amount by which Jincreases for a given increase in M. For example, we observe in
Figure3-10that Jincreasesfrom8.9to7.1percent(therise)when Mincreasesfrom
0 to 10.0 percent (the run). Thus, b, the beta coefficient, can be measured as follows:
.
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 M. For example, if M15%, we would predict J
8.9% 1.6(15%) 15.1%. However, the actual return would probably differ from
the predicted return. This deviation is the error term, eJ, for the year, and it varies
randomly from year to year depending on company-specific factors. Note, though,
r r r
bBeta
Rise
Run
DY
DX
7.1(8.9)
10.00.0
16.0
10.0
1.6
r r
r r
128 CHAPTER 3 Risk and Return
FIGURE 3-10 Calculating Beta Coefficients
Year Market (M) Stock J (J)
1 23.8% 38.6%
2 (7.2) (24.7)
3 6.6 12.3
4 20.5 8.2
5 30.6 40.1
Average 14.9% 14.9%
̄r 15.1% 26.5%
r
r r
r
_
∆
Historic Realized Returns
on Stock J, rJ (%)
Historic Realized Returns
on the Market, rM (%)
10 20 30
aJ = Intercept = – 8.9% r = 8.9% + 7.1% = 16%
= 10% b =Rise
Run
= =^16
10
J = 1.6
M
J
J
–10 0
–20
–10
10
20
30
40
7.1
Year 3
Year 4
Year 1 Year 5
_ _
rJ = aJ + bJ rM + e_J
= – 8.9% + 1.6 rM + eJ
Year 2
_
_
M
_
∆
r
_
∆
r
_
∆
126 Risk and Return
