SOLUTION NOTES FOR EXERCISES ON FUNCTIONS
4. Code for Standard Deviation of Cash Flows
7.2 ESTIMATING BETA COEFFICIENTS
To estimate betas, it is best to work with log returns (in fact log of excess returns). The sheet Beta illustrates how the beta coefficient for a share is estimated by regression using monthly returns on ShareA and Index (columns B and C in Figure 7.1). Here, the index is the FTSE100 and there are 60 monthly returns for both ShareA and Index. The beta (strictly speaking, the ‘unadjusted’ beta) is simply the slope estimate from the regression of share excess returns on index excess returns.
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A B C D E F G H I J
Estimating Beta Coefficients using Ln Xs Returns
Mth ShareA Index Fitted Residuals Excel regression functions 1 -0.1326 -0.0534 -0.0817 -0.0509 INTERCEPT -0.0013
2 0.0671 0.0868 0.1295 -0.0624 SLOPE 1.5065
3 0.0939 0.0182 0.0261 0.0678 RSQ 0.4755
4 -0.1339 -0.0800 -0.1218 -0.0121 STEYX 0.0595 5 -0.0951 -0.0666 -0.1016 0.0065
6 -0.0381 -0.0455 -0.0698 0.0317 Alpha -0.0013 7 0.0882 0.0925 0.1381 -0.0499 Alpha (SE) 0.0078 8 0.1140 0.0393 0.0579 0.0561
9 0.0694 0.0422 0.0623 0.0071 Beta 1.5065
10 0.0527 0.0362 0.0533 -0.0006 Beta (SE) 0.2077 11 -0.1298 -0.0006 -0.0022 -0.1276
12 0.0395 0.0223 0.0323 0.0072 Sp Risk from STEYX 13 -0.0358 0.0071 0.0094 -0.0452 Monthly 5.95%
14 -0.0179 -0.0150 -0.0239 0.0060 Annualised 20.60%
15 0.1251 0.0092 0.0126 0.1125 via fn 20.60%
Figure 7.1 Regression of returns on ShareA on market index to estimate beta
Asset Pricing 127 Excel provides a number of alternative ways to estimate the slope coefficient. The formula in cell I7 DSLOPE(B6:B65,C6:C65) uses Excel’s SLOPE function to estimate beta (ˇi). Similarly, the INTERCEPT function (in cell I6) calculates the intercept (˛i) for the regression equation. Substituting the estimates of ˛i andˇi into the equation gives:
‘Fitted’ returnD 0.0013C1.5065index return
Column E contains the ‘fitted’ values for each index return and column F shows the
‘residuals’, i.e. actual returns from which the ‘fitted’ returns have been differenced. The size of the residuals is captured in the residual standard error (cell I9) with another Excel function, STEYX, which has the same inputs as the SLOPE function. Figure 7.2 shows share returns plotted against index returns with the above regression line superimposed.
The scatter of returns about the line (and hence not related to the index) is called the specific risk of the share. It is the best estimate of the quantity earlier denoted byei).
Beta Regression
-0.20 -0.15 -0.10 -0.05 0.00 0.05 0.10 0.15 0.20 0.25
-0.10 -0.08 -0.06 -0.04 -0.02 0.00 0.02 0.04 0.06 0.08 0.10 0.12
Index Returns
Share Returns
Figure 7.2 Share returns versus market returns with regression line superimposed
Another way to generate all the regression results (albeit in a static form) is to use the commands Tools then Data Analysis then Regression, to get the summary output below cell K6 in Figure 7.3. The intercept and slope are in cells L22:L23 and the residual standard error is in cell L12.
A more concise set of regression results is produced by Excel’s LINEST function. This function has the advantage of giving results still dynamically linked to the data, rather than the static data dump produced by the Regression command. LINEST is an array formula, here with five rows and two columns, so a 5 by 2 cell range must be selected prior to entering the formula:
=LINEST(B6:B65,C6:C65,TRUE,TRUE)
to get the array of regression results (not forgetting the subsequent CtrlCShiftCEnter keystroke combination). These are displayed in range O6:P10 of Figure 7.3 and explana- tory labels have been attached in columns N and Q.
Individual items in the LINEST array can be accessed using the INDEX function (see cells I11, I12, I14 and I15). For instance, in cell I14, the beta coefficient is obtained using the formula:
=INDEX(LINEST(B6:B65,C6:C65,TRUE,TRUE),1,1)
The standard error of the beta coefficient, in cell I15, is obtained using the formula:
=INDEX(LINEST(B6:B65,C6:C65,TRUE,TRUE),2,1)
Specific risk (expressed as an annualised standard deviation) is given in cell I19 (from STEYX) and also in cell I23 (from the sum of squared residuals divided by n2) as can be confirmed in the Beta sheet. The annualised risk measures are simply the monthly measures multiplied byp
12.
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K L M N O P Q
Output from Tools / Data Analysis / Regression Output from Linest
SUMMARY OUTPUT Beta 1.5065 -0.0013 Alpha
Beta (SE) 0.2077 0.0078 Alpha (SE)
Regression Statistics RSQ 0.4755 0.0595 STEYX
Multiple R 0.6896 F 52.5873 58.0000 N-2
R Square 0.4755 Regression SS 0.1860 0.2051 Residual SS
Adjusted R Square 0.4665
Standard Error 0.0595
Observations 60
ANOVA
df SS MS F Significance F
Regression 1 0.1860 0.1860 52.5861 0.0000
Residual 58 0.2051 0.0035
Total 59 0.3911
Coefficients Standard Error t Stat P-value Lower 95% Upper 95%
Intercept -0.0013 0.0078 -0.1628 0.8712 -0.0169 0.0144
X Variable 1 1.5062 0.2077 7.2516 0.0000 1.0905 1.9220
Figure 7.3 Output from Analysis ToolPak Regression command
In routine beta estimation, the usual practice is to ‘adjust’ the beta before it can be used as a forecast measure. In the sheet, the beta of 1.51 is a sample estimate from 60 monthly observations. The adjustment assumes that the value-weighted true beta of the population over all shares must be equal to 1.00 and that high sample betas are likely to be lower in the period we are trying to forecast, and vice versa for low sample betas. Therefore sample betas are adjusted towards the population mean of 1.0. The scale of adjustment depends on the size of the variance of the sample beta (calculated as the square of the standard error of the sample beta) relative to the size of the variance of the population beta (assumed to be 0.32). The ratio is called the mean reversion factor. In our example, set out in Figure 7.4, the sample beta from the regression (1.51) is adjusted by a mean reversion factor of 32%, which moves it towards the population mean. The result is an
Asset Pricing 129 adjusted beta shown in cell O31 of 1.34. The mean reversion factor is also used to adjust the standard error of the sample beta. As the spreadsheet extract shows, user-defined functions provide an easy way to implement this adjustment of the sample betas.
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H I J K L M N O
Adjusting sample beta for mean reversion
Population beta 1.00 Sample beta 1.51
Population beta (SE) 0.30
Adjusted beta 1.34
v1 (variance of sample beta) 0.04 via fn 1.34
v2 (variance of population beta) 0.09
mean reversion factor 32% Sample beta (SE) 0.21
Adjusted beta (SE) 0.17
via fn 0.17
Figure 7.4 Adjusting sample beta for use in forecasting