Asset Pricing When Returns Are Nonnormal: Fama-French Factors vs. Higher-order Systematic Co-Moments*

Transcription

1 Asset Prcng When Returns Are Nonnormal: Fama-French Factors vs. Hgher-order Systematc Co-Moments* Y. Peter Chung Unversty of Calforna, Rversde Herb Johnson Unversty of Calforna, Rversde Mchael J. Schll Unversty of Vrgna February 2004 Forthcomng n the Journal of Busness Abstract A growng lterature contends that, snce returns are not normal, hgher-order co-moments matter to rsk-averse nvestors. Fama and French (1993, 1995) fnd that non-market rsk factors based on sze and book-to-market rato are prced by nvestors. We test the hypothess that the Fama-French factors smply proxy for the prcng of hgher-order co-moments. Usng portfolo returns over varous tme horzons, we show that addng a set of systematc comoments (but not standard moments) of order 3 through 10 reduces the explanatory power of the Fama-French factors to nsgnfcance n almost every case. * We are grateful for helpful comments and suggestons from Warren Baley, Javer Estrada, Wayne Ferson, Campbell Harvey, Raymond Kan, Tae-Hwy Lee, Bruce Lehmann, Davd Mayers, Mark Rubnsten, Serge Sarkssan, Mke Stutzer, and an anonymous referee. Chung acknowledges fnancal support from an A. Gary Anderson Graduate School of Management Summer Research Grant.

2 I. Introducton Accordng to the Captal Asset Prcng Model (CAPM), nvestors only prce market rsk. However, a growng lterature dentfes many non-market rsk factors that appear to be prced. In partcular, Fama and French (1993, 1995) fnd that the non-market rsk factors SMB (the return on a portfolo of small stocks less the return on a portfolo of large stocks) and HML (the return on a portfolo of hgh book-to-market-value stocks less the return on a portfolo of low book-to-market-value stocks) are statstcally mportant n explanng the cross-secton of equty returns. There s, however, substantal debate regardng the economc meanng of SMB and HML. Fama and French (1993, 1996) suggest that book-to-market and sze are proxes for frm dstress. Lakonshok, Shlefer, and Vshny (1994) argue that book-to-market proxes for nvestor bas n earnngs-growth extrapolaton. Danel and Ttman (1997) fnd that SMB and HML pck up the co-movements of stocks wth smlar characterstcs, so t s the characterstcs, not the co-movements, that explan cross-sectonal return varaton. Rolph (2003) and Ferguson and Shockley (2003) argue that the Fama-French factors proxy for leverage effects. Berk (1995), Kothar, Shanken, and Sloan (1995), and Ferson, Sarkssan, and Smn (1999) argue that the explanatory power of SMB and HML are spurous. In ths paper, we propose an alternatve explanaton. If the CAPM holds, only the second-order systematc co-moment (beta) should be prced. Although the CAPM can be derved under varous sets of assumptons, we argue that normalty of returns s the crucal assumpton: Wthout normalty, the CAPM s unlkely to hold. Returns are not n general normal (see, e.g., Mandelbrot (1963), Ane and Geman (2000), and Aparco and Estrada (2001)), and tend toward lognormalty for longer ntervals. We 1

3 examne returns for daly, weekly, monthly, quarterly, and sem-annual ntervals. We fnd evdence that normalty s rejected for all fve cases. In addton, we fnd that factors besdes the CAPM beta explan the cross-secton of returns. The Fama-French (F-F) factors SMB and HML jontly provde statstcally sgnfcant explanatory power across almost all the sample return horzons. We argue that SMB and HML proxy for measures of market rsk not captured by the CAPM. In the CAPM, nvestors care only about two moments mean and varance for portfolo returns and one co-moment covarance for securty returns. In general, however, nvestors may care about hgher moments skewness, kurtoss, and so on and hgher comoments co-skewness, co-kurtoss, and so on. Scott and Horvath (1980) show that nvestors should have a negatve preference for even moments and a postve preference for odd ones. Rubnsten (1973) derves an equaton for the expected return n terms of an arbtrary number of co-moments. We test whether SMB and HML proxy for these co-moments. Others have looked at co-skewness, co-kurtoss, or both to explan returns (see, e.g., Bonsal and Vswanathan (1993), Dttmar (2002), Frend and Westerfeld (1980), Harvey and Sddque (2000), Hung et al. (2003), Kan and Wang (2001), Kraus and Ltzenberger (1976), Lm (1989), Perez-Quros and Tmmerman (2000), and Sears and We (1985)). We argue that there s no reason to stop wth the fourth moment. Rsk-averse nvestors are presumably very concerned about the rsk of run. Furthermore, the popularty of lotteres, sweepstakes, out-of-the-money optons, etc. mples that nvestors are also concerned about the rght tal of the dstrbuton. Varance, skewness, and kurtoss tell us somethng about the tals of the dstrbuton, but they fall far short of specfyng the tal precsely. To see the mportance of hgher moments, consder two dstrbutons: (1) a standard normal and (2) a smple mxture of dstrbutons wth a probablty, p, ( 0 < p < 1) of drawng 2

4 from a blateral exponental wth densty 1 x αe α 2 and a probablty 1 p of drawng from a standard normal. Snce both dstrbutons are symmetrc, the odd moments are all zero for both. For approprate choces of α and p we should be able to match, at least approxmately, the varance and kurtoss for both. Thus, we can match the frst fve moments, yet the tals of the two dstrbutons are completely dfferent: The normal falls off much more quckly than the blateral exponental. Ths example shows that knowng the frst few moments does not mply that the shape of the tals s known. Hgher-order moments have been crtczed for beng unrelable and lackng ntuton. We beleve that both crtcsms can be answered by lookng at several co-moments. Each comoment may ndvdually be unrelable, but the set of co-moments, taken together, should not be. The dosyncratc varatons n ndvdual co-moment estmates can be largely dversfed away by lookng at several co-moments. We have to expect multcollnearty, but our prmary concern s to measure the effect of the entre set of hgher co-moments. Smlarly, whereas the ntuton for an ndvdual co-moment may not be obvous, the ntuton for the entre set s clear: The set of co-moments s a measure of the lkelhood of extreme outcomes, a matter of great mportance to rsk-averse nvestors. Usng a set of co-moments to estmate the tals s rather lke usng a set of lagged values of nflaton to estmate future nflaton: In both cases we are not partcularly nterested n the ndvdual members of the set but n the set as a whole. There are other statstcal estmators we mght use nstead of co-moments, such as cumulants and the Hll estmator, but t s not clear to us that there s any advantage n usng these alternatves nstead of co-moments. When returns are normal, the mean and varance suffce to descrbe the dstrbuton completely. However, n general, to specfy the tals perfectly requres an nfnte number of 3

5 moments. We do not go that far, but we do keep moments up through order ten. We fnd that ncludng systematc co-moments 3 through 10 almost always causes SMB and HML to become nsgnfcant, and always causes ther t-statstcs to drop dramatcally. As a check, we re-run our tests usng standard unvarate moment estmates nstead of co-moments. We fnd that SMB and HML usually reman sgnfcant even f standard moments 3 through 10 are ncluded. Thus, t does not appear that our results are beng drven by smply addng more explanatory varables. In the next secton we ntroduce the hgher-order systematc co-moment equaton. In Secton III we present our data and perform the tests. Secton IV summarzes and gves our conclusons. II. Hgher-Order Co-Moments Assumng perfect markets, a rsk-free asset, and homogeneous expectatons, Rubnsten (1973) derves the followng equaton for E (R j ), the expected return on securty j: where σ n s the n th co-moment: ~ E( R j ) = R f + θ n n= 2 σ ( R n 1 σ n ( R j, W ) = E[ R j E( R j )] [ W E ( W )], ~ n j ~, W ), (1) ~ W ~ ( n) U s ndvdual 's future wealth, θ n = ~, and U ( W ~ ) s 's utlty of wealth. ( n 1)! E[ U ( W )] We can nterpret σ n and θ n as ndvdual s measures of securty rsk and rsk averson, respectvely. Rubnsten notes that an ndvdual adjusts hs portfolo untl the expected return equals the rsk-free rate plus a rsk premum equal to a weghted sum of co-moments. Intutvely, nvestors are concerned about rsk, and rsk must be measured n terms of the entre 4

6 probablty dstrbuton, whch n turn can be measured wth the moments of the dstrbuton. Only n very specal cases, such as quadratc utlty or normalty of returns, can we gnore the hgher moments and focus on just mean and varance. For an ndvdual securty, the contrbuton to the rsk of the portfolo s measured wth co-moments. For the case of separable cubc utlty Rubnsten derves the market relaton: ER ( j) = Rf + λ2covr ( j, Rm) + λ3cosr ( j, Rm, Rm) (2) where Cos s co-skewness, m denotes the market portfolo, and λ 2 and λ 3 are market measures of rsk averson. Kraus and Ltzenberger (1976) derve an equaton of the same form. They state that "t s trval to extend the model to ncorporate any number of hgher moments" (p. 1087). We are wllng to make whatever restrctve assumptons are needed to extend equaton (2) to the case of n co-moments. For example, f we assume dentcal nvestors, then equaton (1) mples the market relaton: N E( R ) = R + λ b (3) j F = 2 j where b j s the th order systematc co-moment between R j and R m, and λ s the market measure of rsk averson for the th co-moment. Ths follows from Rubnsten s equaton (5): E ( R j ) = I E ( R I k ) + n= 2 θ σ ( R n n j I R k ~, W ) where I s the number of ndvduals. Lettng k be the rsk-free asset, assumng homogeneous expectatons, notng that each sum over reduces to I tmes the summand, and equatng functon of the future value of the market portfolo (snce n equlbrum all assets must be held), we obtan equaton (3). Equaton (3) s the generalzaton of the securty market lne: Instead of one market co-moment, there are several. W ~ to a 5

7 These models suggest that only market-rsk factors should matter to nvestors. Yet Fama and French fnd that ther non-market SMB and HML factors are also mportant. In the next secton we test whether the SMB and HML factors are ndependent rsk factors n ther own rght or merely proxes for hgher-order market factors. III. Data and Emprcal Tests A. The Fama-French Model We explore the cross-sectonal return characterstcs of portfolos based on sze and also based on book-to-market value over the 1930 to 1998 sample perod. Our emprcal tests are n the sprt of Fama and MacBeth (1973). Fama and MacBeth test the CAPM wth a twopass procedure that frst sorts stocks nto portfolos based on hstorcal beta estmates and then estmates the mean cross-sectonal relatonshp between the portfolo returns and portfolo betas for each perod. By sortng on beta they are able to maxmze the cross-sectonal varaton n the varable of nterest. We perform ths same experment for both Fama-French factors. At the end of each calendar year, we rank all ordnary common stocks ncluded on the Center for Research n Securty Prces (CRSP) fle by market captalzaton and dvde the sample nto 50 portfolos of equal sze. The sze portfolos ncrease from about 15 stocks per portfolo n the 1930s to about 140 stocks n the 1990s. Usng ths sze-based portfolo defnton, we compute a tme seres of equal-weghted and rebalanced portfolo returns. We subtract the 30-day Treasury bll yeld to obtan the excess portfolo return, r(j,t), where j denotes the portfolo and t denotes the tme perod. We repeat ths procedure for the stratfcaton usng book-to-market value. For ths sort, we use the unverse of both CRSP and Compustat frms. The portfolos are sorted by the begnnng-of-perod book-to-market rato and contan all CRSP and Compustat-lsted ordnary common equtes from 1970 to 1998 for 6

8 all return ntervals. To reterate, by sortng ndependently by sze and book-to-market value, we are able to maxmze the varaton n the rsk factor at hand. For each perod, t, we estmate a cross-sectonal regresson of the perod portfolo returns on the loadngs on SMB and HML and systematc co-moment factor loadngs as n the followng equaton: n 1 + asmbs( j, t) + ahmlh( j, t) + ab(, j, t) + e( j, t) = 2 r( j, t) = a (4) where s ( j, t) and h ( j, t) are the factor loadngs for SMB and HML, respectvely, and b (, j, t) s the th systematc co-moment. For example, the 2 nd systematc co-moment s the CAPM beta and the 3 rd systematc co-moment s the Kraus and Ltzenberger (1976) systematc coskewness factor. Repeatng ths process for all perods n the sample perod produces T sets of coeffcent estmates. We then average the T estmates to produce a sample Fama-MacBeth coeffcent estmate. In Table 1 we report the results for the Fama-French three-factor (SMB, HML, and the 2 nd systematc co-moment) model. Panel A presents results for sze-sorted portfolos over the perod 1930 to 1998 and Panel B has the results for book-to-market-rato-sorted portfolos over the perod 1970 to Snce daly returns on the CRSP fle do not begn untl July 1962, the sample perod n Panel A s shorter for the daly and weekly ntervals ( ). For each perod, portfolo returns are regressed on three factor loadngs: b, s, and h. These loadngs are computed by regressng portfolo returns over the past fve years on the SMB, HML, and market factors, respectvely. We fnd n both panels that at least one of the Fama-French factor loadngs, s and h, has sgnfcant explanatory power for portfolo returns for each of the return frequences. Not surprsngly, s does better n Panel A and h does better n Panel B. Market beta has sgnfcant explanatory power every tme, so all three factors appear to be mportant 7

9 for explanng returns. We perform a Wald test of the jont sgnfcance of SMB and HML coeffcents. The χ 2 -statstc tests whether the coeffcents on s and h are jontly dfferent from zero. The Wald test s sgnfcant at the 1-percent level for all tme ntervals and sortng crtera. The results n Kem (1983), Knez and Ready (1997), and Chan, Karcesk, and Lakonshok (1998) emphasze the mportance of January n understandng asset returns. We look (Panel C) at monthly returns of sze-sorted portfolos by calendar month. The results are strkng. For every month except January the coeffcents for b and h have the normal sgn and usually are sgnfcant, whereas the coeffcent for s s nsgnfcant. However, n January b and h are nsgnfcant and have the opposte sgn, whereas s s strongly sgnfcant. B. The Dstrbuton of Portfolo Returns Table 2 provdes summary statstcs for the dstrbutons of the portfolo returns used n Table 1. Panel A gves the results for sze-sorted portfolos. The return-dstrbuton skewness ncreases wth the length of the return nterval, ncreasng from for daly ntervals to 5.1 and 2.6 for quarterly and sem-annual ntervals, respectvely. The kurtoss statstc s consstently greater than the corrected standard normal dstrbuton value of 0, varyng from 4.5 for weekly returns to 58.9 for quarterly returns. Panel B gves the results for book-tomarket-rato-sorted portfolos. Agan, skewness ncreases wth the return nterval, and kurtoss s always somewhat hgher than that for the normal. To nvestgate the normalty assumpton, we provde test statstcs for two standard normalty tests. The Jarque-Bera statstc tests the thrd and fourth sample moments aganst those of a normal dstrbuton. In Panel A, for all ntervals except weekly, the Jarque-Bera statstc for the portfolo returns strongly rejects normalty. In Panel B, ths statstc does not reject normalty except for daly returns. Portfolo 8

10 return normalty s, however, strongly rejected n every case by the Kolmogorov statstc. Ths statstc measures the maxmum and mnmum dscrepancy between the normal cumulatve densty functon and the emprcal cumulatve densty functon generated by the data. Because we have argued that skewness and kurtoss are not suffcent to descrbe the dstrbuton, we consder the Kolmogorov statstc to be the approprate one for our purposes. C. Hgher-Order Systematc Co-moments Snce returns are not normal, we expect hgher-order co-moments to matter to rskaverse nvestors concerned about extreme outcomes. We compute systematc co-moment estmates of the 3 rd to 10 th order usng the past fve years of portfolo returns. Non-centered systematc co-moments b(,j,t) are computed n the followng manner: b (, jt, ) = T τ = 1 τ = 1 ( r m t τ ) r( j, t τ) (, ) T ( rmt (, τ )) 1 (5) where denotes the order of the co-moment and r (m,t) s the return for the CRSP valueweghted portfolo. (The CRSP portfolo covers NYSE and AMEX stocks untl 1973 when NASDAQ stocks are also added.) We compute a seres of systematc co-moment estmates from the 3 rd to the 10 th order for each of our fve return ntervals. To llustrate, for the monthly nterval we compute monthly co-moment estmates usng the past 60 months of return data. In ths manner estmates are produced for each portfolo for each month from January 1930 to December The result s a monthly tme seres of co-moment estmates for each portfolo. The estmaton procedure s smlar for the other return ntervals (e.g., the daly nterval procedure produces a tme seres of daly co-moment estmates). Snce the estmaton nterval s constant (fve years), the daly nterval co-moment estmates are computed from 9

11 about 1,260 daly portfolo-return observatons, whereas the sem-annual estmates are computed from only ten portfolo-return observatons. We also performed the experments n ths paper usng centered co-moment estmates by demeanng each seres by the sample mean. We report the results based on the noncentered seres snce centerng causes the denomnator of odd co-moment estmates to tend to zero so that the centered estmates can vary dramatcally. To get a sense of the correlaton between the F-F loadngs and the hgher order comoments, we regress the F-F loadngs on the second- to tenth-order monthly systematc comoment estmates. We fnd that the systematc co-moments can explan (loadngs for) SMB and HML wth farly hgh 2 R (.86 to.92 for sze-sorted portfolos and.81 to.93 for book-tomarket-sorted portfolos). We can also explan SMB and HML wth standard moments, but wth less success: The 2 R are consstently lower. We fnd that 2 R ncreases as we add comoments and also ncreases as we use co-moments nstead of moments. Ths result motvates the followng emprcal queston: Are SMB and HML true rsk factors or do they merely proxy for hgher-order systematc co-moments? D. Emprcal Tests The mean coeffcent estmates from equaton (4) for the F-F factors and systematc comoments are reported n Table 3 for each return frequency nterval. Panel A reports the results usng sze sortng. For daly returns, the SMB loadng remans sgnfcant, but ts t-statstc drops sgnfcantly as co-moments are added, whereas the HML loadng mantans sporadc weak sgnfcance. For weekly returns, SMB becomes nsgnfcant when the tenth comoment s added and HML becomes nsgnfcant when the seventh s added. For monthly returns, both SMB and HML become nsgnfcant when co-moments greater than the fourth 10

12 order are ncluded. For quarterly returns, the hghly sgnfcant HML loadng from Table 1 becomes nsgnfcant or has the wrong sgn when any hgher-order co-moments are ncluded. The same s usually true for sem-annual returns, although the HML loadng returns to sgnfcance n some specfcatons. Except for the daly return nterval, the Χ 2 -statstc, whch tests the jont sgnfcance of the SMB and HML coeffcents, s sgnfcant only when few hgher-order co-moments are ncluded. Panel B presents the results for book-to-market sortng. The results are smlar to those n Panel A. Thus, n every case, the prema on the SMB and HML loadngs become nsgnfcant or much less sgnfcant as systematc co-moments are added. 1 The fact that the adjusted R 2 statstcs generally rse as we add co-moments suggests that the addtonal regressors are addng explanatory power. In some cases co-skewness or co-kurtoss or both are suffcent to elmnate the sgnfcance of both SMB and HML. In other cases hgher-order co-moments are requred. It mght be argued that almost any set of varables would reduce the sgnfcance of the Fama-French factors f enough of them are ncluded. To check ths, we substtute standard unvarate moments (skewness, kurtoss, etc.) for our systematc co-moments (co-skewness, co-kurtoss, etc.): T 1 m (, jt, ) = (, ) T τ = 1 ( r jt τ ) (6) 1 A lower sgnfcance level may be requred as the rejecton crteron for large samples, such as our daly (8,561) and weekly (1,774) observatons. See Lndley (1957). Most of the F-F factor loadngs that reman sgnfcant at the conventonal levels of sgnfcance, such as 5% or 1%, become nsgnfcant when we use the 0.1% level nstead. 11

13 where m(,j,t) s the th order standard moment for portfolo j at tme t. We re-run the regresson replacng the co-moment estmates wth the standard hgher-order moment estmates over orders 3 through 10. r( jt, ) = a + a s( jt, ) + a h( jt, ) + ab(2, jt, ) + am (, jt, ) + e( jt, ) (7) 1 SMB HML 2 = 3 The results, n Table 4, show that wth sze sortng the F-F factors mantan hghly sgnfcant explanatory power for portfolo returns n every case (wth the excepton of the HML loadng wth the quarterly return specfcaton) even when moments 3-10 are ncluded. For book-to-market sortng, the SMB premum estmates are nsgnfcant, but, wth the excepton of daly returns, the s estmates are also nsgnfcant when no hgher moments or comoments are ncluded (see Table 1, panel B). The HML estmates reman consstently sgnfcant when the hgher moments are ncluded. Thus, systematc co-moments reduce the sgnfcance of the F-F factors, but the standard moments do not. 10 E. Errors-n-Varables Concerns One mportant concern wth our emprcal approach s that the two-pass Fama-MacBeth method may be based snce the rght-hand sde varables n the second-pass cross-sectonal regresson are the estmates from the frst-pass tme-seres regresson (Shanken (1992), Km (1995, 1997), and Jagannathan and Wang (1998)). To test for the errors-n-varables (EIV) bas we recalculate all standard errors usng the Shanken adjustment. 2 The adjusted t-statstcs 2 The adjusted covarance matrx s defned as: adj.var( a) = V 1 + a '( Z ) 1 a + Z where V s the k-factor-by-k-factor covarance matrx of the monthly demeaned Fama-MacBeth coeffcent estmates, a s the T-perods-by-k matrx of mean coeffcent estmates, and Z s the k-by-k covarance matrx of monthly rsk factors. The respectve rsk factors are the standard Fama-French SMB and HML factors, the CRSP value-weghted market-portfolo 12

14 are slghtly lower for specfcatons wth just a few hgher-order co-moments. However, when addtonal hgher-order co-moments are added, the standard errors ncrease substantally, causng the t-statstcs to approach zero. 3 We are concerned that the Shanken adjustment bases the test results too much aganst the Fama-French factors. Because of the way our hgher-order rght-hand sde varables are created, the adjustment appears to be napproprate for specfcatons that nclude such varables. Omttng the Shanken adjustment appears to nflate the explanatory power of the Fama-French factors and bas our tests aganst the Rubnsten hypothess. 4 We perform an addtonal robustness test to nvestgate further EIV concerns n our test specfcatons. To test for senstvty to the estmaton wndow, we use the orgnal weekly and monthly return data to generate two dfferent sets of loadng estmates wth the return data one from the early part of the rollng estmaton wndow (years t-5 to t-3) and the other from return less the 30-day Treasury Bll yeld (RMRF) for the covarance factor, and RMRF rased to the -1 power for the hgher-order co-factors. 3 The ncrease n standard errors s due to the mechancs of the adjustment. The covarance matrx, Z, s estmated as F *F, where F s the matrx of demeaned factors. Snce the hgherorder factors n F are estmates as excess portfolo returns rased to hgher powers, they have very small values. When they are multpled aganst each other, the elements of the Z matrx become even smaller as hgher-order factors are added. Snce the multplcatve factor n the adjustment ncludes the nverse of Z, the adjusted covarance matrx becomes extremely large as hgher-order factors are added. For the sze-sorted weekly return nterval, for example, the unadjusted t-statstc on the s coeffcent wth the Fama-French specfcaton s 6.28 whle the Shanken-adjusted t-statstc s Wth co-moments 3 through 10 ncluded, the unadjusted t- statstc s 1.48 whle the respectve adjusted t-statstc drops to The other test specfcatons experence smlar large ncreases n the Shanken-adjusted standard errors. 4 We also consdered the Km (1995, 1997) EIV correcton method, whch s defned for a setup where one of the regressors s measured wth error (e.g., beta) and the other rght-hand sde varables are error-free. Our experment setup, however, nvolves multple rght-hand sde varables measured wth error (the SMB and HML loadngs, as well as beta and the hgherorder co-moments, are all estmates). Extendng the EIV correcton procedure to a multvarate framework appeared to be suffcently problematc that we were unable to account explctly for the correcton. 13

15 the late part of the rollng estmaton wndow (years t-2 to t-1). We then use the two addtonal sets of rght-hand sde varables to re-estmate the regressons n the paper: once wth the loadngs based on the early-wndow returns and once wth the loadngs based on the latewndow returns. If our results are based on errors n the estmates, we would expect that usng two unque sets of rght-hand sde varables based on the dfferent estmaton wndows should provde a bass to test for the mportance of such bas. We fnd that the coeffcent estmates and pattern n test statstcs are roughly consstent across dfferent estmaton perods. In no case are the results substantally dfferent from what s reported n the paper. The ablty of the F-F factor loadngs to mantan explanatory power does not appear to be senstve to the rghthand sde varable estmaton wndow. We therefore conclude that although our estmates are lkely to nclude some EIV bas, we do not beleve that the bas s large enough to negate our overall conclusons. F. Robustness Checks So far, all the returns have been computed dscretely. We tred contnuous returns. Ths change caused the skewness to change sgns, but otherwse the results were smlar: Normalty can be rejected, and F-F factors start out sgnfcant but become nsgnfcant when co-moments (but not moments) are ncluded. We also tred the CRSP equal-weghted portfolo as the market proxy and obtaned smlar results. As another robustness check, we sorted stocks based on momentum (performance over the prevous sx months). The results were very smlar to those for book-to-market sortng. We also looked at January returns, and got smlar results. The Kolmogorov statstc strongly rejects normalty for January returns. Introducng co-moments gradually reduces the t-statstc for s from 7 (Table 1, Panel C) to wth co-moments 3 through 10 ncluded. 14

16 Introducng moments nstead of co-moments s a bt less successful at reducng the t-statstc (t=2.675 wth moments 3 through 10 ncluded). If we had found a case where returns were normal, we would have expected the F-F factors to be nsgnfcant. Note that the converse need not hold: Insgnfcance of the F-F factors does not mply returns are normal. We also tested f returns are lognormal. Because contnuous-compoundng factors are exponentals, contnuous returns are addtve, so we mght expect the Central Lmt Theorem, under certan assumptons, to hold. Then contnuous returns would be asymptotcally normal, and dscrete returns would be asymptotcally lognormal. However, we found we could reject lognormalty as well as normalty. We also tred sortng on the Kolmogorov statstc for normalty. One mght suppose that the sub-sample for whch normalty cannot be rejected would obey the CAPM, but ths s not what we found. Perhaps nvestors dd not expect ths sub-sample to exhbt normalty ex ante and so the F-F factors were prced anyway. We note that, whereas our results seem to be robust, the F-F factors seem to lack robustness: one or the other s nsgnfcant n almost every case (Table 1) and addng comoments causes the F-F factors n most cases to become nsgnfcant (Table 3). Fnally, note that, for some cases, the coeffcents for SMB and HML do not change much as co-moments are added: The t-statstcs declne because the estmated standard devatons ncrease. Thus, one may be tempted to argue that our results are drven by mprecson n estmaton caused by addng co-moments. However, two facts strongly suggest ths s not the case. Frst, for the longer ntervals, many of the coeffcents (SMB and HML for quarterly and sem-annual) change sgns. Second, usng moments nstead of co-moments does not lead to a declne n sgnfcance levels. 15

17 IV. Summary and Conclusons We can reject normalty of returns for daly, weekly, monthly, quarterly and semannual ntervals. In the absence of normalty, nvestors should be very concerned wth the shape of the tals of the dstrbuton of portfolo returns, whch can be measured wth a set of hgher-order co-moments. Our results suggest that the Fama-French factors proxy for hgherorder co-moments, as the F-F loadngs generally become nsgnfcant when hgher-order systematc co-moments are ncluded n cross-sectonal regressons for portfolo returns. Thus, we fnd evdence for a model of the sort gven n Rubnsten (1973),.e., measurng rsk requres more than just measurng covarance. Hgher-order co-moments matter to rsk-averse nvestors concerned about extreme outcomes. It s theoretcally satsfyng to have a logcal explanaton for the success of SMB and HML at explanng securty returns. In prncple, we would expect hgher-order co-moments to be better for such practcal matters as measurng portfolo performance. However, t s concevable that the SMB and HML loadngs are such good proxes for the hgher-order comoments that, gven problems of estmatng hgher-order co-moments, the Fama-French factors could be superor n actual use. 16

18 References Ane, T., and Geman, H Order flow, transacton clock, and normalty of asset returns. Journal of Fnance 55: Aparco, F., and Estrada, J Emprcal dstrbutons of stock returns: European securtes markets, European Journal of Fnance 7:1-21. Berk, J A crtque of sze related anomales. The Revew of Fnancal Studes 8: Bonsal, R., and Vswanathan, S No arbtrage and arbtrage prcng. Journal of Fnance 48: Chan, L.; Karcesk, J.; and Lakonshok, J The rsk and return from factors. Journal of Fnancal and Quanttatve Analyss 33: Danel, K., and Ttman, S Evdence on the characterstcs of cross sectonal varaton n stock returns. Journal of Fnance 52:1-34. Dttmar, R Nonlnear prcng kernels, kurtoss preference, and evdence from the crosssecton of equty returns. Journal of Fnance 57: Fama, E., and French, K Common rsk factors n the returns on stocks and bonds. Journal of Fnancal Economcs 33:3-56. Fama, E., and French, K Sze and book-to-market factors n earnngs and returns. Journal of Fnance 50: Fama, E., and French, K Multfactor explanatons of asset prcng anomales. Journal of Fnance 51: Fama, E., and MacBeth, J Rsk, return and equlbrum: emprcal tests. Journal of Poltcal Economy 81: Ferguson, M., and Shockley, R Equlbrum Anomales. Journal of Fnance 58: Ferson, W.; Sarkssan, S.; and Smn, T The alpha factor asset prcng model: a parable, Journal of Fnancal Markets 2: Frend, I., and Westerfeld, R Co-skewness and captal asset prcng. Journal of Fnance 35: Harvey, C., and Sddque, A Condtonal skewness n asset prcng tests. Journal of Fnance 55:

19 Hung, D. C.; Shackleton, M.; and Xu, X CAPM, hgher co-moment and factor models of UK stock returns. Journal of Busness, Fnance, and Accountng. Forthcomng. Jagannathan, R., and Wang, Z An asymptotc theory for estmatng beta-prcng models usng cross-sectonal regresson. Journal of Fnance 53: Kan, R., and Wang, K Nonlnear APT versus the condtonal CAPM: an emprcal comparson. Unversty of Toronto. Workng paper. Kem, D Sze-related anomales and stock return seasonalty. Journal of Fnancal Economcs 12: Km, D The errors n the varables problem n the cross-secton of expected stock returns. Journal of Fnance 50: Km, D A reexamnaton of frm sze, book-to-market, and earnngs prce n the crosssecton of expected stock returns. Journal of Fnancal and Quanttatve Analyss 32: Knez, P., and Ready, M On the robustness of sze and book-to-market n cross-sectonal regressons. Journal of Fnance 52: Kothar, S. P.; Shanken, J.; and Sloan, R Another look at the cross-secton of expected returns. Journal of Fnance 50: Kraus, A., and Ltzenberger, R Skewness preference and the valuaton of rsk assets. Journal of Fnance 31: Lakonshok, J.; Shlefer, A.; and Vshny, R Contraran nvestment, extrapolaton, and rsk. Journal of Fnance 49: Lm, K A new test of the three-moment captal asset prcng model. Journal of Fnancal and Quanttatve Analyss 24: Lndley, D. V A statstcal paradox. Bometrka 44: Mandelbrot, B The varaton n certan speculatve prces. Journal of Busness 36: Perez-Quros, G. and Tmmermann, A Frm sze and cyclcal varatons n stock returns. Journal of Fnance 55, Rolph, D. S Co-skewness, frm-level equty returns and fnancal leverage. Seattle Unversty. Workng paper, Rubnsten, M The fundamental theorem of parameter-preference securty valuaton, Journal of Fnancal and Quanttatve Analyss 8:

20 Scott, R. and Horvath, P On the drecton of preference for moments of hgher order than the varance. Journal of Fnance 35: Sears, R. S., and We, K Asset prcng, hgher moments, and the market rsk premum: a note. Journal of Fnance 40: Shanken, J On the estmaton of beta-prcng models. The Revew of Fnancal Studes 5:

21 TABLE 1 Fama-MacBeth regresson results for the Fama-French Model Panel A. Sze-sorted portfolos Return frequency Perods b s h Mean adj r 2 [Jont test] Daly ** (6.66) ** (4.37) (-0.74) [20.0 ** ] Weekly ** (13.57) ** (6.28) ** (-5.20) [59.5 ** ] Monthly ** (8.35) * (1.68) ** (-6.82) [57.9 ** ] Quarterly ** (6.11) (0.78) ** (-7.29) [57.1 ** ] Sem-annual ** (3.56) (-0.50) ** (-5.01) [25.3 ** ] Panel B. Book-to-market-rato-sorted portfolos Return frequency Perods b s h Mean adj r 2 [Jont test] Daly ** (10.90) ** (2.81) ** (-7.68) [77.0 ** ] Weekly ** (15.70) (-1.91) ** (-13.96) [201.0 ** ] Monthly ** (8.93) (-0.36) ** (-16.07) [283.6 ** ] Quarterly ** (5.47) (0.47) ** (-11.72) [137.5 ** ] Sem-annual ** (6.37) (1.08) ** (-8.77) [71.2 ** ] 20

22 TABLE 1 (Contnued) Fama-MacBeth regresson results for the Fama-French Model Panel C. Monthly returns of sze-sorted portfolos by calendar month Month Perods b s h January (-1.25) February (0.90) March ** (3.01) Aprl (1.17) May ** (2.52) June ** (2.61) July ** (2.39) August ** (3.33) September ** (2.38) October ** (2.91) November ** (5.25) December ** (4.98) ** (7.00) (1.38) (-0.09) (-0.79) (0.06) (-2.26) (0.38) (0.30) (1.18) (-2.23) (-0.46) (-1.48) (3.20) * (-1.91) ** (-2.70) ** (-2.50) (-0.72) (-0.99) ** (-2.72) * (-2.08) * (-2.05) ** (-3.78) ** (-5.05) ** (-4.94) The sample conssts of daly, weekly, monthly, quarterly, and sem-annual returns of 50 equal-szed portfolos. In Panel A and C, the portfolos are sorted by begnnng-of-perod sze and contan all CRSP lsted ordnary common equtes from 1930 to 1998 for the monthly, quarterly, and sem-annual return ntervals and from 1965 to 1998 for the daly and weekly return ntervals. In Panel B, the portfolos are sorted by begnnng-of-perod book-to-market rato and contan all CRSP and Compustat-lsted ordnary common equtes from 1970 to 1998 for all return ntervals. For each calendar perod, portfolo returns are regressed on three factor loadngs: b, s, and h. These loadngs are computed by regressng portfolo returns over the past fve years on the market, SMB, and HML factors, respectvely. SMB represents the return on a portfolo of small stocks less the return on a portfolo of large stocks whle HML represents the return on a portfolo of hgh book-to-market-value stocks less the return on a portfolo of low book-to-market-value stocks. The mean coeffcent estmates across the sample perod are reported wth ther t-statstcs. The jont test s a Wald test of the jont sgnfcance of the s and h estmates. * and ** denote one-tal sgnfcance at the 5-percent and 1-percent level, respectvely. 21

23 TABLE 2 Summary statstcs of portfolo returns Panel A. Sze-sorted portfolos Daly Weekly Monthly Quarterly Sem-annual Number of portfoloperod observatons 428,050 88, ,393 13,798 6,899 Mean Varance Skewness Kurtoss Jarque-Bera statstc ** ** ** ** Kolmogorov statstc ** ** ** ** ** Panel B. Book-to-market-rato-sorted portfolos Daly Weekly Monthly Quarterly Sem-annual Number of portfoloperod observatons 353,750 75,700 17,400 5,800 2,900 Mean Varance Skewness Kurtoss Jarque-Bera statstc ** Kolmogorov statstc ** ** ** ** ** The sample conssts of daly, weekly, monthly, quarterly, and sem-annual returns of 50 equal-szed portfolos. In Panel A, the portfolos are sorted by begnnng-of-perod sze and contan all CRSP lsted ordnary common equtes from 1930 to 1998 for the monthly, quarterly, and sem-annual return ntervals and from 1965 to 1998 for the daly and weekly return ntervals. In Panel B, the portfolos are sorted by begnnng-of-perod book-to-market rato and contan all CRSP and Compustat-lsted ordnary common equtes from 1970 to 1998 for all return ntervals. * and ** denote sgnfcance at the 5-percent and 1-percent level, respectvely. 22

24 TABLE 3 Fama-MacBeth regresson results for Fama-French (F-F) factors and systematc co-moments Panel A. Sze-sorted portfolos Daly (perods=8561) Weekly (perods=1774) Monthly (perods=828) Quarterly (perods =276) Sem-annual (perods=138) System. comoments s h adj r 2 [Jont] s h adj r 2 [Jont] s h adj r 2 [Jont] s h adj r 2 [Jont] s h adj r 2 [Jont] 2 nd to 3 rd.0006 ** (7.14) (-1.00) [3.77 * ] ** (8.06) ** (-2.67) [69.0 ** ] ** (3.83) * (-1.70) [19.0 ** ] (0.24) (-0.83) [2.06] (-1.30) (-1.48) [3.45] 2 nd to 4 th.0007 ** (6.96) * (-1.98) [1.96] ** (4.75) ** (-3.63) [30.8 ** ] * (1.89) * (-1.89) [3.23] (0.69) (-0.51) [4.04] (0.31) * (-2.28) [3.64] 2 nd to 5 th.0007 ** (6.79) * (-1.86) [1.79] ** (5.07) ** (-3.63) [33.0 ** ] (-0.38) (-0.86) [0.78] (-0.05) (0.39) [2.46] (-2.19) * (-1.92) [8.05 * ] 2 nd to 6 th.0008 ** (6.23) * (-1.96) [1.64] ** (4.78) * (-1.99) [22.7 ** ] (0.26) (-0.53) [0.44] (-0.30) (2.19) [1.50] (-0.01) (0.28) [1.97] 2 nd to 7 th.0008 ** (5.27) (-1.49) [1.54] ** (4.59) (-1.49) [19.3 ** ] (-0.25) (-0.39) [1.19] (-0.90) ( [0.51] (-1.23) (-0.87) [5.88] 2 nd to 8 th.0007 ** (3.61) (-1.56) [1.53] ** (3.03) (-0.78) [8.92 * ] (-0.43) (-0.33) [3.27] (-0.58) (0.57) [0.11] (-0.62) (-0.61) [2.26] 2 nd to 9 th.0005 * (2.33) * (-1.98) [1.50] ** (2.50) (-0.93) [6.59 * ] (0.39) (-0.18) [0.10] (-1.15) (1.53) [1.03] (0.22) (-1.12) [2.63] 2 nd to 10 th.0007 ** (3.13) (-1.07) [1.47] (1.48) (-0.26) [2.85] (0.69) (-0.30) [0.37] (-1.81) (2.09) [3.13] (0.65) * (-1.68) [2.36]

25 TABLE 3 (Contnued) Fama-MacBeth regresson results for Fama-French (F-F) factors and systematc co-moments Panel B. Book-to-market-rato-sorted portfolos Daly (perods=7075) Weekly (perods=1514) Monthly (perods=348) Quarterly (perods =116) Sem-annual (perods=58) System. comoments adj r 2 adj r 2 adj r 2 adj r 2 adj r 2 s h [Jont] s h [Jont] s h [Jont] s h [Jont] s h [Jont] 2 nd to 3 rd.0009 ** (5.27) ** (-7.26) [1.93] ** (4.09) ** (-7.57) [72.7 ** ] ** (2.83) ** (-6.15) [50.0 ** ] (-2.56) ** (-4.38) [31.7 ** ] (-2.07) (-1.01) [6.44 * ] 2 nd to 4 th.0010 ** (5.65) ** (-6.68) [1.73] ** (2.78) ** (-5.91) [33.4 ** ] (1.53) ** (-3.97) [16.3 ** ] (-1.09) ** (-3.76) [23.4 ** ] (-0.81) (-0.92) [2.62] 2 nd to 5 th.0009 ** (4.66) ** (-5.27) [1.64] * (1.71) ** (-3.43) [13.7 ** ] (-0.65) ** (-3.13) [17.2 ** ] (-2.68) (-1.29) [19.0 ** ] (-1.92) (-0.95) [5.15] 2 nd to 6 th.0010 ** (4.84) ** (-2.88) [1.58] (1.62) * (-1.99) [5.05] (-0.94) (-0.44) [5.54] (-1.66) (0.32) [9.40 ** ] (-1.70) (0.20) [4.18] 2 nd to 7 th.0004 * (1.94) ** (-3.33) [1.45] (1.51).0002 (0.20) [1.04] (-1.33) (-0.21) [7.32 * ] (-2.46) (0.63) [13.9 ** ] (-2.16) (0.34) [9.29 ** ] 2 nd to 8 th.0006 * (2.28) ** (-3.47) [1.36] (0.56) (-0.08) [0.17] (-0.84) (-0.16) [6.04 * ] (0.04) (-0.98) [2.55] (-2.26) (-0.66) [4.07] 2 nd to 9 th.0006 * (2.30) ** (-2.38) [1.35] (0.23) (1.37) [1.54] (-0.83) (0.94) [0.69] (-1.47) (1.21) [2.49] (-2.17) (0.96) [5.43] 2 nd to 10 th.0008 ** (2.72) (-1.57) [1.29] (0.41) (1.41) [0.03] (-0.23) (0.20) [1.11] (-1.52) (1.17) [1.18] (-2.03) (0.65) [10.2 ** ] The sample conssts of daly, weekly, monthly, quarterly, and sem-annual returns of 50 equal-szed portfolos. In Panel A, the portfolos are sorted by begnnngof-perod sze and contan all CRSP lsted ordnary common equtes from 1930 to 1998 for the monthly, quarterly, and sem-annual return ntervals and from 1965 to 1998 for the daly and weekly return ntervals. In Panel B, the portfolos are sorted by begnnng-of-perod book-to-market rato and contan all CRSP and Compustat-lsted ordnary common equtes from 1970 to 1998 for all return ntervals. The F-F factor loadngs s and h are computed from the three-factor model estmates usng portfolo returns over the past fve years. The systematc co-moments are estmates usng the same rollng fve-year portfolo return data wth noncentered return data. For each perod the portfolo return s regressed on the F-F loadngs and the respectve number of systematc co-moments. The mean coeffcent estmates across the sample perod are reported. The jont test s a Wald test of the jont sgnfcance of the s and h estmates. * and ** denote one-tal sgnfcance at the 5-percent and 1-percent level, respectvely, wth respect to the emprcal sgn on SMB and HML generated n the tests n Table 1. 24

26 TABLE 4 Fama-MacBeth regresson results for Fama-French (F-F) factors and standard moments of order 3 through 10 Return frequency Sze-sorted portfolos s h Mean adj r 2 [Jont test] Book-to-market-rato-sorted portfolos s h Mean adj r 2 [Jont test] Daly ** (4.23) ** (-2.74) [1.69] (-0.57) ** (-5.33) [31.5 ** ] Weekly ** (4.18) ** (-4.70) [31.5 ** ] (-1.01) ** (-8.39) [83.5 ** ] Monthly ** (4.56) * (-1.97) [18.5 ** ] (0.13) ** (-7.98) [45.8 ** ] Quarterly ** (2.48) (0.43) [7.59 * ] (0.38) ** (-6.25) [18.4 ** ] Sem-annual ** (2.50) ** (-2.68) [14.4 ** ] (-1.23) ** (-4.01) [16.6 ** ] The sample conssts of daly, weekly, monthly, quarterly, and sem-annual returns of 50 equal-szed portfolos. As denoted, the portfolos are sorted n two ways. The sze-sorted portfolos are sorted by begnnng-of-perod sze and contan all CRSP-lsted ordnary common equtes from 1930 to 1998 for the monthly, quarterly, and sem-annual return ntervals and from 1965 to 1998 for the daly and weekly return ntervals. The book-tomarket-rato-sorted portfolos are sorted by begnnng of perod book-to-market rato and contan all CRSP and Compustat lsted ordnary common equtes from 1970 to 1998 for all return ntervals. The F-F factor loadngs s and h are computed from the three-factor model estmates usng portfolo returns over the past fve years. The standard moments are estmates usng the same rollng fve-year portfolo return data. For each perod the portfolo return s regressed on the F-F factor loadngs, beta, and moments three through ten. The mean coeffcent estmates across the sample perod are reported. The jont test s a Wald test of the jont sgnfcance of the s and h estmates. * and ** denote one-tal sgnfcance at the 5-percent and 1-percent level, respectvely. 25