THE MARKET PORTFOLIO MAY BE MEAN-VARIANCE EFFICIENT AFTER ALL

Transcription

1 THE ARKET PORTFOIO AY BE EA-VARIACE EFFICIET AFTER A OSHE EVY and RICHARD RO ABSTRACT Testng the CAP bols down to testng the mean-varance effcency of the market portfolo. any studes have examned the meanvarance effcency of varous l market proxes by employng the ple return parameters, and have concluded that these proxes are neffcent. Employng dfferent shrnkage correctons does not help n ths regard. These fndngs cast grave doubt about one of the cornerstone models of modern fnance. In ths study we take a reverseengneerng approach to the problem. Gven a market proxy, we fnd the mnmal varaton of the ple parameters requred to ensure that the proxy s mean-varance effcent. Surprsngly, we fnd that slght varatons of the ple parameters, well wthn the estmaton error bounds, suffce to make the proxy effcent. Thus, conventonal market proxes are shown to be perfectly consstent wth the CAP. Keywords: ean-varance effcency, CAP, portfolo optmzaton, beta. evy s from the Jerusalem School of Busness Admnstraton at the Hebrew Unversty, and Roll s from the Anderson School of anagement at UCA.

2 ITRODUCTIO 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]). 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 the market proxy effcent. Thus, the emprcally measured return parameters and the market portfolo weghts are perfectly consstent wth the CAP. How s ths possble, and how can t be reconcled wth the many prevous studes that have shown that the market proxy s 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 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 evy [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.

3 take a reverse approach: we look for the return parameters that ensure that the market proxy s effcent, and that are as "close" as possble to ther ple counterparts. Surprsngly, we fnd that parameter sets that make the market proxy effcent can be found very close to the ple parameter set. Ths paper s organzed as follows. In the next secton we ntroduce the methodology employed. Secton II descrbes the data and the results. In Secton III we dscuss mplcatons for asset prcng. Secton IV concludes. I. ETHODOOGY Gven a market proxy, m, we look for the mnmal varaton of the ple parameters that makes ths proxy 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 and, respectvely. C denotes the ple covarance matrx, and ρ denotes the ple correlaton matrx. We are lookng for 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 the sake of smplcty, when consderng the covarance matrx C we restrct ourselves only to varaton of the standard devatons,.e. we assume that the correlaton matrx s the e as the ple correlaton matrx:

4 3 () = C ρ O O Allowng the correlatons to vary as well ntroduces techncal dffcultes, but can only make our 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: ( ) ( ) ( ) = = + ) (,,, D α α, () where 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 reason that we dvde the devatons n () by - we want to "punsh" devatons n the parameters of assets wth low standard devatons more heavly than smlar devatons n assets wth hgher standard devatons. We would lke to stress that the ultmate test of whether a set 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 these devatons. For

5 4 example, to consder a parameter set as "reasonably close" to the ple parameters we requre that 95% or more of the parameters are wthn the 95% confdence ntervals of the ple parameters. 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: () = z z z m m m r r r q x x x ρ O O 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 + varables n the optmzaton: s ', s ', q and r z. Any set of these + parameters satsfyng () makes the proxy portfolo mean-varance effcent (see, for example erton [97] and Roll [977]). We are lookng for the set of

6 5 parameters ( ), that satsfy ths condton and are closest to the ple parameters 4. 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 Subject to: () = z z z m m m r r r q x x x ρ O O () = ' m x () ρ = m m x ' x O O. 4 The optmzaton problem solved s smlar n sprt to Sharpe's [7] "reverse optmzaton" problem. evy [7] employs a smlar technque to fnd mean-varance effcent portfolos that have all-postve weghts.

7 In the next secton we solve these optmzaton problems wth emprcal stock data n order to fnd out how large the devatons from the ple parameters must be n order to ensure mean-varance effcency. II. DATA AD RESUTS Our 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 average ple correlaton s.4. The market proxy weghts are gven by the December 6 market captalzatons of these stocks: market cap of frm x. m = j= market cap of frm j The proxy portfolo and the ple mean-varance fronter are descrbed by the trangle and thn lne n Fgure, 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. To fnd the set of parameters that make the proxy portfolo effcent and are closest to the ple parameter we solve Optmzaton Problem. To numercally solve ths constraned mnmzaton problem we employ atlab's fmncon functon, whch s based on the nteror-reflectve ewton 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 6

8 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 [ ]. 5 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 are sgnfcantly dfferent than ther ple counterparts. (Please nsert Table I and Fgure about here) To conform that the parameters (, ) make the proxy portfolo meanvarance effcent, we 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 ( 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 s s gven by 9. < 9 s < 5. 6 or:. 758 < s <. 66. Alternatvely, ths range can be also stated as. 79 < s <. 39. n 7

9 very close to the trangle), they do have a bg effect on the shape of the fronter. Recall that we do not modfy any of the correlatons. What s the reason for the modfed fronter beng much flatter than the ple fronter? The explanaton can be found n Fgure, whch shows the adjustment to the expected return,, as a functon of the ple 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 horzontcal lne). Fgure shows that the correctons to the ple means mpled by our optmzaton are n the sprt of standard statstcal shrnkage methods. However, unlke the standard shrnkage methods whch do not make the proxy portfolo effcent, the methodology employed here ensures that ths s ndeed the case. (Please nsert Fgure about here) 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 of 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. 8

10 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 ). These values came out as a result of the optmzaton problem (and the gven 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 whch are consstent wth these belefs. For example, suppose that we would lke to fnd a varaton of the vectors and such that the proxy portfolo s effcent and has an expected return of and a standard devaton of = % and = 4%. Can we fnd such a varaton of the return parameter vectors that wll be consstent (n a statstcal sense) wth the ple parameters? To answer ths queston, we can solve Optmzaton Problem wth = % and = 4%. We wll consder the soluton (, ) consstent 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. For a more complete pcture we would lke to scan the meanvarance plane and to map the range of proxy portfolos' return parameters, and, whch can be made consstent wth the CAP and the stocks' 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 consstent wth the ple parameters, (, ), the pont s marked as a flled crcle; f not, t s left transparent. For example, the pont ( = %, = 4% ), (ndcated by a lttle arrow n Fgure 3), s ndeed consstent wth the ple parameters and the proxy beng 9

11 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 very 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%. When we repeat the analyss for the 5 largest stocks (nstead of the ) we obtan very smlar results, as shown n Fgure 4. Thus, the results seem robust to the dentty of stocks and to the number of stocks employed n the analyss. (Please nsert Fgures 3 and 4 about here) III. IPICATIOS FOR ASSET PRICIG The Securty arket ne (S) formula s probably the most wdespread method for estmatng the cost of captal and for prcng rsky assets. Usng beta and the S 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, as t s not clear how well the S relatonshp emprcally holds, there are serous doubts about employng betas for prcng. 6 Whle we can not prove that the S 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: 6 Ths s, of course, one of the major debates n fnance. See, for example, Renganum [98], evy [98], akonshok and Shapro [986], Chen, Roll, and Ross [986], Fama and French [99], and Roll and Ross [994].

12 xmj j ρj j= = xm' Cxm β (3a) xmj j ρj j= = xm' C xm β, (3b) where x m are the market portfolo weghts. The true cost of equty of frm s we employ the observable β. If n the S formula nstead of the correct β, how well wll our cost of captal estmate be? In other words, how close are β and β? The answer s shown n Fgure 5. The fgure reveals that the dfference between and β β s very small. The reason for ths s that both the denomnators and the numerators of (3a) and (3b) are very smlar. The varance of the market proxy s very smlar whether the true parameters or the ple parameters are employed (compare the horzontal locaton of star and the trangle n Fgures 3 and 4). As for the covarance n the numerator, note that, and n addton, the devatons tend j to cancel each other out n the summaton, as n some cases >, whle n others < (see Column 7 n Table I). j j j As the market proxy s effcent wth the true parameters (, ), the followng relatonshp holds exactly: = r + β ( r ), (4) f where rf s the approprate rsk-free rate. As m f β β, employng the S wth the ple beta, as s very commonly done n practce, provdes a very good estmate for the true expected return: [ r + β ( r )] = β ( r ) β ( r ) f m f m f m f j j. (5) So t s not only the case that the estmates of betas are statstcally more stable than those of the expected return. In addton, f the CAP holds n a way that s

13 consstent wth the ple parameters, the dfference between the ple betas and the true betas s much smaller than the dfference between the ple average returns and the true expected returns. (Please nsert Fgure 5 about here) IV. COCUSIO 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 ths percepton. We show that small varatons of the ple parameters, well wthn the range of estmaton error, can make the market proxy mean-varance effcent. Whle these parameter varatons are smlar n sprt to the dea of "shrnkage", they are dfferent than those obtaned wth the standard statstcal shrnkage methods: they are the result of a "reverse optmzaton" problem. In ths reverse optmzaton we look for return parameters whch make the market proxy mean-varance effcent, and are as "close" as possble 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 leads us drectly to these parameter sets.

14 These fndngs mply that the CAP s completely consstent wth the emprcally observed return parameters and the market proxy portfolo weghts. oreover, the ntutve dea that shrnkage correctons should ncrease the emprcal valdty of the CAP s shown to be vald - wth the rght correctons, whch are small, the CAP holds perfectly. Our analyss also shows that n ths framework employng the ple betas provdes an excellent estmate of the true expected returns. 3

15 REFERECES Fama, E. F. and K. R. French, 99, The cross-secton of expected stock returns, Journal of Fnance 47, 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, Jagannathan, R., and T. a, 3, Rsk reducton n large portfolos: a role for portfolo weght constrants, Journal of Fnance 58, Jobson, J.D. and B. Korke, 98, Potental performance and tests of portfolo effcency, Journal of Fnancal Economcs, Kandel, S., and R.F. Stambaugh, 987, On correlatons and nferences about mean-varance effcency, Journal of Fnancal Economcs 8, 6-9. akonshok, 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. evy, H., 98, A test of the CAP va a confdence level approach, Journal of Portfolo anagement. evy, H., 983, The captal asset prcng model: theory and emprcsm, Economc Journal 93, evy,., 7, Postve portfolos are all around, Hebrew Unversty Workng Paper. ntner, J., 965, Securty prces, rsk, and the maxmal gans from dversfcaton, Journal of Fnance, (4), acknlay, A.C., and.p. Rchardson, 99, Usng generalzed method of moments to test mean-varance effcency, Journal of Fnance 46, erton, R. C., 97, An analytc dervaton of the effcent portfolo fronter, Journal of Fnancal and Quanttatve Analyss 7, Reganum,. R., 98, A new emprcal perspectve on the CAP, Journal of Fnancal and Quanttatve Analyss 6, 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, 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,

16 Shanken, J., 985, ultvarate tests of the zero-beta CAP, Journal of Fnancal Economcs 4, Sharpe, W. F., 964, Captal asset prces: A theory of market equlbrum under condtons of rsk, Journal of Fnance 9, Sharpe, W. F., 7, Expected utlty asset allocaton, Fnancal Analysts Journal 63, 8-3. Zhou, G., 98, Small ple tests of portfolo effcency, Journal of Fnancal Economcs 3, 65-9.

17 Table I The Sample Parameters and Closest Parameters Ensurng that the arket Proxy s ean-varance Effcent For the sake of brevty we report here only the parameters of 3 (out of the 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 (ths s also true for the 7 other stocks not shown n the table). Column (7) reports the rato between the varances ( ) and the ple varances. The 95% confdence nterval for ths rato s [ ] (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 [ ] )

18 Fgure : The Effcent Fronter and arket Proxy wth the Sample and the Adjusted Return Parameters. The thn lne and the trangle (partly hdden behnd the star) show the meanvarance 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 meanvarance fronter and the market proxy wth the adjusted parameters (, ). Wth these parameters the market proxy s mean-varance effcent.

19 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 n the sprt of statstcal shrnkage methods.

20 Fgure 3: The Set of Proxy Portfolo Parameters Consstent wth ean- Varance Effcency and the Sample Parameters stocks. For each pont on the mean-standard devaton plane, (, ), we solve Optmzaton Problem. The resultng parameter set, (, ), 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 whch 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.

21 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 (, ) are consstent wth the effcency of the market proxy. Ths suggests that the results are robust to the number of stocks employed n the analyss.

22 Fgure 5: The Relatonshp Between Sample Betas and the True Betas. The true parameters are those 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 true expected returns.