1.beta stability

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H ow to determine the:stability of beta valuesA critique of earlier methods and demonstrationofa better way.

Arthur A . Eubank, J r . and 1.Kenton Zumwalt

Athough the subject of beta stability has

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been examined in several previous studies, the focusof these studies has been more upon the stability ofbeta rankings rather than the stability of actual betavalues. The purpose of this paper is to examine thestability of security betas in different risk classes, usingan equal class interval beta classification scheme basedupon numerical beta values instead of beta ranks.

We use numerical beta class intervals becausethe numerical beta value is more important than thebeta rank for portfolio management purposes.

Although the stability of beta ranks is of interest , betarankings of individual securities may change, with nochange occurring in the actual beta values. Con-versely, a change in the numerical values can occurwith no change in the beta ranks. Previous studiesusing decile and pentile rankings imply that betas inboth high and low beta groups are more stable thanbetas in the middle groups. This study examines therelative stability of betas in extreme beta groups com-pared to middle beta groups by comparing the ,stabilityof individual risk classes using, first, a decile

classification method and, second, a n equal class in-terval classification method.

5cl

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TESTS OF BETA STABILITY

The two primary methods used in earlierstudies of the stability of betas have been correlationanalyses and transition matrices. Correlation analysiswas used by Blume and Levy to examine the stabilityof security and portfolio betas in successive non-overlapping time periods.' Both studies found that

1. Footnotes appear at the e nd of the article.

increases in the size of portfolios were associated withincreases in the correlation of portfolio betas betweensuccessive non-overlapping time periods. 'Thesestudies also found that security and portfolio betastended to regress toward a mean of 1.0 over time. Theprincipal difference between the two studies WAS theuse of different estimation intervals in the calculationof betas. Blume's study utilized seven-year beta esti-mation intervals, whereas Levy used 13, 26, arid 52-week estimationintervals. Levy found th.at the use of alonger estimation nterval tended to increase the corre-

lation of beta groups between successive non-overlapping time periods.

While the use of correlation analysis indicatesthe degree to which betas are in thesame group in suc-cessive time periods, it does not indicate into whichgroups betas tend to change over time. In order i o de-termine how betas change over time, a study bySharpe and Cooper and a later study by Baesel utilizedtransition matrices to examine the temporal stability ofindividual security betas.2 In the study by Sharpe an dCooper, individual security betas were ranked by

magnitude and placed into decile groups. The stabilityof individual security betas was examined by notingthe change in decile groupings over time. Sharpe an dCooper concluded that individual securitie, were re-sonably stable over time. In addition, they found thatbetas in the extreme decile groups tended to be morestable than betas in the middle decile groups.

Baesei, using different length estimation inter-vals in the beta calculation, concluded that individualbetas were less stable than portfolios, but he rioted thattheir stability improved as longer estimation intervalswere used. He also observed that extreme group betas



appeared to be more stable than those in the middlegroups. The differences in beta stability among ex-treme versus middle decilelpentile groups are some-what surprising, since extreme betas tend to regresstoward the mean and, intuitively, it would seem thatsecurities with more extreme betas would exhibit lessstability, i.e., a faster rate of regression toward themean, than those nearer the mean of 1.0. A possible

explanation of empirical results showing betas in moreextreme groups to have greater stability over time maybe the grouping methodology used in the studies.

ALTERNATIVE BETA G ROUP ING METH ODS

The use of a decilegrouping method with equalnumbers of securit ies n each group on the basis of se-curity beta ranks instead of beta values results in widenumerical beta ranges for extreme groups and narrowbeta ranges for middle decile groups. For example, if asecurity had a beta of 2.5 in one period and a beta of 2.0

in the following period, in all probability the securitywould still be ranked in the highest risk class groupbecause of the wider range of betas in the more ex-treme groups as a result of the limited number of betasgreater than 2.0. On the other hand, a security in themiddle range with a beta of 1.1 n the first period and abeta slightly lower, say, 1.0, in the next period couldactually shift from the first period beta group to a lowerbeta group in the folIowing period. It might appearthat the security with a middle group beta in the firstperiod had substantially decreased its risk by movingfrom one risk group to another in the next period,while the larger reduction in beta of the more extremesecurity would not result in a risk group change fromone period to the next. Indeed, it is not the ranking (orranking change) that investors are concerned with, butthe change in risk because of the change in the numeri-cal value of beta.

An alternative to the decile grouping method,whereby equal numbers of securities are placed intoeach group, is the equal c lass in tervalgrouping method,where securities are placed into beta groups basedupon equal beta ranges to define the groups on the

basis of numerical beta values. This grouping schemeprovides more information concerning the actualnumerical value of beta and the stability of its numeri-cal value over time than does a decile groupingmethod with equal numbers of securities in eachgroup without regard to the actual numerical beta val-ues of the securities. Thus, an equal class intervalgrouping method provides the basis for a more mean-ingful examination of the stabil ity of security betasover time than do decile or pentile groups, where theprimary concern is only with changes in beta ranksand not with changes in numerical beta values.

DATA A ND METHODOLGY

The security betas used in th is study were calcu-lated using the monthly return relatives from theCRSP return file. The market return used for the re-gressions was the Fisher Index. Sixty months of datawere used in the calculation of the betas for each five-year period, and the fifty-year time period from

January, 1926 through December, 1975 resulted in tennonoverlapping five-year time period^.^ The pairs ofcompanies having valid data for consecutive non-overlapping time periods resulted in 319 pairs of five-year betas for the first comparison period (1926-30 and1931-35), and as successive time periods were used,the number of observations for each pair of non-overlapping periods increased to 876 pairs of five- yearbetas for the last comparison period (1966-70 and

After computing the security betas for com-

panies with available data for a particular comparisonperiod, the betas for each nonoverlapping timeperiod were ranked in ascending order. Two groupingmethods were then used to classify the betas. First,following the procedure reported earlier by Sharpeand Cooper using deciles, the betas were divided intoten groups with equal numbers of securities (deciles),and transition matrices were used to determine theproportion of the securities either remaining in thesame risk class in periods t and t + 1, or changingclasses from periods t to t + 1.j

A second grouping method using an equal classinterval grouping scheme was then applied to the betapairs in each time period. By this method, betas weregrouped into risk classes on the basis of their numeri-cal values rather than their ranks. Ten classes were es-tablished for grouping the security betas, whereby se-curities with beta values less than or equal to .2 wereassigned to group one, those greater than .2 and lessthan or equal to .4 were assigned to group two, etc.This procedure of grouping securities according totheir betas was continued using increments of .2 untilall securities were placed into groups, with group ten

containing those securities with betas greater than 1.8.This grouping scheme resulted in eight groups withequal class intervals (groups one and ten were open-ended), but with unequal numbers of securities ineach group. Transition matrices were again used to ex-amine how the security betas changed over time.

In order to examine further the stability of se-curity betas in successive time periods, the meansquare error and the mean absolute deviation werecalculated for each of the nine comparison intervals forthe complete set of securities, and subsequently foreach beta group using the decile grouping method an d

1971-75).4



the equal class interval group ing method." Finally, thebias, inefficiency, and random error components ofthe mean square error were calculated in order to shedadditional light on the source of any instability thatmay occur over time.7 The components of the meansquare error are reported for the complete set of secu-rities over all time periods as well as for each set of thedifferent risk classes. The mean square error compo-

nents for the different risk classes were calculated inorder to examine the difderential effects of the bias, in-efficiency, and random error associated with betas inmiddle groups compared to betas in the more extremegroups.

EMPIRICAL RESULTS

The transition matrix results for the groupingprocedure utilizing deciles are reported in Table 1.Risk class one represents securities with the lowestbetas while group ten represents those securities with

the highest betas. The values in the transition matrixrepresent the proportion of securities remaining in aparticular risk class in periods t and t + 1. For example,the value .391 in element (1,l) ndicates that 39.1% ofthe betas in the decile with the lowest betas were alsoin the lowest beta decile group in period t + 1. Simi-larly, element (1,2) indicates 24.6% of those betas inthe lowest beta group in time period t were one betadecile group higher in period t + 1. The values on themain diagonal of the matrix are the propor tions ofbetas which did not change risk classes from time

period t to period t + 1.These results are similar to those reported ear-lier by Sharpe and Cooper and Baesel in that the pro-portions remaining in the same groups are higher forthe more extreme risk classes compared to the middleclasses. This finding tends to indicate that extremebetas are more stable than those near the mean of 1.0.However, the apparent greater stability of extremebetas results primarily from the decile groupingmethod used to place the securities into their respec-tive risk classes. That is, the apparent stability of ex-

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10

TABLE 1

.063 ,250 ,208 .250 ,104 ,104 ,000 ,000 .021 ,000,066 ,251 ,372 .187 .073 ,030 ,009 ,003 .009 .M10.010 ,139 .306 ,287 ,160 .055 .025 ,012 .004 ,001,012 .061 ,194 ,267 ,230 .145 .060 ,022 ,005 .004.003 ,024 . lo4 ,230 ,249 .218 ,096 ,050 ,018 ,009.001 ,009 .058 .147 .251 ,234 .164 ,075 ,032 ,029.004 ,005 ,029 .114 ,151 .226 .219 ,125 ,073 ,054.OOO .004 .013 .052 ,112 ,226 ,244 ,177 ,078 ,092.OOO ,004 .008 .025 ,119 .169 ,202 .156 .128 .189,004 ,000 ,017 .025 .038 ,169 ,186 ,169 ,195 ,195

R i s k C l a s s i n P er i od T v s . R i s k C l a s s i n P er i od 1 + 1

Betas Grou p ed by D e c i l e s

R i s k Class i n R is k C l a s s i n P e r i od t + 1P e r i o d t

1 2 2 4 5 6 7 8 !) 10I

I234567

89

10

,391.241,141.083.058,024,032.015.010.006-

,246 ,121 .083.222 ,166 ,136,173 ,162 ,146.138 .146 .155.058 ,131 ,119,063 ,104 ,111,048 .068 ,108.029 ,056 .071.011 .031 .053,015 ,016 .019

.053,078,115.120,170,151,127,100.056.031

,046,056. l o 6.099.136.141,132.116.098,070

,019.046,058,098,125.135,161.126,147,084

.021

.018,053,072,091,128.135.168.156,155

.010,024,034,050,079,083.114,182,215.209

.010,013,014,038,034,059,074,137,223,395

treme betas is associated with stability in a relativesense in that stocks with the highest or lowest betas inone period also tend to have high or low betas in asubsequcmt period.

Table 2 shows the distribution of the number ofbetas falliing into equal class intervals. As can be seen,the extreme beta intervals have far fewer observationsthan those near the mean. In order for a secuiity to

change from one equal class interval group to another,the numerical value of the beta must change, ratherthan its beta rank. As a result, a given change Ln thenumerical value of a security beta will have the sameimpact upon determining its subsequent group re-gardless of the group it is in, except for open-endedgroups m e and ten.

TA B L E 2

The transition matrix results for the equal class

interval grouping scheme are reported in Table 3. Incomparing the main diagonals of Tables 1 an d 3, it canbe seen that there is a substantial difference betweenthe relative stability of extreme beta groups comparedto middle groups indicated by the two groupingmethods. In fact, Table 3 shows that the middle groupsare more stable than those exhibited in Table 1,whereas the extreme groups are much less stable.Also, the distribution around the main diagonal ismore compact in Table 3 than under the decile group-ing scheme reported in Table 1.

TA B L E 3

R i s k C l a s s i n P er i o d T v s . R i s k C l a s s i n P er i o d T + 1

Betas G rouped by Equal Class I n t e r v a l s

R i s k C l a s s i r R i s k Class i n P e r i od t + 1P e r i o d t

1 2 4 5 6 7 8 9 1 0r I



A summary of the beta stability results for thetwo grouping alternatives is given in Table 4. Underthe decile grouping method, the securities appea r tobecome more stable as the beta moves farther awayfrom the middle risk classes, while the middle classesare more stable under the equal interval groupingscheme. Again, this apparent conflict results from thedifference in the ranges of the numerical values of the

betas falling into each of the respective risk classes foreach of the two grouping methods. The decile methodcauses the ranges of betas in the middle risk classes tobe narrow while the ranges of beta values in the moreextreme classes are wider. Hence, a g ivenbeta changein a more extreme decile group may have less impact interms of causing a security to change from one class toanother than a similar beta change in a middle decileclass. On the other hand, the equal class intervalgrouping method does not have this characteristic,since a given change in the numerical value of a betawill cause the same change in beta risk class regardlessof the security’s beta group (except for the highest andlowest open-ended risk classes).

Betas Grouped by Oeciler

.361 ,575 ,214

,545 ,677 . I 3 2

,667 ,770 ,103

. _Period T Period T + 1 Chanqe

.778 ,857 ,079

,886 ,939 ,053

,010988 ,998

1.100 1.050 -.0501.230 1.146 - .084

1.409 1.256 -.153

1.792 1.419 -.373

TABLE 4

B e t a S t a b i l i t y U s i ng A l t e r n a t i v e G r o u pi n g M et ho ds

Betas Grouped by Equal In terva lsPeriod T Period T + 1 Chanqe

,115 .595 ,480

,325 ,533 ,208,510 ,663 ,153

.900 .9y1 ,024

1.064 -.034

1.183 -.110

1 . a 2 - . I88

1.690 1.421 -.E9

,701 . ¶9 .m e

1.098

1.293

1.490

1.468 - . 584.052

R i s k C l a s s

1

2

3

4

56

7

89

1 0

P r o p o r t i o n Wi t h i n O n eR i s k C l a s s

EqualI n t e w a 1sec i 1es

,637 , 3 1 3

,629 ,689

,481 ,132

,421 ,691

, 4 2 5 , 6 9 1, 4 2 7 , 6 4 9

, 4 2 8 , 5 7 0

, 4 7 6 , 4 9 9

,594 ,473

, 6 0 4 , 3 9 0

P r o p o r t i o n i n SameR i s k C l a s s

EqualD e c i l e s I n t e r v a l s

.391 ,063

.222 .251

, 1 6 2 ,306

.155 ,267

, 1 7 0 . 2 4 9.141 ,234

, 1 6 1 , 2 1 9

, 1 6 8 , 1 7 7

, 2 1 5 . 1 2 8

, 3 9 5 , 1 9 5

REGRESSION TENDENCIES OF BETA GROUPS

The mean betas of each risk class associatedwith each of the two grouping methods were calcu-lated for periods t and t + 1 and are presented in Table5. As expected, the mean betas of the different risk

classes regress toward a mean of1.0

from time periodst to t + 1. There is a noticeable difference in the mag-nitude of the regression tendency between the twogrouping schemes, however. For example, when thesecurities are grouped by deciles, the mean of the low-est decile group increased from .361 to .575, an in-crease of .214. However, when the equal intervalscheme is used, the mean of the lowest beta class in-creases from a much lower .115 to .595, a change of.480. Typically, the farther a group mean is from themiddle, the greater is the numerical shift of the groupmean over time. Also, the numerical shift of each

group mean is typically less for the decile groupingmethod than for the equal interval method. This is at-tributable to the smaller difference among the meanbetas for each decile group compared to the larger dif-ferences between the mean betas for the equal classinterval groups.

TABLE 5

Beta Group Means f o r Periods T and T t 1

Risk Class

8

9

10

The forecast errors were calculated for each betagroup between beta estimates for periods t and t + 1for both grouping methods. The mean square error(MSE) and the mean absolute deviation (MAD) arepresented in Table 6. The forecast errors for the middlegroups for both grouping methods tend to be similar,although the more extreme groups appear to be lessstable (less predictable) when actual numerical valuebeta changes are used instead of changes in beta ranks.Contrary to the results of earlier studies showing morestability for extreme groups, this s tudy shows that themore extreme groups based upon equal class intervalstend to contain betas that are substantially less stable.

TABLE 6

F o r e c a s t E r r o r s B e t w ee n B e t a Es t i m a t e s f o r P e r i o d s T and T + 1

Betas Grouped by Oeci les 1 Betas Grouped by Equal In ter va lsRi sk Clas s MSE MA0 I MSE MAD

1

2

3

4

5

6

7

8

9

10

,128 .263

,095 ,226

,098 ,235

. l o 5 ,252

.096 ,240

,107 ,245

,132 ,268

,128 ,283

,154 ,316,359 .480

,330 ,486

,113 .249

.097 ,227

,099 ,239

,097 ,242

.128 ,266

,144 ,303

,176 , 335

,257 ,415

, 550 ,614

Finally, the components of the forecasting erroras measured by the MSE are presented in Table 7. A scan be seen, the bias components, indicating a ten-dency to incorrectly estimate both high betas a nd lowbetas, is much higher for the extreme beta groups an dis relatively small for the middle beta groups. This in-dicates a greater amount of stability for betas falling ineither middle decile groups or middle equal class in-terval groups. The greatest amount of estimation in-efficiency is associated with group ten of both group-



Risk C lass

1234567

89

10

Risk Class

1

23

45

6

789

10

l a b l e 7

Components o f Mean Square E rr o r

Betas Grouped by Deciles

RandomBi as Ineff iciency _ _rror

-SE

.04580 ,00579 ,07614 ,12773

.01766 ,00131 .07648 .09544

.01067 ,00079 .OB674 .09820

.00616 .00140 .09788 .lo543

.00279 .00026 ,09297 .09602

.00011 .00009 .lo687 .lo707,00251 .00028 .12890 .13170.00696 .00076 .12033 .12806,02329 .00150 .12897 .15376.13913 .04949 ,17056 .35918

Betas Grouped by Equal Intervals

RandomBias Ineff iciency _ _ _rror ~ MS E-

.23045

.04331

.02338

.00952,00119.00113.01214.03521.07275.34026

.00018

,00038.00158

. 0008

.00036

.00024

.00024

.00064

.00012

.05961

.09965

.06912

.07214

.08930,09585.12645.13175.13981.le381.15027

,33028

.11281

.09709

.09890

.09739

.12782,14413.17566,25668.55014

ing methods. The random error component is largerfor the op en-en ded equal class interval groups, as ex-pected, an d smaller for the decile groups.

SUMMARY AND CONCLUSIONS

Numerical beta chang es are of more interest forinvestment analysis than changes in beta rankings. Incontrast to the findings of other studies, this studyshows that betas associated with extreme beta group sare less predictable th an t hose closer to a m ea n beta of

1.0. The apparent greater stability of betas associatedwith extreme group s reported in earlier studi es is a re-

sult of the grouping schemes used in those studies andis no t because of inh erent ly greater stability of extreme

beta securities or portfolios. Tr ansi tion matrices,changes in group means, an d the magnitude of fore-cast errors provide evidence of greater instability ofbetas falling into more extreme beta groups basedupo n a n equal class interval grouping metho d versus adecile grouping metho d.

I M. E. Blume, ”On the Assessment of Risk,” Journal of Fi-nance (March, 1971); R. A. Levy, ”On the Short-Term Sta-tionarity of Beta Coefficients,” Financial Analysts Journal(NovemberDecember, 1971).

W. F. Sharpe and G . M. Cooper, “Risk-Return Classes ofNew York Stock Exchange Common Stocks, 1931-1967,”Financial Analysts journal (MarchlApril, 1972); J. B. Baesel,

”On the Assessment of Risk Some Further Considera-tions,” Jor,iraZ of Finance (December, 1974).Individual: security betas, pi's, were calculated using the fol-lowing regression equation

Rit = a , + PIRm, t E , , where

R,t is the natural logarithm of the security return rdativein period t (including cash dividends and adjusted forstock splits and stock dividends), and R,, is the naturallogarithm of the Fisher Index return relative in period t.

Each comparison period includes two consecutive, non-overlapping five-year periods. The first five years of eachsuccessive comparison period is equivalent to the last fiveyears of the preceding period. This procedure of selectingcomparison periods for examining the stability of securitybetas between the first five-year period of the cornpoxisonperiod and second five-year period resulted in nine ten-year comparison periods.

Period t refers to the first five years and period t + 1 efersto the second five years of each respective comparisonperiod.

The mean square error (MSE) was calculated using the fol-

lowing equation:N

i = l

MSE = -N ,

where N refers to the number of securities with availabledata for both five-year periods of each of the nine compari-son periods and P li and PZ i efer to the ithsemTitiy’s etas forthe first five years and second five years of each comparisonperiod, respectively. The mean absolute deviation (MAD)was calculated using the following equation:

N

i s 1MAD = _ _ _ _ _ _

N

A description of the MSE components is given by :R. C.Klemkosky and J. D. Martin, ”The Adjustment of Beta Fore-casts,”JournuZ ofFinance (September, 1975). The co:mpo:nentsof the MSE: are determined as follows:

MSE = B2 - + (1 -- bJ2 Sg, + (1 - t, 9,- + w L - 2

Total MSE Bias Inefficiency Random Error

where

P I

PP

b,

S&

Si 2r&,81 = the coefficient of determination for the regressi,on of

The “bias” term indicates the over- 01’ nder-prediction of P2by PI. The ”inefficiency” term indicates the tendency ofprediction errors to be positive at low predicted values andnegative at high predicted values. The random error termrepresents i random disturbance element which is iinrelatedto the value of the predictor, p,, or the predicted,. P2.

-

-= the mean of all period t betas across all compar-

= the mean of all period t +5 betas across all compar-

= the slope coefficient of PZi n pli,= the sample variance of PI,= the sample variance of P2, and

ison periods,

ison periods,

-

p2 on p , .

1.beta stability

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