THE ARKET PORTFOLIO AY BE EAN-VARIANCE EFFICIENT AFTER ALL OSHE LEVY and RICHARD ROLL January 4, 9 ABSTRACT Testng the CAP bols down to testng the mean/varance effcency of the market portfolo. Numerous studes have examned the mean/varance effcency of varous market proxes by employng ple parameters, and have concluded that these proxes are neffcent. Shrnkage methods do not seem to help. These fndngs cast doubt about one of the cornerstones of modern fnance. Ths study adopts a reverseengneerng approach: gven a partcular market proxy, we fnd the mnmal varatons n ple parameters requred to ensure that the proxy s mean/varance effcent. Surprsngly, slght varatons n parameters, well wthn estmaton error bounds, suffce to make the proxy effcent. Thus, many conventonal market proxes could be perfectly consstent wth the CAP and useful for estmatng expected returns. Keywords: ean-varance effcency, CAP, portfolo optmzaton, beta. Levy s from the Jerusalem School of Busness Admnstraton at the Hebrew Unversty, and Roll, the correspondng author, s from the Anderson School at UCLA, 3-85-68, rroll@anderson.ucla.edu.
INTRODUCTION Testng the Captal Asset Prcng odel (CAP) s equvalent to testng the mean/varance effcency of the market portfolo (see Roll [977] and Ross [977]). The effcency of the market portfolo has very mportant mplcatons regardng the debate over passve versus actve nvestng, and regardng the use of betas for prcng rsky assets. any studes that have examned the mean/varance effcency of varous market proxes have found that these proxes are neffcent, and typcally far from the effcent fronter. oreover, t s well known that portfolos on the effcent fronter typcally nvolve many short postons, whch mples, of course, that the postve-bydefnton market portfolo cannot be effcent. These results hold both when the ple return parameters are employed, and when the return parameters are adjusted by varous shrnkage methods. 3 Ths consttutes a very dark cloud hangng over one of the most fundamental models of modern fnance. In lght of ths evdence, should the CAP be taken serously by fnancal economsts, or s t just a pedagogcal tool for BA classes, grossly nconsstent wth the emprcal evdence? Ths paper shows that a small varaton of the ple parameters, well wthn ther estmaton error bounds, can make a typcal market proxy effcent. Thus, the emprcally measured return parameters and the market portfolo weghts are perfectly consstent wth the CAP usng a typcal proxy. How s ths possble, and how can t be reconcled wth the many prevous studes that have shown that the market proxy s See, for example, Ross [98], Gbbons [98], Jobson and Korke [98], Shanken [985], Kandel and Stambaugh [987], Gbbons, Ross, and Shanken [989], Zhou [99], and acknlay and Rchardson [99]. As shown, for example, by Levy [983], Green and Hollfeld [99], and Jagannathan and a [3]. 3 Jagannathan and a [3] show that constranng the weghts of the mnmum-varance portfolo to be non-negatve s equvalent to modfyng the covarance matrx n a way whch typcally shrnks the large elements of the covarance matrx. When ths shrnkage s employed, however, only a small number of assets are held n postve proportons (and the rest have weghts of zero). Ths s, agan, not an encouragng result for the hope of fndng an effcent market portfolo by employng shrnkage technques.
neffcent? Whle most studes suggest varous varatons of the return parameters relatve to the ple parameters and check whether these varatons lead to an effcent market proxy, we take a reverse approach: we frst requre that the return parameters ensure that the market proxy s effcent. Gven ths requrement, we look for parameters that are as close as possble to ther ple counterparts. Surprsngly, parameters that make the market proxy effcent can be found very close to the ple parameters. Hence, mnor changes n estmaton error reverse prevous negatve and dsappontng fndng for the CAP. We hasten to add that the effcency, or lack thereof, for a market proxy can never be a defntve test of the macro CAP, whch requres the market portfolo of all assets, ncludng real-estate, human captal, etc. Nonetheless, t would be reassurng f typcal proxes could be shown to be less neffcent than prevously beleved. Ths paper s organzed as follows. The next secton ntroduces the methods employed. Secton II descrbes the data and the results. Secton III dscusses mplcatons for asset prcng. Secton IV concludes. I. ETHODS Gven a market proxy, m, we look for the mnmal varaton of ple parameters that would make t mean/varance effcent. Denote the vector of market proxy portfolo weghts by x m, and denote the vector of ple average returns and the vector of ple standard devatons by the ple covarance matrx, and and, respectvely. C denotes ρ denotes the ple correlaton matrx. 3
4 The objectves beng sought are an expected return vector and a covarance matrx C that on the one hand make portfolo m mean/varance effcent, and on the other hand are as close as possble to ther ple counterparts. For smplcty, when consderng the covarance matrx C we allow varaton only n the standard devatons, whle retanng the e ple correlatons: = N N C ρ L O L L O L () Allowng the correlatons to vary as well ntroduces techncal dffcultes, but can only make the results stronger, as t allows more degrees of freedom n the optmzaton procedure descrbed below. In order to obtan the parameters ( ), that are closest to ther ple counterparts, ( ),, we defne the followng dstance measure D between any parameter set ( ), and the ple parameter set: ( ) ( ) ( ) = = + N N N ) ( N,,, D α α, () where N s the number of assets, and α s a parameter determnng the relatve weght assgned to devatons of the means relatve to devatons of the standard devatons. Recall that the larger the standard devaton of a gven asset s returns, the larger the statstcal errors nvolved n estmatng ths asset s parameters, and the larger the confdence ntervals for these parameters. Ths s the ratonale for dvdng the devatons n () by - the resultng dstance measure punshes devatons n the parameters of assets wth low standard devatons more heavly than smlar devatons n assets wth hgher standard devatons. The ultmate test of whether a set
5 of parameters ( ), can be consdered as reasonably close to the ple parameters s the proporton of parameters that devate from the standard estmaton error bounds around ther ple counterparts, and the sze of those devatons. Intutvely, a parameter set s reasonably close when 95% or more of the parameters are wthn the 95% confdence ntervals of the ple parameters (Below we also employ the more formal Bonferron [935] multple-comparson test). The choce of the dstance measure D n eq.(), and ts mnmzaton n the optmzaton problem descrbed below, are desgned to mnmze the statstcal sgnfcance of the devatons between and and ther ple counterparts. To fnd the set of parameters ( ), that make the proxy m mean/varance effcent and are closest to the ple parameters, we solve the followng optmzaton problem: Optmzaton Problem : nmze ( ) ( ) ( ),,, D Subject to: () = ρ z N z z mn m m N N r r r q x x x L O L L O L where q> s the constant of proportonalty, and r z s the zero-beta rate. Both q and r z are free varables n the optmzaton. Thus, there are N+ varables n the optmzaton: N s ', N s ', q and r z. Any set of these N+ parameters satsfyng () makes the proxy portfolo mean/varance effcent (see, for example Roll [977]). We
are lookng for the set of parameter vectors (, ) that satsfy ths mean/varance effcency condton and are closest to the ple parameters 4. Our approach s dfferent from the approach employed n prevous studes, such as Black, Jensen, and Scholes [97] and Gbbons, Ross, and Shanken [989], for example, n two man regards. Frst, we are not requred to assume the exstence of a rsk-free asset. Second, and more mportantly, the standard approach looks at the adjustment to the emprcal average returns requred to make the market proxy effcent (.e. the stocks' alphas), and asks whether these adjustments are statstcally plausble. In contrast, we are lookng at smultaneous adjustments to the average returns and the standard devatons (and could, n prncple, nclude adjustments to the correlatons as well). Thus, whle the standard approach examnes the statstcal plausblty of a sngle vector of alphas, we examne a multtude of vectors of average return and standard devaton adjustments. Ths allows us many more degrees of freedom relatve to the standard approach, and explans why we fnd that only small adjustments are requred to make the market proxy effcent. In some stuaton one may have belefs about the proxy portfolo s ex-ante mean and standard devaton, and would lke to fnd the set of parameters that are closest to the ple parameters, and at the e tme ensure that the proxy portfolo s mean/varance effcent wth the pre-specfed mean and standard devaton. Denotng the pre-specfed mean and standard devaton by and, respectvely, the optmzaton problem solved n ths case s: Optmzaton Problem : ( ) nmze D (, ),(, ) 4 Ths optmzaton problem s smlar n sprt to Sharpe's [7] "reverse optmzaton" problem. Levy [7] employs an analogous technque to fnd mean/varance effcent portfolos that have all-postve weghts. Ths approach was frst used n a very nnovatve paper by Best and Grauer [985]. 6
7 Subject to: () = ρ z N z z N N r r r q L O L L O L m x () = ' m x () ρ = m N N m x ' x L O L L O L, where, agan, x m s the vector of a gven proxy s portfolo weghts. The next secton presents solutons to these optmzaton problems wth emprcal equty data n order to ascertan how large the devatons from the ple parameters must be n order to ensure mean/varance effcency. II. DATA AND RESULTS Our demonstraton ple conssts of the largest stocks n the U.S. market (accordng to December 6 market captalzatons), whch have a complete monthly return records over the perod January 997 - December 6 ( return observatons). Columns () and (4) n Table I report the ple average returns and standard devatons for 3 of these stocks (the complete nformaton for all stocks s gven n Table AI n the Appendx). The average ple correlaton s.4.
Followng prevous research (e.g., Stambaugh [98]), we examne a market proxy whose weghts are market captalzatons, n ths case of the stocks as of December 6, market cap of frm x. m = j= market cap of The proxy portfolo and the ple mean/varance fronter are shown n Fgure by the trangle and thn lne, respectvely. As the fgure llustrates, the proxy portfolo s far from the effcent fronter when the ple parameters are employed. Ths s consstent wth prevous studes. frm To solve Optmzaton Problem numercally, we mplement atlab s fmncon functon, whch s based on the nteror-reflectve Newton method and the sequental quadratc programmng method. The soluton (, ) s gven n Columns (3) and (5) of Table I. The t-values for the expected returns are gven n Column (6). These t-values reveal that the j dfference between the ple average return,, and s non-sgnfcant at the 95% level for all stocks (ths s true not only for the 3 stocks shown n the table, but for the other 7 stocks as well). Column (7) provdes the rato ( ) ( ) for each stock. The 95% confdence nterval for ths rato s the range [.79-.39]. 5 ( n )s 5 The rato s dstrbuted accordng to the χ n dstrbuton, where s the populaton varance, s s the ple varance (or ( ) n the notaton used n ths paper), and n s the number of observatons. We have monthly return observatons, hence n=. As we are lookng for the 95% confdence nterval for s, we need to fnd the crtcal values c and c for whch ( χ > c ) 5, and P ( < c ) = 5 P 9 =. χ. For large n, χ n can be approxmated by the 9. standard normal dstrbuton. Thus, the crtcal values c and c satsfy c 9 =. 96 and c 9 =. 96, whch yeld: c = 5. 6 and c = 9.. Thus, the 95% confdence nterval for n 8
The values n Column (7) reveal that for all stocks the rato ( ) ( ) s well wthn ths range (and ths s also true for the 7 stocks not shown n the table). Thus, the soluton (, ) to the optmzaton problem s very close to the ple parameter set n the sense that none of the parameters s sgnfcantly dfferent from ts ple counterpart. ore formally, as we have N= parameters, we are smultaneously testng hypotheses (each statng that the gven parameter s not dfferent than ts ple counterpart at the 5% sgnfcance level). The Bonferron [935] test states that we should reject the multple-comparson hypothess at the 5% level f any one of the parameters s sgnfcantly dfferent than ts ple counterpart at the (5/)% level (see also ller [99]). As none of our parameters s sgnfcantly dfferent at the 5% level, of course none s sgnfcant at the much lower (5/)% level, and we cannot reject the multple comparson hypothess. (Please nsert Table I and Fgure about here) To confrm that the parameters (, ) make the proxy portfolo mean/varance effcent, one can examne the effcent fronter and the locaton of the proxy portfolo n the mean-standard-devaton plane wth these parameters. These are llustrated by the bold lne and the star n Fgure. The fgure shows that wth the parameters (, ) the proxy portfolo les on the effcent fronter. It s nterestng to note that whle the modfed parameters (, ) do not have a bg mpact on the expected return or the standard devaton of the proxy portfolo (the star s located very close to the trangle), they do have a bg effect on the shape of the fronter. Why s the modfed fronter much flatter than the ple fronter? s s gven by 9. < 9 s < 5. 6 or:. 758 < s <. 66. Alternatvely, ths range can be also stated as. 79 < s <. 39. 9
The explanaton can be found n Fgure, whch shows the adjustment to the expected return,, as a functon of the ple average return,. The fgure reveals that hgh ple returns tend to get negatve correctons ( ) <, whle the opposte holds for low ple returns. Thus, the cross sectonal varaton of s smaller than the cross sectonal varaton of, whch explans why the fronter s flatter (recall that n the lmtng case where all expected returns are dentcal, the fronter becomes completely flat t s a horzontal lne). Fgure shows that the correctons to the ple means mpled by the optmzaton are remnscent of standard statstcal shrnkage methods. However, unlke the standard shrnkage methods, the method employed here ensures that the proxy s mean/varance effcent. 6 (Please nsert Fgure about here) There s excellent ntuton behnd such a result when one recalls two facts (a) the effcent fronter tself s the result of an optmzaton; t gves the mnmum varance for each level of mean return, and (b) ple parameter estmates are equal to true populaton parameters plus estmaton errors. An effcent fronter computed usng ple estmates optmzes wth respect to plng errors n addton to true parameters, so assets wth over-estmated means are lkely to be weghted too heavly n fronter portfolos and vce versa for assets wth under-estmated means. Ths suggests that an effcent fronter computed usng populaton parameters, f they were only known, would fall well nsde the fronter computed usng ple estmates, at 6 One may wonder whether the adjustment s smlar for stocks that are relatvely hghly correlated wth one another. In order to check ths, we calculate the ple return correlaton for each par of stocks (,j), and examne the relaton across pars between ths ple correlaton and the dfference between the adjustments of the two stocks,.e. ( ) ( j j ). We fnd no such relaton (R =.9),.e. pars that are more hghly correlated are not more lkely to have smlar adjustments.
least at most ponts. The man excepton would be near the global mnmum varance portfolo, whose weghts do not depend on mean returns; ndeed, such a relaton s exactly what we see depcted n Fgure. The mplcaton of these results s qute strkng. In contrast to common wsdom, they show that the emprcal proxy portfolo parameters are perfectly consstent wth the CAP f one allows for only slght estmaton errors n the return moments. The reason that most prevous studes have found that the market proxy s neffcent, even when varous standard shrnkage methods have been employed, s that the varaton of the parameters necessary to make the proxy portfolo effcent s very specfc. Whle ths varaton s n the sprt of shrnkage, t s specfcally desgned to ensure the effcency of the proxy portfolo, and thus t s fundamentally dfferent than the standard statstcal shrnkage methods. Wth the soluton (, ) to Optmzaton Problem the proxy portfolo has a monthly expected return of.4% and a standard devaton of 4.6% (see Fgure ), whch are very close to ts ple values,.5% and 4.6%. These values were produced by the optmzaton (gven the proxy portfolo weghts). In some stuatons one may have belefs about the proxy portfolo s ex-ante return parameters, and may wsh to look for solutons that are consstent wth these belefs. For example, suppose one would lke to fnd vectors and such that the proxy portfolo s effcent and has an expected return and a standard devaton of = % and = 4%, respectvely. Are such ndex values compatble (n a statstcal sense) wth the ple parameters and wth a mean/varance effcent ndex? To answer ths queston, Optmzaton Problem can be solved wth = % and = 4%. We wll consder the soluton (, ) compatble wth the ple parameters f 95% or more of the parameters are wthn the 95% confdence ntervals of ther ple counterparts.
Of course, = % and = 4% s just one example. A more complete pcture would scan the mean/varance plane and map the range of proxy portfolos return parameters, and, that are compatble wth the CAP and the ple returns and market proxy weghts. Fgure 3 shows the results of ths analyss. For each combnaton of prespecfed proxy portfolo parameters (, ) we solve Optmzaton Problem. The ponts scanned are shown by the crcles n the mean/varance plan. If the resultng optmal parameter set (, ) s found to be statstcally compatble wth the ple parameters, (, ), and wth the CAP (mean/varance effcency of the ndex), the pont s marked as a flled crcle; f not, t s left transparent. For example, the pont ( = %, = 4% ), (ndcated by a down arrow n Fgure 3), s ndeed consstent wth the ple parameters and the proxy beng effcent. In contrast, the pont to the left, ( = %, = 3.95% ), s not consstent. The fgure shows that the range of possble proxy portfolo return parameters that can be made smultaneously consstent wth the CAP and the ple parameters s n fact qute large. The proxy portfolo expected return can be as small as.5% or as large as %, and the standard devaton can be as small as 4% or as large as 5%. These are percent per month. (Please nsert Fgure 3 about here) It would be nterestng to redo ths analyss usng ndexes wth even more ndvdual assets, but there are techncal dffcultes. When the number of assets exceeds the number of tme seres observatons, the correlaton matrx s sngular, whch produces some nstabltes n the optmzaton problem. We can, however, partally nvestgate ths ssue by varyng the number of assets for N< and lookng
for any trend n the range of proxy portfolo return parameters consstent wth the CAP. For example, repeatng the analyss for the 5 largest stocks (nstead of the ) yelds the results shown n Fgure 4. These results are comparable to those obtaned wth stocks n the range of proxy portfolo return parameters consstent wth the CAP s roughly smlar. To nvestgate n more detal possble systematc effects of the number of assets, we repeat ths analyss for N=,,, stocks. For each value of N we measure the area of admssble proxy portfolo parameters (estmated by the polygon contanng the admssble ponts, see, for example the polygon n Fgure 4). The results are shown n Fgure 5. Although the area s an approxmaton of the precse area of admssble ponts, because of the dscreteness of the ponts, (and as ndcated by the error bars n Fgure 5), Fgure 5 shows that the area does not seem to change systematcally wth the number of assets. Thus, the results seem robust to the dentty of stocks and to the number of stocks contaned n any market ndex proxy. (Please nsert Fgures 4 and 5 about here) III. IPLICATIONS FOR ASSET PRICING AND PRACTICAL USE OF THE CAP The Securty arket Lne (SL) formula s probably the most wdespread method for estmatng the cost of captal and for prcng rsky assets. Usng beta and the SL formula for estmatng the expected return, rather than employng the ple average return drectly, s usually justfed on the bass that the statstcal estmaton of beta s more stable than that of the average return. However, when there are questons about how well the SL relatonshp holds emprcally, there are serous doubts about 3
employng betas for prcng. 7 Whle we cannot prove that the SL relatonshp holds emprcally wth the ex-ante parameters, our analyss does provde another reason for employng betas for estmatng the cost of captal. Suppose that the CAP holds wth the true ex-ante parameters (, ), and that the emprcally measured parameters are (, ). The true and ple betas of stock are gven respectvely by: N xmj j ρj j= = xm' Cxm β (3a) N xmj j ρj j= = xm' C xm β, (3b) where x m denotes the market portfolo weghts. The true cost of equty of frm s If one employs the observable β. n the SL formula nstead of the correct β, how accurate wll be the resultng cost of captal estmate? In other words, how close are β and β? The answer s shown n Fgure 6, where the parameter set (, ) employed s the soluton to Optmzaton Problem. The fgure reveals that the dfference between β and β s very small. The reason s that both the denomnators and the numerators of (3a) and (3b) are very smlar. The varance of the market proxy s qute close whether the optmzed parameters or the ple parameters are employed (compare the horzontal locaton of star and the trangle n Fgures 3 and 4). As for the covarances n the numerator, note that, and n addton, the devatons tend to cancel each other out n the summaton, as n some cases >, whle n others < (see Column 7 n Table I). 8 j j j j j j 7 Ths s, of course, one of the major debates n fnance. See, for example, Renganum [98], Levy [98], Lakonshok and Shapro [986], Chen, Roll, and Ross [986], Fama and French [99], and Roll and Ross [994]. 8 Fgure 6 shows the relaton between the β 's and the β 's when we use a value of α =. 75 n the dstance measure D (see eq.()). When a hgher value of α s employed, the ' s are closer to ther 4
Snce the market proxy s effcent wth the true parameters (, ), the followng relatonshp holds exactly: = r + β r ), (4) z ( m where r z s the expected return on the zero-beta portfolo for ndex m. Common practce substtutes a rskless rate, r f, for r z, but ths s approprate only when f and z z have the e mean return. Snce β β, employng the SL wth the ple beta, as s commonly done n practce, provdes an excellent estmate for the true expected return (assumng r f = r z ): [ r + β ( r )] = β ( r ) β ( r ). (5) f m f m The above argument s based on takng the true ex-ante parameters as the (, ) vectors solvng Optmzaton Problem,.e. the parameters ensurng the CAP that are closest to the ple parameters. What f, nstead, we take another set of parameters that ensures the effcency of the proxy and s consstent wth the ple parameters? For example, suppose that we take as the true parameters those that solve Optmzaton Problem wth = % and = 4.5% (see pont A n Fgure 3). It turns out that wth these parameters the β s and the β f m f s are stll very close see Panel A n Fgure 7. Ths s also true for other ponts wth very dfferent proxy portfolo expected returns and standard devatons see Panels B,C, and D n Fgure 7, correspondng to the ponts B,C, and D n Fgure 3. Ths s a strong result: f the CAP holds n a way that s consstent wth the ple parameters, the dfferences between ple betas and true betas are gong to be small. Thus, f the SL formula for prcng, whch mples that the CAP holds ple counterparts, and the ' s are more dstant from ther ple counterparts. As a result, the dfferences between the β 's and the β 's also ncrease. Yet, even wth a very hgh value ofα =. 97 the β 's are stll very close to the β 's, wth a correlaton of.96. 5
wth the ex-ante parameters, one can be confdent about usng the ple betas, and should not worry about estmaton errors n the betas. Ths concluson s reached because we are not just lookng at the statstcal estmaton error of a sngle asset s beta n solaton, as s typcally done, but rather at the error n beta gven that the CAP holds n a way that s consstent wth the ple parameters (, ). (Please nsert Fgures 6 and 7 about here) From a practcal perspectve, snce ple betas are qute close to betas that have been adjusted to render the market proxy mean/varance effcent, mproved estmates of expected returns can be obtaned from ple betas alone. Sample mean returns should be gnored! To llustrate, Fgure 8, Panel A shows the cross-sectonal relaton between ple mean returns and ple betas for our stocks whle Fgure 8, Panel B shows the analogous relaton for adjusted means and betas. Clearly, the ple means n Panel A are not closely related at all to ple betas but the adjusted means n Panel B are perfectly related to adjusted betas. 9 (Please nsert Fgure 8 about here) Consequently, to obtan an mproved expected return for any stock, frst calculate the adjusted mean return for the market ndex proxy and for ts correspondng zero-beta portfolo. Pluggng these numbers along wth the ple beta (because t s close to the adjusted beta) nto the usual CAP formula delvers the mproved estmate of expected return. akng the market ndex proxy mean/varance effcency produces useful betas for many practcal purposes such as estmaton of the cost of equty captal for a frm or of the dscount rate for a rsky project. 9 The slght devatons from lnearty n Fgure 8, Panel B are caused by roundng error. For most proxes, the ple means wll be close to the adjusted means. 6
IV. CONCLUSION arket proxy portfolos are typcally very far from the ple effcent fronter. any studes have tred varous adjustments to the ple parameters to make the market proxy mean/varance effcent, wthout success. Thus, the common wsdom s that the emprcal return parameters and market portfolo weghts are ncompatble wth the CAP theory. In ths paper we hope to change that percepton. We show that small varatons of the ple parameters, well wthn the range of estmaton error, can make a typcal market proxy mean/varance effcent. Whle such parameter varatons are remnscent of shrnkage, they dffer from those obtaned wth the standard statstcal shrnkage methods: they are the result of reverse optmzaton. In ths reverse optmzaton, return parameters are derved to make the market proxy mean/varance effcent whle beng close to ther ple counterparts. The fact that we fnd many such parameter sets, together wth the fact that many prevous attempts to vary the return parameters n order to obtan an effcent proxy were unsuccessful, seem to ndcate that such parameter sets may be very rare n parameter space t s very unlkely to stumble onto one of them by concdence. Yet, the reverse optmzaton problem delvers them smply and drectly. These fndngs suggest that the CAP (.e., ex ante mean/varance effcency of the market ndex proxy) s consstent wth the emprcally observed return parameters and the market proxy portfolo weghts. Of course, ths does not consttute a proof of the emprcal valdty of the model, but t shows that the model can not be rejected, n contrast to the wdespread belef n our professon. The ntutve dea that 7
shrnkage correctons should ncrease the emprcal valdty of the CAP s shown to be vald - wth the rght correctons, whch are small, the ndex proxy s perfectly effcent. The analyss also shows that n ths framework employng the ple betas provdes an excellent estmate of the true expected returns. 8
REFERENCES Best,.J. and R.R. Grauer, 985, Captal asset prcng compatble wth observed market value weghts, Journal of Fnance 4, 85-3. Black, F.,.C. Jensen,. and. Scholes, 97, The captal asset prcng model: Some emprcal tests, n Studes n the Theory of Captal arkets..c. Jensen, Ed., New York: Praeger, 79-. Bonferron, C.E., 935, Il calcolo delle asscurazon su grupp d teste. In Stud n Onore del Professore Salvatore Ortu Carbon. Rome: Italy. 3-6 Fama, E. F. and K. R. French, 99, The cross-secton of expected stock returns, Journal of Fnance 47, 45-465. Gbbons,. R. 98, ultvarate tests of fnancal models: A new approach, Journal of Fnancal Economcs, 3-7. Gbbons,. R., S.A. Ross, and J. Shanken, 989, A test of the effcency of a gven portfolo, Econometrca 57, -5. Green, R. C. and B. Hollfeld, 99, When wll mean/varance effcent portfolos be well dversfed? Journal of Fnance 47, 785-89. Jagannathan, R., and T. a, 3, Rsk reducton n large portfolos: a role for portfolo weght constrants, Journal of Fnance 58, 65-684. Jobson, J.D. and B. Korke, 98, Potental performance and tests of portfolo effcency, Journal of Fnancal Economcs, 433-466. Kandel, S., and R.F. Stambaugh, 987, On correlatons and nferences about mean/varance effcency, Journal of Fnancal Economcs 8, 6-9. Lakonshok, J., and A.C. Shapro, 986, Systematc rsk, total rsk and sze as determnants of stock market returns, Journal of Bankng and Fnance, 5-3. Levy, H., 98, A test of the CAP va a confdence level approach, Journal of Portfolo anagement. Levy, H., 983, The captal asset prcng model: theory and emprcsm, Economc Journal 93, 45-65. Levy,., 7, Postve portfolos are all around, Hebrew Unversty Workng Paper. Lntner, J., 965, Securty prces, rsk, and the maxmal gans from dversfcaton, Journal of Fnance, (4), 587-65. acknlay, A.C., and.p. Rchardson, 99, Usng generalzed method of moments to test mean/varance effcency, Journal of Fnance 46, 5-57. ller, R. G., 99. Smultaneous Statstcal Inference. New York: Sprnger-Verlag. Renganum,. R., 98, A new emprcal perspectve on the CAP, Journal of Fnancal and Quanttatve Analyss 6, 439-46. 9
Roll, R., 977, A crtque of the asset prcng theory s tests; Part I: On past and potental testablty of the theory, Journal of Fnancal Economcs 4, 9-76. Roll, R., and S. Ross, 994, On the cross sectonal relaton between expected returns and betas, Journal of Fnance 49, -. Ross, S., 977, The captal asset prcng model (CAP), short-sale restrctons and related ssues, Journal of Fnance 3, 77-83. Shanken, J., 985, ultvarate tests of the zero-beta CAP, Journal of Fnancal Economcs 4, 37-348. Sharpe, W. F., 964, Captal asset prces: A theory of market equlbrum under condtons of rsk, Journal of Fnance 9, 45-44. Sharpe, W. F., 7, Expected utlty asset allocaton, Fnancal Analysts Journal 63, 8-3. Stambaugh, R. F., 98, On the excluson of assets from tests of the two-parameter model: a senstvty analyss, Journal of Fnancal Economcs, 37-68, Zhou, G., 98, Small ple tests of portfolo effcency, Journal of Fnancal Economcs 3, 65-9.
Table I The Sample Parameters and Closest Parameters Ensurng that the arket Proxy s ean/varance Effcent For the sake of brevty, ths table reports only 3 of the stocks (the complete table s gven n the appendx). The ple parameters are gven n the second and fourth columns. The expected returns and standard devatons whch are closest to these parameters and ensure that the market proxy s effcent (.e. the parameters that solve Optmzaton Problem ) are gven n columns (3) and (5). The t-values for the expected returns are gven n column (6), whch shows that none of these values are sgnfcant at the 95% level (ths s also true for the 7 other stocks not shown n the table). Column (7) reports the rato between the optmzed varances ( ) and the ple varances. The 95% confdence nterval for ths rato s [.79-.39] (see footnote 5). All of the ratos n the table, as well as the ratos for all other 7 stocks not shown here, fall well wthn ths nterval. These results are obtaned wth a value of α =. 75 n the mnmzed dstance measure D (see eq.()). Hgher values of α reduce the varaton n the expected returns (at the expense of ncreasng the devatons n the standard devatons). () () (3) (4) (5) (6) (7) Stock # () t-value for ( ) / ( ) (the 95% confdence nterval for ths value s [.79-.39] ).4.8.65.67 -.43.9..9.5.5 -.7.3 3..7.6.4.588.963 4.9.3.58.6 -.444.8 5.39..5.56 -.8.77 6.5..75.73.95.953 7.7.3.7.7.938.94 8...5.5 -.433.8 9.3.5.7.69.86.978.6.8.99.98.85.986..3.67.66.344.977.6.9.9.93 -.89.5 3.5..7.7 -.67.35 4.9... -.7.34 5...6.6 -.9.6 6.3.4.59.6 -.5.44 7.3.5.58.57.45.99 8.4..46.47 -.3.6 9...86.85.99.988.7..67.66.477.979...65.65.8.996.8.6.8.8 -.5.8 3..8.67.68 -.65.3 4.3.4.59.59 -.533.995 5.7.4.88.88 -.36. 6.4.3.8.8 -.8.7
7.6..77.75.8.955 8.8..77.78 -.58.44 9...87.86.76.989 3...65.64.55.999 Fgure : The Effcent Fronter and arket Proxy wth the Sample and the Adjusted Return Parameters. The thn lne curve and the trangle (partly hdden behnd the star) show the mean/varance fronter and the market proxy wth the ple parameters. As typcal of other studes, the market proxy s very far from the effcent fronter when the ple parameters are employed. The bold lne and the star show the mean/varance fronter and the market proxy wth the adjusted parameters,. Wth these parameters the market proxy s mean/varance effcent. ( )
Fgure : The Correcton to the Expected Returns and a Functon of the Sample Average Return. For stocks wth hgh ple average returns, the correcton n the expected return tends to be negatve. The opposte holds for stocks wth low ple average returns. Thus, the correctons produced by the soluton to the optmzaton problem are remnscent of statstcal shrnkage methods. 3
Fgure 3: The Set of Proxy Portfolo Parameters Consstent wth ean/varance Effcency and the Sample Parameters stocks. Optmzaton Problem s solved for each pont on the mean-standard devaton, s consdered consstent wth the ple parameters f 95% or more of the parameters are wthn the 95% confdence ntervals of ther ple counterparts. The (, ) ponts that are consstent wth the mean/varance effcency of the proxy portfolo and wth the ple parameters are ndcated by the flled crcles. For example, the proxy portfolo can be made mean/varance effcent wth a standard devaton of 4% and a mean return of %, but not wth a standard devaton of 3.95% and a mean return of %. The fgure shows that gven a set of ple parameters and proxy portfolo weghts, the proxy portfolo can be made mean/varance effcent wth a large range of possble mean and standard devaton combnatons. As n Fgure, the trangle and the star represent the market proxy wth the ple parameters and wth the parameters solvng Optmzaton Problem, respectvely. plane, ( ),. The resultng parameter set, (, ) 4
Fgure 4: The Set of Proxy Portfolo Parameters Consstent wth ean/varance Effcency and the Sample Parameters 5 stocks. Ths fgure s the e as Fgure 3, but t s constructed wth only the 5 largest stocks (rather than ). Agan, a wde range of ( ), s consstent wth the effcency of the market proxy. The area of a polygon drawn through the outer consstent ponts s an approxmaton to the range of consstency. 5
Fgure 5: The Area of Admssble Proxy Portfolo Parameters as a Functon of the Number of Assets. For each value of N, startng wth the largest ten stocks, the area of a consstency polygon s computed analogous to the one shown n Fgure 4. Ths area measures the range of proxy portfolo return parameters consstent wth the CAP and the gven proxy portfolo. Ths s an approxmaton of the precse area, because t depends on a fnte set of parameter ponts n the V plane. The error bars reflect ths possble estmaton error. The fgure shows that the area of admssble parameters does not change systematcally wth the number or the dentty of the stocks ncluded n the market ndex proxy. 6
Fgure 6: The Relaton Between Sample Betas and the True Betas. The true parameters are those that solve Optmzaton Problem and satsfy the CAP: (, ). The ple parameters are (, ). The true and ple betas are gven by eq.(3). The fgure shows that the ple betas are very close to the true betas, and thus yeld excellent estmates of the expected returns. 7
Fgure 7: The Relaton Between Sample Betas and True Betas. The true parameters are those that satsfy the CAP and solve Optmzaton Problem. Each panel corresponds to a dfferent combnaton of values of the pre-specfed expected return and standard devaton of the proxy portfolo, and. (The ponts correspondng to these four panels are ndcated by A,B,C, and D, respectvely, n Fgure 3). The true and ple betas are gven by eq.(3). The fgure shows that the ple betas are very close to the true betas, and thus yeld excellent estmates of the true expected returns, even when and are not close to the values obtaned wth the ple parameters. 8
Panel A. Sample means and betas 4.5% 4.% 3.5% 3.% ean Sample Return.5%.%.5%.%.5%.%.5..5..5..5 3. Sample Beta 3.% Panel B. Adjusted eans and Betas.5%.% Adjusted ean Return.5%.%.5%.%.5..5..5..5 3. Adjusted Beta Fgure 8: The Securtes arket Lne Scatter for Sample vs. Adjusted eans and Betas Sample estmates of means and betas for our stocks are plotted aganst each other n Panel A. Panel B plots the correspondng adjusted means and betas that are obtaned from optmzaton problem. 9
Appendx- Table AI The Sample Parameters and Closest Parameters Ensurng that the arket Proxy s ean/varance Effcent Ths s the complete verson of Table I gven n the text, where here the data s provded for all stocks. The ple parameters are gven n the second and fourth columns. The expected returns and standard devatons whch are closest to these parameters and ensure that the market proxy s effcent (.e. the parameters that solve Optmzaton Problem ) are gven n columns (3) and (5). The t-values for the expected returns are gven n column (6), whch shows that none of these values are sgnfcant at the 95% level. Column (7) reports the rato between the varances ( ) and the ple varances. The 95% confdence nterval for ths rato s [.79-.39] (see footnote 5). All of the ratos n the table fall well wthn ths nterval. () () (3) (4) (5) (6) (7) Stock # () t-value for ( ) ( ) / (the 95% confdence nterval for ths value s [.79-.39] ).4.8.65.67 -.43.9..9.5.5 -.7.3 3..7.6.4.588.963 4.9.3.58.6 -.444.8 5.39..5.56 -.8.77 6.5..75.73.95.953 7.7.3.7.7.938.94 8...5.5 -.433.8 9.3.5.7.69.86.978.6.8.99.98.85.986..3.67.66.344.977.6.9.9.93 -.89.5 3.5..7.7 -.67.35 4.9... -.7.34 5...6.6 -.9.6 6.3.4.59.6 -.5.44 7.3.5.58.57.45.99 8.4..46.47 -.3.6 9...86.85.99.988.7..67.66.477.979...65.65.8.996.8.6.8.8 -.5.8 3..8.67.68 -.65.3 4.3.4.59.59 -.533.995 5.7.4.88.88 -.36. 6.4.3.8.8 -.8.7 7.6..77.75.8.955 8.8..77.78 -.58.44 9...87.86.76.989 3...65.64.55.999 3..3.86.85.47.99 3.9.6.8.8 -.46.4 33.6.9.8.83 -.86.6 34.7.6.77.78 -.46.8 3
35...7.7.43.984 36.9.3.64.6.658.954 37...64.64 -.8. 38.6.3.3.4 -.4.6 39...65.65 -.95.9 4.6..87.85.749.96 4..5.5.4.48.978 4.6.7.9.9.. 43.8.3..99 -.65.986 44.3.7.5.4.364.976 45.9.3.88.87.499.974 46.6.4.85.8.67.93 47.3.8.4..49.978 48...84.83.57.997 49.8.9.77.77..998 5.7..8.84 -.884.36 5..4.8.8.65.984 5..8.5.6 -.77.9 53.6..7.73 -.57.3 54..4.6.5.8.984 55...74.74.8.993 56.7.3.76.74.889.945 57..3.7.7.379.975 58.4.9...58.95 59.3.6.89.9 -.87.56 6.4.8.9.88.489.96 6...7.7 -.95.5 6...93.93 -.95. 63.8..75.74.436.979 64..9.6.7 -.7. 65.6.3.77.78 -.336.8 66.3.4.74.74..993 67.6.7.76.75.3.988 68..8.5.5 -.6. 69...34.33.9.994 7.4.4.76.76.9.997 7..3.94.94.346.983 7.5..7.7 -.56.8 73.8.3.88.89 -.658.33 74..4.96.98 -.934.49 75..7.59.59 -.75.8 76.5.3.83.8.3.937 77.7.3.83.8.78.957 78.5.3.83.8.3.938 79.3.4.86.86.8.997 8.6.5.9.9 -.46.6 8..5.74.7.39.964 8..3.7.69.9.983 83...7.6.99.99 84.9.9.89.88 -.4.993 85.8..98. -.8.9 86.3..73.73 -.8. 87...3.3.3.996 88.7.6.95.9.968.939 89.... -.9.9 9.4..93.99 -.35.5 9.34.5.6.64 -.74.46 9.3.7.7.7 -.63.4 3
93..4.86.86.3.98 94.3..8.8 -.4.9 95.3.3.3.33 -.579.45 96.6..47.47 -.45.9 97.7..87.88 -.53.4 98.7.7...35.997 99..4.89.9 -.74.4..3.87.89 -.997.57 3