CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS
CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS
PROBLEM SETS
1. Using the regression feature of Excel with the data presented in the text, the
first-pass (SCL) estimation results are:
Stock: A B C D E F G H I
2. The hypotheses for the second-pass regression for the SML are:
3. The second-pass data from first-pass (SCL) estimates are:
Average
Excess
Return Beta
A 5.18 -0.47
B 4.19 0.59
CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS
S
The second-pass regression yields:
Regression Statistics
Multiple R0.7074
Coefficients
Standard
Error
t Statistic
for β=0
t Statistic
for β=8.12
4. As we saw in the chapter, the intercept is too high (3.92% per year instead of 0) and
5. Arranging the securities in three portfolios based on betas from the SCL estimates,
the first pass input data are:
Year ABC DEG FHI
1 15.05 25.86 56.69
2 -16.76 -29.74 -50.85
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CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS
The first-pass (SCL) estimates are:
ABC DEG FHI
R-square 0.04 0.48 0.82
Grouping into portfolios has improved the SCL estimates as is evident from the
higher R-square for Portfolio DEG and Portfolio FHI. This means that the beta
(slope) is measured with greater precision, reducing the error-in-measurement
problem at the expense of leaving fewer observations for the second pass.
The inputs for the second pass regression are:
Average
Excess
Return Beta
The second-pass estimates are:
Regression Statistics
Coefficients
Standard
Error
t Statistic
for β =0
t Statistic
for β =8.12
Despite the decrease in the intercept and the increase in slope, the intercept is now
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CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS
6. Roll’s critique suggests that the problem begins with the market index, which is
not the theoretical portfolio against which the second pass regression should hold.
7.
Except for Stock I, which realized an extremely positive surprise, the CML shows
8. The first-pass (SCL) regression results are summarized below:
A B C D E F G H I
R-square 0.07 0.36 0.11 0.44 0.24 0.84 0.12 0.68 0.71
Observations 12 12 12 12 12 12 12 12 12
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ABC
Market
DEG
C
B
A
D
E
F
G
H
FHI
I
0
5
10
15
20
25
0 10 20 30 40 50 60 70
Standard Deviation
Average Return
CML
CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS
9. The hypotheses for the second-pass regression for the two-factor SML are:
The intercept is zero.
10. The inputs for the second pass regression are:
Average
Excess
Return Beta MBeta F
A 5.18 -0.47 -0.35
B 4.19 0.58 2.33
The second-pass regression yields:
Regression Statistics
Multiple R0.7234
Coefficients
Standard
Error
t Statistic
for β =0
t Statistic
for β =8.12
t Statistic
for β =0.6
These results are slightly better than those for the single factor test; that is, the
intercept is smaller and the slope of M is slightly greater. We cannot expect a great
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CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS
improvement since the factor we added does not appear to carry a large risk
11. When we use the actual factor, we implicitly assume that investors can perfectly
replicate it, that is, they can invest in a portfolio that is perfectly correlated with the
factor. When this is not possible, one cannot expect the CAPM equation (the second
pass regression) to hold. Investors can use a replicating portfolio (a proxy for the
Proxy Portfolio for Factor F (PF)
Weights on
Universe
Stocks Year
PF Holding
Period
Returns
A -0.14 1 -33.51
B 1.00 2 62.78
This proxy (PF) has an R-square with the actual factor of 0.80.
We next perform the first pass regressions for the two factor model using PF
instead of P:
A B C D E F G H I
R-square
0.08
0.55
0.20
0.43
0.33
0.88
0.16
0.71
0.72
Intercept
9.28
-2.53
-1.35
-4.45
-0.23
-3.20
4.99
-2.92
5.54
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CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS
Beta PF -0.06 0.42 0.16 -0.13 0.21 -0.29 0.21 0.11 0.08
t-intercept
0.72
-0.21
-0.12
-0.36
-0.02
-0.55
0.27
-0.33
0.58
Note that the betas of the nine stocks on M and the proxy (PF) are different from
those in the first pass when we use the actual proxy.
The first-pass regression for the two-factor model with the proxy yields:
Average
Excess
Return
Beta M Beta PF
A 5.18 -0.50 -0.06
B 4.19 0.80 0.42
The second-pass regression yields:
Regression Statistics
Multiple R0.71
Coefficients Standard
Error
t Statistic
for β =0
t Statistic
for β =8.12
t Statistic
for β =0.6
Intercept 3.50 2.99 1.17
We can see that the results are similar to, but slightly inferior to, those with the
actual factor, since the intercept is larger and the slope coefficient smaller. Note
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CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS
12. We assume that the value of your labor is incorporated in the calculation of the rate
of return for your business. It would likely make sense to commission a valuation of
your business at least once each year. The resultant sequence of figures for
percentage change in the value of the business (including net cash withdrawals from
CFA PROBLEMS
1. (i) Betas are estimated with respect to market indexes that are proxies for the true
market portfolio, which is inherently unobservable.
(ii) Empirical tests of the CAPM show that average returns are not related to beta
2. a. The basic procedure in portfolio evaluation is to compare the returns on a
where rf is the risk-free rate, E(rM ) is the expected return for the unmanaged
b. The benchmark error might occur when the unmanaged portfolio used in the
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CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS
c. Your graph should show an efficient frontier obtained from actual returns, and
d. The answer to this question depends on one’s prior beliefs. Given a consistent
track record, an agnostic observer might conclude that the data support the
e. The question is really whether the CAPM is at all testable. The problem is that
even a slight inefficiency in the benchmark portfolio may completely
3. The effect of an incorrectly specified market proxy is that the beta of Black’s
portfolio is likely to be underestimated (i.e., too low) relative to the beta calculated
Proxy
2
Proxy
Cov( , )
βσ
Portfolio Market
Portfolio
Market
r r
=
An incorrectly specified market proxy is likely to produce a slope for the security
market line (i.e., the market risk premium) that is underestimated relative to the true
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