adopt the procedure n Pastor and Stambaugh (00a). 11

19 months of data and compute the OLS estmate of b for each fund. 14 Then we set b equal to the sample mean of the OLS estmates and V b equal to the covarance matrx of the OLS estmates. Each OLS regresson also yelds σ ˆu, the estmate of we ntroduce the frst and second moments of σ u. To explan how we specfy σ u. Based on Zellner (1971, p ), s and ν, E σ ( u) νs = (9) ν Var ( σu) = νs (10) ( ν ) ( ν 4) By substtutng, (9) nto (10), we can express ν as ν ( E( σ )) = 4 + (11) Var( ) u σu We nsert the cross sectonal mean and varance of σ ˆu nto the rght-hand sde of (11) and evaluate that expresson. ν s set equal to the next largest nteger of the resultng value on the rght-hand sde of (11). Once we have solved for ν, we use that value, the cross sectonal mean of σ ˆu and (9) to solve for s. By combnng the pror and lkelhood, we obtan the posteror dstrbuton of the regresson parameters (see Appendx B for the dervatons). For the jth model, the Bayesan estmate of alpha s the mean of the posteror dstrbuton of α, E( α D, M ). j 3. Bayesan model averaged alpha The Bayesan model averaged alpha of fund s 14 In secton 4, we provde detals of the mutual fund nvestment objectves. 1

20 M E( α D) = E( α D, M ) p( M D) (1) j= 1 j j where pm ( D ) s the posteror model probablty of the jth model (see Appendx B). For j example, f the model n queston s the four-factor model of Carhart (1997), then E( α D, M ) s the posteror mean of the ntercept term n ths model. For brevty, we shall refer to E( α D) as fund s BMA alpha. The BMA alpha ncorporates model uncertanty by weghng the alpha of each model by ts respectve probablty. In ths way, the BMA alpha combnes nformaton contaned n dfferent model alphas n an ntutvely appealng manner. It places hgher (lower) weghts on the alphas of models wth hgher (lower) posteror model probabltes. Roughly speakng, models that ft the fund data better have hgher posteror model probabltes and ther alphas receve bgger weghts. Ths makes sense because f a model fts the data better than other models, ts alpha estmate should contan more nformaton about future returns. j 4. Data We obtan US mutual fund data through December 003 from the CRSP Mutual Fund Database. Our sample conssts of three types of mutual funds: equty, bond and balanced funds. We dentfy mutual funds usng nvestment objectve nformaton from Wesenberger, ICDI and Strategc Insght (avalable n the CRSP Database). To dentfy balanced funds, we also use the POLICY varable n the CRSP Database. Equty mutual funds nclude funds wth the followng nvestment objectves: small company growth, other aggressve growth, growth, ncome, growth and ncome, and maxmum captal gan. Bond funds consst of funds wth the followng objectves: government bonds, mortgage-backed securtes and corporate bonds. We exclude ndex funds from our sample. 13

21 Snce 1980, many mutual funds started offerng multple share classes to nvestors. In a mult-class fund, the underlyng portfolo of assets s common to all share classes. Share classes dffer n terms of loads (sales charge) and fees (Red and Rea, 003). The CRSP Mutual Fund Database contans nformaton on every share class of the same fund. In ths study, our basc unt of analyss s a specfc fund, not a specfc share class. When a fund has multple share classes, we consoldate them nto one fund. Furthermore, for mult-class funds, we compute valueweghted monthly net returns, expenses, loads, 1b-1 fees and turnover (Wermers, 000; Nanda, et al, 004). Each share class s weght s ts total net assets dvded by the sum of the total net assets of all share classes. For fund characterstcs reported on an annual bass (expense rato, turnover, varous load fees and 1b-1 fees), the value-weghted characterstc s computed usng the calendar year-end total net assets. We compute value-weghted monthly net returns usng monthly total net assets when avalable. The CRSP Mutual Fund Database reports total net assets on an annual bass between 1961 and 1969, on a quarterly bass between 1970 and 1991 and on a monthly bass startng from Gven ths reportng pattern, we obtan monthly total net assets n the followng manner: when total net assets are reported on an annual bass, we assgn that total net assets fgure to every month n that year. When total net assets are reported on a quarterly bass, we assgn the quarter end total net assets fgure to the other months n the same quarter. Our study requres the factors from all return generatng models to be avalable for the same perod of tme. Ths turns out to be from 1/1980-1/003, a perod of 88 monthly observatons. Thus, we restrct our data set to ths nterval. Our emprcal analyss uses fund net returns (net of fees and expenses). We retan funds wth at least 37 months of returns and wth 15 In the CRSP Database, not all funds swtched to monthly reportng of total net assets startng from In our own sample, two funds reported only quarterly total net assets n Selgman Fronter Fund and Lazards 14

22 avalable data on expense, turnover and load. We need 36 months of returns for estmaton and at least 1 month of returns for the forecastng analyss. Our selecton process yelds a fnal sample of 3,619 funds between 1/1980 and 1/003. Of these, 56 are balanced funds,,35 are equty funds and 1,18 are bond funds. 5. Predctng mutual fund returns To examne the predctablty of BMA alphas, we adopt a portfolo approach. We sort funds nto decle portfolos based on ther BMA alphas estmated usng data from the prevous 36 months and then we observe subsequent fund returns over post-rankng perods rangng from 1 month to 60 months. Wth the 1-month post-rankng perod, at the end of every month, we sort funds nto decles based on ther past BMA alphas. Decle 1 contans funds wth the lowest BMA alphas and Decle 10 contans funds wth the hghest BMA alphas. We then compute the equally-weghted monthly BMA alpha of each decle portfolo durng the next month. By repeatng ths process tll the end of the sample perod, we obtan the tme seres of monthly BMA alphas for each decle portfolo startng n January 1983 and endng n December 003. We form the frst set of decle portfolos at the end of December 198 and the last set of decle portfolos at the end of November 003. We also form the 10-1 portfolo, whch s long Decle 10 and short Decle 1. We employ the same procedure wth the 3-, 6-, 1-, 4-, 36-, 48-, and 60-month post-rankng perods, except that we rebalance the decle portfolos every 3, 6, 1, 4, 36, 48, and 60 months respectvely. For each post-rankng month t, a fund s BMA alpha s the posteror model probablty weghted average rsk-adjusted return. Specfcally, for fund and model j, the rsk-adjusted return n post-rankng month t s calculated as Funds:Equty Portfolo. 15

23 α = r x b (13) jt,, t, jt,, j, where jt,, α s model j s rsk-adjusted return for fund n post-rankng month t, r t, s fund s excess return (n excess of the rskfree rate), x jt,, s the vector of explanatory varables specfc to model j durng post-rankng month t and b j, s the posteror mean of the regresson coeffcents (excludng the ntercept) obtaned n the decle formaton month. We then calculate fund s BMA alpha n post-rankng month t as M M t, = α 1 jt,, pm ( j D) = ( r 1 t, x jt,, bj, ) pm ( j D) j= (14) j= α where pm ( D ) s the posteror model probablty of model j and all other varables have been j defned above. We compute the decle equally-weghted BMA alpha n post-rankng month t by averagng the BMA alphas of funds belongng to that decle. When the post-rankng perod spans multple months (e.g., a post-rankng perod of 3 months), we calculate each month s modelspecfc rsk-adjusted return ( α jt,, ) usng the fund return ( r t, ) and explanatory varables ( x jt,, ) from that month, but we apply the posteror mean of the regresson coeffcents obtaned n the decle formaton month. Smlarly, n calculatng the BMA alpha, we apply the posteror model probabltes obtaned n the decle formaton month. Thus, the BMA alpha s an out-of-sample measure of rsk-adjusted return. We summarze post-rankng perod performance by calculatng the tme seres average BMA alphas (n percent per month) for each decle and the 10-1 long-short portfolo. If past BMA alphas contan nformaton about future returns, then decle 10 wll outperform decle 1 durng the post-rankng perod. The return of the 10 1 portfolo wll therefore be postve and 16

24 statstcally sgnfcant. 16 Furthermore, we would expect post-rankng perod performance to mprove as we move from Decle 1 to Decle 10. To test ths statstcally, we compute the nonparametrc Spearman rank correlaton. Measurng post-rankng perod performance usng BMA alpha allows us to account for model uncertanty n fund returns durng the post-rankng perod and ensures consstency across the rankng and post-rankng perods. In the subsequent dscusson, we wll use BMA alpha, alpha, rsk-adjusted return and return nterchangeably Balanced funds The ratonale for usng BMA alphas s that they ncorporate nformaton from a range of mutual fund return generatng models, ncludng models of stock and bond fund returns. Snce balanced funds hold stocks and bonds, ther returns should be most amenable to predcton by BMA alphas. Our results ndcate that ths s ndeed the case. BMA alphas demonstrate predctve ablty over varyng horzons and under dfferent estmaton specfcatons. In Table 1, we present the return predctablty results of balanced funds where we mpose a skeptcal pror belef n skll (pror standard devaton of alpha s set to 0.01). For brevty, we 16 Throughout the paper, we use Newey and West (1987) heteroskedastcty-and-autocorrelaton consstent (HAC) standard errors to compute p-values. The lag length s set to 6 months for computng the HAC covarance matrx. We also experment wth lag lengths of 3, 9, and 1 months and fnd that our fndngs are qualtatvely unchanged. Hamlton (1994, p.8-83) descrbes the computaton of the HAC covarance matrx and standard errors. 17 Besdes model averagng rsk-adjusted returns, we also measure future fund returns n two dfferent ways. For each decle and the 10 1 portfolo, we calculate the average excess return (n excess of the rsk-free rate) and a rskadjusted return (alpha) defned wth respect to a specfc return generatng model. For balanced funds, the rskadjusted return must account for the fact that such funds can nvest n both equtes and bonds. Therefore, we employ the model n Elton, et al (1996a) and Gruber (1996) because t accounts for rsks n these two asset classes (see equaton 0 n Appendx A.1). For equty funds, the rsk-adjusted return s the alpha defned wth respect to the fourfactor model of Carhart (1997) (equaton 17 n Appendx A.1). For bonds funds, the rsk-adjusted return s the alpha defned wth respect to the model n Blake, Elton and Gruber (1993) (equaton 45 n Appendx A.). Usng these alternatve measures of future fund returns, we fnd evdence of return predctablty for balanced, equty and bond funds. Ths suggests that our fndngs are not due to the way we measure future fund returns. These results are avalable from the author upon request. 17

25 report results for Decles 1 and 10, and the 10-1 long-short portfolo. 18 When the post-rankng perod s one month (Table 1 Panel A), we fnd that past BMA alphas are able to predct future BMA alphas of balanced funds. Hgh BMA alphas forecast hgh future BMA alphas and vce versa; the average monthly BMA alphas ncrease as we move from Decle 1 to Decle 10. In addton, the 10 1 portfolo earns a rsk-adjusted return of bass ponts per month or approxmately.64% per year. The non-parametrc Spearman rank correlaton s 0.84 wth a p- value smaller than 0.001, ndcatng that past performance s closely correlated wth future performance across the decles. [ Insert Table 1 here ] Our forecastng results suggest that BMA alphas can predct balanced fund returns over a one-month horzon. Next, we address the queston of whether BMA alphas contan nformaton about future returns over longer horzons. Lookng across Table 1 Panel A, we see that BMA alphas relably predct future returns up to 1 months after decle formaton. Wth a 1-month post-rankng perod, the 10 1 portfolo generates an average monthly BMA alpha of 13 bass ponts (approxmately 1.56% per year). The Spearman rank correlaton s 0.95 and hghly statstcally sgnfcant, whch ndcates predctablty. Evdence of predctablty weakens when we extend the post-rankng perod beyond twelve months. For example, wth a 4-month postrankng perod, the 10-1 portfolo does not generate any postve alpha and the Spearman correlaton drops to In general, evdence of predctablty tends to weaken as the postrankng perod lengthens. As the post-rankng perod ncreases from 1 month to 60 months, both the 10-1 portfolo return and the Spearman correlaton decrease. Ths s a recurrng pattern n our return predctablty results. 18 Results for the ntermedate portfolos are avalable from the author upon request. 18

26 When we use a less skeptcal pror belef n skll to estmate past BMA alphas, we contnue to fnd evdence of predctablty (Table 1 Panel B). In addton, across varous postrankng perods, the less skeptcal belef produces larger 10-1 portfolo spreads and Spearman correlatons compared to the skeptcal belef. For example, when the post-rankng perod s 1 month, the 10-1 portfolo return s 7 bass ponts per month and the Spearman correlaton s under the less skeptcal belef. For the same post-rankng perod, the skeptcal pror produces a 10-1 portfolo return of bass ponts per month and a Spearman correlaton of One possble explanaton s that a less skeptcal pror belef n skll allows estmated rankng alphas to be more dspersed. Ths helps to dentfy the worst and best out-of-sample performers and leads to both larger 10-1 spreads and hgher Spearman correlatons. Fnally, BMA alphas estmated wth the less skeptcal pror belef can predct future returns over a longer horzon. Specfcally, past BMA alphas can predct future returns up to 48 months after decle formaton. Over ths horzon, the 10-1 portfolo return s 13 bass ponts per month (sgnfcant at the 1% level) and the Spearman correlaton s Wth BMA alphas based on the skeptcal belef (Panel A), the 10-1 portfolo return s statstcally nsgnfcant and the Spearman correlaton s only We repeat the analyss usng 60 months of data to estmate BMA alphas. The results are presented n Table 1 Panel C (wth the skeptcal pror belef) and Panel D (wth the less skeptcal pror belef). We contnue to fnd evdence of predctablty when we use a longer estmaton wndow to estmate BMA alphas. The best past performers contnue to outperform the worst past performers up to 60 months after decle formaton. For example, when funds are sorted by past BMA alphas estmated wth the less skeptcal pror (Panel D), the 10-1 portfolo generates an 19

27 average monthly return of 7 bass ponts 60 months after decle formaton. The Spearman correlaton s 0.71 and sgnfcant at the 5% level. Holdng pror belef constant, the longer estmaton wndow ncreases 10-1 spreads across all post-rankng perods wthout producng a correspondng effect on Spearman correlatons. Ths suggests that a longer estmaton provdes more nformaton about the best and worst performers, but does not necessarly provde more nformaton about the relatve performance of the ntermedate decles (.e., Decles through 9). Wth the skeptcal pror belef (Panel C), the 10-1 portfolo generates postve returns up to 60 months after decle formaton. Ths ndcates that past BMA alphas can sort the best and worst performng funds up to 60 months nto the future. In contrast, wth an estmaton wndow of 36 months, BMA alphas can only dstngush the best and worst performng funds up to 1 months nto the future (Panel A). Comparng Spearman correlatons n the two panels, we see that usng the 60 month estmaton wndow actually produces lower Spearman correlatons over the 6-, 1- and 48-month post-rankng perods. Ths mples that over these post-rankng perods, usng a longer estmaton wndow does not make BMA alphas more nformatve about the subsequent relatve performance of ntermedate decles. These conclusons reman f we examne results based on the less skeptcal pror belef (.e., compare Panels B and D). 5. Equty funds Table Panel A reports the forecastng performance of BMA alphas for equty funds where we mpose a skeptcal pror belef n skll (pror standard of alpha s set to 0.01) and use 36 months of data to estmate BMA alphas. We fnd that BMA alphas are able to predct future equty fund returns up to 60 months after decle formaton. Decle return ncreases as we move from Decle 1 to Decle 10 suggestng that hgh BMA alphas forecast hgh future BMA alphas and vce versa. 0

28 Furthermore, the 10 1 portfolo earns an average BMA alpha rangng from 15 to 38 bass ponts per month. Ths translates nto a range of 1.80% to 4.56% per year. Agan, we observe that a lengthenng of the post-rankng perod s accompaned by a narrowng of the return dfferental between the top and bottom decles and a reducton n Spearman correlaton. We fnd stronger evdence of return predctablty by usng a less skeptcal pror belef n skll (Table Panel B). Wth the less skeptcal pror belef, 10-1 portfolo returns and Spearman correlatons tend to be larger. For example, for the 1-month post-rankng perod, the skeptcal belef 10-1 portfolo generates an average return of 6 bass ponts per month and a Spearman correlaton of The less skeptcal belef 10-1 portfolo generates an average return of 31 bass ponts per month and a Spearman correlaton of The effect of the less skeptcal pror remans f we use an estmaton wndow of 60 months. Comparng Panels C and D reveals that 10-1 spreads and Spearman correlatons tend to be larger when past BMA alphas are estmated wth the less skeptcal pror. We contnue to fnd evdence of predctablty when we use a 60-month estmaton wndow to estmate BMA alphas. The results are presented n Table Panel C (wth the skeptcal pror belef) and Panel D (wth the less skeptcal pror belef). The best past performers contnue to outperform the worst past performers up to 60 months after decle formaton. For the same pror belef n skll, a longer estmaton wndow provdes addtonal nformaton for dentfyng the best and worst future performers for many post-rankng perods. In addton, the longer wndow usually provdes more nformaton about the relatve performance of ntermedate decles n the post-rankng perod. For example, comparng Panels B and D, we observe that a 60-month estmaton wndow produces 10-1 portfolo returns that equal and often exceed 10-1 portfolo returns based on a 36-month estmaton wndow. Furthermore, Spearman correlatons are often 1

29 hgher wth a 60-month estmaton wndow. Ths suggests that a longer estmaton wndow s assocated wth stronger correlatons between past and future performance across all fund decles. [ Insert Table here ] 5.3 Bond funds We repeat our analyss for bond funds and report the results n Table 3. We fnd that BMA alphas are able to predct future bond fund returns for varous combnatons of pror belef and estmaton wndow. Past BMA alphas are able to predct future bond fund returns up to 60 months after decle formaton. Nevertheless, evdence of predctablty weakens as the postrankng perod lengthens; 10-1 portfolo return and Spearman correlaton tend to declne as the post-rankng perod ncreases from 1 month to 60 months. Holdng constant the estmaton wndow, mposng the less skeptcal pror belef generally ncreases 10-1 portfolo returns. For example, comparng Panels A and B reveals that the less skeptcal pror produces larger 10-1 portfolo return across all post-rankng perods except one (60-month). [ Insert Table 3 here ] The key fndng n ths secton s that BMA alphas can predct the future returns of balanced, stock and bond funds over varyng horzons and under dfferent estmaton specfcatons. Although BMA alphas are potentally useful to nvestors, t s not clear whether nvestors actually respond to the nformaton contaned n BMA alphas. We examne ths ssue n the next secton. 6. Investors response to model averaged alphas Extant research on mutual fund flows shows that mutual fund nvestors respond to past performance, whch s typcally measured as raw returns or alphas defned by ndvdual models

30 (see, e.g., Chevaler and Ellson (1997), Srr and Tufano (1998)). Ths approach s restrctve because t assumes that nvestors behave as f they use a sngle mutual fund return generatng model to measure past performance. A more plausble assumpton s that nvestors behave as f they employ a varety of models to measure past performance. In the aggregate, we would expect fund flows to respond to a performance measure that combnes nformaton contaned n a varety of mutual fund return generatng models. The BMA alpha s such a performance measure because t s a weghted combnaton of alphas from a range of models. To nvestgate whether aggregate flow behavor s consstent wth fund nvestors usng a range of models to evaluate fund performance, we relate past BMA alphas to subsequent fund flows. We proceed by calculatng monthly flow nto a mutual fund as Flow t, TNA TNA (1 + r ) t, t, 1 t, = (15) TNA t, 1 where r,t s fund s month t return (wthout subtractng the rsk-free rate), and TNA,t (TNA,t-1 ) s fund s total net assets at the end of month t (t 1). To gauge nvestors respond to BMA alphas, we agan apply a portfolo approach. At the end of every month, we sort funds nto decles based on ther BMA alphas estmated over the prevous 36 months. Decle 1 contans funds wth the lowest BMA alphas and Decle 10 contans funds wth the hghest BMA alphas. We form the frst decle portfolos at the end of December 198 and the last decle portfolos at the end of November 003. We then compute the equallyweghted monthly flow nto each decle portfolo durng the next month. Thus, the post-rankng perod s 1 month. By repeatng ths process tll the end of the sample perod, we obtan the tme seres of monthly flows nto each decle portfolo. The sample perod over whch cash flows are calculated s January 1983 through December 003. To nvestgate the extent to whch BMA 3

31 alphas are related to future flows, we consder post-rankng perods of 1, 3, 6, 1, 4, 36, 48 and 60 months. That s, we reform the decle portfolos after ntervals of ncreasng lengths. For each rebalancng scheme, we make sure that the tme seres of monthly flows nto each decle portfolo starts n January 1983 and ends n December 003. Wth a post-rankng perod of 60 months, for example, we form the frst decle portfolos at the end of December 198, compute the equally weghted monthly flows nto each decle over the next 60 months and reform the decles at the end of December In ths case, we form the last decle portfolos at the end of December We conduct separate analyss for balanced, equty and bond funds. 6.1 Balanced funds We begn wth the dscusson of results for balanced funds. Table 4 reports the flows of decle portfolos constructed usng BMA alphas estmated wth a skeptcal pror belef n skll (pror standard devaton of alpha s set to 0.01). For brevty, we report results for Decles 1 and 10 and the 10-1 long-short portfolo. 19 When the post-rankng perod s one month (Panel A), we fnd that past BMA alphas strongly predct flows nto balanced funds. Hgh BMA alphas forecast subsequent nflows and low BMA alphas forecast subsequent outflows. Decle 10, whch contans funds wth the hghest past BMA alphas, receves an average monthly nflow of 1.9%. Decle 1, whch contans funds wth the lowest past BMA alphas, receves an average monthly outflow of 0.99%. The 10 1 portfolo has an average monthly normalzed cash flow of.83%. The Spearman rank correlaton s ndcatng that BMA alphas are almost perfectly correlated wth average subsequent flows of the decle portfolos. In other words, average monthly flows ncrease as we move from Decle 1 to Decle 10. Comparng Decles 10 and 1, we fnd that flows nto good past perfomers exceed flows out of poor past performers n magntude. 19 Results for the ntermedate portfolos are avalable from the author upon request. 4

32 Decle 1 s average monthly flow s -0.99% whle decle 10 s average monthly flow s 1.90%. Ths pattern s consstent wth the asymmetrc relaton between flows and past performance as documented by Srr and Tufano (1998), Chevaler and Ellson (1997), Huang, We and Yan (007), among others. BMA alphas are related to subsequent flows over perods longer than one month. Decle 1 experences statstcally sgnfcant outflows 48 months after formaton whle Decle 10 experences statstcally sgnfcant nflows 60 months after formaton. For example, wth a postrankng perod of 48 months, Decle 1 s outflow s 0.49% per month, whle Decle 10 s nflow s 0.99% per month. The Spearman rank correlaton s and the 10-1 portfolo s flow s 1.49%. Although flows respond to past performance up to four years after decle formaton, t s clear that the response weakens as the post-rankng perod lengthens. Lookng across Table 4 Panel A, we see that flows decrease n magntude as the post-rankng perod lengthens. Ths holds for Decle 1, Decle 10 and the 10-1 portfolo. In addton, the Spearman correlaton also declnes as the post-rankng perod lengthens. [ Insert Table 4 here ] Next, we address the queston of whether nvestors cash flow response s senstve to the choce of pror belef n skll. Table 4 Panel B reports the flows of decle portfolos constructed usng BMA alphas estmated wth a less skeptcal pror belef n skll (pror standard devaton of alpha s set to 0.03). For post-rankng perods rangng from 1 month to 4 months, past BMA alphas strongly predct cash flows nto balanced funds. Funds wth good performance experence subsequent nflows whle funds wth poor performance experence subsequent outflows. For example, when the post-rankng perod s one month, Decle 1 s average monthly flow s -0.89% whle Decle 10 s average monthly flow s 1.8%. The Spearman correlaton s ndcatng 5

33 that average monthly flow ncreases as we move from Decle 1 to Decle 10. We contnue to observe an asymmetrc response of fund flows to past performance. As n Panel A, the response of flows to past performance weakens over longer post-rankng perods. Results based on the less skeptcal pror are generally consstent wth those based on the skeptcal pror. Nevertheless, a comparson of Panels A and B reveals stronger and more persstent flows when BMA alphas are estmated wth the skeptcal pror belef n skll. Take, for nstance, flows over the 36 month postrankng perod. Wth the skeptcal pror, Decles 1 and 10 have statstcally sgnfcant flows. The 10-1 portfolo has an nflow of.10% per month and the Spearman correlaton s In contrast, wth the less skeptcal pror, decle 1 s outflow s not statstcally sgnfcant. The 10-1 portfolo s monthly nflow decreases to 1.44% and the Spearman correlaton s lower at Fnally, we consder whether the length of the estmaton wndow affects our fndngs. We repeat our analyss usng BMA alphas estmated wth 60 months of data rather than 36 months of data. Panel C reports results based on a skeptcal pror belef n skll whle Panel D reports results based on a less skeptcal pror belef n skll. For both pror belefs, we fnd that BMA alphas predct future flows up to 48 months after decle formaton. We document outflows from the worst performng funds and nflows nto the best performng funds. The response of flow to performance s asymmetrc as the magntude of nflow nto Decle 10 s usually larger than the magntude of outflow from Decle 1. Thus, usng a longer return hstory to estmate BMA alphas does not change our fndngs. 6. Equty and bond funds We repeat the cash flow analyss for equty funds and report the results n Table 5. Table 5 Panel A contans results based on a skeptcal pror belef n skll (pror standard of alpha s 0.01) whle Table 5 Panel B contans results based on a less skeptcal pror belef n skll (pror standard 6