The value of coskewness in mutual fund performance evaluation

Transcription

1 The value of coskewness n mutual fund performance evaluaton Davd Moreno, Rosa Rodríguez * Department of Busness Admnstraton, Unversty Carlos III, C/Madrd, 126, Getafe 28903, Span abstract Recent asset prcng studes demonstrate the relevance of ncorporatng coskewness n asset prcng mod els, and llustrate how ths component helps to explan the tme varaton of ex ante market rsk prem ums. Ths paper analyzes the role of coskewness n mutual fund performance evaluaton and fnds evdence that addng a coskewness factor s economcally and statstcally sgnfcant. It documents that coskewness s sometmes managed and shows persstence of the coskewness polcy over tme. One of the most strkng results s that many negatve (postve) alpha funds, measured relatve to the CAPM rsk adjustments, would be reclassfed as postve (negatve) alpha funds usng a model wth coskewness. Therefore, performance rankng based on rsk adjusted returns wthout consderng coskewness could generate an erroneous classfcaton. Moreover, some fund characterstcs, such as turnover rato or cat egory, are related to the lkelhood of managng coskewness. Ó 1. Introducton After more than 40 years of research, the problem of how to evaluate actve portfolo management remans largely unresolved. Classc performance measures proposed by Treynor (1965), Sharpe (1966), Jensen (1968) were developed assumng a normal dstrbu ton of returns. Durng the 1970s, others realzed that these perfor mance measures dd not evaluate fund performance accurately because the dstrbuton of fund returns was not Gaussan. Thus, Klemkosky (1973) and Ang and Chua (1979) demonstrated that gnorng the thrd moment of the return dstrbuton would gener ate a bas n the performance evaluaton. Ths bas could affect nvestors drectly by leadng them to create portfolos wth a sub optmal asset allocaton. 1 From the asset prcng lterature, several authors also pont out the convenence of usng models wth hgher moments. Kraus and Ltzenberger (1976) document the mportance of consderng the thrd moment (skewness) of returns. They develop a model n whch nvestors are compensated for holdng systematc rsk and coskewness rsk, requrng a hgher (lower) return whenever the * Correspondng author Tel ; fax E mal addresses jdmoreno@emp uc3m es (D Moreno), rosa rodrguez@uc3m es (R Rodríguez) 1 Along the same lnes, other authors such as Prakash and Bear (1986) or Leland (1999) have developed performance measures ncorporatng skewness, and Stephens and Profftt (1991) generalze the performance measure to account for any number of moments All these results ndcate that gnorng hgher moments could have a sgnfcant mpact on the performance rankngs of these funds systematc rsk s hgher (lower) and the coskewness rsk s lower (hgher). The negatve prce of rsk n the second component nd cates that nvestors dslke assets wth negatve coskewness that requres hgher returns. For those unfamlar wth coskewness, an asset wth negatve coskewness s an asset that, when ncorporated nto a portfolo, adds negatve skewness, ncreasng the probablty of obtanng undesrable extreme values (n the left tal of the dstrbuton). Gven ths agreement regardng the mportance of hgher mo ments from both the mutual fund and asset prcng lteratures, re cent studes have started to develop new performance measures. As hedge funds can employ dynamc strateges such as leverage, short sellng, and nvestment n llqud assets, t seems clear that return dstrbutons wll be non normal and that therefore these measures wll have a sgnfcant effect (see Ranaldo and Favre, 2005; Dng and Shawky, 2007). It s less evdent that these mea sures also generate changes n the performance of common mutual funds unable to use those strateges. 2 Moreno and Rodríguez (2006) have taken the frst step of ntroducng coskewness n mutual fund evaluaton for the Spansh case. They fnd some performance dffer ences when these new measures are taken nto consderaton. The major contrbuton of ths paper s to provde emprcal ev dence about the role of coskewness n evaluatng mutual funds. Ths evdence s not yet addressed n the publshed lterature. It 2 It must be noted that fund managers could use, for example, dervatves (see Kosk and Pontff, 1999; Frno et al, 2009), basng the dstrbuton of fund returns to the left or rght and generatng coskewness n the return dstrbuton 1

2 s provded by examnng: (a) the changes n the average fund per formance; (b) the varatons n the rankng of mutual fund manag ers; (c) the relatonshps between characterstcs such as portfolo turnover, sze, category, and coskewness management; and (d) the possblty that some fund managers may proft from the coskew ness spread. Hereafter, managng coskewness refers to havng a specfc polcy regardng the assets ncorporated nto the fund s portfolo to acheve hgher or lower portfolo coskewness. In order to thoroughly study the relevance of ncludng the thrd co moment of asset returns n performance evaluaton, two dffer ent multfactor asset prcng models are consdered: the CAPM and the Carhart (1997) four factor model. In both models, a coskew ness factor s added and the best adjustment of rsk 3 s sought. The use of a coskewness factor s based on the results from the asset prcng lterature, whch has demonstrated the convenence of usng coskewness models nstead of the popular Fama and French (1993) three factor model. Thus, Harvey and Sddque (2000) test the three moment CAPM s mplcaton that a stock wth a negatve coskewness wth the market wll earn a hgher rsk premum. They form a coskewness factor followng the methodology that Fama and French use n constructng the SMB and HML factors and fnd that coskewness s economcally sgnfcant. Barone Ades et al. (2004) use a quadratc model and fnds that addtonal varables rep resentng portfolo characterstcs (such as those consdered n the Fama and French model) have no explanatory power for expected re turns when coskewness s taken nto account. Chung et al. (2006) suggest that hgher order co moments are mportant for rsk averse nvestors concerned about extreme outcomes. The authors also fnd that the rsk factors of Fama and French approxmate these hgher order co moments especally when usng low frequences. Vanden (2006) also ponts out that SMB and HML measure coskewness rsk, but that they are mperfect proxes. More recently, Smth (2007) fnds that whle the condtonal two moment CAPM and the cond tonal Fama and French three factor model are rejected, a model that ncludes coskewness s not rejected by the data. 4 Ths paper yelds revealng results. Frst, t fnds that the coskewness factor s both economcally and statstcally sgnf cant. Second, the average fund performance wll change when coskewness s taken nto account; ths change s greater when compared wth the CAPM alpha (the average alpha for all equty funds s moved to the left sde by more than double) than wth a Carhart model (the average alpha s modfed by approxmately 6%). Thrd, n general, these movements n the alpha mght affect categores of equty mutual funds n dfferent ways, so that n ths sample, the Aggressve Growth funds are made to look better whle the rest look worse. Fourth, as those varatons n performance wll have a dfferent sgn dependng on the loadng on the coskewness factor, a rankng based on rsk adjusted returns wthout consder ng coskewness mght result n a msleadng classfcaton of the funds. Moreover, one of the most strkng results s that many neg atve (postve) alpha funds measured relatve to the CAPM rsk adjustments would be reclassfed as postve (negatve) alpha 3 An alternatve way to take nto account the skewness of the return dstrbuton could be n an equlbrum framework lke that of Leland (1999). However, ths performance measure would requre two assumptons: the rate of return on the market portfolo must be ndependently and dentcally dstrbuted and perfect markets must exst. Moreover, many of the econometrc problems related to the estmaton of the CAPM alpha wll also be present n estmatng ths performance measure, ncludng fndng an approprate proxy for the market portfolo (as mentoned n Leland (1999), footnote 22). In contrast, multfactor prcng models, such as the ones proposed n ths paper, are not subject to those problems. 4 Coskewness s also consdered relevant n some other economc areas. For example, Vnes et al. (1994) study the mportance of coskewness n the prcng of real estate and Chrste-Davd and Chaudhry (2001) examne t n explanng the returngeneratng process n futures markets. Bal et al. (2008) nvestgates the role of condtonal skewness n the estmaton of condtonal VaR. Post et al. (2008) focuses on the 3MCAPM and the economc meanng of the coskewness premum. funds usng the CAPM plus coskewness. Ffth, those managers usng a specfc polcy for managng coskewness repeat the same polcy over tme, thus persstence n coskewness polcy appears n the majorty of the tme perods. Sxth, fund turnover and fund category are related to havng a specfc polcy of managng coskewness. The remander of ths paper s organzed as follows: the follow ng secton descrbes the coskewness measure and the models used to analyze ts effect on performance evaluaton. Secton 3 presents the database of mutual funds and the benchmarks used. Secton 4 provdes the emprcal evdence. Summares and conclusons are presented n Secton The effect of the coskewness factor on performance evaluaton 2.1. The coskewness Kraus and Ltzenberger (1976) extend the CAPM to ncorporate the effect of skewness n asset prcng, developng the three mo ment CAPM (3MCAPM). Thus, n equlbrum, the expected returns of a rsky asset satsfy: R R f ¼ k 1 b þ k 2 c ; ð1þ where R s one plus the expected return of the rsky asset, R f de notes one plus the return of the rsk free asset, b s the systematc rsk, and c ndcates the systematc skewness (standardzed coskewness) of the asset, a measure of the asset s coskewness rsk. 5 The rsk premums of each rsk factor are k 1 and k 2. Therefore, nves tors are compensated by the expected excess returns for bearng the relatve rsks measured by beta and gamma. Gven that the nvestor requres hgher returns for securtes wth hgher betas, a postve rsk premum, k 1 > 0, s expected. However, there would be a nega tve rsk premum for assets wth postve systematc skewness, k 2 <0. From an emprcal pont of vew, asset prcng models can be tested through the restrctons that they mpose on the coeffcents of the return generatng process. Thus, the return generatng pro cesses consstent wth the CAPM and the 3MCAPM are the market model and the quadratc model, respectvely. Whereas the market model assumes that the return of a rsky asset s lnearly related to the return of a stock ndex representatve of the market, the qua dratc model establshes a nonlnear relatonshp expressed as: R ;t f ;t ¼ c 0 þ c 1 ½R M;t R f ;t Šþc 2 ½R M;t R M;t Š 2 þ m t : ð2þ The estmaton of c 2 n the quadratc model (2) gves a coskewness measure. Through the use of a parttoned regresson argument (Frsch Waugh Lovell theorem) t s easy to verfy that c 2 s equal to Eðe ;tþ1 e 2 M;tþ1 Þ=Vðe2 M;tþ1 Þ, where e ;tþ1 represents the resduals from the regresson of the excess return on the contemporaneous market excess return and e M;tþ1 represents the resduals of the excess mar ket return over ts mean. Harvey and Sddque (2000) compute the standardzed uncon dtonal coskewness as Eðe ;tþ1e 2 M;tþ1 S ¼ q Þ ; ð3þ Eðe 2 Þ ;tþ1 Eðe 2 M;tþ1 Þ 5 Accordng to Kraus and Ltzenberger (1976) the expressons are: b E½ðR R ÞðRM RM ÞŠ E½ðR and c R ÞðRM RM Þ 2 Š E½ðRM RM Þ 2 Š, where E½ðRM RM Þ 3 c s defned as the rato of the Š coskewness of that asset s return and the market s return to the market s skewness. In the same way that the covarance (the numerator of beta) represents the margnal contrbuton of an asset to the varance of the market portfolo, the coskewness (the numerator of gamma) represents the asset s margnal contrbuton to the skewness of the market portfolo. 2

3 where e M;tþ1 and e ;tþ1 are defned as above. The nformaton gven by the coskewness allows a rsk factor to be constructed n the same way that the Fama and French (1993) factors are constructed. Ths rsk factor can be replcated by a portfolo of assets. In order to elab orate on ths factor, t s necessary to compute the coskewness mea sure for each asset and then use t to rank the assets. The assets form two portfolos: one contans the 30% of the assets that have the most negatve coskewness (S ) and another contans the 30% of the assets wth the most postve coskewness (S + ). The return spread of the two portfolos (S S + ) and the return spread of the portfolo S and the rsk free rate (S R f ) are the coskewness rsk factors (CSK) The models In order to analyze the effect of the coskewness factor on per formance evaluaton, the standard CAPM and the Carhart (1997) four factor model are used as base cases. The Carhart (1997) four factor model uses the three factors of Fama and French (1993) plus one that captures the momentum effect. Here, the four factor model (FF4) s used to adjust the performance of the fund for the regulartes found n fnancal returns. Thus, the mod els are R ;t R f ;t ¼ a þ b m ½R M;t R f ;t Šþe ;t ; ð4þ R ;t R f ;t ¼ a þ b m ½R M;t R f ;t Šþb smb SMB t þ b hml HML t þ b wml WML t þ e ;t ; ð5þ where fr M;t R f ;t ; SMB t ; HML t ; WML t g represents the market, sze, book to market value, and momentum factors. When a coskewness factor s ncluded, R ;t R f ;t ¼ a þ b m ½R M;t R f ;t Šþb csk CSK t þ e ;t ; ð6þ R ;t R f ;t ¼ a þ b m ½R M;t R f ;t Šþb smb SMB t þ b hml HML t þ b wml WML t þ b csk CSK t þ e ;t : ð7þ To study the effect of addng ths new factor to the tradtonal Jen sen s alpha, the coskewness rsk must be consdered n the same way as the systematc market rsk. Just as greater returns are re qured for portfolos (and thus managers) wth larger systematc rsks (betas), n a model that ncludes coskewness, greater returns are requred for portfolos (managers) wth larger coskewness rsks. To llustrate, evaluate two mutual funds wth an annual abnormal return of 3% as measured by the classc Jensen s alpha, a A a B Manager A has over weghted the portfolo wth as sets that have negatve coskewness and has therefore obtaned a spread by coskewness, whereas Manager B has not gven specal consderaton to coskewness. Accordng to (6), the loadng parame ter that captures the coskewness rsk must be postve for Manager A (e.g. b CSK 0.20) and zero for Manager B. Because nvestors dslke negatve coskewness assets that requre greater returns, the coskewness factor must have a postve mean (e.g. 0.10), so the fnal alpha for Manager A wll be lower than the alpha for Manager B. Therefore, the abnormal return obtaned would be a CSK,A a A b 2 (CSK) (0.10) 0.01 for Manager A and a CSK,B a B b 2 (CSK) 0.03 for Manager B. In ths example, the manager who tres to proft from the coskewness spread acheves the worse performance. The reason s that alpha s a performance measure that consders rsk adjusted returns, so that just as greater returns are requred of a manager assumng larger market rsk (and therefore a greater b m, reducng hs Jensen s alpha), greater returns are requred of the manager assumng a larger systematc rsk of skewness. To complete the argument, t s mportant to note that Manager A adds negatve skewness to the portfolo by ncorporatng negatve coskewness assets, whch s an undesrable stuaton for nvestors. Thus, a cor rect measure of abnormal returns must penalze ths strategy Condtonal performance evaluaton The models explaned above use uncondtonal expected re turns and are based on the assumpton that factor loadngs are constant. However, f expected returns and rsks vary over tme, such an uncondtonal approach may gve based results. Chen and Knez (1996) and Ferson and Schadt (1996) advocate cond tonal performance evaluaton (CPE). In ther one and mult factor models, factor loadngs (betas) are condtoned on publc nforma ton varables. The resultng condtonal Jensen s alphas represent the average dfference between the return of a fund and the return of the dynamc strateges based on publc nformaton. 6 Therefore, n ths paper, the prevous models are analyzed n a condtonal and an uncondtonal framework, gven the exstng evdence that asset prcng models need to be condtonal snce ex pected returns vary over tme. Ths analyss contrbutes to cond tonal performance evaluaton lterature by presentng the dfferences between condtonal and uncondtonal performance evaluaton when coskewness s ncluded. 3. Data and benchmarks 3.1. Fund returns The database used n ths study conssts of monthly returns for 6819 US equty mutual funds between January 1962 and December 2006, obtaned from the Center for Research n Securty Prces (CRSP). These mutual funds are classfed nto three categores: Aggressve Growth, Growth Income Funds, and Long Term Growth Funds. Table 1 provdes a complete economc and statstcal descrp ton of the database. Presented n rows for each category are the annualzed mean return, rsk (standard devaton), kurtoss, mn mum and maxmum monthly return durng the entre sample per od, and the percentage of funds for whch the null hypothess of normalty, usng a Jarque Bera test, s rejected. The table also shows the total number of mutual funds n each category n nter vals of 36 84, , , , more than 288, and more than 432 observatons. The fgures n Table 1 ndcate that the kurtoss s, on average, hgher than 3 (a value under the null of a normal dstrbuton) and the null hypothess of normalty s rejected for approxmately 48% of mutual funds (51% of Aggressve Growth, 53% of Growth In come Funds, and 44% of Long Term Growth Funds). Accordng to ths result, the use of a performance measure based on normalty should be questoned Benchmark portfolos Ths study uses the CRSP NYSE/AMEX/NASDAQ value weghted ndex as the market portfolo. The monthly seres of SMB, HML, and WML factors obtaned from Kenneth French s webste 7 s used to capture the effects of sze, book to market value, and momentum. The short term rsk free securty s the 1 month Treasury bll (from Ibbotson Assocates). The predetermned varables used as nstru ments n the condtonal models are: (1) the lagged level of the 6 Chrstopherson et al. (1998) propose a refnement of the condtonal performance evaluaton. Introducng tme varaton n alpha makes t possble to determne whether manageral performance s ndeed constant or vares over tme as a functon of the condtonal nformaton. 7 lbrary.html 3

4 Table 1 Summary statstcs of mutual funds: January 1962 March Mean return Standard devaton Kurtoss Max. losses Max. returns Test normalty Aggressve Growth Growth/Income Long-Term Growth All Funds Number of funds >288 >432 Aggressve Growth Growth/Income Long-Term Growth All Funds The table reports summary statstcs for 6819 mutual funds n the database categorsed by nvestment objectves: Aggressve Growth, Income, Growth and Income and Long-Term Growth. A fund s ncluded n the nvestment unverse f t contans at least 36 consecutve monthly return observatons. For each category, we present n columns the annualzed mean return (as a %), standard devaton (as a %), kurtoss, monthly mnmum return (Max. Losses) and monthly maxmum return (Max. Return) durng the entre sample perod, and the percentage of funds for whch the null hypothess of normalty of a Jarque Bera test s rejected at the 10% level of sgnfcance. The table also shows the total number of mutual funds n each category n ntervals of 36 84, , , and more than 288 and 432 observatons. one month Treasury bll yelds; (2) the lagged dvdend yeld of the CRSP value weghted NYSE/AMEX/NASDAQ stock ndex; (3) a lagged measure of the slope of the term structure; and (4) a lagged corpo rate spread on the corporate bond market. The term spread s a con stant maturty 10 year Treasury bond yeld mnus the 3 month T bll yeld. The corporate bond default yeld spread s Moody s BAA rated corporate bond yeld mnus the AAA rated corporate bond yeld. These varables have fgured most promnently n studes of mutual fund performance (see Ferson and Schadt, 1996; Ferson and Warther, 1996). Table 2 presents summary statstcs on the rsk factors. The mean, medan, and standard devaton return data are n annual percentages. In addton, the monthly maxmum and mnmum re turn, and the p value of the Jarque Bera test are shown. The mar ket factor s the excess return on the value weghted ndex; SMB s the factor mmckng portfolo sze; HML s the factor mmckng portfolo book to market value; WML s the factor mmckng port folo 1 month return momentum; and (S S + ) and (S R f ) are the coskewness factors. Monthly US equty returns from CRSP NYSE/AMEX/NASDAQ fles from December 1957 to December 2006 are used to compute Table 2 Summary statstcs of rsk factors and nstruments. Market SMB HML WML S S + S R f Mean Medan Maxmum Mnmum Std. dev Skewness Kurtoss Jarque Bera Cross correlatons Market SBM HML WML S S + S R f EXRM SBM HML WML S S S Rf Ths table reports summary statstcs on the rsk factors. The mean, medan and std. dev. are represented n annual percentages. We show the monthly maxmum and mnmum return, and the p-value of the Jarque Bera test. The Market factor s the excess return on the value-weghted CRSP NYSE/AMEX/Nasdaq portfolo; SMB s the factor mmckng portfolo for sze; HML s the factor mmckng portfolo for book-to-market, WML s the factor mmckng portfolo for the 1-month return momentum; S S + and S R f are the coskewness factors. Table 2 also presents the contemporaneous correlatons between the factors ncluded n the models. the coskewness factor, ncludng ordnary common stocks and excludng real estate nvestment trusts, stocks of companes ncor porated outsde of the Unted States, and closed end funds. For robustness, varous specfcatons of the coskewness factor are nvestgated to ensure that t s not senstve to ts constructon methodology. 8 Thus, there are varous cut off defntons (bottom 15% top 15% and bottom 20% top 20%) n addton to Harvey and Sddque s (2000) factor (bottom 30% top 30%). Moreover, these fac tors are computed by employng the parameter c 2 from the quadratc model (2) to sort the common stocks nstead of usng the standard zed uncondtonal coskewness (3) proposed by Harvey and Sddque (2000). The rsk premum for all these coskewness factors s postve, becomng hgher as the cut off s more extreme. The rsk premums for the (S S + ) are 3.44%, 3.19%, and 2.33% annually for the 15 15, 20 20, and cut offs respectvely, and range from 8.9% to 8.30% annually n the case of the (S R f ) coskewness factor. Con structng the coskewness factor from the quadratc model, the pre mums for the dfferent cut offs are 3.52%, 2.74%, and 2.56% respectvely for the (S S + ), and 9.94%, 9.01%, and 8.73% for the (S R f ) factor. In order to analyze the potental mpact of the factors when they were added to the models, the correlaton coeffcent among all of them s computed. The correlaton for the same coskewness factor usng dfferent cut offs s very hgh (e.g. for the (S S + ) and the correlaton s 0.94), allowng the concluson that the factor s not senstve to the selecton of dfferent cut offs. In addton, the correlaton computed usng dfferent measures of coskewness s also hgh, e.g. for the (S S + ) from the Harvey and Sddque measure (2000) and the measure from the quadratc model t s Therefore, the coskewness factor s not senstve to the dfferent ways of measurng the coskewness. Table 2 also presents the contemporaneous correlatons be tween the factors ncluded n the models. Observe that these cor relatons are generally small, rangng from 0.40 to But there s one case n whch correlaton s not neglgble: when the exstng correlaton between the market factor and the coskewness factor (S R f ) s In order to avod possble multcollnearty problems, ths coskewness factor s orthogonalzed wth respect to the market factor. The correlaton wth the market then changes to zero, and the correlaton wth the other coskewness factor (S S + ) changes to 0.90 (whereas before t was only 0.37), corroborat ng the robustness of the coskewness factor used here. Conse quently, the rest of the paper shows results only consderng the 8 The authors are grateful to an anonymous referee for comments on ths secton, whch have led to substantal mprovements n the paper. 4

5 Table 3 Skewness and coskewness of mutual funds. All funds Aggressve Growth Growth/Income Long-Term Growth Panel A: uncondtonal skewness Mean Medan Postve and Sgn. at 5% Negatve and Sgn. at 5% Panel B: uncondtonal coskewness Mean Medan Postve and Sgn. at 5% Negatve and Sgn. at 5% Panel C: C 2 Mean Medan Postve and Sgn. at 5% Negatve and Sgn. at 5% t-statstc q The uncondtonal skewness s computed as the thrd central moment over the mean. The uncondtonal coskewness s defned as Eðe ;tþ1 e 2 M;tþ1 Þ= Eðe2 ;tþ1 Þ Eðe 2 M;tþ1Þ, where e,t+1 s the resdual from the regresson of the excess return on the contemporaneous market excess return and e M,t+1 s the resdual of the excess market return over ts mean. c 2 s an alternatve measure of coskewness from the quadratc model consstent wth the three-moment CAPM, R ;t R f ;t c 0 þ c 1 ½R M;t R f ;t Šþc 2 ½R M;t R M;tŠ 2 þ m t. Sgnfcance levels for uncondtonal skewness and coskewness are computed based on bootstrap percentles methodology (for a more detaled descrpton on ths methodology see Efron and Tbshran (1993)), and we fnd that at 5% they are 0.40 and 0.40 for the uncondtonal skewness and 0.20 and 0.20 for the uncondtonal coskewness. factor (S S + ), where coskewness s computed usng (3) and wth the ntermedate cut off Emprcal results 4.1. The coskewness of mutual funds Table 3 reports some summary statstcs that compare the coskewness measures across the three categores of funds analyzed n ths paper. Panel A shows the uncondtonal skewness computed as the thrd central moment over the mean. The results ndcate that, consderng all funds jontly, half of the funds have a negatve skewness, sgnfcant at the 5% level. 10 For each of the categores, these percentages are 42%, 61%, and 49%, respectvely. Ths result shows that the skewness of the funds s sgnfcant and that the thrd moment of the return dstrbuton should not be gnored. Panel B presents the results of measurng the uncondtonal coskewness of the mutual funds. The mean value for all equty mutual funds s neg atve ( 0.013) and the proporton of funds that have a sgnfcant coskewness s, on average, 19.63%. Moreover, t can be observed that each category has a dfferent standardzed uncondtonal coskew ness, wth the Aggressve Growth beng the only one havng a nega tve average coskewness ( 0.063) and further, havng the greatest number of funds wth negatve uncondtonal coskewness (16%). G ven that a mutual fund s a portfolo of assets, ths fndng ndcates that approxmately 16% of those mutual funds are nvestng n assets wth negatve coskewness and that therefore the requred return of these funds accordng to the 3MCAPM should be hgher. When the quadratc model (2) s estmated as an alternatve measure of coskewness, the results are smlar to those found n Panel B. The mean value estmated for the parameter c 2, shown n Panel C, s neg atve when all categores are consdered together and the percentage of funds wth a sgnfcant parameter s around 10%. Agan, the Aggressve Growth category shows a negatve value ( 0.531) on 9 The results for the rest of the specfcatons of the coskewness factor are very smlar and are avalable upon request. 10 Sgnfcance levels for uncondtonal skewness and coskewness are computed based on bootstrap percentles methodology (for a more detaled descrpton of ths methodology see Efron and Tbshran, 1993). Here, they are 0.40 and 0.40 for the uncondtonal skewness and 0.20 and 0.20 for the uncondtonal coskewness at the 5% level. average and the Growth Income and Long Term Growth funds pres ent a postve one. The fgures n Table 3 mght gve the mpresson that very few funds exhbt sgnfcant coskewness and that, as a result, the m pact of coskewness could be margnal. Gven that the man goal of the paper s to analyze the mportance of coskewness as an add tonal factor n performance evaluaton, t s nterestng to report the funds exposure to the coskewness factor rather than just the coskewness measures. The senstvtes to the coskewness factor depend on the factor and the funds. Snce, n general, the average betas dffer sgnfcantly from zero for all categores and factors, usng ths (S S + ) factor, between 58% and 64% of the funds are sta tstcally sgnfcant. 11 Hence, these results suggest that coskewness could play an mportant role n explanng the performance evaluaton of mutual funds and that dsregardng t wll create a bas perhaps a sgnf cant one n assessng performance evaluaton. Ths hypothess s tested n the followng secton Performance evaluaton of mutual funds The results of the tme seres estmaton for the models are re ported n Table 4 (the frst panel reportng all funds jontly and subsequent panels reportng each category of mutual funds). For each panel, Rows 1 and 2 show the uncondtonal estmaton of the CAPM and the CAPM plus the coskewness factor. The Carhart four factor model (1997) and the same model plus the coskewness factor are n Rows 3 and 4. For each model, alpha, beta(s), the ad justed R 2 of the regressons, and the lkelhood rato test are reported. An nterestng result from Table 4 s that, n general, for all the categores of funds used n ths paper and for all models, the aver age coeffcent obtaned for the CSK factor s statstcally dfferent from zero. The values range from 0.09 to 0.10, dependng on the category and the model analyzed. In addton to statstcal sg nfcance, there s the economc sgnfcance. As the coskewness factor s an excess return, an approxmate value of the coskewness rsk premum can be calculated by multplyng the loadngs on the factor by the sample average return of the coskewness portfolo 11 These fgures are calculated by regressng the excess return of each fund on the returns on the (S -S + ) portfolo. 5

6 Table 4 Measures of performance usng models wth and wthout coskewness. Alpha Rm SMB HML WML CSK R 2 ADJ (%) LR-test Panel A: All Funds CAPM (1.28) (21.47) CAPM + CSK (1.23) (21.42) (2.54) FF (1.14) (21.64) (3.70) (3.64) (3.16) FF4 + CSK (1.15) (21.90) (3.64) (3.28) (3.05) (1.89) Panel B: Aggressve Growth CAPM (1.20) (14.54) CAPM + CSK (1.12) (14.42) (2.37) FF (1.14) (16.14) (5.75) (4.00) (3.10) FF4 + CSK (1.12) (16.09) (5.72) (3.83) (2.95) (1.78) Panel C: Growth Income CAPM (1.23) (28.81) CAPM + CSK (1.27) (29.21) (2.94) FF (1.13) (29.47) (2.85) (3.54) (3.71) FF4 + CSK (1.19) (30.28) (2.92) (3.04) (3.64) (2.34) Panel D: Long-Term Growth CAPM (1.36) (22.36) CAPM + CSK (1.29) (22.12) (2.44) FF (1.14) (21.29) (2.73) (3.45) (2.92) FF4 + CSK (1.16) (21.49) (2.60) (3.02) (2.81) (1.72) Ths table shows the OLS estmates of the models (4) (7) for equty mutual funds. Alpha s n monthly unts and n percentages. The absolute t-statstcs are n parentheses. LR test s the medan rght-tal probablty value of a standard lkelhood rato test n order to determne whether there s a statstcally sgnfcant dfference between the explanatory power of the model wth or wthout the coskewness factor. (3.19%). Thus, the average coskewness rsk premum ranges from 0.29% to 0.32%. In Panel A of Table 4, where all funds are analyzed jontly, there s a slght ncrease n the R 2 value when a coskewness factor s used as an addtonal explanatory varable; ths ncrease beng from 0.76 to 0.78 when the coskewness factor s ncorporated n the uncon dtonal CAPM, and from 0.84 to 0.85 when t s ncluded n the uncondtonal FF4 model. The last column n Table 4 reports a standard lkelhood rato (LR) test n order to determne whether there s a statstcally sg nfcant dfference between the explanatory power of the new model wth coskewness and the prevous model. The ntroducton of ths extra factor leads to an ncrease n log lkelhood, ndcatng the relevance of the coskewness. The explanatory power of the coskewness model ncreases sgnfcantly over each correspondng model wthout ths new factor, especally relevant n the case of the Aggressve Growth and Growth Income funds for every model, and n the Long Term Growth category only for the CAPM. There fore, not ncorporatng the effect of the systematc skewness may create a potental problem of specfcaton that bases the rsk ad justed return obtaned by mutual funds. Table 4 also reports the average alphas and ther t statstc. In general they are close to zero and negatve except for the estma ton of the CAPM for the Aggressve Growth category where the al phas are postve. Although the average alphas are not statstcally sgnfcant n any case, the economc sgnfcance of the effect of coskewness on the performance of a fund s not neglgble. For example, from Table 4, when comparng the mean alpha from the CAPM and that from the CAPM + CSK, alpha decreases by percent per month, from to 0.041, after coskew ness s controlled. Thus, the net effect of coskewness on alpha s approxmately 0.28% ( ) annually. However, t must be noted that the average change n perfor mance, measured by alpha, s not unform across categores and models. Frst, when coskewness s ncluded n the CAPM, the change n alpha s always greater than when t s ncluded n the FF4 model. Second, when comparng performance among dfferent categores, accountng for the coskewness generally makes the funds belongng to the Growth Income and Long Term Growth cat egores look worse, but the funds n the Aggressve Growth cate gory look better. Furthermore, t s mportant to note that lookng only at average alphas may erroneously lead to the conclu son that the economc mpact of coskewness on performance s neglgble. Ths s because wthn a fund sample, there may be managers wth a postve beta for the coskewness factor (whch would mply a decrease n alpha, as these funds have greater expo sure to assets wth negatve coskewness whch s undesrable for the nvestor) and managers wth a negatve beta for the coskew ness factor (whch would mply an ncrease n alpha). Therefore, coskewness may have a neglgble mpact on the average alpha, even f ts effect on ndvdual alphas s sgnfcant. Ths argument suggests that a more detaled analyss s needed to assess the m 6

7 Table 5 Measures of performance usng condtonal models wth and wthout coskewness. Alpha CSK R 2 ADJ (%) LR-test Pval F Alpha CSK R2 ADJ (%) LR-test Pval F Panel A: All Funds Panel B: Aggressve Growth CAPM (1.21) (1.06) CAPM + CSK (1.20) (2.24) (1.01) (2.10) FF (1.10) (1.03) FF4 + CSK (1.14) (1.81) (1.05) (1.69) Panel C: Growth Income Panel D: Long Term Growth CAPM (1.26) (1.28) CAPM + CSK (1.33) (2.57) 0.10 (1.25) (2.16) FF (1.19) (1.10) FF4 + CSK (1.27) (2.25) (1.13) (1.66) Ths table shows the OLS estmates of the condtonal models (4) (7) for all funds analyzed n ths paper. We use the lagged level of the 1-month Treasury bll yeld, the lagged dvdend yeld of the CRSP value-weghted NYSE/AMEX and NASDAQ stock ndex, a lagged measure of the slope of the term structure and a lagged corporate spread on the corporate bond market as nstruments. Alpha s n monthly unts and n percentages and CSK s the coskewness loadng factor. The absolute t-statstcs are n parentheses. LR test s the medan rght-tal probablty value of a standard lkelhood rato test n order to determne whether there s a statstcally sgnfcant dfference between the explanatory power of the model wth or wthout the coskewness factor. Pval F s the medan rght-tal probablty value of the F-test for the margnal sgnfcance of the term ncludng the nstruments. pact of coskewness on performance. In partcular, the analyss should pay specal attenton to the coskewness management strat egy mplemented by fund managers. In the followng subsecton, the funds wll be grouped accordng to ther senstvty to the coskewness factor. The estmaton of the condtonal models presented n Table 5 shows that the sgns for the loadngs on the coskewness factor are the same as n the uncondtonal estmatons, and also that they are statstcally sgnfcant. Accordng to the LR test, n general there s a statstcally sgnfcant ncrement n the explanatory power of the model wth coskewness, clearer n the CAPM models than n the FF4 models, and clearer for the Growth Income cate gory. An F test s performed for the margnal explanatory power of condtonng nformaton n the models. In ths sample, the F test shows that, consdered together, the nstruments are not sg nfcant at the 5% level. 12 Therefore, gven that the condtonal mod els seem not to contrbute sgnfcantly, the rest of the paper focuses on the uncondtonal models. Table 6 dsplays the dstrbuton of the t statstcs for alpha coeffcents to analyze whether the coskewness factor sgnfcantly changes the dstrbuton of alphas. The fgures n each column of the body of the table are the percentages of mutual funds n whch the t statstcs for the alphas fall wthn the range of values nd cated n the far left hand column. Panel A of Table 6 reports the uncondtonal models usng the CAPM as a base case and Panel B the uncondtonal FF4 model. In general, when systematc skew ness s consdered, the dstrbuton of the alphas moves slghtly to the left, ndcatng that the coskewness factor makes the average performance of fund managers look worse. If the coskewness factor s added to the uncondtonal CAPM, the percentage of negatve and sgnfcant alphas ncreases by be tween 1% and 6%, dependng on the category of funds. The larger ncreases from 21% to 27% are obtaned n the Growth Income funds. In the case of addng the coskewness factor to the FF4 model the results depend a lot on the category. Whle n Growth Income Table 6 Dstrbuton of t-statstcs for the alpha coeffcents. CAPM CAPM + CSK ALL AG GI LTG ALL AG GI LTG Panel A Bonferron p-value t < < t < < t < < t < <t < < t < < t < t > Bonferron p-value FF4 FF4 + CSK ALL AG GI LTG ALL AG GI LTG Panel B Bonferron p-value t < < t < < t < < t < <t < < t < < t < t > Bonferron p-value The numbers n each column of the table are the percentages of mutual funds for whch the t-statstcs for the alphas fell wthn the range of values ndcated n the far-left-hand column. In Panel A we show the models usng the CAPM as a base case. In Panel B the base case s the FF4 model. Insde each type of model, n columns, we present each of the categores of funds: ALL (All Funds), AG (Aggressve Growth Funds), GI (Growth Income Funds) and LTG (Long-Term Growth Funds). The Bonferron p-value ndcates the p-values based on the Bonferron nequalty. Ths s computed as the p-value (one-taled) assocated wth the maxmum or mnmum t- statstc, multpled by the number of funds. It tests the hypothess that all the alphas are zero aganst the alternatve that at least one s postve (maxmum value) or negatve (mnmum value). 12 Ferson and Schadt (1996) fnd p-values of 0.06 for ths test n a dfferent sample, from 1968 to the negatve and sgnfcant alphas ncrease from 21% to 23%, n the Aggressve Growth category they decrease from 20% to 17%. 7

8 Table 7 The sgnfcance of coskewness n performance. b CSK CAPM (a%) CAMP + CSK (a%) b CSK) FF4 (a%) FF4 + CSK (a%) All Funds Q ** ** Q ** Q ** Q ** Q ** ** Aggressve Growth Q ** ** Q ** * Q Q ** Q ** Growth Income Q ** Q Q ** Q ** * Q ** ** Long-Term Growth Q ** ** Q ** Q Q ** Q ** ** Ths table presents the average estmated alphas n the dfferent models once funds have been classfed nto quntles based on the t-statstc of the beta coeffcent of the coskewness factor (S S + ). The b CSK column presents the value of the beta coeffcent of the coskewness factor n each quntle. CAPM (a%) ndcates the average alpha usng the CAPM n each quntle. CAMP + CSK (a%) s the average alpha as a usng the CAPM wth an addtonal coskewness factor, FF4 (a%) s the average alpha form the Carhart model, and FF4 + CSK (a%) s the average alpha form a Carhart model ncludng the coskewness factor. * Means sgnfcance at the 10% level for the Wlcoxon test of dfferences n alpha dstrbuton between the model wth and wthout coskewness. ** Means sgnfcance at the 5% level for the Wlcoxon test of dfferences n alpha dstrbuton between the model wth and wthout coskewness The effect of coskewness and the coskewness polcy As stated n the prevous secton, the net effect of consderng coskewness and mutual fund performance cannot be analyzed only by the changes n the average alpha, gven that dfferent mutual funds have dfferent exposure to the coskewness factor. Funds wth negatve senstvty to the coskewness factor add assets wth pos tve coskewness to ther portfolos and therefore nvestors wll de mand lower returns for these funds. The average alpha (adjusted by the rsk of coskewness) should then be hgher. However, those funds that ncorporate negatve coskewness assets must present a postve beta of coskewness and nvestors wll demand hgher re turns due to the hgher rsk of coskewness; n that case, the ad justed alpha should be lower. Gven ths, the net effect over the average alpha may seem small because the effects of both types of funds are mutually balanced. It could be erroneously concluded that coskewness has a neglgble effect on mutual fund perfor mance when n fact, as Table 6 shows, ths s not the case. Table 7 shows the average estmated alpha n the dfferent models once the funds have been classfed nto quntles accordng to the sgnfcance of the beta to the coskewness factor. Ths table shows that there are clearly opposte effects on a mutual fund s al pha dependng on the sgn of the loadng factor of coskewness. For example, consderng all funds jontly n the frst quntle (that s, the 20% of the funds wth the lowest exposure to the CSK factor), the alpha changes from a negatve value of 0.14 to a postve va lue of 0.04; ths varaton n means s statstcally sgnfcant at the 5% level. For the FF4 model the change n alpha s also sgnfcant usng all the funds jontly. Smlarly, for the 20% of funds wth the greatest senstvty to the CSK factor (Q5), the effect on alpha s the opposte, movng from a postve value of 0.21 to a value of 0.00, and the varaton n means s also statstcally sgnfcant. Logcally, these changes are not statstcally sgnfcant for the cen tral quntles formed by funds that do not manage coskewness. Therefore, usng the coskewness factor allows correctng for the performance of funds managng coskewness whle not affectng those not managng coskewness. A second outcome observed n Table 7 s that a rankng based on rsk adjusted returns wthout consderng coskewness mght result n a contrary classfcaton for the funds n the extreme quntles, where losers would be consdered wnners and vce versa. For example, n the CAPM the mean alphas for quntles 1 and 5 are 0.14% and 0.21% respectvely, but when the coskewness factor s consdered the mean alphas change to 0.04% and 0.00%, respec tvely. Moreover, the Wlcoxon test shows how the average alphas are sgnfcantly dfferent n these extreme quntles. Analyzng categores, the conclusons are dentcal. However, the dfferent effect obtaned n the Aggressve Growth and Growth Income funds must be emphaszed. Takng nto account the coskewness for the former n a FF4 model, the change n alphas s only statstcally sgnfcant n the frst and second quntles, ndcatng a better performance for ths category of funds. How ever, n the case of the Growth Income category, the change n mean alphas s only statstcally sgnfcant n the last two quntles (Q4 and Q5), generatng a worse performance for ths category. It must be noted that ths result s n accordance wth the movement n alphas observed for the FF4 model n Table 6. In concluson, there s evdence of sgnfcant changes n mutual fund performance when the systematc skewness s consdered; t s a dfferent sgn dependng on the fund s exposure to the coskew ness factor. Moreover, these changes n performance are statst cally sgnfcant n 80% of the mutual funds sampled (Q1, Q2, Q4, and Q5) when the coskewness s ntroduced n the CAPM, and are statstcally sgnfcant for between 20% and 40% n the FF4 model. 13 Once more, these conclusons are consstent ndependent of the coskewness factor employed here. Consequently, these results 13 The lower mpact of coskewness n the FF4 model ndcates that, as Chung et al. (2006) and Vanden (2006) ponted out, Fama and French rsk factors may be proxyng hgher order co-moments. 8

9 mght have serous effects on other mutual fund research where per formance rankng s requred, such as persstence studes or studes of nvestors selecton ablty and flows of mutual funds Persstence managng coskewness Thus far, the analyss of coskewness has used a 44 year sample. In such a long perod, t s qute lkely that the coskewness of the funds has vared over tme. 14 Instead of assumng that the coskew ness betas have remaned constant over the whole perod, t would be nterestng to estmate them over shorter perods to apprecate, by categores, whether the coskewness beta changes n magntude and sgn between perods. In ths secton, the models presented n Table 4 are estmated agan, splttng the sample nto three subsam ples: , , and Table 8 reports these estmatons. The alpha wth and wthout coskewness and the beta of the CSK factor are presented for each model to analyze the sgn and sgnfcance of ths parameter n each subsample. The results show that the coskewness polcy does not seem con stant, gven that the beta for the CSK factor has vared over tme. Thus, for example, from subperod 1 to 2, every category changes from a negatve and sgnfcant beta to a postve one (e.g. Aggres sve Growth goes from to 0.175) when the CAPM s consd ered. When the FF4 model s consdered, there s also a change n sgn from subperod 1 to 2 for the majorty of categores. From sub perod 2 to 3, there s only a change n sgn of the coskewness beta for the Aggressve Growth and Growth Income categores when the FF4 model s used. Hence, these results hghlght the need to consder coskewness when evaluatng the performance of mutual funds because, dependng on the tme perod, coskewness affects them n dfferent ways. 15 In addton, Table 8 shows that the coskewness factor s especally sgnfcant n the thrd tme perod for all models and categores. 16 On the other hand, after the prevous analyss, t s unknown f the coskewness polcy of a partcular manager remans constant over tme, because the funds have been aggregated n categores. However, Table 7 shows that wthn the same category there are funds wth postve senstvty and funds wth negatve senstvty to the coskewness factor. Therefore, from an economc pont of vew t would be nterestng to fnd out f certan managers mght be keepng a constant coskewness polcy over tme and whether they may be proftng from the spread of coskewness. A non parametrc methodology based upon contngency tables s used to study ths queston: a contngency table of funds called postves and negatves, where a fund s termed postve f ts senstvty to the coskewness factor s postve, and negatve f t s not. The analyss s smlar to persstence performance studes. However, n ths context, persstence ndcates those funds that are postve n two consecutve perods, denoted by PP, or negatve n two consecutve perods, denoted by NN. Smlarly, postve (negatve) n the frst perod and negatve (postve) n the second perod, denoted by PN (NP), ndcates a reversal behavor. Ths con tngency analyss requres dvson of the sample nto subperods, as well as funds that exst n two consecutve perods. Here, there are subperods of 3 years, although n a context of two perods. Thus, perod 62/64 65/67 ndcates that the beta of the factor coskewness s consdered for perod 62/64 and s compared wth that obtaned n the 65/67 perod. 17 A Cross Product Rato (CPR) s used to detect persstence n managng coskewness. 18 The CPR reports the odds rato of the num ber of managers that repeat to the number of those that do not re peat, that s, (PP NN/PN NP). The null hypothess that the coskewness polcy n the frst perod s unrelated to the coskewness polcy n the second corresponds to an odds rato of one. Table 9 re ports the test statstc for the odds rato test. 19 In Panel A the analyss s carred out usng all mutual funds n the database, and n Panel B only those funds that really manage the coskewness are consdered (funds wth a statstcally sgnfcant beta of coskewness). Independent of the coskewness factor used n Panel A, n gen eral there are some cases n whch there s a persstence n manag ng coskewness (ths number s hgher when the CAPM s used) but also a smlar number of cases for reversals (e.g. usng the CAPM, the proporton of cases of persstence aganst reversals s 14/1, and usng the FF4 model, t decreases to 7/7). Thus, t could be erro neously concluded that there s not a persstence behavor from mutual fund managers n managng coskewness. As mentoned above, reversals appear when managers change ther coskewness polcy. However, these reversals could also be generated unnten tonally, that s, when non sgnfcant betas are changng from po stve to negatve or vce versa, but are not statstcally sgnfcant. Ths would be the case of fund managers who have no specfc coskewness polcy. In order to verfy ths ssue, those funds wth a sgnfcant beta of coskewness are analyzed exclusvely. Panel B presents the results for those mutual funds wth a coskewness beta statstcally sgnfcant at 5%. Once the funds that truly take a polcy of coskewness are studed separately, n pract cally all cases ndependence s rejected and the reversal pattern dsappears. Therefore, the results ndcate that fund managers wth a certan polcy of managng coskewness tend to mantan t over tme and that ths persstence seems to be more relevant n the la ter tme perods (t could also be due to the very low number of funds at the begnnng of the sample). Moreover, that persstence behavor s senstve to the model used, beng clearer when ntro ducng the coskewness factor n a CAPM Mutual fund characterstcs and coskewness These results suggest that some fund managers are managng the coskewness of ther portfolos. The next logcal objectve would be to nvestgate the characterstcs of those funds. Are they the largest or the smallest funds? Do they have a hgher or lower ex pense rato? Do they have a hgher or lower turnover rato? To shed some lght on these questons, two dfferent analyses are per formed: a unvarate analyss of the mean of some characterstcs after separatng the funds nto three dfferent groups accordng to ther coskewness, and then a multnomnal logt model to est mate the probablty of a fund havng a sgnfcant coskewness con dtonal on the explanatory varables. Now, group (S ) ncludes the 15% of funds wth the most nega tve uncondtonal coskewness, (S + ) the 15% wth the most postve coskewness, and (S 0 ) the rest of the funds. Gven the results of Ta ble 3, ths result s smlar to separatng the funds wth sgnfcant coskewness (especally negatve coskewness) from the funds wth out sgnfcant coskewness. The characterstcs consdered are To 14 Smth (2007) fnds evdence that coskewness s tme-varyng and rejects the null hypothess of constant coskewness. 15 The analyss has also been repeated for only two subsamples ( and ) and the results are very smlar; they are avalable upon request. 16 In the second subsample (January 1977 December 1991) the absolute t-statstcs for the coskewness beta are not statstcally sgnfcant on average, but ths s because of the extreme return on the crack of October If the models are estmated wthout ths date, then all the t-statstcs for beta coskewness are statstcally sgnfcant, as n the thrd subsample. 17 In addton to subperods of 3 years, the analyss has also been repeated wth subperods of 5 years and the results and conclusons are dentcal. 18 There s also a v 2 test comparng the observed frequency dstrbuton of PP, PN, NP, and NN for each fund wth the expected frequency dstrbuton. Gven that the conclusons are dentcal, they not shown to save space, but are avalable upon request. 19 The statstcal sgnfcance of the CPR s determned by usng the standard error of the natural logarthm of the CPR (see Chrstensen (1990) for more detals). 9