Downside risk aversion, fixed income exposure, and the value premium puzzle

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1 Downsde rsk averson, fxed ncome exposure, and the value premum puzzle GUIDO BALTUSSEN, THIERRY POST AND PIM VAN VLIET* Ths verson: 13 November ABSTRACT The value premum substantally reduces for downsde rsk averse nvestors wth a substantal fxed ncome exposure. Growth stocks are attractve to these nvestors because they offer a good hedge aganst a bad bond performance. Ths result holds for evaluaton horzons of around one year. Our fndngs cast doubt on the practcal relevance of the value premum for nsttutonal nvestors such as nsurance companes and penson funds, and reterates the mportance of the choce of the relevant test portfolo, rsk measure and nvestment horzon n emprcal tests of market effcency and equlbrum. Key words: downsde rsk, sem-varance, nterest rates, fxed ncome, value premum, asset prcng, behavoral fnance JEL: G11, G1 * Baltussen s at the Department of Fnance, Stern School of Busness, New York Unversty, and the Erasmus School of Economcs, Erasmus Unversty of Rotterdam. Post s at the Cass Busness School of the Cty Unversty, London, and the Unversty of Wales at Bangor, Van Vlet s at Robeco Asset Management. E-mal: gbaltuss@stern.nyu.edu, gtpost@hotmal.com, p.van.vlet@robeco.nl. Fnancal support by Tnbergen Insttute, Erasmus Research Insttute of Management, Erasmus Trustfonds, Erasmus Center of Fnancal Research and Robeco s gratefully acknowledged. We thank Kenneth French, Stefan Nagel, Laurens Swnkels, and Wel Zhou for useful comments and suggestons. Any remanng errors are our own.

2 Downsde rsk averson, fxed ncome exposure, and the value premum puzzle ABSTRACT The value premum substantally reduces for downsde rsk averse nvestors wth a substantal fxed ncome exposure. Growth stocks are attractve to these nvestors because they offer a good hedge aganst a bad bond performance. Ths result holds for evaluaton horzons of around one year. Our fndngs cast doubt on the practcal relevance of the value premum for nsttutonal nvestors such as nsurance companes and penson funds, and reterates the mportance of the choce of the relevant test portfolo, rsk measure and nvestment horzon n emprcal tests of market effcency and equlbrum. The value premum refers to stocks whch have a hgh fundamental value relatve to ther market value (value stocks) earnng hgher average stock returns than stocks wth a relatvely low fundamental value (growth stocks). Ths fndng manfests tself n several forms. For example, stocks of frms that rank hgh on earnngs-to-prce rato (E/P; Basu, 1977, 193, Jaffe, Kem and Westerfeld, 199), debt-equty rato (D/E; Bhandar, 19), dvdend-to-prce rato (D/P; Kem, 193), cash flow-to-prce rato (C/P; Chan, Hamao, and Lakonshok, 1991, Lakonshok, Shlefer and Vshny, 199), and rato of book value of common equty to market value of common equty (B/M; Rosenberg, Red and Lansten, 195, De Bondt and Thaler, 197) perform hstorcally substantally better than frms that rank low on these measures. 1 Moreover, Fama and French (199, 1993) show that the CAPM (Sharpe, 19, Lntner 195, Mossn, 19) cannot explan the value premum, snce value stocks have hgher average returns than growth stocks, but do not have hgher equty betas. Furthermore, the effects do not seem to be the result of data mnng, as suggested by Black (1993) and Lo and Macknlay (199), and are manfested n multple countres and sub-perods. 3 Thus, there seems to be convncng and robust evdence that 1 See Chan and Lakonshok () and Lettau and Wachter (7) for a recent update of ths evdence. Fama and French (1993) document two common factors n the returns of stocks sorted on sze and B/M that are unrelated to the market return (SMB and HML). Fama and French (199) show that these sze and value factors explan the strateges based on E/P, B/M, fve-year sales growth, and three to fve year past returns documented by De Bondt and Thaler (195). In addton, Fama and French (1995) fnd that the SMB and HML factors can partly be traced to common factors n the earnngs and sales of frms. 3 For example, Chan, Hamao and Lakonshok (1991) fnd a value effect n the Japanese stock market,dmson, Nagel and Qugley (3) n the UK market. Also, Capaul, Rowley and Sharpe (1993), Fama and French (199, a) and Grffn () provde further nternatonal evdence. In addton, Davs (199), Davs, Fama and French () show a value premum for the pre-193 perod. Barber and Lyon (1997) fnd a value premum usng fnancal frms only, whle Km (1997) fnds a value premum after correctng for the possble selecton bases of Compustat data. Furthermore, Basu (193), Jaffe, Kem and Westerfeld (199) and Fama and French (199) fnd that the value premum to be dfferent from the sze premum documented by Banz (191). 1

3 nvestors can enhance ther portfolos by overweghtng value stocks and underweghtng growth stocks. Ths evdence consttutes a major challenge to advocates of market effcency and passve nvestment strateges. Ths study challenges the evdence for the value premum by questonng three mantaned assumptons n the emprcal tests. Frst, these studes generally compare the performance of value and growth portfolos relatve to an all equty market portfolo. However, a substantal part of an nvestor s portfolo s lkely to be ted up n fxedncome nstruments, or assets that are hghly correlated to fxed-ncome nstruments. For nstance, consumer loans and mortgages represent clams to resdental real estate, consumer durables and human captal, household assets that consttute an mportant part of the total portfolo of many nvestors. Fgure 1 shows that large nsttutonal nvestors lke nsurance companes and penson funds nvest heavly n fxed ncome nstruments. In prncple, ths concern can be addressed by addng fxed ncome assets to the benchmark or market portfolo. For nstance, Stambaugh (19) and Shanken (197) found that for beta, ndustry and sze sorted portfolos nferences about the CAPM are ndependent of the ncluson of bonds n the market ndex. Stll, they dd not nclude value sorted portfolos n the analyss and also ponted out that nferences about asset prcng theory crtcally depend on the test assets ncluded. However, the ncluson of bond returns n the benchmark portfolo can have substantal effects on the especally value sorted portfolos as bond returns and ts predctors correlate less wth growth stocks than wth value stocks and tend to predct better growth returns f bonds are expected to perform relatvely bad (see among others Baker and Wurgler,, and Kojen, Lustg and Van Newerburgh, ). [Insert Fgure 1 about here] Second, studes of the value premum tend to assume, ether mplctly or explctly, that nvestors equate rsk wth varance. A well-known lmtaton of varance s that t places the same weght on upward and downward devatons. However, already n hs semnal work on the mean-varance portfolo theory Markowtz (1959) suggested that nvestors are only concerned wth losses and that sem-varance may be a better measure of rsk than varance. In fact, n hs Nobel Lecture Markowtz (1991, Although Loughran (1997) suggest that the value premum s less mportant for money managers, snce; () t s only present n the smallest frms whch represent % of the total market value, () s drven by a January seasonal for large frms, and () exceptonally low returns on small young growth stocks outsde January, whch are hard to short. However, Fama and French (a) fnd that a value premum s present n both small and large frm portfolos sorted on E/P and n nternatonal value sorted portfolos of stocks. Moreover, the bad performance of small growth stocks n the B/M sort are manly due to bad performance of small frms wth negatve earnngs.

4 p. 7) ponts out that: Sem-varance seems more plausble than varance as a measure of rsk, snce t s concerned only wth adverse devatons. Ths conjecture s supported by numerous psychologcal works on the way people perceve and deal wth rsk, rangng from students to busness managers and professonal nvestors. For nstance, n ther revew on many of these studes Slovc (197) and Lbby and Fshburn (1977) conclude that varance seems to be a bad descrptve measure of manageral rsk preferences. Instead, a model that trades off expected return wth rsk defned by below target return (lke sem-varance) seems the most approprate. Moreover, Cooley (1977) fnds that nsttutonal nvestors are manly concerned wth downsde rsk. Kahneman and Tversky (1979) and Tversky and Kahneman (1991, 199) show that people care dsproportonably more about losses than gans, a fndng they term loss averson. Furthermore, Ang, Chen and Xng () show the relevance of systematc downsde rsk for the cross-secton of stock returns. In fact, results reported by Petkova and Zhang (5) suggest that downsde rsk averson may especally have a large nfluence on value sorted portfolos f an nvestor s portfolo s also subject to a substantal fxed ncome exposure. Ther results show that value stocks have a hgher downsde beta than growth stocks wth respect to varables known to predct bond returns (as for example documented by Kem and Stambaugh, 19 and Fama and French, 199). Interestngly, downsde rsk averson may also help to explan why a substantal fracton of nvestable wealth s nvested n fxed ncome nstruments, despte the szeable equty premum (see Benartz and Thaler, 1995, Barbers and Huang, 1, Barbers, Huang and Santos, 1, and Berkelaar, Kouwenberg and Post, 5). 5 Thrd, most studes rely on monthly returns to calculate the systematc rsk of value stocks. However, the nvestment horzon of most nvestors s lkely to exceed one month. Bernartz and Thaler (1995) argue that an annual evaluaton perod s most approprate because most fnancal reportng takes place on an annual bass (e.g. fnancal statements, tax fles, update retrement accounts). Interestngly, Campbell and Vcera (5) show that the varance and correlaton structure of real returns can change dramatcally across nvestment horzons. For example, they fnd that meanreverson n stock returns decreases the volatlty per perod of real stock returns at long horzons, whle renvestment rsk ncreases the volatlty per perod of real T-bll returns. We wll study the senstvty of the value premum to these mantaned assumptons. To examne the role of fxed ncome exposure, we consder varous hypothetcal mxtures of equty and fxed ncome as well as the actual fxed ncome exposures of nsttutonal nvestors. To account for downsde rsk, we use the mean- 5 In fact, Barbers and Huang (1) show that the value premum naturally emerges n an economy n whch nvestors are; () loss averse, () less rsk averse after gans and more rsk averse after losses, and () care about fluctuatons n the outcomes of each asset held (nstead fluctuatons n ther portfolo). By contrast, we wll study the mportance of the value premum for ratonal nvestors that only care about downsde fluctuatons n ther portfolo, whle havng a certan fxed ncome exposure. 3

5 sem-varance crteron (see for example, Mao, 197, and Hogan and Warren, 197), as well as non-parametrc stochastc domnance crtera, and compare ts performance wth the classcal mean-varance crteron (see for example, Markowtz, 1959). Fnally, to study the effect of the nvestment horzons we consder horzons varyng from one month to two years. Table I gves a frst llustraton of our fndngs. Panel A shows the returns n the three worst years for equtes: 1973, 197 and, years durng whch the stock market plummeted by more than %. A rsk-averse all equty nvestor would want to hedge aganst such losses by holdng stocks that perform relatvely well durng such years. However, growth stocks performed worse than value stocks durng these crtcal years; the HML return n 1973, 197 and was 11.7% on average. Ths demonstrates the dffculty of ratonalzng the value premum for a rsk-averse all equty nvestor. Panel B shows the returns n the three worst years for bonds: 199, 1979 and 19, years durng whch bonds lost more than 1% of ther real value. Stocks generally performed well durng these years, lmtng the losses for nvestors who mx stocks and bonds, and llustratng the advantages of dversfcaton over asset classes. Interestngly, growth stocks performed substantally better than value stocks dd durng these years; the average HML return over 199, 1979 and 19 was -11.3%. Clearly, growth stocks offer a better hedge aganst a bad bond performance than value stocks. [Insert Table I about here] Ths study wll show that the value premum s severely reduced for nvestors wth substantal bond exposures (larger than about %), an averson to downsde rsk, and an medum evaluaton horzon (of around one year). For -9% fxed ncome exposure the spread n CAPM alpha between value and growth stocks s reduced from.% to 5.%-3.%. Furthermore, the assumpton that nvestors care only about downsde rsk reduces the value premum agan by about two percent to 1.% and becomes statstcally nsgnfcant. These results hold for evaluaton horzons of sx to 1 months, whle the value premum s unaffected for shorter evaluaton horzons. The results are robust to a number of factors, such as the use of actual portfolo weghts of nsttutonal nvestors, the use of the relevant data sets and sortng procedure, and the precse specfcaton of the downsde rsk measure. These fndngs cast doubt on the practcal relevance of the value premum for nvestors wth a substantal fxed ncome exposure. In fact, growth stocks are attractve because they offer a better hedge aganst a bad bond performance than value stocks do. Our fndngs also have a number of other nterestng mplcatons. Frst, our results demonstrate the effect of non-normal asset returns and the need to nclude rsk measures that dffer from varance. Levy and Markowtz (1979) report that the mean-

6 varance crteron generally gves a good approxmaton for general expected utlty maxmzers. By contrast, we demonstrate that the mean-varance crteron and the mean-sem-varance crteron gve very dfferent results for value sorted portfolos. Second, the sgnfcant effect of addng fxed ncome nstruments to the analyss contrasts wth the robustness reported by Stambaugh (19) and Shanken (197) and reterates the mportance of Roll s (1977) crtque and the choce of the relevant test portfolo. Thrd, our results reveal the mportance of the nvestment horzon used to study portfolo effcency. The remander of ths paper s structured as follows. Secton I ntroduces prelmnary notaton, assumptons and concepts. Secton II and Secton III subsequently dscuss our emprcal methodology and data set, respectvely. Next, Secton IV dscusses the test results and the robustness wth respect to the data set and methodology. Fnally, Secton V gves concludng remarks and suggestons for further research. I. Theoretcal framework It s hard to fnd assets that provde rskless long-term real returns. For example, even one-month T-blls yelded real returns of less than -3% n 197 and 1979 due to unexpectedly hgh nflaton. Nowadays, Treasury Inflaton-Protected Securtes (TIPS) promse rskless real yelds-to-maturty. However, such nstruments have been ntroduced n the US as late as 1997 and were not avalable durng the largest part of our sample perod (193-7). Also, the TIPS market remans relatvely llqud n terms of outstandng amounts and tradng actvty. For ths reason, we analyze portfolo effcency wthout a rskless asset. We consder a smple sngle-perod, portfolo-based model of nvestment n a perfect captal market. The nvestment unverse conssts of N rsky assets. The returns x x... 1 xn and are treated as random varables wth jont cumulatve dstrbuton functon G : P N [,1], where the doman P R s to the rsky assets are denoted by ( ) T nonempty, closed and convex. Investors may dversfy between the assets, and the N portfolo possbltes are represented by the polyhedron Λ { λ R : Τ λ = 1} 1. The evaluated portfolo s denoted by τ Λ. Investors choose nvestment portfolos to maxmze the expected value of an ncreasng and concave utlty functon u : P R that s defned over the return of ther N N Throughout the text, we wll use R for an N-dmensonal Eucldean space, and N R and N R denote the + negatve and postve orthants. To dstngush between vectors and scalars, we use a bold font for vectors and a Τ regular font for scalars. Further, all vectors are column vectors and we use x for the transpose of x. Fnally, N and 1N denote a (1xN) zero vector and a (1xN) unty vector. 5

7 portfolos. The mean-varance nvestor can be represented by the followng standardzed, one-parameter, quadratc utlty functon: 7 u MV ( x, θ ) = (1 θe[ + x x Τ τ ]) x.5θ (1) wth θ for the rsk averson parameter. The utlty functon for the downsde rsk averter, or mean-sem-varance nvestor, s quadratc for losses and lnear for gans: u MS ( x, θ ) = (1 θe[( x Τ Τ x τ )1[ x τ ]]) x +.5θ 1[ x ] () wth θ. Under the above assumptons, the nvestor s optmzaton problem can be summarzed as max λ Λ s. t. 1 Τ u( x λ) dg( x) Τ N λ = 1 u { u MV, u MS } (3) The evaluated portfolo τ Λ s effcent or the optmal soluton for some utlty functon u f and only f the frst-order optmalty condton apples: Τ E[ u ( x τ, θ ) x] = γ 1 () N Τ where γ s the shadow prce of the budget restrcton 1 λ = 1 or the shadow prce of not havng a rskless asset avalable for lendng and borrowng. A negatve shadow prce mples that the nvestor would lke to nvest n a rskless asset (rskless lendng) f such an asset were avalable; a postve shadow prce mples that rskless borrowng s desred. Volatons of the optmalty condton or alphas are defned as N Τ α ( θ, γ ) E[ u ( x τ, θ ) x] γ 1N (5) 7 Τ Ths utlty functon s standardzed such that u (, θ ) = and E [ ( x τ, θ )] = 1. Maxmzng the expectaton of ths utlty functon s equvalent to maxmzng a trade-off between mean E [x] and varance Var[ x] x = E[ x ] E[ ] : E[ u MV ( x, θ )] = {1 θe[ Τ τ ] +.5θE[ x]} E[ x] +.5θVar[ x] MV u MV x. The varable 1[ x ] s a dummy varable that takes the value 1 f x and else. The utlty functon s Τ standardzed such that u (, θ ) = and E [ ( x τ, θ )] = 1. Maxmzng the expectaton of ths utlty MS u MS functon s equvalent to maxmzng a trade-off between mean and sem-varance Τ Τ SVar [ x] = E[ x 1[ x ]] / Pr[ x ] : E[ u MS ( x, θ )] = {1 θe[ x τ1[ x τ ]]} E[ x] +.5θ Pr[ x ] SVar[ x].

8 Effcency occurs f and only f α ( θ, γ ) =. If ( θ, γ ) >, asset s underweghted N and ts weght n the portfolo should be ncreased relatve to τ n order to acheve effcency. Smlarly, f ( θ, γ ) <, the asset s overweghed and ts weght n the α portfolo should be decreased. We may further reformulate the optmalty condton as the followng trade-off between mean return and beta or systematc rsk of the evaluated portfolo: α E[ x ] γ 1 + ρ( θ ) β ( θ ) () = N wth Τ Τ ρ( θ ) Cov [ u ( x τ, θ ),( x τ )] (7) Τ Cov[ u ( x τ, θ ), x] β ( θ ) Τ Τ Cov[ u ( x τ, θ ),( x τ )] () The varable ρ (θ ) s the rsk premum for every unt of beta rsk. Due to rsk averson ( θ ), margnal utlty s a decreasng functon of the portfolo return and hence the rsk premum s postve, that s, ρ ( θ ). In the case of mean-varance nvestors, we obtan the followng expressons for the rsk premum and the betas: ρ ( ) [ Τ MV θ = θvar x τ ] (9) Τ Cov[ x, x τ ] β MV ( θ ) = (1) Τ Var[ x τ ] In case of downsde rsk averters, the followng expressons apply Τ Τ Τ ρ MS ( θ ) = θcov[( x τ ),( x τ )1( x τ )] (11) Τ Τ Cov[ x,( x τ )1( x τ )] β MS ( θ ) = (1) Τ Τ Τ Cov[( x τ ),( x τ )1( x τ )] The above analyss apples for every sngle-perod, portfolo-orented model of nvestment n a perfect captal market; every nvestor s portfolo needs to be effcent accordng to the effcency crteron assocated wth hs or her preferences over money. 7

9 The model can also be generalzed to an equlbrum model of captal markets. In representatve nvestor models, captal market equlbrum can be descrbed by the optmzaton problem of a sngle, representatve nvestor. In these models, the valueweghted market portfolo s the optmal soluton for the representatve nvestor. Equaton () becomes the equlbrum condton wth τ equal to the relatve market captalzaton of the assets and u (x) equal to the utlty functon of the representatve nvestor. The representatve nvestor s margnal utlty functon u (x) then represents a prcng kernel and the alphas represent prcng errors or devatons from equlbrum. For the mean-varance specfcaton, the equlbrum model s equvalent to Black s (197) zero-beta model wth no lendng and borrowng at the rskless rate of nterest, and for the mean-sem-varance specfcaton, we obtan a zero-beta varant to the equlbrum model by Hogan and Warren (197) and Bawa and Lndenberg (1977). However, we stress the need to be cautous wth market portfolo effcency tests, because relable nformaton about the market value of all captal assets currently s not avalable due to, for example, measurement problems for non-traded assets such as human captal and the problem of double-countng multple fnancal clams on the same underlyng assets (see Roll, 1977). II. Emprcal methodology In practce, we cannot drectly gauge portfolo effcency, because the return dstrbuton of the assets (G) s unknown. However, we can estmate the return dstrbuton usng tme-seres return observatons and employ statstcal tests to determne f effcency s volated to a sgnfcant degree. Throughout the text, we wll represent the observatons by can construct the followng emprcal alphas: Τ x t ( x 1t Lx Nt ), t 1, L, T =. Usng the observatons, we T 1 ˆ α ( θ, γ ) T u ( x τ, θ )x γ 1 (13) t= 1 Τ t t N In the sprt of the Generalzed Method of Moments (Hansen, 19), we can use the followng aggregate procedure to test effcency: JT mnt ˆ( α θ, γ ) Τ Wαˆ( θ, ) (1) γ θ, γ

10 wth W for an approprately chosen weghtng matrx. The JT-statstc thus selects the rsk averson parameter θ and the shadow prce γ that mnmze a weghted average of the squares and cross-terms of the alphas. 9 In ths study, we wll follow the recommendatons of Cochrane (1) and employ an one-stage GMM procedure wth the dentty matrx as weghtng matrx,.e. W = I N. In ths case, mnmzng the JT statstc s equvalent to maxmzng the R- squared of a cross-sectonal regresson between sample means and sample second moments and the estmaton s almost smlar to the classcal cross-sectonal Fama and MacBeth (1973) procedure. Use of the dentty matrx as weghtng matrx nstead of the optmal weghng matrx, or the emprcal covarance matrx of the frst-stage alphas, allows the comparson of our non-nested models and avods the emprcal ptfall of maxmzng the volatlty of the alphas nstead of truly mnmzng the alphas. However, we stress that usng another common pre-specfed weghtng matrx, namely the nverse of the sample second moment matrx of returns proposed by Hansen and Jagannathan (1997), yelds smlar conclusons. 1 In addton to the R-squared, we also report the p-values of each alpha. These p- values requre the emprcal covarance matrx of the alphas, whch may be poorly estmated n our analyss. Ths s caused by the large number of moments relatve to the number of tme seres observaton, makng the estmates of ths matrx possbly unstable. Instead, we compute the p-values by means of 1,99 bootstrap draws of the current sample and calculate the standard errors of the alphas over these bootstrap realzatons. 11 Although, the R-squared s ntutve, t has one potental weakness as model comparson crteron. It gves equal weght to each alpha, even though some assets are more volatle than others. To surmount ths statstcal shortcomng, we follow Campbell and Vuolteenaho () and also compute the followng composte test-statstc: Τ ˆ 1 CV ˆ α( θ, γ ) Ω ˆ( α θ, γ ) (15) Where CV s the Campbell and Vuolteenaho test statstc, ˆ α( θ, γ ) are the estmated alphas for value sorted portfolos, and Ωˆ s a dagonal matrx wth estmated return volatltes on the man dagonal. The CV-test statstc places less weght on more volatle observatons, yet allows a clean model comparson, snce t employs the same weghtng matrx for dfferent models. In addton, t provdes us wth a test on the jont 9 See Cochrane (1) and Jagannathan and Wang () for the effcacy of the GMM procedure, as well as a comparson and equvalence between dfferent GMM, cross-sectonal and tme seres regressons approaches. 1 These results are not tabulated, yet avalable form the authors upon request. 11 However, bootstrappng the t-values or the asymptotc p-value yelds smlar conclusons. More detals are avalable from the authors upon request. 9

11 equalty of all value-sorted-portfolo-alphas to zero. 1 Lke Campbell and Vuolteenaho (), we avod usng a freely estmated varance-covarance matrx of test asset returns for Ωˆ, snce the nverse of ths matrx may be poorly behaved wth a large number of test assets relatve to tme-seres observatons. The p-values for the CV-test statstc are produced by bootstrappng 1,99 observatons from the sample n whch the test asset returns are adjusted to yeld alphas equal to zero, gven the orgnal parameter estmates. The above methodology assumes serally ndependently and dentcally dstrbuted (IID) returns and does not condton on the state-of-the-world. Some studes provde evdence n favor of tme-varyng rsk and tme-varyng rsk averson, and propose condtonal asset prcng models that explan the value premum (see among others, Jagannathan and Wang, 199, Lettau and Ludvgson, 1b, Lustg and Van Neuwerburgh, 5, Petkova and Zhang, 5, and Santos and Verones, ). Ths condtonal rsk based approach typcally measures rsk as the covarance of returns wth margnal utlty of consumpton or returns. Stocks are rsky f they pay out less n bad tmes (n whch the margnal utlty s hgh), and vce versa for good tmes. Unfortunately, condtonal models ental several problems. There s lttle theoretcal gudance for selectng the approprate specfcaton and the results can be very senstve to specfcaton errors (see for example, Ghysels, 199). Furthermore, the models may lack statstcal power due to the use of addtonal free parameters. There s also no guarantee that the model s consstent wth rsk averson and no-arbtrage n all states of the world (see for example, Wang and Zhang, ). Moreover, f a condtonal approach captures the value premum, t s explaned by the co-varaton of value and growth wth a scaled verson of the market return. For example, Lettau and Ludvnson (1b) argue that value stocks earn hgher returns than growth stocks snce the value stocks have a hgher correlaton wth consumpton growth and the market rsk premum n bad tmes, characterzed by a hgh level of ther aggregate consumpton-towealth rato. However, as ponted out by Lewellen and Nagel (), condtonal models are unlkely to explan the value premum for two major reasons. Frst, the co-varaton between the condtonal expected return on the market and the condtonal market betas of value and growth stocks s not hgh enough, and often has the wrong sgn. 1 Another possble test statstc, provded by Cochrane (1, p. ), s: Τ T α ( θ, γ ) [( I ) W ) S ( I Τ 1 Τ Τ 1 Τ Τ 1 ˆ N d ( d Wd ) d N d ( d Wd ) d W ) ] ˆ α ( θ, γ ) where d contans the dervatves of the moment condtons wth respect to the parameters, W = I N, and S ) s the estmated emprcal covarance matrx of the alphas that s sngular and hence has to be pseudo-nverted. Assumng that the tme-seres observatons are serally ndependently and dentcally dstrbuted (IID) random draws, ths test statstc obeys an asymptotc ch-squared dstrbuton wth (N-) degrees of freedom. However, ths test statstc has two serous drawbacks for our analyss. Frst, S ) may be unstable. Second, ths statstc s not comparable across dfferent models, snce the squared alphas are weghted dfferently over varous models (.e. S ) s dfferent for the dfferent models). 1

12 Second, the betas of value stocks ncrease n bad tmes, but by too lttle to generate sgnfcant uncondtonal alphas, a fndng also shown by Petkova and Zhang (5). In fact, the analyss of Lewellen and Nagel () reveals that tme varaton n rsk or rsk prema should have a relatvely small mpact on cross-sectonal asset prcng tests. Stll, the uncondtonal approach wth a mxed market proxy, as employed n ths study, may partly capture possble tme varaton n the rsk premum and/or rsk loadngs of value and growth stocks. For example, Baker and Wurgler () and Kojen, Lustg and Van Newerburgh () show that growth stocks correlate less wth nomnal bond returns and ts predctors. Smlarly, the results reported by Fama and French (199), Ferson and Harvey (1999), Petkova and Zhang (5) and Petkova () suggest that the varables related to good tmes and a relatvely good performance of growth stocks over value stocks, are also closely lnked to a bad performance of fxed ncome nstruments. And precsely these perods could generally be classfed as bad tmes n whch margnal utlty s hgher, especally for an nvestor who nvests substantal amounts of hs portfolo n fxed ncome. We wll explore ths lnk n more detal n Secton IV.G. III. Data We consder yearly real returns on stocks and bonds. 13 As dscussed n Benartz and Thaler (1995, p.3), one year s a plausble choce for the nvestor s evaluaton perod, because ndvdual nvestors fle taxes annually, receve ther most comprehensve reports from ther brokers, mutual funds, and retrement accounts once a year, and nsttutonal nvestors also take the annual reports most serously. Another reason for focusng on annual returns rather than hgher-frequency returns s that hgher-frequency returns are affected by heteroskedastcty and seral correlaton to a sgnfcant degree. These statstcal problems cast doubt on the use of statstcal procedures whch assume serally IID returns (such as the procedure descrbed n Secton II). Heteroskedastcty and seral correlaton also have an mportant economc effect, because nvestors wth an annual nvestment horzon want to be protected especally from a seres of monthly losses that translate nto annual losses. For these reasons, annual returns seem the most approprate choce. Stll, we wll also use monthly returns to nvestgate the monthly return dynamcs that determne the shape of the hgher frequency return dstrbutons. Moreover, we wll also test our fndngs for a range of other return frequences rangng from monthly to b-annual returns. Our sample starts n 193 and ends n 7 (5 annual observatons). There are two reasons for startng n 193 and omttng the pre-193 data. Frst, pror to 193, the Compustat database s affected by survvorshp bas caused by the back-fllng procedure excludng delsted frms, whch typcally are less successful (Kothar, 13 However, the results are not materally affected by usng nomnal returns. The nomnal results are avalable from the author upon request. 11

13 Shanken and Sloan, 1995). Further, from June 19, AMEX-lsted stocks are added to the CRSP database, whch ncludes only NYSE-lsted stocks before ths month. Snce AMEX stocks generally are smaller than NYSE stocks, the relatve number of small caps n the analyss ncreases from June 19. Snce the value effect s most pronounced n the small-cap segment, the post-june-19 data set s most challengng. The nvestment unverse of stocks s proxed by ten value weghted portfolos constructed on B/M. We choose ten portfolos rather than a larger number, because ths guarantees a mnmum number of stocks n every portfolo whle stll havng substantal varaton n returns on value sorted portfolos. We wll demonstrate the robustness of our results to the benchmark set by usng portfolos sorted on E/P and C/P, as well as portfolos constructed at the ntersecton of two groups formed on market captalzaton of equty, or sze, and three groups formed on B/M. 1 Furthermore, n the sprt of Fama and French (1993) we wll employ a hgh-mnus-low hedge portfolo that buys the hghest two value portfolos and shorts the lowest two, to summarze the value effects. Followng Dttmar (), we complete the nvestment unverse by addng a portfolo consstng of one-month Treasury blls, whch has a relatvely low return and beta. Incorporatng the moment condton for ths portfolo n our estmaton procedure enforces the shadow prce to le near the real one-month Treasury bll rate, thereby preventng extreme negatve shadow prces and extremely hgh rsk prema. The stock market portfolo s proxed by the CRSP all-share ndex, a valueweghted average of common stocks lsted on NYSE, AMEX, and NASDAQ. The bond ndex s defned as the average of the Long Term Government bond ndex (LTG), Long Term Corporate bond ndex (LTC) and Intermedate Term Government bond ndex (ITG) mantaned by Ibbotson Assocates. 15 We wll also analyze the robustness of our fndngs wth respect to usng ths partcular ndex. Bond data s obtaned from Ibbotson Assocates, Consumer Prce Index nflaton data from the U.S. department of Labor and the stock portfolo data from Kenneth French s onlne data lbrary. Table II shows some descrptve statstcs for our data set. Partcularly puzzlng are the low returns on growth stocks. The lowest two value sorted stock portfolo earned an average annual real return of.% (.% and 7.%), 5.9% less than the 1.7% (11.% and 13.5%) for the two hghest value sorted portfolo. At frst sght, t seems dffcult to explan away ths premum wth rsk because growth stocks actually have 1 In these sorts, stocks wth negatve B/M, E/P or C/P are excluded. These stocks typcally have hgh returns and hgh market betas. However, ths excluson s unlkely to nfluence our results, because t only nvolves a small number of frms that have a relatvely low market cap (see Jaffe, Kem and Westerfeld, 199, and Fama and French, 199). 15 These bond ndces are constructed as follows; the LTG ndex ncludes U.S. government bonds wth remanng maturty closest to years or longer, the LTC ndex ncludes nearly all U.S. Aaa or Aa rated corporate bonds wth an average maturty of approxmately years, and the ITG ndex ncludes U.S. government bonds wth a remanng maturty closest to 5 years or longer. 1

14 almost the same standard devaton as value stocks. However, as suggested n Table I, growth stocks provde the best hedge durng bad bond market years. [Insert Table II about here] IV. Emprcal results Ths secton dscusses our emprcal fndngs. Secton A frst dscusses the man results for the ten B/M sorted stock portfolos and the one-month Treasury blls, usng annual returns. Next, we wll analyze the robustness of our fndngs wth respect to the use of actual portfolo weghts for four types of nsttutonal nvestors (secton B), the choce of the stock portfolos (Secton C), the return frequency (Secton D), the bond ndex (Secton E), and the choce of parameterzaton (Secton F). Fnally, we wll lnk our fndngs to the lterature on tme-varyng rsk (Secton G). A. Man results Table III summarzes our man results. Panel A shows the mean-varance results for an annual nvestment horzon. Consstent wth exstng evdence, the meanvarance model gves a poor ft for the all equty ndex, wth an alpha of.3% (p-value =.) for the growth portfolo and an alpha of 3.7% (p-value =.) for the value portfolo. The presence of a value premum s captured by the alpha of the VMG hedge portfolo; ts alpha s substantal (.%) and sgnfcantly dfferent from zero (p-value =.). Moreover, the overall R-squared s 7% and the p-value of the CV-test s.. Usng a market proxy wth a substantal fracton nvested n bonds helps to mprove the ft. When bonds represent % of the portfolo, the growth stock alpha falls to -3.3% (p-value =.11), the value stock alpha falls to.% (p-value =.), and the alpha of the value premum portfolo (VMG) becomes 5.1% (p-value =.). The overall R-squared ncreases to 3% and the CV-test statstc falls to.113. Moreover, when bonds represent 9% of the portfolo, the growth stock alpha falls to -.9% (pvalue =.7), the value stock alpha falls to.9% (p-value =.5), and the VMG alpha falls to 3.3% (p-value =.). The overall R-squared ncreases further to % and the CV-test statstc falls to.59 wth an assocated p-value of.9. Stll, some portfolos may have a negatve alpha for a mean-varance 9% bond nvestor. For nstance, the alpha of the second lowest B/M portfolo s -1.97% wth an assocated p-value of.9. 1 As shown n Panel B, the results further mprove for the mean-sem-varance crteron. In lne wth the fndngs of Ang, Chen and Xng () the value premum remans present for an mean-sem-varance all equty nvestor, wtnessng for example 1 Note that the Tbll portfolo s slghtly msprced. However, restrctng the alpha of the Tbll portfolo to equal zero does not materally affect our results. Stll, we choose to present to current results snce we do not want to mpose restrctons on the shadow prce that are not warranted. 13

15 the VMG alpha of 5.7% (p-value=.3). However, wth % nvested n bonds, the growth stock alpha falls to -.77% (p-value =.7), the value stock alpha falls to.9% (p-value =.1), and the alpha of the value premum portfolo (VMG) falls to.59% (pvalue =.1). The overall R-squared becomes 7% and the CV-test p-value.3. When bonds represent 9% of the portfolo, the growth stock alpha falls to -.37% (p-value =.9), the value stock alpha falls to 1.37% (p-value =.5), and the alpha of the value premum portfolo (VMG) falls to 1.5% (p-value =.). The overall R-squared ncreases further to 95% and the CV-test statstc falls to.1 wth an assocated p- value of.99. [Insert Table III about here] Fgure llustrates the same pattern usng the alphas of the VMG hedge portfolo, the R-squared, and CV-test p-value. Clearly, the VMG alpha crtcally depends on the percentage bonds ncluded n the market portfolo. But the choce between the mean-varance and the mean-sem-varance effcency crteron has mportant consequences as well. Roughly, for portfolos n whch bonds consttute % or more of the portfolo, the value premum s severely reduced, and for the mean-semvarance model portfolos n whch bonds consttute to 9% of the portfolo t approaches zero. [Insert Fgure about here] Fgure 3 further llustrates our fndngs by means of mean-beta plots for meanvarance and mean-sem-varance nvestors who nvest ether % or % n bonds (and hence 1% or % n equty). The mean-beta lne shows the ftted expected return for varous values of (downsde) beta. The ftted returns are computed usng the estmated parameter values for ether the mean-varance or mean-sem-varance model specfcaton. The dots show the tme-seres averages of the returns on the 1 sorted portfolos on ncreasng values of B/M (n that order, G= growth, V= value) and the onemonth Treasury bll (Tbll), gven ther (downsde) beta. If the portfolos are n lne wth a gven nvestor s mean-varance or mean-sem-varance preferences, the dots should le on the straght mean-beta lne. The upper two fgures show the results for the all equty nvestors. Clearly, the returns on the B/M sorted portfolos are dffcult to reconcle wth these nvestor s preferences. Most notably, the value portfolos have a hgher return than the growth portfolo, whle ts (downsde) beta s smlar. By contrast, the lower left fgure shows that the 1 B/M sorted portfolos algn more wth the preferences of mean-varance nvestors who nvests % of hs wealth n bonds. Portfolos wth a hgher return generally have a hgher beta (although the second lowest B/M portfolo stll seems 1

16 rather anomalous ). 17 The results mprove further for the mean-sem-varance nvestor (see lower rght fgure), for whch all 1 portfolos le almost on the mean-beta lne. [Insert Fgure 3 about here] Hence, the value premum s severely reduced for downsde rsk averters holdng relatvely low to ntermedate fractons of ther portfolos n equtes. As dscussed n the ntroducton, for many nsttutonal nvestors, ths s representatve of ther actual equty exposure durng our sample perod (193-7). For example, at the begnnng of 7, US lfe nsurance companes had $1,35bn nvested n corporate equtes and $19bn n mutual fund shares. The combned amount of $1,51bn represents roughly 3% of the total fnancal assets of $,5 bn, whch consst prmarly of money-market fund shares ($179bn) and credt-market nstruments ($,9bn). Moreover, durng most of the sample perod (193-7), the nvestment n equty was substantally smaller than n. For example, n 193, equtes represented just 5% of the fnancal assets held by lfe-nsurers. 1 B. Usng nsttutonal nvestor s portfolo weghts The results n the prevous secton assume a certan fxed and constant dstrbuton of portfolo weghts between equty and bonds. Although, ths dstrbuton may be representatve for many nvestors, we check the robustness of our results to the portfolos of groups of actual nvestors wth tme varyng bond/equty exposures. To accomplsh ths we pck four large groups of nsttutonal nvestors who nvested n both equtes and bonds over the entre sample perod,.e. lfe nsurance companes (lfe ns), property-casualty nsurance companes (other ns), prvate penson funds (prv. pen), and state and local government employee retrement funds (state ret), and nfer ther portfolo composton from the Federal Reserve Board s Flow of Funds Accounts. 19 Lke the analyss n the prevous secton, we assume that each nsttutonal nvestor type dvded her money between the CRSP all-share equty ndex and the equal weghted bond ndex. We compute the annual fractons nvested n equtes as the sum of the amounts outstandng n corporate equtes and equty mutual funds, dvded by the 17 In addton, as wll be shown n Secton F, the mean-varance nvestors nvestng % of ther wealth n bonds are volatng the basc regularty condton of non-sataton. In fact, the utlty functon s decreasng on a large part of the observed return range, castng doubt on the economc meanng of the mean-varance results. For example, volatons of non-sataton can lead to the non-exstence of a general equlbrum n the mean-varance CAPM wthout a rskless asset (see Nelsen, 199) and volate the no-arbtrage condton (Harrson and Kreps, 1979). 1 See the Federal Reserve Board Flow of Funds Quarterly Summary Report. 19 The relevant data can be found n tables L.11 tll L.119 of the Flow of Funds Accounts of the Unted States. 15

17 total fnancal assets mnus mscellaneous assets, reported at the end of last quarter of the prevous year. Table IV summarzes the results for the four nsttutonal nvestor types, as well as for the all equty nvestor. Panel A shows the mean-varance results. Consstent wth the results n the prevous secton, addng a substantal fracton of bonds to the portfolo decreases the value premum and helps to mprove the ft. Most notably, for lfe nsurance companes (who nvested on average 5% n bonds) the growth stock alpha falls to -.1% (p-value =.35), the value stock alpha falls to.1% (p-value =.), and the alpha of VMG hedge portfolo falls to.39% (p-value =.11). The overall R-squared ncreases to 73% and the CV-test p-value to.7. However, the meanvarance nvestor stll has some outperformance possbltes, snce the alpha of the second lowest B/M portfolo s -.9% wth an assocated p-value of.3. Smlar results are obtaned for the other nsurance companes, who nvest on average 7% n bonds. By contrast, prvate penson funds and state retrement fund, who have nvested a relatvely low fracton of ther portfolo n bonds (on average 39% and 3% respectvely), dsplay only a slght ncrease n the ft and decrease n the alpha of the value, growth and VMG hedge portfolos. As shown n Panel B, the results for the mean-sem-varance crteron show a further decrease n the value premum for the nsttutonal nvestors wth the hghest fxed ncome exposure, confrmng the earler fndngs. Most notably, for lfe nsurance companes, the growth stock alpha falls to -1.35% (p-value =.1), the value stock alpha falls to 1.% (p-value =.), and the alpha of the value premum portfolo (VMG) falls to.% (p-value =.3). The overall R-squared becomes %, and the CV-test p-value becomes.9. Moreover, the alpha of the second lowest B/M portfolo now reduces to - 1.5% wth an assocated p-value of.5. In sum we fnd that the value premum becomes smaller for retrement nvestors and becomes nsgnfcant for nsurance nvestors whch have larger fxed ncome exposures. [Insert Table IV about here] C. Choce of stock portfolos We may ask f our results are specfc to the B/M sorted portfolos. To check that our results also hold for other value measures we rerun our analyss usng 1 E/P and C/P sorted portfolos. Fgure shows the results, usng the alphas of the VMG hedge portfolo (that buys the top two value decles and shorts the bottom two), the R-squared, and CV-test statstc. Clearly, the sub-fgures on the left show that the E/P based VMG alpha substantally falls and the R-squared substantally rses for the mean-varance nvestor who nvest substantal amounts n bonds. For example, the VMG alpha falls In addton, for other nsurance companes we subtract trade recevables from the reported total fnancal assets. However, the mscellaneous assets and trade recevables categores are generally neglgble and have therefore almost no mpact on our results. 1

18 from.5% (p-value =.1) for an all equty nvestor, to.57% (p-value =.1) for an nvestor who nvests % n bonds, to 1.% (p-value =.5) for an nvestor who nvests 9% n bonds. The results further mprove for the mean-sem-varance nvestor. For 9% nvested n bonds the VMG alpha decreases to.% (p-value =.93). [Insert Fgure about here] Roughly smlar results are obtaned f the 1 C/P rato sorted portfolos are used (see the rght sub-fgures). Here the mean-sem-varance assumpton s more mportant than the percentage nvested n bonds. The rse n the R-squared and fall n the VMG alpha are relatvely small for the mean-varance nvestor. For example, the VMG alpha falls from 5.5% (p-value =.1) for an all equty nvestor, to.9% (p-value =.3) for an nvestor who % nvests n bonds, to 3.7% (p-value =.1) for an nvestor who 9% nvests n bonds. Smlarly, ts CV-test p-value rses to.9, only margnally nsgnfcant at a 5% level. By contrast, the results for the mean-sem-varance nvestor are n lne wth the other value measures. For % nvested n bonds the VMG alpha decreases to 3.51% (p-value =.1) and for 9% nvested n bonds the VMG alpha decreases further to 1.3% (p-value =.). Moreover, the R-squared ncreases substantally from 57% to % for the 9% bond mean-sem-varance nvestor. Gven the evdence of a larger value premum n the small cap segment (see for example Fama and French, 199, and Loughran, 1997), we may ask f our results also hold for the sx double-sorted portfolos formed on sze and B/M. Fgure 5 contans the results. Shown are the HML hedge portfolo of Fama and French (1993) that buys the hghest and shorts the lowest B/M portfolos n both sze segments, the correspondng portfolo for the bg cap (BV-BG) and small cap (SV-SG) segment, the R-squared and the CV-test p-value. [Insert Fgure 5 about here] Agan, some ntrgung results appear. The alpha of the HML hedge portfolo falls from 7.5% (p-value =.) for a mean-varance all equty nvestor, to 5.95% and.5% (p-value =.1 and.3) for a mean-varance nvestor who nvests % or 9% n bonds, to 5.75% and.% (p-value =. and.3) for a mean-sem-varance nvestor who nvests % or 9% n bonds. Smlarly, the R-squared goodness of ft measure ncreases from %, to 77%, and the CV-test p-value ncreases from.1, to.7. However, some unreported, but anomalous results reman. For nstance, for the meanvarance-9%-bond nvestor the alpha of the small value portfolo equals.% (p-value =.). Smlarly, the alpha of the bg growth stocks actually ncreases to -3.9% (pvalue =.5). By contrast, the mean-sem-varance-9%-bond model has a better 17

19 performance; the alpha of small value stock portfolo falls to.91% (p-value =.19), and of the bg growth portfolo to -1.9% (p-value =.37). These results hold both n the bg and small cap segments. In the bg cap segment the value premum falls from.79% (p-value =.) for a mean-varance all equty nvestor, to.75% (p-value =.3) for a mean-varance-%-bond nvestor, to 1.% (p-value =.3) for a mean-sem-varance-%-bond nvestor. In the small cap segment the alpha of the SV-SG portfolo goes from 1.1% (p-value =.) for a mean-varance all equty nvestor, to 5.1% (p-value =.1) for a mean-varance-%- bond nvestor, to 5.1% (p-value =.17) for a mean-sem-varance-%-bond nvestor. As before, the best ft s acheved for the mean-sem-varance nvestor who nvests roughly between the % and 9% n bonds. In fact, for the mean-sem-varance nvestor wth 9% nvested n bonds the alpha of the BV-BG portfolo just equals.1% (p-value =.9), whle the alpha of the SV-SG portfolo equals.33% (p-value =.7), a reducton of respectvely 97% and 77% compared to the classcal mean-varance all equty nvestor. Overall, the results are very smlar to those obtaned wth the 1 B/M sorted stock portfolos. These fndngs show that our results are robust wth respect to the value defnton of the cross-secton, and hold n the small cap segment as well. D. Choce of return frequency Followng Benartz and Thaler (1995), our analyss reles on annual returns. To analyze f our results are affected by the return frequency, we rerun our analyss usng monthly, quarterly, sem-annual, 1.5 yearly, and b-annual real returns. Fgure shows the results. The sub-fgures show the annualzed alphas of the VMG hedge portfolo for the varous evaluaton horzons. Addng bonds to the portfolo has lttle mpact on the value premum for horzons up to a quarter. Smlarly, for a sem-annual horzon the value premum s practcally unchanged for the mean-varance nvestor. By contrast, for the sem-annual mean-sem-varance nvestor the annualzed alpha of the VMG portfolo s reduced from.33% (p-value =.), to 3.17% (p-value =.) for a 9% bond nvestor. A smlar pattern s found n the (unreported) R-squared; t ncreases from 3% for a mean-varance all equty nvestor, to % for the mean-sem-varance- 9%-bond nvestor. The 1.5 yearly evaluaton horzon yeld smlar results as the semannual results; the alpha of the VMG hedge portfolo for the mean-varance nvestor s largely unchanged for varous percentages nvested n bonds, whle t substantally decreases for the mean-sem-varance nvestor. By contrast, the b-annual horzon gves some dfferent fndngs. Although, the alpha of the VMG hedge portfolo decreases for nvestors who nvest substantal percentages n fxed ncome, the effect of downsde rsk averson over an averson to varance s absent for an nvestor who nvest roughly % or more n fxed ncome. Moreover, unreported results reveal that the b-annual horzon yelds sgnfcant alphas of roughly.5% to 3.% for the one- and two-but-hghest 1

20 B/M portfolos, for both the mean-varance and the mean-sem-varance nvestor who nvests % or 9% n bonds. Hence, generally the alphas decrease for mean-varance nvestor wth an annual evaluaton horzon, and for mean-sem-varance nvestor wth nvestment horzons rangng between and 1 months. For shorter nvestment horzons (1-3 months), the value premum remans large and sgnfcant, rrespectve of the composton of the benchmark portfolo. Consequently, smlar to a term-structure of the rsk-return tradeoff (see Campbell and Vcera, 5), there s a close connecton between the nvestment horzon and the value premum for nvestor wth a substantal fxed ncome exposure. [Insert Fgure about here] The strkng dfferences for dfferent nvestment horzons are presumably caused by the dfferent shapes of the return dstrbuton for monthly returns and other lower frequency returns. For example, losses on the / mxed portfolo occur roughly n 35% of the months but n 3% of the years. 1 Wthout pretendng to forward the correct dynamc specfcaton for monthly returns, t s nsghtful to consder the followng regresson model wth four betas: xt Τ Τ + Τ Τ = α + β x τ1 ( x τ ) + β x τ1( x τ > ) t t t t L Τ Τ + Τ Τ + β l, x t 1τ 1( xt 1τ ) + β l, xt 1τ 1( xt 1τ > ) + ε t (1) l= 1 l, Ths model ncludes separate betas for downsde market movements ( β and β ) and upsde market movements ( β and β + + l, ). In case of a symmetrc response to market movements, the upsde and downsde betas wll be dentcal. Also, the model ncludes separate betas for the nstantaneous response ( β and β + ) and lagged + responses ( β and β l, l, ). If returns are serally IID, then the lagged betas wll be zero. If there s a sgnfcant lagged response, then the long-term market exposure wll dffer from the short-term exposure. We estmate equaton (1) usng OLS regresson analyss for the monthly returns to the 1 B/M sorted portfolos relatve to the CRSP all equty ndex and relatve to the bond ndex. We estmate the model wth no lags and wth lagged betas up to a quarter, half year, year, and 1 months. Table V summarzes our estmaton results. 1 Also, the monthly returns are affected by heteroskedastcty and seral correlaton to a sgnfcant degree. These statstcal problems cast doubt on the use of statstcal procedures whch assume serally IID returns (such as the procedure descrbed n Secton II) as well as the representatveness of the monthly return dstrbuton for annual returns. 19