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  • 1. A New Empirical Perspective on the CAPMAuthor(s): Marc R. ReinganumReviewed work(s):Source: The Journal of Financial and Quantitative Analysis, Vol. 16, No. 4, Proceedings of16th Annual Conference of the Western Finance Association, June 18-20, 1981, Jackson Hole,Wyoming (Nov., 1981), pp. 439-462Published by: University of Washington School of Business AdministrationStable URL: http://www.jstor.org/stable/2330365 .Accessed: 21/08/2012 09:08Your 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 ofcontent in a trusted digital archive. We use information technology and tools to increase productivity and facilitate new formsof scholarship. For more information about JSTOR, please contact support@jstor.org.. University of Washington School of Business Administration is collaborating with JSTOR to digitize, preserve and extend access to The Journal of Financial and Quantitative Analysis.http://www.jstor.org
  • 2. JOURNAL OF FINANCIAL AND QUANTITATIVE ANALYSISVolume XVI, No. 4, November 1981 A NEW EMPIRICAL PERSPECTIVE ON THE CAPM Marc R. Reinganum* Introduction The adequacy of the capital asset pricing models (CAPM) of Sharpe [27],Lintner [17], and Black [4] as empirical representations of capital market equili?brium is now seriously challenged (for example, see Ball [1], Banz [2], Basu [3],Cheng and Graver [8], Gibbons [15], Marsh [18], Reinganum [22], and Thompson [20]).Yet, the influence of earlier empirical studies (such as Black, Jensen, and Scholes[5] and Fama and MacBeth [11]) still remains; the current consensus seems to bethat a securitys beta is still an important economic determinant of equilibriumpricing even though it may not be the sole determinant. In light of the recentempirical evidence, however, the claim that a securitys beta is an importantdeterminant of equilibrium pricing should be reexamined. The purpose of this paper is to investigate empirically whether securitieswith different estimated betas systematically experience different average ratesof return. While the statistical tests are designed to assess the cross-sectionalimportance of beta, cross-sectional regressions are not employed, so that someof the problems which plagued earlier research are avoided. The test resultsdemonstrate that estimated betas are not systematically related to average returnsacross securities. The average returns of high beta stocks are not reliably dif?ferent from the average returns of low beta stocks. That is, portfolios withwidely different estimated betas possess statistically indistinguishable averagereturns. Thus, estimated betas based on standard market indices do not appear toreliably measure a "risk which is priced in the market." These findings, alongwith the evidence on empirical "anomalies," suggest that the CAPM may lack sig?nificant empirical content. University of Southern California. The author wishes to thank FischerBlack, Victor Canto, Kim Dietrich, Doug Joines, Terry Langetieg, Dick Roll, andAlan Shapiro. Any errors that remain are the authors responsibility. 439
  • 3. II. The Beta Hypothesis and Test Design The development of the CAPM is well known and can be found elsewhere (forexample, see Fama [10]). Depending on the particular set of assumptions, thepricing relationships which emerge from the CAPM can be expressed as either:(1) E(R.) i = E(R ) + 3. [E(R ) - E(R )] om i m om(2) E(R.) = R + 3. [E(R ) - Rl l F l m Fwhere: E(R.) = expected return on asset i; E(R ) E expected return on an asset whose return is uncorrelated with the market return; E(R ) E expected return on the market portfolio; E cov(R.,R E the beta of asset and 3. i i m)/var(R m) i; R? E risk-free rate of interest. FThe two forms of the CAPM share an important implication. Namely, two assetswith different betas possess different expected returns. Thus, a necessary con?dition for the data to be consistent with the CAPM is that variations in esti?mated betas must be systematically related to variations in average returns.While Roll [23] questions the testability of the theoretical CAPM, the concernof this paper is the common empirical representation of the paradigm. The betahypothesis is that assets with different estimated betas experience differentaverage rates of return. Confirmation of the hypothesis would offer evidencethat supports the contention that betas matter in equilibrium pricing. Evidencethat rejected the hypothesis would seem to indicate that the risk premia associatedwith betas are economically insignificant. A straightforward, two-step strategy is employed to test the beta hypothesis.First, in period A, individual security betas are estimated, and securities areplaced into one of ten portfolios based upon the relative rank of their estimatedbeta. Then, in period B, the returns of the ten beta portfolios are calculatedby combining with equal-weights the returns of the component securities withineach portfolio. With the time-series of ten portfolio returns in hand, a multi?variate statistical procedure is invoked to test whether or not the ten port?folios possess significantly different average returns. 440
  • 4. The composition of each beta portfolio is periodically updated. The fre?quency of the revisions depends upon the data base being analyzed. When analyzingthe daily returns of the New York Stock Exchange and American Stock Exchangefirms (1963-1979), the beta portfolios are revised annually. Thus, the 1964 betaportfolios are created based upon security betas estimated with 1963 daily re?turns. Similarly, betas estimated with 1964 data are used to identify the secu?rities within the 1965 beta portfolios. With monthly return data for NYSE firms(1930-1979), the beta portfolios are updated every five years. For example,security betas estimated with data from 1930-1934 are used to form the member?ship of the 1935-1939 beta portfolios. Regardless of the frequency of updates,betas are estimated in the period prior to the one in which portfolio returnsare measured. Three different estimators are used to compute beta estimates. First, forboth daily and monthly return data, betas are calculated using ordinary leastsquares. Security returns are regressed against the CRSP value-weighted marketreturns, and the computed coefficient on the market is the estimated beta. Recentresearch, however, indicates that this "market model" estimator may be inappro?priate for daily returns because of nonsynchronous trading problems. To assessthe impact of this potential problem, the estimators proposed by Scholes andWilliams [25] and Dimson [9] also are used to calculate security betas. Hence,with daily data, the sensitivity of the results to different beta estimators canbe investigated. III. Empirical Tests of the Beta Hypothesis This section reports the results of tests designed to determine if port?folios with different estimated beta experience statistically different averagereturns. The section is divided into three parts. In the first part, the dataand sample selection criteria are described. In the next part, the test resultsbased on the daily returns of NYSE and AMEX companies during 1964 through 1979are presented. The final part contains evidence based on 45 years of monthlyreturns for NYSE companies.A. The Data and Sample Selection Criteria Stock return data used in this analysis are gathered from the University ofChicagos Center for Research in Security Prices (CRSP) monthly and daily stockreturn files as of December 1979. The daily file contains the daily stock re?turns (capital gains plus dividends) of all companies that have traded on theNew York Stock Exchange or the American Stock Exchange from July 1962 throughDecember 1979. Unlike the daily file, the monthly file contains informationonly on NYSE companies; however, the stock return information on the monthly file 441
  • 5. dates back to January 1926. Each time security betas are estimated and the composition of the ten betaportfolios is revised, the sample of firms changes. With the daily data, thesample changes yearly. The only restriction placed on securities is that theyhave at least 100 one-day returns during the beta estimation period. No otherrestriction, such as survival through the portfolio holding period, is imposed.If a firm is delisted during the holding period, any funds returned are held incash until the end of the year. In any one given year, the number of NYSE andAMEX firms that qualified for inclusion in the sample ranged between 2,000 and2,700. The selection criterion with monthly data differs from the above criteriononly because portfolios are updated every five years. During the beta estima?tion period, a firm is excluded only if it fails to have at least 40 one-monthreturns. The number of NYSE firms included in the monthly sample ranged from678 in the 1930s to 1296 in the 1970s.B. The Test Results with Daily Returns: 1964-1979 The years from 1964 through 1979 represent a good period in which to studythe cross-sectional relationship between returns and estimated betas for at leasttwo reasons. First, these years are primarily outside the time periods of thepivotal studies by Black, Jensen and Scholes [5], and Fama and MacBeth [11],which ended in December, 1965 and June, 1968, respectively. Second, unlike theearlier studies, the hypotheses can be tested with AMEX firms as well as withNYSE companies. The first stage of the test involves estimating betas and placing securi?ties into one of ten beta portfolios. Three different beta estimates are usedto create three sets of ten beta portfolios. As explained above, these betasare computed using the "market model," Scholes-Williams, and Dimson estimatorswith a value-weighted NYSE-AMEX market index. In the next period, the dailyreturns of the ten beta portfolios are calculated by combining with equal-weightsthe daily returns of the component securities within the portfolios. If betasmatter in the way the theory suggests, then one ought to observe a positive rela?tionship between betas and returns and be able to reject the hypothesis that themean returns of the ten beta portfolios are equal. Tables 1 through 3 present the daily return statistics for the ten betaportfolios created with the different beta estimates. Table 1 contains information For the Scholes-Williams and OLS estimators, betas were also calculatedusing an equal-weighted market index. The results were not significantly dif?ferent from those reported in the text. Dimson betas were not calculated withthe equal-weight index. 442
  • 6. 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 Autocorrelations Portfolio Beta Skewness Kurtosis 1 2 3 .05 -.101 5.601 .48 .28 .25 .33 .131 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 using 4009 trading day returns from 1964through 1979. Mean daily returns are multiplied by 1000 for reporting purposes.Standard errors are in parentheses. Skewness and kurtosis measures are based onmoments of the normal distribution. 2 The estimated portfolio beta is just the linear combination (equal weights)of security betas. These betas are estimated in the year prior to the portfolioholding period. 443
  • 7. on the beta portfolios formed with betas computed using the "market model" esti?mator. The null hypothesis that the mean returns of the ten portfolios areidentical can be formally tested using Hotellings T-squared test. This testtakes into account contemporaneous correlations between the ten portfolio re?turns. The test statistic has an F(9,4000) distribution under the null hypothe?sis. At the one and five percent levels, the values of F(9,??) are 2.41 and 1.88,respectively. For the data in Table 1, the computed value of this test statis?tic is 2.99. Thus, even at the one percent level, the hypothesis of identicalmean returns would be rejected. This rejection should not be interpreted as evi?dence in support of the CAPM because the average daily return of the low betaportfolio actually exceeds the average daily return of the high beta portfolioby .03 percent. One must be cautious in interpreting the exact statistical significance ofthe results because of the apparent departures from normality. In particular,one observes that the portfolio returns seemed to be both skewed and leptokurtic.The skewness and kurtosis measures, however, are particularly sensitive to out?liers. Examination of the daily returns revealed that on May 27, 1970, the mar?ket experienced about a six percent gain; the returns of the high beta portfolioswere about ten standard deviations above their means. If this one observationis deleted from the sample, the skewness and kurtosis measures for the high betaportfolios are vitually the same as those associated with the normal distribu?tion; the low beta portfolios remain somewhat leptokurtic. One also observesin Table 1 that the daily returns of the ten beta portfolios are positively auto-correlated. With autocorrelation, the variance-covariance matrix of portfolioreturns is estimated consistently, but not efficiently. There is no reason,however, to suspect that the tests are biased in favor of rejecting the nullhypothesis of identical mean returns. Despite the potential statistical problems associated with constructing anappropriate confidence region, the results in Table 1 are not consistent withthe predictions of the CAPM: low beta portfolios actually experience greateraverage returns than those of the high beta portfolios during the period 1964-1979. While one might be able to accept this result in any one year, the factthat it can be detected during a 16-year period reduces the probability thatthis phenomenon is a fluke. After all, 16 years represents nearly 30 percentof the time for which CRSP has collected data. Furthermore, this is the onlyperiod in which computer readable data are systematically available for all Ameri?can Stock Exchange companies as well as those that trade on the New York StockExchange. Thus, the data analyzed in these tests would not seem to constitutea "small" sample. 444
  • 8. One potential criticism of the results presented in Table 1 is that thebeta portfolios are created using ordinary least-squares estimates of securitybetas based on daily data. If nontrading is a serious problem, then this mightlead to biases in estimation which could affect the results. This possibilityis now explored. Table 2 contains the daily returns statistics for portfoliosconstructed with Scholes-Williams estimated betas. The numbers presented inthis table are similar to those reported in Table 1. Even using the Scholes-Williams estimator, the low beta portfolios experience higher average returnsthan do the high beta portfolios. If one tests the hypothesis of identical meanreturns using Hotellings T-squared technique, the appropriate F-test takes ona value of 1.85. Hence, one would not reject the null hypothesis at the fivepercent level. Of course, the statistical caveats discussed above apply heretoo. The results in Table 2 seem to indicate that, at best, the average returnsof the ten beta portfolios are indistinguishable from each other. This corro-borates the evidence in Table 1; positive differences in estimated betas are notreliably associated with positive differences in average returns. Dimson recently argued that even the Scholes-Williams estimator might bebiased and inconsistent if nontrading is a serious enough problem. Dimson sug?gested that one use an aggregated coefficients method for estimating securitybetas with daily data. The idea behind this estimation technique is to regresslagged and leading (as well as the contemporaneous) market returns on securityreturns. Thus, instead of a simple regression, one runs a multiple regression.The estimated security beta is simply the sum of the estimated slope coefficients.In this paper, regressions are calculated using 20 lagged and five leading mar?ket returns. This procedure is virtually identical to those used in Roll [24]and Reinganum [21]. In Table 3, the ten portfolios are constructed with estimated betas basedon Dimsons aggregated coefficients methodology. One observes that, except forportfolio P9, the mean daily returns of all the portfolios are between .06 per?cent and .08 percent. Furthermore, the mean daily return of the high beta port?folio exceeds the mean daily return of the low beta portfolio by only .001 per?cent with an associated t-value of 0.09. As with the portfolio returns reportedin the previous tables, there is no immediately evident association between Dim?son betas and average portfolio returns. One would reject, however, the hypo?thesis of equality between means for all ten portfolios jointly considered atthe .01 level, but the average returns of the two extreme beta portfolios arestatistically indistinguishable. Furthermore, for the intermediate portfolios,higher estimated betas are not always associated with higher average returns. The evidence analyzed in Tables 1 through 3 is based on 16 years of daily 445
  • 9. 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 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 lA mean return is calculated using 4009 trading day returns from 1964through 1979. Mean daily returns are multiplied by 1000 for reporting purposes.Standard errors are in parentheses. Skewness and kurtosis measures are based onmoments of the normal distribution. 2The estimated portfolio beta is just the linear combination (equal weights)of security betas. These betas are estimated in the year prior to the portfolioholding period. 446
  • 10. 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 .35 .09 .11 A mean return is calculated using 4009 trading day returns from 1964 through1979. Mean daily returns are multiplied by 1000 for reporting purposes. Standarderrors are in parentheses. Skewness and kurtosis measures are based on momentsof the normal distribution. The estimated portfolio beta is just the linear combination (equal weights)of security betas. These betas are estimated in the year prior to the portfolioholding period. 447
  • 11. return data. One potential problem with drawing inferences from such a timeseries could be that the statistical distribution may not be sufficiently station?ary, especially since the portfolios are revised yearly; however, the year-by-year portfoliosresults can be examined to gauge whether this is a serious prob?lem. Perhaps the most succinct way to convey the results of this analysis isto present the differences in average returns between the high and low beta port?folios. Table 4 reports these differences for security betas calculated withthe different beta estimators. The year-by-year results also reveal no significant relationship betweenestimated portfolio betas and average returns. For example, when security betasare calculated with the "market model" estimator, the average return of the highbeta portfolio exceeds the average return of the low beta by two standard errorsin only one of the 16 years. For the other 15 years, the differences in averagereturns between the high and low beta portfolios are not statistically signifi?cant. In nine of these years, the point estimate of the mean return of the lowbeta portfolio exceeds that of the high beta portfolio. Using the Scholes-Williams estimator, none of the differences in averagereturns between the high and low beta portfolio are more than two standard er?rors from zero. In seven of the 16 years, the average return of the low betaportfolio is greater than the average return of the high beta portfolio. For portfolios created with Dimson betas, the average return of the highbeta portfolio exceeds the average return of the low beta portfolio by twostandard errors in two of the 16 years. The average return, however, of thelow beta portfolio exceeds the average return of the high beta portfolio in oneof the years as well. In the remaining years, the differences are not statisti?cally significant. Hence, the year-by-year results corroborate the findingspresented in Tables 1 through 3. Thus, the danger of interpreting the resultsin Tables 1 through 3 as illustrating the average effects throughout the 16-year period does not seem great. Another possible explanation for the above results is that the portfoliobetas are really not different in the year in which portfolio returns are meas?ured. Recall that portfolios are formed based on security betas estimated inthe prior year. Since these estimated betas are ranked, the extreme beta port?folios in particular may contain securities whose betas are estimated with thelargest error. This possibility can be investigated by computing the betas ofthe ten portfolios in the year in which average returns are measured. Table 5 compares the grouping period betas with the holding period betasfor the three estimators during the entire 16-year sample. For each set of tenportfolios, holding period betas are computed with the "market model," Scholes- 448
  • 12. TABLE 4 MEAN DIFFERENCES IN DAILY RETURNS BETWEEN THE HIGH AND LOW BETA PORTFOLIOS ON A YEARLY BASIS (Betas Computed with Daily Returns) Beta Estimator Mean differences in daily returns are multiplied by 1000 for reporting pur?poses. T-values are in parentheses. Each year contains approximately 250 trad?ing days. 449
  • 13. TABLE 5 COMPARISON OF NYSE-AMEX PORTFOLIO BETAS ESTIMATED IN GROUPING PERIODS AND HOLDING PERIODS GROUPING PERIOD ESTIMATOR Aggregated Scholes-Williams Coefficients GP MM SW AC GP MM SW AC .07 .43 .53 1.07 -.50 .72 .82 1.26 .41 .53 .64 1.06 .29 .71 .80 1.14 .59 .65 .77 1.21 .62 .76 .86 1.20 .75 .76 .88 1.28 .89 .81 .91 1.25 .91 .84 .96 1.36 1.15 .87 .97 1.35 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 1.73High Beta 2.25 1.69 1.67 1.95 2.25 1.60 1.62 1.95 3.77 1.32 1.39 1.89 GP stands for the estimated beta of the portfolio during the groupingperiod. Grouping period betas are shown for portfolios created with "marketmodel" estimates, Scholes-Williams estimates, and Dimsons aggregated coefficientsestimates. For each set of portfolios, three estimated holding period betas areshown: MM ("market model"); SW (Scholes-Williams); and AC (Dimsons aggregatedcoefficients method). In a grouping period, a portfolio beta is just the equal-weighted combination of estimated security betas within that portfolio. Thegrouping period betas reported above are the averages of the portfolio betas overthe 16 grouping periods from 1963 through 1978. Holding period betas are calcu?lated by analyzing 16 years of daily portfolio and market returns (1964-1979). 450
  • 14. Williams, and Dimsons aggregated coefficient estimators, even though each setof portfolios is created with betas based on only one of these estimators. Thus,for example, "market model," Scholes-Williams, and Dimson holding period betasare presented for the portfolios formed on the basis of "market model" betas alone.In Table 5 one observes attenuation in the estimated betas of the high and lowbeta portfolios. For example, the estimated beta of the lowest "market model"beta portfolio rises from .05 to .40; similarly, the estimated beta of the highest"market model" beta portfolio drops from 2.25 to 1.69. The attenuation in esti?mated betas of the Scholes-Williams beta portfolios is similar to the attenuationexhibited by the "market model" beta portfolios. The estimated betas of the ag?gregated coefficients portfolios, however, reveal severe attenuation. For example,the beta of the lowest AC portfolio rises from -.50 to 1.26; the beta of the high?est AC portfolio drops from 3.77 to 1.89. Thus, the spread in Ac betas betweenthe AC portfolios is smaller than the spread in "market model" and Scholes-Williamsholding period betas for portfolios created with those two estimators. Table 5 also reveals that each estimator almost perfectly preserves the rankordering of estimated betas for each set of ten portfolios during the holdingperiods, regardless of the estimator used to create the ten portfolios. Considerportfolios formed on the basis of "market model" betas. During the holding periods,the Scholes-Williams estimates of the betas of these portfolios are perfectly rankordered with the "market model" estimates. Furthermore, the spread between thehigh and low beta portfolios during the holding periods is 1.16 based on the Scholes-Williams estimates, and 1.29 based on the "market model" estimates. The spreadfor these portfolios based on Dimson betas is .88. Thus, the other estimatorsnot only tend to preserve the rank ordering of estimated betas, but also seem toexhibit spreads roughly equivalent to those of the "market model" estimator whichis used to form these portfolios. One discovers in Table 5 that similar conclu?sions can be drawn for portfolios created on the basis of Scholes-Williams betasand Dimsons aggregated coefficients betas. Hence, one may feel confident in con-cluding that the portfolios analyzed in Tables 1 through 4 possess widely differ?ent estimated betas during the portfolio holding periods.C. The Test Results with Monthly Returns: 1935-1979 The purpose of this section is to investigate whether the "beta does notmatter" result is specific to the 1964-1979 period or whether, in fact, it appearsto hold over a longer time horizon. Indeed, evidence from the work of Black,Jensen, and Scholes [5] may be consistent with the proposition that portfolioswith widely different estimated betas possess statistically indistinguishableaverage returns. For example, in their Table 2, the mean excess return, (R -R ),of the low beta portfolio is within the two standard error confident interval 451
  • 15. about the mean excess return of the high beta portfolio. Furthermore, Black,Jensen, and Scholes note that the intercepts in their "market model" regressionsare negative for portfolios with high estimated betas (3 > 1) and positive forportfolios with low estimated betas (3 < 1). This inverse relationship betweenbetas and the intercepts is precisely what one would expect if portfolios withdifferent estimated betas had statistically indistinguishable average returns. The two-stage test procedure used with the monthly data is similar to theone employed in the previous section except that the initial estimation and port?folio holding periods are five years instead of one year. In the first period,security betas are estimated using ordinary least squares and membership in theten beta portfolios is established. In the next period, the monthly returns ofthe ten beta portfolios are computed by combining with equal-weights the monthlyreturns of the securities within the portfolio. The first grouping period isfrom 1930 through 1934; the last portfolio holding period is from 1975 through1979. If estimated betas matter, then the ten beta portfolios ought to have sig?nificantly different mean returns. Table 6 presents the monthly return statistics for the ten beta portfoliosformed by grouping securities on the basis of their OLS betas estimated with theCRSP equal-weighted NYSE market index. The statistics in these tables are basedupon 45 years of monthly return data from 1935 through 1979. At first glance,the evidence seems to indicate that estimated betas matter. For example, theaverage return of the high beta portfolio is about 1.5 percent per month, whereasthe average return of the low beta portfolio is only .9 percent. In addition,the rank ordering of average returns corresponds to the rank ordering of estimatedbetas. One cannot, however, draw inferences from point estimates alone. Thefact that the high beta portfolios possess higher average returns than the lowbeta portfolios does not mean that the differences are reliable or statisticallysignificant. Indeed, while the mean difference between the returns of the highand low beta portfolios in Table 6 is .580 percent per month, the standard errorof the difference is .294 percent. Thus, the mean difference between the averagereturns of the high and low beta portfolios is less than two standard errors fromzero. Furthermore, since these statistics are computed with 540 observations,one could argue that, taking into account the power of the test, a three standarderror confidence region might be more appropriate than the conventional two stan?dard error interval. One can also formally test whether the ten beta portfoliospossess identical mean returns using Hotellings T-squared test. Under the nullhypothesis of identical mean returns, the test statistic assumes an F (9,531) dis?tribution. Based on the data analyzed in Table 6, the value of the test statisticis 1.22; one clearly cannot reject the null hypothesis of identical mean returns 452
  • 16. 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 Skewness Kurtosis 1 2 3 -.287 4.961 .11 -.00 .04 -.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 .00 .827 10.000 .02 .08 -.00 .453 6.290 .02 .10 .01 .876 8.422 .03 .08 -.01 .948 5.642 .03 .09 -.02 Mean returns are multiplied by 100 for reporting purposes. A 1.0 equals1%. Standard errors are in parentheses. The statistics are based on 540 monthlyobservations from 1935 through 1979. 2The estimated portfolio beta is the equal-weighted combination of securitybetas. These betas are estimated in the five-year periods prior to the portfolioholding periods. 453
  • 17. even at the .05 level. The data analyzed in Table 7 are similar to the data analyzed in Table 6except that security OLS betas are computed with a value-weighted NYSE market in?dex. Again, the high beta portfolio experienced an average return greater thanthe low beta portfolio, but in this case the difference is about .4 percent permonth rather than .85 percent; the t-statistic associated with this differenceis only 1.59. In addition, the null hypothesis of identical mean returns for theten beta portfolios still would not be rejected at the .05 level, although thevalue of the test statistic, 1.88, is just slightly less than the critical valuefor the F(9,531) distribution. Unlike the data in Table 6, however, the averageportfolio returns and estimated betas are not perfectly rank correlated in Table7. Thus, based on the evidence in these two tables, there does not appear to bea statistically reliable relationship between average portfolio returns and esti?mated portfolio betas. While one might tentatively conclude that betas computed with standard methodsand market indices do not seem to be reliably related to average portfolio returns,three additional issues should be addressed. First, are the results within thesubperiods consistent with the findings based on the analysis of 45 years of monthldata? Secondly, are the estimated betas of the ten portfolios during the holdingperiods similar to the grouping period betas? Finally, given that the empiricaldistribution of monthly returns appears nonnormal (refer to the skewness and kur?tosis measures in Table 6 and 7), are the conclusions drawn from test statisticsbased upon multivariate normality still valid? An analysis of the subperiod results for the data summarized in Tables 6 and7 is important because the returns distributions of the ten beta portfolios areprobably not stationary over the entire 45-year period, especially since the com?position of each beta portfolio changes every five years. Table 8 contains themean differences between the monthly returns of the high and low beta portfoliosin each of the nine five-year subperiods. These data corroborate the findingthat a strong systematic relationship between estimated betas and average port?folio returns does not exist. For example, with portfolios formed using betasestimated with the equal-weighted index, the mean difference is more than twostandard errors from zero in only one subperiod. In three of the other eightsubperiods, the average return of the low beta portfolio exceeds the averagereturn of the high beta portfolio. When grouping is based on security betasestimated with the value-weighted index, the mean difference in average returnsbetween the high and low beta portfolios does not exceed two standard errors inany of the nine subperiods. One possible explanation for the above results is that the holding period 454
  • 18. 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 Autocorrelations Portfolio Beta Skewness Kurtosis 1 2 3 .44 .175 7.818 .11 .01 .05 .69 -.171 4.259 .03 .03 .04 .84 -.166 5.585 .02 .06 .02 .98 .190 6.444 .00 .08 .01 1.10 .605 8.797 .03 .10 -.01 1.23 .568 8.087 .01 .10 .01 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 purposes. A 1.0 equals1%. Standard errors are in parentheses. The statistics are based on 540 monthlyobservations from 1935 through 1979. 2 The estimated portfolio beta is the equal-weighted combination of securitybetas. These betas are estimated in the five-year periods prior to the portfolioholding periods. 455
  • 19. 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, Market Model, Period Equal-Weighted Value-Weighted Overall .579 .416 (1.97) (1.59) 1/35 - 12/39 1.297 1.172 (0.78) (0.76) 1/40 - 12/44 1.690 1.364 (1.42) (1.34) 1/45 - 12/49 .295 .287 (0.42) (0.45) 1/50 - 12/54 .688 .692 (1.32) (1.45) 1/55 - 12/59 -.049 .062 (-.11) (0.14) 1/60 - 12/64 -.384 -.408 (-.95) (-1.12) 1/65 - 12/69 .757 .529 (1.47) (1.12) 1/70 - 12/74 -.853 -1.009 (-1.06) (-1.38) 1/75 - 12/79 1.775 1.053 (2.30) (1.88) Mean differences in monthly returns are multiplied by 100 for reportingpurposes. T-values are in parentheses. Results for the overall period arebased on 540 months. 456
  • 20. betas of the ten portfolios do not differ from each other; however, this possibilityseems ruled out by evidence contained in Table 9. This table presents a compari?son of the grouping period betas with the holding period betas. One observes at?tenuation in the estimated betas of the high and low beta portfolios; that is, theholding period betas of these portfolios are closer to 1.0 than are the groupingperiod betas. Nonetheless, the difference in estimated holding period betas be?tween the two extreme portfolios is still greater than .9. Furthermore, the hold?ing period betas preserve the rank ordering established by the grouping periodbetas. Thus, this evidence indicates that there are significant differences inthe holding period betas of the ten portfolios. Since the empirical distributions of monthly returns do not appear to con-form to the normal distribution, a proper concern is whether test statistics basedon normality might lead to inappropriate interpretations of the data. One noticesin Tables 6 and 7 that the monthly portfolio returns of the ten beta portfoliostend to be skewed and leptokurtic relative to the normal distribution; one alsoobserves that, unlike daily portfolio returns, the monthly returns do not sufferfrom severe autocorrelation. A nonparametric test can be employed to test for abeta effect if one believes that the skewness and kurtosis in monthly returnsmight seriously affect Hotellings T-squared test. To avoid the assumption ofnormality, Friedmans [13] rank test for a beta effect is performed. Under thenull hypothesis, any one ranking of the ten portfolio returns (from 1 through 10)in a given month is assumed to be as likely as any other ranking. The null hypo?thesis does not imply that each set of ten monthly returns is drawn from the samepopulation; however, independence between monthly returns is assumed. It is im?portant to note that each set of monthly observations may differ tremendously withrespect to location, dispersion, or both. Hence, skewness and kurtosis relativeto the normal distribution will not invalidate this test. The test is only de?signed to detect any systematic tendency for the monthly returns of one portfolioto exceed or be smaller than the same-month returns of other portfolios. Underthe null hypothesis, the appropriate test statistic is distributed approximatelyas chi-square with nine degrees of freedom. Table 10 presents the chi-square test statistics for the two sets of tenportfolios during the overall period as well as during each of the nine five-year subperiods. At the .01 significance level, the null hypothesis of identicalreturns could not be rejected for either set of ten beta portfolios during theoverall period. Indeed, with 540 observations, the .01 level may not be too a criterion against which to test the hypothesis. Furthermore, at thestringent .05 significance level, the hypothesis of identical returns could not be rejectedfor the portfolios created with security betas estimated with the value-weighted 457
  • 21. 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 In a grouping period, a portfolio beta is just the equal-weighted combina?tion of estimated security betas within that portfolio. The grouping periodbetas reported above are the averages of portfolio betas over the nine five-yeargrouping periods from 1930 through 1974. 2 The holding period betas are calculated by regressing monthly portfolioreturns against market returns from 1935 through 1979. Standard errors, whichare rounded to two significant digits, are reported in parentheses. 458
  • 22. TABLE 10 CHI-SQUARE STATISTICS BASED ON FRIEDMANS NONPARAMETRICRANK TEST FOR A BETA EFFECT Beta Estimator, NYSE Market Index Market Model Market Model Period Equal-Weighted Value-Weighted Overall 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 The chi-square statistics presented in this table are distributed withnine degrees of freedom. The values of the 1 and 5 percent limits for thisdistribution are 21.65 and 16.93, respectively. 459
  • 23. NYSE market index. The subperiod results seem to indicate that a systematic rela?tionship between estimated betas and portfolio returns was not present. For port?folios formed with betas computed with the equal-weighted index, the hypothesisof identical returns would be rejected at the .01 level in three of the nine sub?periods. In one of these three subperiods, however, the average return of thelow beta portfolio actually exceeded the average return of the high beta port?folio. For portfolios formed with betas calculated against a value-weighted in?dex, the null hypothesis would be rejected at the .01 level in only one of thenine subperiods, but in this subperiod the low beta portfolio experienced higherreturns than the high beta portfolio. The nonparametric tests do not seem to detect a strong, persistent and sys?tematic relationship between estimated betas and portfolio returns. Yet thesetests do yield insights into the nature of the data analyzed in Tables 6 and 7.In those tables, one could not help but notice a monotonic relationship betweenaverage portfolio returns and estimated betas that appeared to be consistent withthe CAPM. But the average returns turned out to be deceptive to the extent thatthey masked the great variability associated with the time-series of portfolioreturns. While the average returns exhibited a rank ordering consistent with theCAPM, the hypothesis test based on Friedmans nonparametric rank test indicatedthat the month-by-month rankings of portfolio returns could not be distinguishedfrom random rankings. This variability in the time-series of portfolio returnsis also the reason why the parametric procedure, Hotellings T-squared test, didnot reject the hypothesis of identical mean returns. IV. Conclusion This paper investigates whether differences in estimated portfolio betas arereflected in differences in average portfolio returns. During 1964 through 1979,the evidence indicates that NYSE-AMEX stock portfolios with widely difrerent esti?mated betas possess statistically indistinguishable average returns. Evidencebased on NYSE stock portfolios dating back to 1935 corroborates this result. Ofcourse, this finding should not be construed to mean that all securities possessidentical average returns. Indeed, during this time period, Banz [2] and Reinganum[22] report that portfolios of small firms experienced average returns nearly 20percent higher than portfolios of large firms. The findings of this study demon?strate that cross-sectional differences in portfolio betas estimated with commonmarket indices are not reliably related to differences in average portfolio re?turns; that is, the returns of high beta portfolios are not significantly differ?ent from the returns of low beta portfolios. In this cross-sectional sense, therisk premia associated with these betas do not seem to be of economic or empiricalimportance for securities traded on the New York and American Stock Exchanges. 460
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