CHAPTER 13: EMPIRICAL EVIDENCE ON SECURITY RETURNS
Note: For end-of-chapter-problems in Chapter 13, the focus is on the estimation procedure. To
keep the exercise feasible the sample was limited to returns on 9 stocks plus a market
index and a second factor over 12 years. The data was generated to conform to a twofactor CAPM so that actual rates of return equal CAPM expectations plus random noise
and the true intercept of the SCL is zero for all stocks. The exercise will give you a feel
for the pitfalls of verifying social-science models. However, due to the small size of the
sample, results are not always consistent with the findings of other studies that are
reported in the chapter itself.
1.
Using the regression feature of Excel with the data presented in the text, the first-pass
(SCL) estimation results are:
Stock:
R Square
Observations
Alpha
Beta
t-Alpha
t-Beta
A
0.06
12
9.00
-0.47
0.73
-0.81
B
0.06
12
-0.63
0.59
-0.04
0.78
C
0.06
12
-0.64
0.42
-0.06
0.78
D
0.37
12
-5.05
1.38
-0.41
2.42
E
0.17
12
0.73
0.90
0.05
1.42
F
0.59
12
-4.53
1.78
-0.45
3.83
G
0.06
12
5.94
0.66
0.33
0.78
H
0.67
12
-2.41
1.91
-0.27
4.51
2.
The hypotheses for the second-pass regression for the SML is that the intercept is zero
and the slope equal to the average return on the index portfolio.
3.
The second-pass data from first-pass (SCL) estimates are:
A
B
C
D
E
F
G
H
I
M
Average Excess Return
5.18
4.19
2.75
6.15
8.05
9.90
11.32
13.11
22.83
8.12
13-1
Beta
-0.47
0.59
0.42
1.38
0.90
1.78
0.66
1.91
2.08
I
0.70
12
5.92
2.08
0.64
4.81
The second-pass regression yields:
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
Intercept
Slope
0.7074
0.5004
0.4291
4.6234
9
Coefficients
Standard Error
3.92
5.21
2.54
1.97
t Stat
for=0
1.54
2.65
t Stat
for =8.12
1.48
The results are reflected in the regression equation above intercept, slope coefficients,
standard errors and t-statistics as shown.
4.
As we saw in the chapter, the intercept is too high (3.92%/year instead of 0) and the slope
is too flat (5.21% instead of a predicted value equal to the sample-average risk premium,
rM  rf = 8.12%). While the intercept is not significantly greater than zero (the t-statistic
is less than 2) and the slope is not significantly different from its theoretical value (the tstatistic for this hypothesis is only 1.48), the lack of statistical significance is probably
due to the small size of the sample.
5.
Arranging the securities in three portfolios based on betas from the SCL estimates, the
first pass input data are:
Year
1
2
3
4
5
6
7
8
9
10
11
12
Average
Std Dev.
ABC
15.05
-16.76
19.67
-15.83
47.18
-2.26
-18.67
-6.35
7.85
21.41
-2.53
-0.30
4.04
19.30
DEG
25.86
-29.74
-5.68
-2.58
37.70
53.86
15.32
36.33
14.08
12.66
-50.71
-4.99
8.51
29.47
FHI
56.69
-50.85
8.98
35.41
-3.25
75.44
12.50
32.12
50.42
52.14
-66.12
-20.10
15.28
43.96
13-2
which result in first-pass (SCL) estimates:
ABC
0.04
12
2.58
0.18
0.42
0.62
R Square
Observations
Alpha
Beta
t-Alpha
t-Beta
DEG
0.48
12
0.54
0.98
0.08
3.02
FHI
0.82
12
-0.34
1.92
-0.06
6.83
Grouping into portfolios has improved the SCL estimates as is evident from the higher Rsquare. 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:
ABC
DEH
FGI
M
Avg. Excess Return
4.04
8.51
15.28
8.12
Beta
0.18
0.98
1.92
which results in second-pass estimates:
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
Intercept
Slope
0.9975
0.9949
0.9899
0.5693
3
Coefficients
Standard Error
2.62
6.47
0.58
0.46
t Stat
for =0
4.55
14.03
t Stat
for=8.12
-3.58
Despite the decease in the intercept and increase in slope, the intercept is now
significantly positive and the slop is significantly less than the hypothesized value by
more than twice the standard error.
6.
Roll’s critique suggests that the problem begins with the market index, which isn’t the
theoretical portfolio against which the second pass regression is supposed to hold.
13-3
Hence, even if the relationship is valid with respect to the true (unknown) index, we may
not find it. As a result, the second pass relationship may be meaningless.
7.
CAPITAL MARKET LINE FROM SAMPLE DATA
25
I
CML
Average Return
20
FHI
15
H
G
10
F
Market
DEG
E
D
A
5
B
ABC
C
0
0
10
20
30
40
50
60
70
Standard Deviation
Except for Stock I, which realized an extremely positive surprise, the CML shows that the
index dominates all other securities, and the three portfolios dominate all individual
stocks. The power of diversification is evident despite the use of a very small sample.
8.
The first-pass (SCL) regression results are summarized below.
R-Square
Observations
Intercept
Beta M
Beta F
t- Intercept
t-Beta M
t-Beta F
A
0.07
12
9.19
-0.47
-0.35
0.71
-0.77
-0.34
B
0.36
12
-1.89
0.58
2.33
-0.13
0.87
2.06
C
0.11
12
-1.00
0.41
0.67
-0.08
0.75
0.71
D
0.44
12
-4.48
1.39
-1.05
-0.37
2.46
-1.08
13-4
E
0.24
12
0.17
0.89
1.03
0.01
1.40
0.94
F
0.84
12
-3.47
1.79
-1.95
-0.52
5.80
-3.69
G
0.12
12
5.32
0.65
1.15
0.29
0.75
0.77
H
0.68
12
-2.64
1.91
0.43
-0.28
4.35
0.57
I
0.71
12
5.66
2.08
0.48
0.59
4.65
0.63
9.
10.
The hypotheses for the two-factor model are:
i.
The intercept should be zero.
ii.
The market-index slope coefficient should equal the market-index average return.
(1 and 2 are as with a single-factor model.)
iii.
The factor slope coefficient should equal the average return on the factor.
The inputs for the second pass regression are:
A
B
C
D
E
F
G
H
I
M
F
Average Excess Return
5.18
4.19
2.75
6.15
8.05
9.90
11.32
13.11
22.83
8.12
0.6
Beta M
-0.47
0.58
0.41
1.39
0.89
1.79
0.65
1.91
2.08
Beta F
-0.35
2.33
0.67
-1.05
1.03
-1.95
1.15
0.43
0.48
The second-pass regression yields:
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
Coefficients
Intercept
Beta M
Beta F
3.36
5.53
0.80
0.72
0.52
0.36
4.88
9
Standard Error
2.88
2.16
1.42
13-5
t Stat
for  =0
1.16
2.56
0.56
t Stat
for =8.12
t Stat
for =0.6
-1.20
0.14
These results are slightly better than the single factor test, that is, the intercept is smaller
and the slope on M is slightly greater. We cannot expect a great improvement since the
factor we added does not appear to carry a large risk premium (average excess return less
than 1%), and its effect on mean returns is therefore small. The data does not reject the
second factor because the slope is close to the average excess return and the difference is
less than one standard error. However, with this sample size, the power of this test is
extremely small.
11.
When we use the actual factor, we implicitly assume that investors can perfectly replicate
it, that is, 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 factor) that maximizes the
correlation with the factor. The CAPM equation is then expected to hold with respect to
the proxy portfolio.
Using the bordered covariance matrix of the 9 stocks and the Excel Solver as in Chapter
8, we produce a proxy portfolio for factor F, denoted PF. To preserve the scale, we
included constraints that require the nine weights to be in the range of [-1,1] and that the
mean equal the factor mean of .60%. The resultant weights for the proxy and period
returns are shown below:
Proxy Portfolio for Factor F (PF)
Weights on
Universe Stocks
A
B
C
D
E
F
G
H
I
-0.14
1.00
0.95
-0.35
0.16
-1.00
0.13
0.19
0.06
Year
1
2
3
4
5
6
7
8
9
10
11
12
Average
PF Holding
Period
Returns
-33.51
62.78
9.87
-153.56
200.76
-36.62
-74.34
-10.84
28.11
59.51
-59.15
14.22
0.60
This proxy, PF, has an R-square with the actual factor of .80.
We next perform the first pass regressions for the two factor model using PF instead of P.
13-6
A
0.08
12
9.28
-0.50
-0.06
0.72
-0.83
-0.44
R-square
Observations
Intercept
Beta M
Beta PF
t- Intercept
t-Beta M
t-Beta PF
B
0.55
12
-2.53
0.80
0.42
-0.21
1.43
3.16
C
0.20
12
-1.35
0.49
0.16
-0.12
0.94
1.25
D
0.43
12
-4.45
1.32
-0.13
-0.36
2.29
-0.97
E
0.33
12
-0.23
1.00
0.21
-0.02
1.66
1.47
F
0.88
12
-3.20
1.64
-0.29
-0.55
6.00
-4.52
G
0.16
12
4.99
0.76
0.21
0.27
0.90
1.03
H
0.71
12
-2.92
1.97
0.11
-0.33
4.67
1.13
I
0.72
12
5.54
2.12
0.08
0.58
4.77
0.78
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:
A
B
C
D
E
F
G
H
I
M
PF
Average Excess Return
5.18
4.19
2.75
6.15
8.05
9.90
11.32
13.11
22.83
8.12
0.6
Beta M
-0.50
0.80
0.49
1.32
1.00
1.64
0.76
1.97
2.12
Beta PF
-0.06
0.42
0.16
-0.13
0.21
-0.29
0.21
0.11
0.08
The second-pass regression yields:
Regression Statistics
Multiple R
R Square
Adjusted R Square
Standard Error
Observations
Coefficients
Intercept
Beta M
Beta PF
3.50
5.39
0.26
0.71
0.51
0.35
4.95
9
Standard Error
2.99
2.18
8.36
13-7
t Stat
for  =0
1.17
2.48
0.03
t Stat
for =8.12
t Stat
for =0.6
-1.25
-0.04
We can see that the results are similar and slightly inferior to those with the actual factor,
since the intercept is larger and the slope coefficient smaller. Note also that we use here
an in-sample test rather than tests with future returns, which is more forgiving than an
out-of-sample test.
12.
(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 in the
manner predicted by the theory. The empirical SML is flatter than the theoretical one.
(iii) Multi-factor models of security returns show that beta, which is a one-dimensional
measure of risk, may not capture the true risk of the stock of portfolio.
13.
a.
[From the CFA Study Guide]
The basic procedure in portfolio evaluation is to compare the returns on a managed
portfolio to the return expected on an unmanaged portfolio having the same risk, via use
of the SML. That is, expected return is calculate from:
E(rp) = rf + P[E(rM) – rf]
where rf is the risk-free rate, E(rM) is the unmanaged portfolio (or the market) expected
return and P is the beta coefficient (or systematic risk) of the managed portfolio. The
benchmark of performance then is the unmanaged portfolio. The typical proxy for this
unmanaged portfolio is some aggregate stock market index such as the S&P 500.
b.
The benchmark error may occur when the unmanaged portfolio used in the evaluation
process is not "optimized." That is, market indices, such as the S&P 500, chosen as
benchmarks are not on the manager’s ex ante mean/variance efficient frontier.
c.
Your graph should show an efficient frontier obtained from actual returns, and a different
one which represents (unobserved) ex-ante expectations. The CML and SML generated
from actual returns do not conform to the CAPM predictions, while the hypothesized
lines do.
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 claim of superiority.
Other observers might start with a strong prior that, since so many managers are
attempting to beat a passive portfolio, a small number are bound to come up with
seemingly convincing track records.
13-8
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 invalidate any test of the
expected return-beta relationship. It appears from Roll’s argument that the best guide to
the question of the validity of the CAPM is the difficulty of beating a passive strategy.
13-9