Calculating Beta Coefficients
The CAPM is an ex ante model, which means that all of the variables represent
before-the-fact, expected values. In particular, the beta coefficient used in the SML
equation should reflect the expected volatility of a given stock’s return versus the
return on the market during some future period. However, people generally cal-
culate betas using data from some past period and then assume that the stock’s
relative volatility will be the same in the future as it was in the past.
To illustrate how betas are calculated, consider Figure 8A.1. The data at the
bottom of the figure show the historical realized returns for Stock J and for the
market over the last five years. The data points have been plotted on the scatter
diagram, and a regression line has been drawn. If all the data points had fallen on
a straight line, as they did in Figure 8.7 in Chapter 8, it would be easy to draw an
accurate line. If they do not, as in Figure 8A.1, then you must fit the line either “by
eye”as an approximation or with a calculator.
Recall what the term regression line,orregression equation, means: The equation
Y = a + bX + e is the standard form of a simple linear regression. It states that the
dependent variable, Y, is equal to a constant, a, plus b times X, where b is the slope
coefficient and X is the independent variable, plus an error term, e. Thus, the rate
of return on the stock during a given time period (Y) depends on what happens to
the general stock market, which is measured by X = rM.
Once the data have been plotted and the regression line has been drawn on
graph paper, we can estimate its intercept and slope, the a and b values in Y = a +
bX. The intercept, a, is simply the point where the line cuts the vertical axis. The
slope coefficient, b, can be estimated by the “rise-over-run”method. This involves
calculating the amount by which rJincreases for a given increase in rM. For
example, we observe in Figure 8A.1 that rJincreases from –8.9 to +7.1% (the rise)
when rMincreases from 0 to 10.0% (the run). Thus, b, the beta coefficient, can be
measured as shown below.
b¼Beta ¼Rise
Run ¼ΔY
ΔX¼7:1−ð−8:9Þ
10:0−0:0¼16:0
10:0¼1:6
Note that rise over run is a ratio, and it would be the same if measured using any
two arbitrarily selected points on the line.
The regression line equation enables us to predict a rate of return for Stock J,
given a value of rM. For example, if rM= 15%, we would predict rJ=–8.9% + 1.6
(15%) = 15.1%. However, the actual return would probably differ from the pre-
dicted return. This deviation is the error term, e
J
, for the year, and it varies ran-
domly from year to year depending on company-specific factors. Note, though,
that the higher the correlation coefficient, the closer the points lie to the regression
line, and the smaller the errors.
In actual practice, monthly, rather than annual, returns are generally used for
rJand rM, and five years of data are often employed; thus, there would be 5 × 12 =
60 data points on the scatter diagram. Also, in practice one would use the least
squares method for finding the regression coefficients a and b. This procedure
minimizes the squared values of the error terms, and it is discussed in statistics
courses.
The least squares value of beta can be obtained quite easily with a financial
calculator. The procedures that follow explain how to find the values of beta and
the slope using either a Texas Instruments or Hewlett-Packard financial calculator;
or a spreadsheet program, such as Microsoft Excel.
WEB APPENDIX 8A WEB APPENDIX 8A WEB APPENDIX 8A WEB APPENDIX 8A WEB APPENDIX 8A WEB APPENDIX 8A WEB APPENDIX 8A
WEB APPENDIX 8A
8A-1
