8A-1
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 calculate 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 8-9
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
, or
regression equation
, means: The equation Y
5
a
1
bX
1
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
5
r
–
M
.
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
5
a
1
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 r
–
J
increases for a given increase in r
–
M
. For example, we observe in Figure 8A-1
that r
–
J
increases from
2
8.9 to
1
7.1% (the rise) when r
–
M
increases from 0 to 10.0% (the run).
Thus, b, the beta coefficient, can be measured as shown below.
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 r
–
M
. For example, if r
–
M
5
15%, we would predict r
–
J
52
8.9%
1
1.6(15%)
5
15.1%.
However, the actual return would probably differ from the predicted return. This devia-
tion is the error term, e
J
, for the year, and it varies randomly 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 r