THE ECONOMICS OF FINANCIAL MARKETS
R. E. BAILEY
Solution Guide to Exercises for
Chapter 6 The capital asset pricing model
1. The following information is provided for a stock market:
μ
j
β
j
Asset 1
6.6%
0.4
Asset 2
9.8%
1.2
Asset 3
12.2%
1.8
Notation:
μ
j
=
expected rate of return on asset
j
;
β
j
=
betacoefﬁcient for asset
j
,
j
= 1
,
2
,
3
.
(a) In the context of the Capital Asset Pricing Model (CAPM), deﬁne the ‘betacoefﬁcient’,
β
j
, corresponding to asset
j
. Discuss how assets’ betacoefﬁcients should be interpreted
and explain how their values can be obtained in practice.
Answer
:
The betacoefﬁcient can be deﬁned in any of the following equivalent ways:
β
j
=
σ
jM
σ
2
M
=
ρ
jM
σ
j
σ
M
σ
2
M
=
ρ
jM
σ
j
σ
M
,
where
σ
jM
is the covariance between the rate of return on asset
j
and the market rate
of return,
σ
M
is the standard deviation of the market rate of return,
σ
j
is the standard
deviation of the rate of return on asset
j
, and
ρ
jM
is the correlation coefﬁcient between
the rate of return on asset
j
and the market rate of return.
An asset’s betacoefﬁcient is a measure of the relationship between its rate of return
and the market rate of return. It can be interpreted as a measure of the asset’s risk, relative
to the market as a whole. An asset’s betacoefﬁcient is formally the slope coefﬁcient on
the excess rate of return on the market in a regression of the excess rate of return on asset
j
on the excess rate of return on the market:
r
j
=
r
0
+ (
r
M

r
0
)
β
j
+
ε
j
,
j
= 1
,
2
, ..., n,
where
ε
j
is an unobserved random error. It is assumed that E
[
ε
j

r
M
] = 0
, that is, the
expected value of the error, conditional upon the rate of return on the market portfolio, is
zero.
Typically (almost always) betacoefﬁcients are estimated from data on past rates of
return (in the regression described above).
(b) Assuming that a riskfree asset is available, explain and interpret the Security Market
Line (SML) in the context of the CAPM. Construct the SML from the given information
and interpret the values of its coefﬁcients.
Answer
:
The CAPM predicts that:
μ
j
=
r
0
+ (
μ
M

r
0
)
β
j
,
where
μ
j
is the expected rate of return on asset
j
,
μ
M
is the expected rate of return on
the market portfolio, and
r
0
is the riskfree rate of return The SML treats
μ
j
as a function
of
β
j
and shows how the expected rate of return on each asset differs according to its
betacoefﬁcient. The slope of the SML is then a measure of the market ‘price’ of risk.
See ﬁgure 1.
The data in the question must satisfy:
0
.
066 =
r
0
+ 0
.
4(
μ
M

r
0
)
,
and
0
.
098 =
r
0
+ 1
.
2(
μ
M

r
0
)
.