Appendix 9A
Empirical Evidence
for the Risk-Return
Relationship
(Question 9)
•
The CAPM is a simple linear model expressed in terms of expected returns and
expected risk.
•
In its ex-ante form:
E  Ri   R f   E  Rm   R f  i
(9A.1)
•
Although many of the aforementioned extensions of the model support this
simple linear form, others suggest that it may not be linear, that factors other
than beta are needed to explain E  Ri  , or that the R f is not the appropriate
riskless rate.
•
The first step necessary to empirically test the theoretical CAPM is to transform
it from expectations (ex-ante) form into a form that uses observed data.
•
On average, the expected rate of return on an asset is equal to the realized rate of
 mt  Rmt - E ( Rmt );
return. This can be written as
E ( mt )  0;
Rit  E  Rit   i mt  eit
eit  a random error term;
where Cov  eit ,  mt  = 0;
Cov  eit , eit 1  = 0; and
it = Cov( Rit Rmt )Var( Rmt ).
•
When CAPM is empirically tested it is usually written in the following form:
Rpt'   0   1 p  ept
•
where
(9A.2)
These relationships can be stated as follows:
1.
 1  Rmt  R ft
Rpt  R pt  R ft
The intercept term  0 should not be significantly different from zero.
2.
Beta should be the only factor that explains the rate of return on a risky
asset. If other terms, such as residual variance, dividend yields, P/E
ratios, firm size, or beta squared are included in an attempt to explain
return, they should have no explanatory power.
3.
The relationship should be linear in beta.
4.
5.
The coefficient of beta, “II” should be equal to Rmt  R ft .
When the equation is estimated over very long periods of time, the rate of
return on the market portfolio should be greater than the risk-free rate.
•
Black et al. observed that cross-sectional tests may not provide direct validation
of the CAPM, and they proceeded to construct a time-series test, which they
considered more powerful.
•
Their results lead them to assert that the usual form of the CAPM does not
provide an accurate description of the structure of security returns.
•
Their results indicate that  s are nonzero and are directly related to the risk
level.
•
Low-beta securities earn significantly more on average than predicted by the
model, and high-risk securities earn significantly less on average than predicted
by the model.
•
They go on to argue for a two-factor model:
•
Rit  (1 - i ) Rzt  i  Rmt   eit
(9A.3)
If E ( Rz )  0 then the S–L–M CAPM would be consistent with this model.
•
Fama and MacBeth (1973) tested
(1)
(2)
•
a linear relationship between return on the portfolio and the portfolio’s
beta
whether unsystematic risk has an effect between portfolio return and a
risk measure in addition to beta
Their basic estimation equation is:
Rit   0t   1t i   2t  i2   3t i2     eit
•
•
(9A.4)
in which  0t is the intercept term,  i is the average of the i for all individual
securities in portfolio i, and  i    is the average of the residual standard
deviations from all securities in portfolio j.
Roll’s critique is a broad indictment of most of the accepted empirical evidence
concerning the CAPM theory.
9A.1 anomalies in the Semi-Strong
Efficient-Market hypothesis
•
Three anomalies in the semi-strong efficient-market hypothesis are noteworthy:
(1)
P/E ratios
(2)
size effects
(3)
the January effect
•
P/E ratios effect is shown more in APT model (see Chapter 11) than CAPM
model.
•
The effect of firm size is that small-company stock returns were higher than largecompany stock returns.
•
The January effect is the phenomenon that over 50% of these excess returns were
realized in January and that 50% of the January excess return occurs in the first
week of January.