A New Empirical Perspective on the CAPM Author(s): Marc R. Reinganum Source: The Journal of Financial and Quantitative Analysis, Vol. 16, No. 4, Proceedings of 16th Annual Conference of the Western Finance Association, June 18-20, 1981, Jackson Hole, Wyoming (Nov., 1981), pp. 439-462 Published by: Cambridge University Press on behalf of the University of Washington School of Business Administration Stable URL: http://www.jstor.org/stable/2330365 Accessed: 13-02-2016 20:36 UTC Your use of the JSTOR archive indicates your acceptance of the Terms & Conditions of Use, available at http://www.jstor.org/page/ info/about/policies/terms.jsp JSTOR is a not-for-profit service that helps scholars, researchers, and students discover, use, and build upon a wide range of content in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new forms of scholarship. For more information about JSTOR, please contact support@jstor.org. University of Washington School of Business Administration and Cambridge University Press are collaborating with JSTOR to digitize, preserve and extend access to The Journal of Financial and Quantitative Analysis. http://www.jstor.org This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSIS Volume XVI, No. 4, November 1981 ON THE CAPM A NEW EMPIRICAL PERSPECTIVE Marc R. Reinganum* Introduction of the capital The adequacy Lintner and Black [17], brium is now seriously Cheng and Graver [5] a security's beta even though it pricing determinant of equilibrium with different estimated While importance of beta, demonstrate betas systematically that The average securities. ferent from the average widely different returns Thus, estimated reliably measure a "risk with the evidence empirical betas an important is beta are different on standard suggest results portfolios market indices that returns These dif? not reliably indistinguishable in the market." "anomalies," are That is, stocks. some so that to average related stocks rates average The test avoided. are securities the cross-sectional not employed, statistically priced whether to assess of high beta possess based of the recent not systematically of low beta which is on empirical research returns betas estimated returns. are betas estimated of equilibrium In light experience regressions earlier seems to be consensus empirically are designed tests and Scholes Jensen, Black, determinant a security's to investigate cross-sectional across nificant is which plagued of the problems economic [3], be reexamined. should paper the statistical of return. as Basu [2], and Thompson [20]). [22], Reinganum (such Banz [1], determinant. that the claim Ball the current remains; an important pricing of this The purpose [18], studies may not be the sole however, evidence, empirical still still is Marsh empirical [11]) see [27], market equili? of capital representations (for example, [15], of earlier (CAPM) of Sharpe models pricing empirical Gibbons and Fama and MacBeth that as [4] challenged [8], the influence Yet, asset with average do not appear findings, the CAPM may lack along sig? content. The author wishes to thank Fischer of Southern California. University and Dick Roll, Kim Victor Joines, Dietrich, Terry Langetieg, Canto, Doug Black, Alan Shapiro. responsibility. Any errors that remain are the author's 439 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions to II. The development pricing Fama [10]). see example, The Beta and Test Hypothesis of the CAPM is well Design known and can be found elsewhere on the particular Depending set which emerge from the CAPM can be expressed relationships (for of assumptions, as the either: ) + 3. [E(R ) - E(R )] i m om om (1) E(R.)i = E(R (2) E(R.) l = R + 3. [E(R ) - Rl F l m F where: = expected E(R.) return E(R E expected ) return R? F E risk-free rate betas for the data of this average [23] to be consistent is rates assets related of the hypothesis supports the contention that that rejected the hypothesis would seem to indicate with betas are economically A straightforward, First, in period placed into beta. Then, each folios possess based B, the returns procedure significantly is that returns. The beta different experience would offer evidence Evidence pricing. the risk estimated, of the ten beta associated premia the beta hypothesis. and securities rank of their portfolios are to test average returns are estimated calculated of the component securities of ten portfolio invoked different are to test upon the relative the returns is employed betas security With the time-series statistical in esti? CAPM, the concern in equilibrium matter strategy with equal-weights portfolio. variate two-step one of ten portfolios by combining betas con? insignificant. A, individual in period variations in average betas that a necessary of the paradigm. representation Confirmation Thus, that to variations estimated two assets Namely, of the theoretical with different of return. and i; returns. with the CAPM is the common empirical that of asset implication. expected the testability questions is paper hypothesis an important must be systematically Roll uncorrelated of interest. different possess is on the market portfolio; E cov(R.,R E the beta i m)/var(R m) The two forms of the CAPM share While whose return 3.i with different mated betas i; ) E expected return on an asset with the market return; E(R dition on asset in hand, within a multi? whether or not the ten port? returns. 440 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions The composition of each of the revisions quency the daily firms turns. are created within the beta (1930-1979), betas security of the 1935-1939 ship are betas estimated are with data beta analyzed. Exchange annually. Thus, the 1964 beta with 1963 daily estimated are to identify used With monthly return every five updated from 1930-1934 portfolios. in the period being When analyzing Stock with 1964 data portfolios. portfolios estimated betas The fre? updated. and American revised upon security estimated the 1965 beta periodically base Exchange are portfolios based betas Similarly, rities upon the data depends the beta (1963-1979), is portfolio of the New York Stock returns portfolios beta are data the secu? for NYSE firms For example, years. to form the member? used of the frequency Regardless of updates, to the one in which portfolio prior re? returns are measured. Three different both daily estimators and monthly return returns Security squares. are betas data, are for daily priate the impact Williams returns of this potential data, [9] "market model" the estimators also used are estimator III. Tests To assess by Scholes and betas. security to different Recent may be inappro? problems. proposed to calculate of the results market beta. the estimated trading problem, the sensitivity least the CRSP value-weighted against for First, ordinary using on the market is this estimates. calculated of nonsynchronous because and Dimson [25] with daily that indicates however, research, are regressed and the computed coefficient returns, to compute beta used beta Hence, can estimators be investigated. This section estimated and sample selection are three contains part different statistically In the first parts. evidence during part, results 1964 through 1979 of monthly on 45 years based average the data the test In the next part, described. if port? to determine designed experience into Hypothesis of NYSE and AMEX companies returns The final are presented. of tests beta divided criteria on the daily based is The section returns. of the Beta the results reports with different folios for NYSE companies. returns A. Empirical The Data Stock and Sample return Center Chicago's return files turns (capital 1979. used for Research as of December New York Stock December data gains plus Exchange Criteria Selection in this Prices in Security 1979. dividends) the daily only on NYSE companies; however, file The daily of all file, (CRSP) Exchange return monthly and daily contains the monthly file the stock from the University gathered companies Stock or the American Unlike are analysis that the daily have traded of stock stock re? on the from July 1962 through contains information information on the monthly file 441 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions to January back dates time security Each is portfolios sample If a firm is delisted AMEX firms that of firms changes. returns survival during during the holding In any one given they No other is imposed. are held in the number of NYSE and year, in the sample that period. period, holding any funds returned period, for inclusion qualified estimation the data, is on securities placed the beta of the ten beta With the daily through the portfolio the end of the year. until and the composition estimated The only restriction 100 one-day such as are the sample yearly. least restriction, cash betas revised, changes have at 1926. between ranged and 2,000 2,700. are updated only because portfolios tion a firm is period, 678 in the 1930's B. excluded five every only if it The Test to 1296 Results the cross-sectional two reasons. studies pivotal which ended earlier with Daily relationship these First, in December, the beta estima? 40 one-month to have at least in the monthly sample from ranged between [11], the unlike Second, respectively. can be tested of the and Fama and MacBeth [5], 1968, for at least betas the time periods outside are primarily 1965 and June, and estimated returns and Scholes in which to study a good period 1979 represent years the hypotheses studies, During 1964-1979 Returns: Jensen by Black, years. criterion in the 1970's. from 1964 through The years from the above fails The number of NYSE firms included returns. differs with monthly data criterion The selection with AMEX firms as well as with and placing securi? NYSE companies. The first ties into one of ten beta to create are of the test stage with a value-weighted returns returns the daily matter tionship between mean returns Tables portfolios betas and returns of the ten beta 1 through created suggests, and be able are the daily with the different beta used betas the daily with equal-weights the portfolios. then one ought to observe portfolios 3 present by combining within of the component securities in the way the theory these above, In the next period, calculated are estimates and Dimson estimators Scholes-Williams, are portfolios beta As explained NYSE-AMEX market index. of the ten beta betas different portfolios. the "market model," computed using estimating Three portfolios. of ten beta sets three involves a positive the hypothesis to reject If betas that rela? the equal. return statistics estimates. Table for the ten beta 1 contains information betas were also calculated For the Scholes-Williams and OLS estimators, dif? were not significantly The results market index. using an equal-weighted with Dimson betas were not calculated ferent from those reported in the text. the equal-weight index. 442 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions TABLE 1 DAILY RETURN STATISTICS FOR THE TEN BETA PORTFOLIOS WITH BETAS ESTIMATED USING DAILY RETURNS, A VALUE-WEIGHTED INDEX, AND THE "MARKET MODEL" ESTIMATOR Estimated Portfolio Beta .05 Autocorrelations Skewness -.101 .33 The estimated of security betas. holding period. 1 2 3 5.601 .48 .28 .25 6.601 .52 .24 .21 .50 -.074 6.125 .47 .19 .18 .64 -.098 5.864 .47 .15 .16 .79 -.105 4.810 .45 .14 .14 .95 .056 6.025 .42 .11 .13 1.13 .012 5.569 .40 .09 .10 1.34 .177 5.730 .38 .07 .09 1.64 .166 5.764 .33 .06 .08 2.25 .314 5.788 .26 .02 .06 A mean return is calculated Mean daily returns through 1979. Standard errors are in parentheses. moments of the normal distribution. 2 .131 Kurtosis using 4009 trading day returns from 1964 are multiplied by 1000 for reporting purposes. measures are based on Skewness and kurtosis combination beta is just the linear portfolio (equal weights) in the year prior to the portfolio These betas are estimated 443 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions on the beta mator. The null into account The test turns. tested tic is 2.99. Thus, the ten portfolio re? under the null hypothe? of F(9,??) and 1.88, of this test statis? of identical as evi? of the low beta return daily of the high beta return daily 2.41 not be interpreted should rejection are the hypothesis level, the average the average exceeds actually portfolio This of the CAPM because in support dence between the values the one percent would be rejected. mean returns test 1, the computed value in Table even at This test. T-squared distribution levels, percent For the data respectively. an F(9,4000) has esti? are of the ten portfolios Hotelling's correlations contemporaneous statistic At the one and five sis. using the "market model" computed using the mean returns that hypothesis can be formally identical takes formed with betas portfolios portfolio by .03 percent. the results of the apparent because one observes that The skewness and kurtosis of the daily a six about ket experienced measures, were about ten standard is from the sample, deleted are portfolios the daily 1 that in Table is estimated to suspect however, confidence that it can be detected this phenomenon is Exchange Exchange. Thus, a "small" sample. the null during a fluke. a 16-year After companies the data 16 years data. collected as well analyzed are as in these nearly Furthermore, this that tests trade 1964- the fact the probability represents systematically those the period in any one year, reduces period all, data result this greater experience during portfolios an with not consistent actually portfolios to accept in which computer readable can Stock no reason, with constructing 1 are in Table of the high beta of the time for which CRSP has period is of portfolio of rejecting associated problems low beta While one might be able 1979. There in favor biased the results region, than those returns average statistical of the CAPM: the predictions are the tests the potential Despite matrix but not efficiently. auto- are positively portfolios mean returns. of identical hypothesis appropriate that observes One also somewhat leptokurtic. the variance-covariance consistently, for the high beta with the normal distribu? associated of the ten beta With autocorrelation, correlated. returns returns portfolios one observation measures and kurtosis those remain If this means. their the mar? 1970, of the high beta the returns gain; the skewness portfolios on May 27, that to out? sensitive are particularly revealed above the same as vitually the low beta tion; returns percent of significance In particular, seemed to be both skewed and leptokurtic. however, deviations statistical from normality. departures returns the portfolio Examination liers. the exact in interpreting One must be cautious that 30 percent is available the only for all Ameri? on the New York Stock would not seem to constitute 444 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions One potential beta to biases data. Table table similar are returns using a value of 1.85. percent level. too. Of course, The results borates in Table associated Dimson recently and inconsistent gested that betas one use with daily and leading lagged returns. In this security ket returns. This and Reinganum [21]. on Dimson's folio beta aggregated is exceeds coefficients return t-value in the previous there and average son betas of equality thesis the .01 higher between indistinguishable. estimated The evidence betas are analyzed is returns. returns not always in Tables used One observes return evident in Roll betas based for .06 per? returns jointly beta Dim? between the hypo? portfolios at are portfolios, average on 16 years This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions per? reported considered 445 port? .001 by only however, with higher based [24] except between for the intermediate 3 is mar? leading association of the two extreme 1 through coefficients. of the high beta portfolio ten portfolios associated regression. that, are One would reject, Furthermore, on security with estimated the portfolios no immediately security to regress slope As with the portfolio means for all but the average level, statistically portfolio of 0.09. Dimson sug? and five to those of the low beta not might be is technique 20 lagged the mean daily Furthermore, are estimator market returns constructed of all corro- betas one runs a multiple methodology. returns cent with an associated tables, are This enough problem. identical virtually returns returns. estimation using here apply the average the sum of the estimated calculated is the mean daily this on at the five method for estimating regression, simply are the mean daily a serious the contemporaneous) of a simple procedure and .08 percent. cent as behind 3, the ten portfolios P9, portfolio well regressions paper, In Table The idea instead Thus, The estimated (as is coefficients mean takes F-test other. even the Scholes-Williams if nontrading of identical in estimated in average returns average above from each in the Scholes- hypothesis at best, that, differences an aggregated data. higher discussed differences 1; positive that argued biased caveats indistinguishable with positive Even using the null 2 seem to indicate in Table 1. the hypothesis technique, the statistical are portfolios the evidence reliably T-squared possibility for portfolios the appropriate one would not reject Hence, might The numbers presented experience If one tests portfolios. Hotelling's of the ten beta portfolios This statistics betas. the that of security then this problem, the results. returns in Table reported the low beta estimator, affect estimated to those than do the high beta a serious the daily 1 is estimates least-squares is which could 2 contains in Table presented ordinary If nontrading with Scholes-Williams constructed Williams using in estimation now explored. this of the results created on daily based lead is are portfolios betas criticism returns. of daily TABLE 2 DAILY RETURN STATISTICS FOR THE TEN BETA PORTFOLIOS WITH BETAS ESTIMATED USING DAILY RETURNS, A VALUE-WEIGHTED INDEX, AND THE SCHOLES-WILLIAMS ESTIMATOR Autocorrelations lA mean return is calculated Mean daily returns through 1979. Standard errors are in parentheses. moments of the normal distribution. 2The estimated betas. of security holding period. Kurtosis 1 2 3 7.228 .48 .25 .23 6.205 .49 .20 .20 5.668 .48 .19 .17 5.265 .44 .15 .15 4.484 .43 .14 .14 5.354 .41 .12 .12 5.945 .39 .09 .11 6.152 .37 .06 .09 6.345 .35 .06 .09 5.935 .30 .05 .08 using 4009 trading day returns from 1964 are multiplied by 1000 for reporting purposes. measures are based on Skewness and kurtosis combination beta is just the linear (equal weights) portfolio in the year prior to the portfolio These betas are estimated 446 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions TABLE 3 DAILY RETURN STATISTICS FOR THE TEN BETA PORTFOLIOS WITH BETAS ESTIMATED USING DAILY RETURNS, A VALUE-WEIGHTED, AND THE DIMSON ESTIMATOR Autocorrelations .43 .16 .16 .43 .13 .15 .43 .13 .13 .43 .13 .13 .42 .12 .13 .41 .11 .13 .41 .11 .12 .40 .09 .11 .08 .10 .09 .11 .35 A 1979. errors of the mean return is calculated using 4009 trading day returns from 1964 through Standard Mean daily returns are multiplied by 1000 for reporting purposes. measures are based on moments Skewness and kurtosis are in parentheses. normal distribution. The estimated betas. of security holding period. combination beta is just the linear (equal weights) portfolio in the year prior to the portfolio These betas are estimated 447 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions return data. series could One potential be that since especially ary, can be examined year portfolios'results lem. the most succinct Perhaps the differences to present beta the different are these results exceeds beta Using rors is portfolio portfolio presented in Tables 1 through in Tables 1 through 3 as does are really the prior in particular folios largest This error. the ten portfolios Table portfolios, the danger portfolio by two in one portfolio not statisti? are the findings corroborate the results of interpreting effects of the however, return, throughout these betas estimated are the 16- possibility can be investigated in which average period during betas period the entire are betas that are meas? returns betas in estimated are port? with the estimated the betas by computing returns of are measured. with the holding 16-year the portfolio the extreme beta ranked, whose betas may contain the grouping is on security securities estimators holding results in which portfolio formed based are in the year 5 compares for the three results in the year portfolios Since year. er? of the high the differences years, for the above explanation not different that Recall ured. in average portfolio. of the high beta the average illustrating signifi? of the low return the average return Thus, in average not seem great. Another possible betas 3. errors of the low beta return of the high beta The average the year-by-year Hence, of the high return by two standard of the low beta return betas are more than two standard with Dimson betas, In the remaining between of the mean return the average return the average exceeds significant. year period with portfolio. portfolio of the 16 years, the average as well. of the years port? when security none of the differences estimator, in two of the 16 years. low beta is calculated not statistically are estimate of the high beta created exceeds errors prob? analysis the differences 15 years, portfolios than the average greater standard cally years, In seven portfolio a serious relationship of the low beta return the high and low beta For portfolios beta betas the average estimator, the point the Scholes-Williams from zero. is of this For example, returns. For the other that exceeds between returns no significant the average of these portfolio the year-by- the high and low beta between reveal the high and low beta between In nine cant. this for security also in only one of the 16 years. returns whether differences with the "market model" portfolio however, the results returns and average betas portfolio calculated beta to gauge yearly; estimators. The year-by-year estimated revised way to convey in average 4 reports Table folios. are the portfolios station? may not be sufficiently distribution the statistical from such a time inferences with drawing problem sample. betas period For each computed with the "market model," 448 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions set of ten Scholes- TABLE 4 MEAN DIFFERENCES IN DAILY RETURNS BETWEEN THE HIGH AND LOW BETA PORTFOLIOS ON A YEARLY BASIS (Betas Computed with Daily Returns) Beta Mean differences are T-values poses. ing days. Estimator in daily returns are multiplied pur? by 1000 for reporting 250 trad? in parentheses. Each year contains approximately 449 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions TABLE 5 COMPARISON OF NYSE-AMEX PORTFOLIO BETAS ESTIMATED IN GROUPING PERIODS AND HOLDING PERIODS GROUPING PERIOD ESTIMATOR Aggregated Coefficients GP MM SW AC .72 .82 1.26 .29 .71 .80 1.14 1.21 .62 .76 .86 1.20 .88 1.28 .89 .81 .91 1.25 .84 .96 1.36 1.15 .87 .97 1.35 Scholes-Williams MM SW GP AC .07 .43 .53 1.07 .41 .53 .64 1.06 .59 .65 .77 .75 .76 .91 -.50 6 .95 .95 1.07 1.46 1.07 .96 1.07 1.46 1.41 .93 1.03 1.40 7 1.13 1.06 1.17 1.54 1.24 1.07 1.17 1.53 1.71 .99 1.09 1.51 8 1.34 1.19 1.28 1.63 1.44 1.18 1.28 1.64 2.06 1.08 1.18 1.59 9 1.64 1.37 1.42 1.75 1.72 1.34 1.40 1.76 2.56 1.18 1.27 1.73 2.25 1.69 1.67 1.95 2.25 1.60 1.62 1.95 3.77 1.32 1.39 1.89 High Beta GP stands for the estimated beta of the portfolio during the grouping created with "market period. Grouping period betas are shown for portfolios model" estimates, Scholes-Williams and Dimson's aggregated coefficients estimates, For each set of portfolios, estimates. three estimated holding period betas are shown: MM ("market model"); SW (Scholes-Williams); and AC (Dimson's aggregated In a grouping period, a portfolio beta is just the equalcoefficients method). of estimated betas within that portfolio. The security weighted combination betas over of the portfolio grouping period betas reported above are the averages the 16 grouping periods from 1963 through 1978. Holding period betas are calcu? 16 years of daily portfolio lated by analyzing and market returns (1964-1979). 450 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions and Dimson's Williams, of portfolios for example, "market model," are presented for the portfolios beta created portfolio "market model" beta portfolio gregated coefficients the beta of the lowest AC portfolio is 5 also Table of estimated periods, regardless of the estimator the Scholes-Williams estimates. with the "market model" for these and 1.29 estimates, exhibit spreads is to form these roughly the rank ordering equivalent and Dimson's cluding the portfolios that betas ent estimated The Test is result to hold over Jensen, and Scholes average specific a longer returns. of the low beta section is For example, portfolio is betas within period Indeed, betas confident 4 possess whether in condiffer? widely the "beta or whether, evidence Table statistically in fact, it not appears that portfolios indistinguishable 2, the mean excess the two standard does from the work of Black, with the proposition possess in their of Scholes-Williams 1935-1979 to the 1964-1979 estimated which periods. to investigate may be consistent seem to estimator conclu? 1 through holding estimators similar 5 that one may feel Hence, in Tables the portfolio time horizon. [5] different betas. with Monthly Returns: of this matter" analyzed during Results The purpose with widely coefficients The spread but also betas, "market model" on the basis created rank the on the Scholes- based 1.16 the other in Table One discovers portfolios. aggregated of the periods, between estimates. Thus, of estimated to those can be drawn for portfolios sions is Consider are perfectly the spread periods .88. is the holding the holding portfolios on the "market model" based on Dimson betas based portfolios not only tend to preserve used during portfolios the rank preserves During Furthermore, the holding and Scholes-Williams during betas. of these of the betas estimates high and low beta C. of "market model" between the ten portfolios. to create used in Ac betas perfectly of ten portfolios set For example, of the high? two estimators. with those estimator of the ag? betas the beta in "market model" almost in esti? attenuation. to 1.26; the spread Thus, created for each betas formed on the basis portfolios to 1.89. for portfolios severe reveal from -.50 of the highest to the attenuation similar The estimated portfolios. rises each is alone. "market model" beta The attenuation to 1.69. than the spread that the estimated portfolios however, from 3.77 reveals ordering beta AC portfolio smaller betas period Williams from 2.25 betas of the high and low of the lowest beta similarly, beta portfolios, drops the AC portfolios ordered .40; drops by the "market model" exhibited holding to betas Thus, betas period of "market model" in the estimated of the Scholes-Williams mated betas est formed on the basis set estimators. and Dimson holding the estimated from .05 rises based attenuation even though each estimators, on only one of these Scholes-Williams, For example, portfolios. beta with betas 5 one observes In Table coefficient aggregated is error return, confident 451 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions (R -R ), interval about the mean excess and Scholes Jensen, are and the intercepts different estimated folio test security betas ten beta portfolios the ten beta are estimated is different nificantly seems to indicate of average the rank ordering betas. One cannot, that is of the difference zero. one could dard argue that, confidence error possess tribution. 1.22; these of identical one clearly Thus, are statistics account test using Hotelling's mean returns, the test cannot analyzed reject in Table the null of estimated The than the low or statistically of the high the returns error per month, the standard less between than two standard the average errors from computed with 540 observations, the power of the test, formally the alone. returns reliable the mean difference is glance, In addition, estimates between percent might be more appropriate on the data are based per month, whereas percent average are For example, .9 percent. from point higher .580 1.5 with the tables to the rank ordering the differences portfolios into mean returns Based sig? portfolios estimated At first 1979. matter. only corresponds 6 is One can also interval. portfolio percent. taking region identical hypothesis is since Furthermore, error .294 about the mean difference of the high and low beta returns ought to have in these betas is possess in Table portfolios is draw inferences portfolios while Indeed, and low beta is period from 1975 through OLS betas The statistics portfolio not mean that does portfolios grouping is of the monthly for the ten beta of their estimated returns however, the high beta significant. that of the low beta return the average period, the monthly returns portfolios from 1935 through data of the high beta return average to the and port? and membership in the period statistics on the basis of monthly return the evidence similar with equal-weights holding the monthly return NYSE market index. CRSP equal-weighted upon 45 years returns. estimation The first then the ten beta matter, securities formed by grouping portfolio is with mean returns. 6 presents Table the portfolio. for between In the first squares computed by combining the last betas least and positive average of one year. ordinary regressions if portfolios the initial In the next period, within 1934; through If estimated 1979. beta are portfolios that Black, relationship with the monthly data instead years using inverse indistinguishable except established. of the securities from 1930 section five (3 > 1) what one would expect used procedure "market model" betas This had statistically are periods in their (3 < 1). precisely in the previous holding returns the intercepts betas Furthermore, portfolio. with high estimated is betas The two-stage one employed fact that with low estimated portfolios betas note for portfolios negative of the high beta return a three two stan? than the conventional whether the ten beta test. T-squared statistic assumes 6, the value hypothesis standard portfolios Under the null an F (9,531) of the test of identical 452 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions dis? statistic mean returns TABLE 6 MONTHLY RETURN STATISTICS FOR THE TEN BETA PORTFOLIOS BASED ON BETAS ESTIMATED USING AN EQUAL-WEIGHTED NYSE INDEX AND THE "MARKET MODEL" ESTIMATOR ?2 Autocorrelations 3 -.00 .04 Kurtosis 1 -.287 4.961 .11 -.388 4.521 .03 .02 .03 -.111 5.131 .00 .07 .04 .206 6.300 .02 .08 .00 .375 7.860 -.01 .10 -.02 .321 7.022 .03 .11 .827 10.000 .02 .08 .453 6.290 .02 .10 .876 8.422 .03 .08 -.01 .948 5.642 .03 .09 -.02 Mean returns are multiplied by 100 for reporting The statistics Standard errors are in parentheses. 1%. from 1935 through 1979. observations 2The estimated These betas betas. holding periods. 2 Skewness .00 -.00 .01 A 1.0 equals purposes. are based on 540 monthly of security combination beta is the equal-weighted portfolio in the five-year are estimated prior to the portfolio periods 453 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions even at the .05 The data that except dex. level. OLS betas security the high beta Again, the low beta portfolio, month rather than is ten beta portfolio 7. based Thus, and estimated betas on the evidence a statistically reliable mated portfolio betas. in these additional three data? be addressed. First, the estimated to the grouping similar in Table tosis measures based upon multivariate probably not stationary position of each folio returns does subperiods, from zero the average of the high beta estimated the high and low beta any of the nine One possible during that within the of monthl the holding the empirical drawn from test These between index, and kur? statistics corroborate data estimated betas portfolios are since the com? In three is the mean difference does not exceed betas eight the average on security based port? more than two of the other exceeds portfolio portfolios the finding and average is the 8 contains formed using the mean difference When grouping portfolios Table years. 6 and in Tables especially with portfolios of the low beta index, summarized of the high and low beta in only one subperiod. with the value-weighted returns, to the skewness period, every five For example, portfolio. methods portfolio of the ten beta 45-year subperiods. return and esti? of 45 years given (refer for the data distributions changes not exist. to be valid? results relationship in Table returns the results are the conclusions the monthly returns five-year value the average not appear to average Finally, nonnormal the entire portfolio systematic errors standard between over with the equal-weighted estimated return beta of the nine a strong still between mean differences that normality the returns because important in each are of the subperiod An analysis betas? appears 6 and 7), does of the ten portfolios betas the although 6, however, on the analysis based period of monthly returns distribution 7 is should with the findings are Secondly, periods related issues for the computed with standard betas that level, portfolio average per difference rank correlated there do not seem to be reliably consistent subperiods between .4 percent than the critical less in Table two tables, than mean returns .05 not perfectly conclude While one might tentatively and market indices the data are relationship at the slightly just Unlike distribution. returns is 1.88, about greater with this of identical hypothesis NYSE market in? return is associated would not be rejected statistic, for the F(9,531) the difference case the t-statistic the null still portfolios an average experienced 6 in Table analyzed computed with a value-weighted but in this .85 percent; of the test value are to the data similar portfolio In addition, 1.59. only 7 are in Table analyzed in average two standard betas returns errors subperiods. explanation for the above results is that the holding 454 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions period in TABLE 7 MONTHLY RETURN STATISTICS FOR THE TEN BETA PORTFOLIOS BASED ON BETAS ESTIMATED USING A VALUE-WEIGHTED NYSE INDEX AND THE "MARKET MODEL" ESTIMATOR Estimated Portfolio Beta Autocorrelations Skewness Kurtosis 1 2 3 .175 7.818 .11 .01 .05 .44 .69 -.171 4.259 .03 .03 .04 .84 -.166 5.585 .02 .06 .02 .01 .98 .190 6.444 .00 .08 1.10 .605 8.797 .03 .10 1.23 .568 8.087 .01 .10 1.36 .064 4.588 .04 .08 -.00 1.52 1.401 14.888 .02 .07 -.00 1.71 .614 6.134 .03 .10 -.01 2.13 .680 5.240 .02 .10 -.01 Mean returns are multiplied by 100 for reporting The statistics 1%. Standard errors are in parentheses. from 1935 through 1979. observations 2 betas. holding The estimated These betas periods. -.01 .01 A 1.0 equals purposes. are based on 540 monthly of security combination beta is the equal-weighted portfolio in the five-year are estimated prior to the portfolio periods 455 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions TABLE 8 MEAN DIFFERENCES IN MONTHLY RETURNS BETWEEN THE HIGH AND LOW BETA PORTFOLIOS DURING THE FIVE-YEAR PERIODS FROM 1935 THROUGH 1979 Beta Estimator, NYSE Market Index Market Model, Equal-Weighted Period Market Model, Value-Weighted Overall .579 (1.97) .416 (1.59) 1/35 - 12/39 1.297 (0.78) 1.172 (0.76) 1/40 - 12/44 1.690 (1.42) 1.364 (1.34) 1/45 - 12/49 .295 (0.42) .287 (0.45) 1/50 - 12/54 .688 (1.32) .692 (1.45) 1/55 - 12/59 -.049 (-.11) .062 (0.14) 1/60 - 12/64 -.384 (-.95) -.408 (-1.12) 1/65 - 12/69 .757 (1.47) .529 (1.12) 1/70 - 12/74 -.853 (-1.06) -1.009 (-1.38) 1/75 - 12/79 1.775 (2.30) 1.053 (1.88) in monthly returns Mean differences are in parentheses. T-values purposes. based on 540 months. are multiplied by 100 for reporting for the overall Results period are 456 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions do not differ of the ten portfolios betas seems ruled out by evidence son of the grouping tenuation holding with the holding betas of the high and low beta betas of these the holding evidence betas period Since is might lead a proper to inappropriate affect might seriously null month is thesis however, population; to note respect to location, each dispersion, will to the normal distribution to detect signed the null as Table year returns overall stringent .05 could degrees the overall .01 At the not be rejected a criterion against level, significance for the portfolios is Under the The null It assumed. hypo? is im? tremendously is The test of other 10) drawn from the same with relative and kurtosis test. of only de? of one portfolio Under portfolios. distributed approximately of freedom. period as during as well which to test the level betas estimated of identical the may not be too Furthermore, returns five- during portfolios the hypothesis. of identical with security .01 of the nine hypothesis of ten beta set for either each the null level, significance of ten for the two sets statistics test the hypothesis created is may differ is statistic with 540 observations, Indeed, period. this for a (from 1 through for the monthly returns test to test ranking. skewness Hence, not invalidate the chi-square 10 presents subperiods. or both. do not suffer the assumption returns monthly returns between tendency one also performed. any other than the same-month returns with nine during as likely is effect of ten monthly returns the appropriate hypothesis, chi-square portfolios any systematic or be smaller to exceed for a beta of monthly observations set portfolios in monthly returns To avoid of the ten portfolio set One notices of the data. and kurtosis test. based statistics can be employed test T-squared independence that portant each test the monthly returns the skewness to be as assumed not imply that does in to con- do not appear to the normal distribution; returns, rank test [13] any one ranking hypothesis, in a given that Hotelling's Friedman's normality, period differences of the ten beta returns A nonparametric if one believes effect beta be? the hold? Furthermore, whether interpretations relative portfolio daily autocorrelation. from severe the is, betas period significant is concern the monthly portfolio tend to be skewed and leptokurtic unlike are at? the grouping by the grouping of monthly returns distributions 6 and 7 that .9. established there than are holding than greater that indicates form to the normal distribution, that, still that of the ten portfolios. the empirical on normality in estimated One observes portfolios; to 1.0 possibility a compari? presents betas. period closer the rank ordering preserve this Thus, are portfolios the difference Nonetheless, betas ing period observes table betas tween the two extreme portfolios in Tables This period period betas. 9. in Table contained this however, other; in the estimated betas. period from each could at the not be rejected with the value-weighted 457 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions TABLE 9 COMPARISON OF GROUPING PERIOD AND HOLDING PERIOD BETAS FOR THE TEN BETA PORTFOLIOS OF NYSE STOCKS (Betas computed with Monthly Returns using the "Market Model" Estimator) NYSE Market Index combina? a portfolio In a grouping period, beta is just the equal-weighted The grouping period tion of estimated betas within that portfolio. security betas reported above are the averages of portfolio betas over the nine five-year from 1930 through 1974. grouping periods 2 The holding period betas market returns returns against are rounded to two significant are calculated monthly portfolio by regressing which Standard errors, from 1935 through 1979. are reported in parentheses. digits, 458 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions TABLE 10 CHI-SQUARE STATISTICS BASED ON FRIEDMAN'S NONPARAMETRICRANK TEST FOR A BETA EFFECT Beta Estimator, Market Model Equal-Weighted Period NYSE Market Index Market Model Value-Weighted 19.74 16.23 1/35 - 12/39 4.94 6.88 1/40 - 12/44 10.70 10.81 1/45 - 12/49 2.81 4.89 1/50 - 12/54 13.66 19.71 1/55 - 12/59 13.43 7.57 1/60 - 12/64 8.58 12.21 1/65 - 12/69 29.32 18.96 1/70 - 12/74 22.69 23.45 1/75 - 12/79 26.75 18.89 Overall with in this table are distributed statistics The chi-square presented for this of the 1 and 5 percent limits The values nine degrees of freedom. are 21.65 and 16.93, distribution respectively. 459 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions The subperiod NYSE market index. between tionship formed with betas folios of identical returns low beta folio. dex, the null nine subperiods, but in this do yield In those one could tables, they masked the great While returns. test This is why the parametric the reason also in differences reflected mated betas course, this identical [22] returns. that portfolios higher that that is, are the returns premia importance for securities during to mean that this betas average betas are In this Of result. securities possess and Reinganum [2] returns with common re? portfolio not significantly cross-sectional 460 This content downloaded from 202.28.191.34 on Sat, 13 Feb 2016 20:36:33 UTC All use subject to JSTOR Terms and Conditions differ? sense, do not seem to be of economic Stock 20 nearly study demon? estimated on the New York and American esti? Evidence this in average 1979, difrerent of this The findings portfolios portfolios. 1964 through Banz time period, to differences related all are betas returns. average in portfolio of high beta traded did test, portfolio to 1935 corroborates firms. of large with these returns T-squared with widely firms experienced differences of low beta associated back of small not reliably ent from the returns risk Indeed, indicated of portfolio During portfolios indistinguishable not be construed than portfolios cross-sectional market indices turns; should finding rank test not be distinguished in estimated returns. portfolio dating portfolios with the consistent Hotelling's differences NYSE-AMEX stock statistically average report percent strate possess on NYSE stock based that indicates the evidence in average of portfolio Conclusion whether investigates paper that mean returns. IV. This with to the extent could returns procedure, of identical the hypothesis not reject appeared in the time-series variability between to be consistent nonparametric of portfolio the month-by-month rankings relationship a rank ordering on Friedman's based 6 and 7. with the time-series exhibited Yet these in Tables analyzed out to be deceptive associated returns that betas and sys? persistent returns. a monotonic but notice turned variability from random rankings. higher experienced portfolio of the data the nature returns the average CAPM, the hypothesis in? in only one of the level and portfolio betas and estimated But the average the CAPM. estimated not help returns portfolio average port? a value-weighted a strong, do not seem to detect into insights .01 of the return of the high beta against at the sub? portfolio. between relationship the average calculated For port? of the nine in three return rela? the hypothesis index, level the low beta subperiod tests The nonparametric tematic the average would be rejected hypothesis .01 however, subperiods, exceeded than the high beta returns that three formed with betas For portfolios was not present. returns at the would be rejected actually portfolio and portfolio a systematic that computed with the equal-weighted In one of these periods. tests betas estimated seem to indicate results the or empirical Exchanges. 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