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CAPM - Do you want fries with that?Investment decision making ultimately comes down to questions of risk . How shouldrisk be assessed? How much risk should we take to obtain a given return? What types ofrisk are rewarded and what types are not?
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CAPM: Do you want fries with that?
Investment decision making ultimately comes down to questions of risk . How should
risk be assessed? How much risk should we take to obtain a given return? What types of
risk are rewarded and what types are not?
The capital asset pricing model (CAPM) is the standard model representing the re-
lationship between risk and return. CAPM states that risk is measured by the variance
in the returns, so that the expected return of an investment represents the reward, while
the variance of returns is the risk. In this representation of reality, given two investments
with the same expected return but different variances, an investor will always choose the
investment with smaller variance. Similarly, given two investments with the same variance
of returns but different expected returns, an investor will always choose the investment
with higher expected return.
Under the CAPM model, all variance is risk, but not all risk is rewarded. For any
asset, risk comes from two sources: effects that come from the specific actions of the
asset manager (which affect only that asset), and marketwide movements (which affect all
assets). Since marketwide effects will affect all assets, they cannot be diversified away. On
the other hand, assetspecific components of risk will cancel out with each other if a large
portfolio of assets is constructed, so under CAPM they are not rewarded. That is, under
CAPM only variability related to market variability (the systematic risk or nondiversifiable
risk) is rewarded.
Under CAPM, the expected return on an asset R can be written as a function of the
riskfree rate Rf (the return on the riskless asset, which has no variance; this is typically
taken to be a short or longterm bond rate, such as the 3month Treasury bills rate) and
the expected return of the market E(Rm):
E(R) = Rf + (E[Rm]Rf )= Rf (1 ) + (E[Rm]), (1)
where is the beta of the asset, the covariance of the assets returns with the market
returns divided by the market return variance. This function is called the security market
line.
c 2011, Jeffrey S. Simonoff 1
The beta of a security is of interest to an investor, as it measures the relative risk
of the security compared with the market (a beta greater than one indicates a riskier
than average security, while a beta less than one is consistent with a safer than average
security). The beta can be estimated using a regression model relating stock returns to
market returns,
Ri = 0 + 1Rmi + i, (2)
with V (i) = 2. Comparing this regression equation to (1) shows that the estimate
of the slope is an estimate of beta. The estimated constant term can be compared to
Rf (1 ) to see how the stock performed relative to the prediction of performance usingCAPM. (Technically, this is called the SharpeLintner version of CAPM; the Black version
replaces Rf (1 ) in equation (1) with E(R0m)(1 ), where R0m is the return on thesocalled zerobeta portfolio, the portfolio that has the minimum variance of all portfolios
uncorrelated with the market portfolio of assets.) The R2 of the regression, which estimates
the proportion of the variability in the security accounted for by the market, estimates the
market (nondiversifiable) risk of the security.
The data examined here are the monthly returns for the McDonalds Food Corpo-
ration. The data cover November 1988 through March 1996, or 89 months. The market
return is measured using the New York Stock Exchange Composite Index. Here are the
values:
Row Date McDonalds return Market return
1 8811 -0.042501 -0.020172
2 8812 0.021505 0.028091
3 8901 -0.001347 0.037379
4 8902 0.079096 0.041268
5 8903 -0.009143 -0.017024
6 8904 0.048028 0.024549
7 8905 0.084656 0.052701
8 8906 0.016789 0.020975
9 8907 0.017058 0.037563
10 8908 -0.016772 0.076022
11 8909 0.011736 0.009559
12 8910 0.018951 -0.048047
13 8911 0.040373 -0.019689
c 2011, Jeffrey S. Simonoff 2
14 8912 0.076532 0.025833
15 9001 -0.054485 -0.028048
16 9002 0.017563 -0.030466
17 9003 -0.025920 0.034622
18 9004 -0.005864 0.007355
19 9005 0.021756 0.019589
20 9006 0.098020 0.015266
21 9007 0.010572 -0.013860
22 9008 -0.197538 -0.088701
23 9009 -0.069716 -0.050428
24 9010 0.002323 -0.024987
25 9011 0.056075 0.012492
26 9012 0.039858 0.038468
27 9101 -0.045752 0.006911
28 9102 0.103667 0.115468
29 9103 0.096968 0.007186
30 9104 0.027639 0.005604
31 9105 -0.025118 0.007187
32 9106 -0.023000 0.018338
33 9107 -0.002770 -0.008401
34 9108 -0.020769 0.013011
35 9109 0.010605 -0.004602
36 9110 0.071503 0.021460
37 9111 -0.016911 0.003192
38 9112 0.022588 -0.029701
39 9201 0.194721 0.095603
40 9202 0.019247 0.006729
41 9203 -0.034140 -0.002026
42 9204 0.006742 -0.000769
43 9205 0.063522 0.023114
44 9206 0.028110 -0.014667
45 9207 -0.009537 -0.005648
46 9208 -0.055916 -0.011874
47 9209 0.038035 -0.001818
48 9210 -0.030287 -0.016921
49 9211 0.087144 0.029031
50 9212 0.046727 0.022980
51 9301 0.000653 0.010092
52 9302 0.020408 0.032486
53 9303 0.050000 0.021991
54 9304 -0.065486 0.007931
55 9305 0.001916 0.000844
56 9306 0.008910 -0.002222
57 9307 -0.011977 0.010511
c 2011, Jeffrey S. Simonoff 3
58 9308 0.100776 0.026065
59 9309 0.000000 0.003056
60 9310 -0.001149 0.004092
61 9311 0.039424 0.012023
62 9312 0.030143 0.014189
63 9401 -0.002184 0.026788
64 9402 0.051041 0.000882
65 9403 0.003107 -0.028628
66 9404 -0.057672 -0.052835
67 9405 0.041512 -0.001732
68 9406 0.006816 0.007713
69 9407 -0.042189 -0.009808
70 9408 -0.075072 0.009913
71 9409 0.025863 0.003630
72 9410 0.004587 -0.017590
73 9411 0.065059 -0.011684
74 9412 -0.020339 -0.017653
75 9501 0.021881 0.039433
76 9502 0.132760 0.024697
77 9503 0.047243 0.014850
78 9504 0.007220 0.037510
79 9505 0.044817 0.026078
80 9506 0.033427 0.030091
81 9507 0.013278 0.053668
82 9508 -0.025370 -0.004127
83 9509 0.052920 0.029195
84 9510 0.012769 0.001223
85 9511 0.085914 0.023503
86 9512 0.045700 0.040412
87 9601 0.016655 -0.005367
88 9602 0.112645 0.050522
89 9603 -0.000628 0.005572
The use of monthly returns is quite typical in CAPM calculations, but the 7 12year time
period is a bit longer than is typical (for example, Value Line and Standard and Poors
use five years of data, while Bloomberg uses two).
CAPM implies a linear relationship between McDonalds returns and market returns,
which looks reasonable here:
c 2011, Jeffrey S. Simonoff 4
-0.1 0.0 0.1
-0.2
-0.1
0.0
0.1
0.2
Market return
McD
on
ald
s r
etu
rn
There is one noteworthy month at the lower left, which is case 22 (August 1990). This was
at the beginning of a recession, and while the market did poorly (a 9% drop), McDonalds
did particularly poorly (a 20% drop). Its not too surprising that a company that specializes
in fast food (hardly a staple item) would suffer in a recession, and McDonalds did; its
longterm debt was $4.4 billion in 1990, its highest value ever up through early 1996.
Here are the results of a regression fit.
Regression Analysis
The regression equation is
McDonalds return = 0.00735 + 1.09 Market return
Predictor Coef SE Coef T P
Constant 0.007351 0.004641 1.58 0.117
Market r 1.0893 0.1503 7.25 0.000
S = 0.04171 R-Sq = 37.7% R-Sq(adj) = 36.9%
Analysis of Variance
c 2011, Jeffrey S. Simonoff 5
Source DF SS MS F P
Regression 1 0.091398 0.091398 52.55 0.000
Error 87 0.151328 0.001739
Total 88 0.242726
The estimate of beta is 1.089; while this is greater than one (indicating a riskier
than average stock), it is not significantly greater than one, as a ttest for the hypothesis
H0 : 1 = 1 is
t =1.0893 1
.1503= .59.
R2 = .377, leaving 62.3% diversifiable risk. This value of market (nondiversifiable) risk is
a bit higher than is typical for U.S. stocks, since market risk averages about 27.0% in the
U.S. market (it averages about 35% for U.K. stocks, 45% for German stocks, and 60% for
the Taiwanese stock market).
Does the least squares model fit these data? Here are some regression diagnostics.
Note that August 1990 is apparently an outlier / leverage / influential point; August 1989,
February 1991, and January 1992 also show up as possibly problematic.
Data Display
Row Date SRES1 HI1 COOK1
1 8811 -0.67611 0.022593 0.005283
2 8812 -0.39749 0.015770 0.001266
3 8901 -1.19775 0.021396 0.015683
4 8902 0.65035 0.024418 0.005293
5 8903 0.04970 0.020305 0.000026
6 8904 0.33654 0.014214 0.000817
7 8905 0.48577 0.035575 0.004352
8 8906 -0.32366 0.012974 0.000688
9 8907 -0.75655 0.021530 0.006297
10 8908 -2.65715 0.068855 0.261047
11 8909 -0.14534 0.011236 0.000120
12 8910 1.57633 0.054091 0.071046
13 8911 1.32082 0.022226 0.019828
14 8912 0.99137 0.014740 0.007352
15 9001 -0.76136 0.029448 0.008794
c 2011, Jeffrey S. Simonoff 6
16 9002 1.05761 0.031876 0.018414
17 9003 -1.71889 0.019493 0.029369
18 9004 -0.51187 0.011290 0.001496
19 9005 -0.16732 0.012583 0.000178
20 9006 1.78572 0.011682 0.018846
21 9007 0.44332 0.018263 0.001828
22 9008 -2.79304 0.136198 0.615005
23 9009 -0.54669 0.057717 0.009153
24 9010 0.53931 0.026592 0.003973
25 9011 0.84681 0.011360 0.004120
26 9012 -0.22787 0.022203 0.000590
27 9101 -1.46207 0.011317 0.012234
28 9102 -0.76968 0.157294 0.055288
29 9103 1.97225 0.011300 0.022228
30 9104 0.34204 0.011424 0.000676
31 9105 -0.97171 0.011300 0.005396
32 9106 -1.21419 0.012272 0.009159
33 9107 -0.02342 0.015352 0.000004
34 9108 -1.01992 0.011405 0.006000
35 9109 0.19962 0.013783 0.000278
36 9110 0.98415 0.013123 0.006440
37 9111 -0.66903 0.011737 0.002658
38 9112 1.15927 0.031091 0.021562
39 9201 2.11255 0.107705 0.269347
40 9202 0.11011 0.011329 0.000069
41 9203 -0.94805 0.012932 0.005888
42 9204 0.00553 0.012580 0.000000
43 9205 0.74825 0.013676 0.003882
44 9206 0.88922 0.018759 0.007558
45 9207 -0.25925 0.014178 0.000483
46 9208 -1.21729 0.017115 0.012901
47 9209 0.78831 0.012871 0.004051
48 9210 -0.46522 0.020234 0.002235
49 9211 1.16446 0.016237 0.011190
50 9212 0.34628 0.013629 0.000828
51 9301 -0.42658 0.011242 0.001035
52 9302 -0.54036 0.018153 0.002699
53 9303 0.45122 0.013293 0.001371
54 9304 -1.96467 0.011264 0.021987
55 9305 -0.15330 0.012187 0.000145
56 9306 0.09605 0.012991 0.000061
57 9307 -0.74217 0.011252 0.003134
58 9308 1.57097 0.014840 0.018588
59 9309 -0.25758 0.011759 0.000395
c 2011, Jeffrey S. Simonoff 7
60 9310 -0.31250 0.011602 0.000573
61 9311 0.45758 0.011325 0.001199
62 9312 0.17691 0.011533 0.000183
63 9401 -0.93543 0.015159 0.006735
64 9402 1.03083 0.012179 0.006551
65 9403 0.65592 0.030016 0.006657
66 9404 -0.18482 0.061532 0.001120
67 9405 0.86994 0.012846 0.004924
68 9406 -0.21549 0.011273 0.000265
69 9407 -0.93920 0.016029 0.007185
70 9408 -2.24787 0.011239 0.028719
71 9409 0.35112 0.011669 0.000728
72 9410 0.39732 0.020697 0.001668
73 9411 1.70344 0.017010 0.025107
74 9412 -0.20497 0.020742 0.000445
75 9501 -0.68952 0.022943 0.005582
76 9502 2.37893 0.014272 0.040971
77 9503 0.57198 0.011621 0.001923
78 9504 -0.99361 0.021492 0.010842
79 9505 0.21884 0.014845 0.000361
80 9506 -0.16210 0.016792 0.000224
81 9507 -1.28342 0.036674 0.031354
82 9508 -0.68139 0.013613 0.003204
83 9509 0.33280 0.016321 0.000919
84 9510 0.09859 0.012105 0.000060
85 9511 1.27870 0.013817 0.011454
86 9512 -0.13765 0.023719 0.000230
87 9601 0.36586 0.014069 0.000955
88 9602 1.22558 0.033186 0.025779
89 9603 -0.33879 0.011427 0.000663
We could now try to address potential model violations relative to the OLS model.
For example, August 1990 might be removed, and we would reanalyze without it. Rather
than do that, however, Id like to raise a different question: is August 1990 really unusual?
Its further from the regression line than we would expect under OLS assumptions, but
there is good reason to doubt one of those assumptions here the assumption of constant
variance of the errors. If August 1990 corresponds to an observation with inherently larger
residual variance, then its observed McDonalds return might not be unusually low at all.
c 2011, Jeffrey S. Simonoff 8
Why might we expect nonconstant variance here? It comes from a crucial CAPM
assumption: that the beta is constant over the entire 7 12year time period. This is unlikely
to be true, as there is ample empirical evidence that betas change over time. If we fit a
model with a constant beta to data consistent with changing beta, this will show up as
nonconstant variance of a specific type.
Lets consider a simple example: say there are two possible beta values for a given
month, 1 + c and 1 c (obviously we could choose 1 and c to represent the two valuesthis way). The true underlying regression relationships are
Ri = 0 + (1 + c)Rmi + i (3a)
with probability .5, and
Ri = 0 + (1 c)Rmi + i (3b)
with probability .5. Under this model, we have
E(Ri) = .5[0 + (1 + c)Rmi] + .5[0 + (1 c)Rmi]= 0 + 1Rmi;
that is, on average the asset returns satisfy the CAPM formula (2). However, what are
the variances of the errors, E[Ri E(Ri) |Rmi]2? For group (3a), we have
V (i) = E[Ri E(Ri) |Rmi]2
= E[0 + (1 + c)Rmi + i {0 + 1Rmi}]2
= E[cRmi + i]2
= c2R2mi + 2.
For group (3b), we have
V (i) = E[Ri E(Ri) |Rmi]2
= E[0 + (1 c)Rmi + i {0 + 1Rmi}]2
= E[cRmi + i]2
= c2R2mi + 2.
c 2011, Jeffrey S. Simonoff 9
That is, if the true beta varies in this way, the variance of the errors is 2+c2R2mi; we have
heteroscedasticity, with the observed variance being a quadratic function of the market
return.
We can look at a plot of the absolute residuals from the OLS fit versus the market
return values to see if nonconstant variance of this form is indicated. Here is a plot, with
a lowess curve superimposed. This curve is an example of what is called a nonparametric
regression estimate. Basically, it puts a smooth curve through the data points to help
suggest structure that might not otherwise show up very clearly (it does this by fitting
straight lines locally, rather than one straight line globally). The quadratic form of the
nonconstant variance is very obvious.
-0.1 0.0 0.1
0
1
2
3
Market return
Ab
so
lute
re
sid
ua
ls
A Levenes test clearly rejects constant variance in favor of a quadratic model for het-
eroscedasticity (see the appendix for discussion of how to identify and handle nonconstant
variance that is related to a numerical predictor, rather than group membership):
c 2011, Jeffrey S. Simonoff 10
Regression Analysis
The regression equation is
Absolute residuals = 0.691 - 1.35 Market return + 119 Markretsquared
Predictor Coef SE Coef T P
Constant 0.69102 0.07033 9.83 0.000
Market r -1.353 2.322 -0.58 0.562
Markrets 118.77 34.53 3.44 0.001
S = 0.5928 R-Sq = 12.7% R-Sq(adj) = 10.7%
Analysis of Variance
Source DF SS MS F P
Regression 2 4.3986 2.1993 6.26 0.003
Error 86 30.2192 0.3514
Total 88 34.6178
Here is a regression to estimate the weights for a WLS fit:
Regression Analysis
The regression equation is
lgsressq = - 1.51 + 1.28 Market return + 262 Markretsquared
Predictor Coef SE Coef T P
Constant -1.5086 0.2430 -6.21 0.000
Market r 1.278 8.023 0.16 0.874
Markrets 261.8 119.3 2.19 0.031
S = 2.048 R-Sq = 6.6% R-Sq(adj) = 4.4%
Analysis of Variance
Source DF SS MS F P
Regression 2 25.337 12.669 3.02 0.054
Error 86 360.871 4.196
Total 88 386.208
The following plot illustrates the quadratic fit being used to estimate these weights:
c 2011, Jeffrey S. Simonoff 11
-0.1 0.0 0.1
-10
-5
0
Market r
lgsre
ssq
Y = -1.50862 + 1.27803X + 261.782X**2
R-Sq = 0.066
Regression Plot
Here is a WLS version of the CAPM fit:
Regression Analysis
Weighted analysis using weights in wt
The regression equation is
McDonalds return = 0.00956 + 0.961 Market return
Predictor Coef SE Coef T P
Constant 0.009556 0.004241 2.25 0.027
Market r 0.9610 0.1904 5.05 0.000
S = 0.07457 R-Sq = 22.6% R-Sq(adj) = 21.8%
Analysis of Variance
Source DF SS MS F P
Regression 1 0.14168 0.14168 25.47 0.000
Error 87 0.48401 0.00556
Total 88 0.62569
c 2011, Jeffrey S. Simonoff 12
Things have changed a bit. The estimated beta for McDonalds is now less than one
(although again, not significantly different from one). Note also that if this regression
model was used to predict the McDonalds return from a given market return, the use
of weights could change things dramatically. One would expect to find that a prediction
interval from the WLS model would be narrower than one from the OLS model for a
prediction for a small (close to zero) market return month, and wider for a prediction for
a large (absolute) market return month, reflecting the inherent difference in variability off
the regression line in these circumstances.
August 1990 is no longer an outlier, since its high variability is accounted for by a
small weight (that is, the assessment of the point as an outlier has changed because our
model for the underlying variability of the observation has changed). Similarly, points
previously flagged as potential leverage points are no longer assessed as problematic.
Row Date SRES2 HI2 COOK2
1 8811 -0.90899 0.0313846 0.0133862
2 8812 -0.38421 0.0216368 0.0016323
3 8901 -1.10109 0.0278343 0.0173564
4 8902 0.67416 0.0301746 0.0070705
5 8903 -0.06580 0.0278675 0.0000621
6 8904 0.38974 0.0193488 0.0014985
7 8905 0.47698 0.0348600 0.0041088
8 8906 -0.34625 0.0172601 0.0010528
9 8907 -0.67121 0.0279507 0.0064772
10 8908 -1.28713 0.0313948 0.0268489
11 8909 -0.19751 0.0133179 0.0002633
12 8910 1.24446 0.0582867 0.0479275
13 8911 1.38654 0.0308341 0.0305820
14 8912 1.09424 0.0201590 0.0123170
15 9001 -0.99147 0.0405856 0.0207919
16 9002 0.98144 0.0433563 0.0218270
17 9003 -1.66055 0.0260398 0.0368614
18 9004 -0.63790 0.0132034 0.0027223
19 9005 -0.17892 0.0165339 0.0002691
20 9006 2.03582 0.0146603 0.0308323
21 9007 0.40715 0.0245613 0.0020871
22 9008 -1.33968 0.0398550 0.0372495
23 9009 -0.67044 0.0592510 0.0141552
24 9010 0.45649 0.0370008 0.0040032
c 2011, Jeffrey S. Simonoff 13
25 9011 0.96311 0.0138295 0.0065039
26 9012 -0.15504 0.0285165 0.0003528
27 9101 -1.75915 0.0132096 0.0207128
28 9102 -0.07833 0.0096077 0.0000298
29 9103 2.28448 0.0132046 0.0349175
30 9104 0.36163 0.0132861 0.0008805
31 9105 -1.17992 0.0132046 0.0093147
32 9106 -1.36401 0.0159271 0.0150562
33 9107 -0.12198 0.0196353 0.0001490
34 9108 -1.19281 0.0139605 0.0100721
35 9109 0.15737 0.0169432 0.0002134
36 9110 1.10382 0.0175263 0.0108677
37 9111 -0.84500 0.0136596 0.0049442
38 9112 1.09992 0.0424880 0.0268421
39 9201 0.76404 0.0201325 0.0059970
40 9202 0.09159 0.0132150 0.0000562
41 9203 -1.20053 0.0155118 0.0113544
42 9204 -0.05963 0.0149358 0.0000270
43 9205 0.83955 0.0184774 0.0066344
44 9206 0.92534 0.0253776 0.0111478
45 9207 -0.39286 0.0176171 0.0013839
46 9208 -1.54221 0.0226415 0.0275492
47 9209 0.86912 0.0154108 0.0059115
48 9210 -0.66391 0.0277551 0.0062915
49 9211 1.25946 0.0222654 0.0180612
50 9212 0.39922 0.0183982 0.0014936
51 9301 -0.52347 0.0133812 0.0018582
52 9302 -0.50161 0.0246045 0.0031735
53 9303 0.51414 0.0178250 0.0023986
54 9304 -2.34118 0.0132102 0.0366878
55 9305 -0.24256 0.0143164 0.0004273
56 9306 0.04283 0.0156088 0.0000145
57 9307 -0.88904 0.0134404 0.0053840
58 9308 1.71500 0.0203083 0.0304847
59 9309 -0.35750 0.0136898 0.0008870
60 9310 -0.41817 0.0134847 0.0011951
61 9311 0.51192 0.0137214 0.0018229
62 9312 0.19265 0.0143000 0.0002692
63 9401 -0.96648 0.0207773 0.0099098
64 9402 1.16619 0.0143037 0.0098676
65 9403 0.56110 0.0412568 0.0067741
66 9404 -0.34721 0.0599191 0.0038420
67 9405 0.96668 0.0153696 0.0072933
68 9406 -0.28770 0.0132056 0.0005538
c 2011, Jeffrey S. Simonoff 14
69 9407 -1.21161 0.0207955 0.0155879
70 9408 -2.65115 0.0133585 0.0475814
71 9409 0.36648 0.0135692 0.0009238
72 9410 0.33529 0.0284856 0.0016481
73 9411 1.90440 0.0224654 0.0416743
74 9412 -0.36314 0.0285548 0.0019381
75 9501 -0.58862 0.0291046 0.0051932
76 9502 2.60271 0.0194413 0.0671540
77 9503 0.64719 0.0145156 0.0030847
78 9504 -0.90140 0.0279175 0.0116676
79 9505 0.26432 0.0203163 0.0007244
80 9506 -0.12688 0.0229809 0.0001893
81 9507 -0.92057 0.0350679 0.0153991
82 9508 -0.89047 0.0166545 0.0067148
83 9509 0.38746 0.0223760 0.0017181
84 9510 0.05847 0.0141909 0.0000246
85 9511 1.41815 0.0187098 0.0191727
86 9512 -0.06134 0.0296836 0.0000575
87 9601 0.35241 0.0174310 0.0011016
88 9602 1.09682 0.0342752 0.0213484
89 9603 -0.44263 0.0132891 0.0013193
The Levenes test is no longer significant, which is consistent with the residual plots,
which all look fine:
The regression equation is
absres = 0.816 - 0.61 Market return - 8.3 Marketsquared
Predictor Coef SE Coef T P
Constant 0.81588 0.07184 11.36 0.000
Market r -0.607 2.371 -0.26 0.799
Marketsq -8.31 35.27 -0.24 0.814
S = 0.6055 R-Sq = 0.2% R-Sq(adj) = 0.0%
Analysis of Variance
Source DF SS MS F P
Regression 2 0.0729 0.0365 0.10 0.905
Residual Error 86 31.5284 0.3666
Total 88 31.6013
c 2011, Jeffrey S. Simonoff 15
-0.1 0.0 0.1
-3
-2
-1
0
1
2
3
Market return
SR
ES
2
-0.1 0.0 0.1
0
1
2
3
Market return
ab
sre
s
c 2011, Jeffrey S. Simonoff 16
10 20 30 40 50 60 70 80
-3
-2
-1
0
1
2
3
Observation Order
Sta
nd
ard
ize
d R
esid
ua
lResiduals Versus the Order of the Data
(response is McDonald)
-3 -2 -1 0 1 2 3
-3
-2
-1
0
1
2
3
Normal Score
Sta
nd
ard
ize
d R
esid
ua
l
Normal Probability Plot of the Residuals(response is McDonald)
A new estimate of the market risk is based on squaring the correlation between the
fits from this model and the observed McDonalds returns:
c 2011, Jeffrey S. Simonoff 17
Correlations (Pearson)
Correlation of McDonalds return and FITS2 = 0.614
That is, R2w1 = .6142 = 37.7% (the Fbased R2 measure is only R2w2 = 22.6%, reflecting
that much of the apparent market risk is driven by months with high volatility). The
riskless rate, as measured by the monthly equivalent rate for the first three month Treasury
bill auction for that month, averaged .0045 over this time period; comparing the observed
constant term to Rf (1 1) gives us an estimate of how McDonalds performed comparedto what CAPM would have predicted for it. Here this equals
.009556 (.0045)(1 .9610) = .009381.
That is, McDonalds outperformed its CAPM prediction by 0.9381% per month, which
converts to an 11.86% annual outperformance of its CAPM prediction [(1.009381)12 =
1.1186].
The value of beta reported by investment analysts is usually rounded off to the nearest
.05. It is also usually shrunk towards one because of regression to the mean (that is,
analysts believe that stocks with unusually high or low betas in the past will probably
be less extreme in the future). So, given our WLS estimate of 0.961, we would probably
report McDonalds beta as 1.00. In fact, at this time the Value Line Investment Survey
reported a beta of 1.00 for McDonalds, so were right in line with established opinion.
One flaw in the previous analysis is that it is difficult to assess whether the observed
unexpected performance (relative to CAPM) could just be due to random fluctuations;
that is, is the 11.86% annual outperformance significantly different from zero? Also, the
comparison of 0 to Rf (1 1) assumes that the riskless rate is constant over the entiretime period, which is not reasonable. We can correct these problems if we use a slightly
different regression model to fit CAPM one based on excess returns. Lets go back to
the original formulation of the CAPM model, but represent it a little differently:
E(R) = Rf + (E[Rm]Rf),
c 2011, Jeffrey S. Simonoff 18
E(R)Rf = (E[Rm]Rf ). (4)
The values E(R) Rf and E[Rm] Rf are the expected excess returns of the asset andthe market, respectively, over the riskless rate; that is, they represent the returns that can
be expected to be gained beyond those that come with zero risk. A regression model based
on (4),
Ri Rfi = 0 + 1(Rmi Rfi) + i,
where the target and predictor values are now excess returns, provides an alternative
way to estimate beta (via the slope in the model). Further, by (4), CAPM implies that
the expected excess return exactly equals beta times the market excess return, so 0
is an estimate of McDonalds performance relative to its predicted CAPM performance
(sometimes called ). A test of whether the observed performance is significantly above or
below the expected performance is then just the usual ttest for the constant term equaling
zero.
Here is an OLS regression using excess returns:
Regression Analysis
The regression equation is
McDonalds excess rate = 0.00773 + 1.09 Market excess rate
Predictor Coef SE Coef T P
Constant 0.007735 0.004481 1.73 0.088
Market e 1.0931 0.1504 7.27 0.000
S = 0.04170 R-Sq = 37.8% R-Sq(adj) = 37.1%
Analysis of Variance
Source DF SS MS F P
Regression 1 0.091861 0.091861 52.83 0.000
Error 87 0.151277 0.001739
Total 88 0.243138
c 2011, Jeffrey S. Simonoff 19
The estimate of beta (1.093) is similar to that from the earlier OLS fit (1.089). The
estimated outperformance of McDonalds from its CAPM prediction is .007735 (9.69%
annualized), and it is not significantly different from zero at a .05 level (p = .088). Residual
plots and a Levenes test (not given here) again indicate heteroscedasticity in the square
of market return, with the following estimated weights:
Regression Analysis
The regression equation is
lgsressq = - 1.51 + 3.75 Market excess rate + 266 Markexsq
Predictor Coef SE Coef T P
Constant -1.5100 0.2446 -6.17 0.000
Market e 3.752 7.761 0.48 0.630
Markexsq 266.3 120.0 2.22 0.029
S = 2.088 R-Sq = 6.5% R-Sq(adj) = 4.4%
Analysis of Variance
Source DF SS MS F P
Regression 2 26.280 13.140 3.01 0.054
Error 86 375.025 4.361
Total 88 401.305
Here is the WLS fit:
Regression Analysis
Weighted analysis using weights in wt
The regression equation is
McDonalds excess rate = 0.00936 + 0.945 Market excess rate
Predictor Coef SE Coef T P
Constant 0.009357 0.004084 2.29 0.024
Market e 0.9454 0.1913 4.94 0.000
S = 0.07487 R-Sq = 21.9% R-Sq(adj) = 21.0%
Analysis of Variance
c 2011, Jeffrey S. Simonoff 20
Source DF SS MS F P
Regression 1 0.13692 0.13692 24.43 0.000
Error 87 0.48765 0.00561
Total 88 0.62457
The estimated beta (.945) is similar to the earlier WLS estimated beta (.961). The esti-
mated outperformance of McDonalds compared to CAPM is .009357 (11.82% annualized),
very similar to the earlier WLS estimate of .009381. Note that from this model fit, how-
ever, we can establish that this outperformance is apparently significantly different from
zero (p = .024), something that the other model fits could not do. That is, CAPM fails for
McDonalds, in the sense that McDonalds performance is significantly better than CAPM
predicts.
An interesting application of WLS in the CAPM context can be found in the paper
Outlier-Resistant Estimates of Beta by R.D. Martin and T.T. Simin (Financial Analysts
Journal, 59(5), 56-69 [2003]). In that paper the authors useWLS to construct an estimator
of beta that is resistant to the long-tailed nature of stock returns by downweighting those
observations in the regression.
Appendix: WLS when the error variance is related to numerical predictors
We have previously discussed how nonconstant variance related to group membership
can be identified using Levenes test, and handled using weighted least squares with the
weights for the members of each group being the inverse of the residual variance for that
group. Another way to refer to nonconstant variance related to group membership is
to say that nonconstant variance is related to the values of a predictor variable, where
that predictor variable happens to be categorical. It is also possible (as was the case
here) that the variance of the errors is related to a (potential) predictor variable that is
numerical (in this case it was effectively related to two variables, Market return and Market
return2). Generalizing the Levenes test for this situation is straightforward; just construct
a regression with the absolute residuals as the response and the potential numerical variable
as a predictor. Note that this also can be combined with the situation with natural
c 2011, Jeffrey S. Simonoff 21
subgroups by running an ANCOVA model with the absolute residuals as the response and
both the grouping variable(s) and the numerical variable(s) as predictors. It is important
to remember that the response variable itself should never be used as a potential predictor
for nonconstant variance, since the (potential) nonconstant variance is already reflected in
that response.
Constructing weights for WLS in this situation is more complicated. What is needed
is a model for what the relationship between the variances and the numerical predictor
actually looks like. An exponential/linear model for this relationship is often used, whose
parameters can be estimated from the data (this model has the advantage that it can only
produce positive values for the variances, which of course is consistent with the actual
situation). The model for the variance of ith error is
var(i) = 2
i = 2 exp
j
jzij
,
where zij is the value of the jth variance predictor for the ith case and 2 is an overall
average variance of the errors. These z variables would presumably be the predictors
that were used above for the Levenes test, and while they would typically be chosen from
the same pool of potential predictors as those for the regression itself (what we typically
call the xs), they dont have to be the same variables (2 could be related to a variable
that isnt related to E(y), and it could be unrelated to a variable that is).
The problem with this formulation is that the j coefficients are unknown, and need
to be estimated from the data. The key is to recognize that since 2i = E(2
i ), by the
model given above
logE(2i ) = log2 +j
jzij 0 +j
jzij .
That is, the logged expected squared errors follows a linear relationship with the z variables.
This suggests that linear regression could be used to estimate the parameters, except that
the expected squared errors are (of course) unknown. The trick is then to say that since
the residuals are the best guesses we have for the errors, the squared residuals should be
reasonable guesses for the expected squared errors, which means that the logged squared
c 2011, Jeffrey S. Simonoff 22
residuals can be used as a response in a regression to estimate the s. The steps are thus
as follows:
(1) Create a variable that is the natural logarithm of the squares of the standardized
residuals (LGSRESSQ, say). This variable can be formed in Minitab using the trans-
formation Let LGSRESSQ = LN(SRES*SRES).
(2) Perform a regression of LGSRESSQ on the variance predictor variables (the z variables),
and record the fitted regression coefficients (dont worry about measures of fit for this
regression).
(3) Create a weight variable for use in the weighted least squares analysis. The weights
are estimates of the inverse of the variance of the errors for each observation. They
have the form WT = 1/ exp(FITS1), where FITS1 is the variable with fitted values from
the regression in step 2.
(4) Perform a weighted least squares regression, specifying WT as the weighting variable.
You should redo a Levenes test to make sure that the nonconstant variance has been
corrected. Remember that all plots and tests must be based on the standardized
residuals, not the ordinary residuals, since the attempts to address nonconstant vari-
ance are accounted for in the standardized residuals.
Just as was the case when doing WLS based on a categorical predictor, the estimated
variance of the error for any member of the population is s/WTi. The value of WTi comes
from the estimated regression function in step 2 above (which is why it is a good idea to
write down that function). So, for example, for the CAPM data the function that defined
the weights was
WT = 1/ exp(1.51 + 1.278 Market return + 261.8 Market return2).
If a prediction for a new trading day for the McDonalds return was desired, and the market
return on that day was .05 (for example), the weight associated with that day would be
1/ exp(1.51 + (1.278)(.05) + (261.8)(.052)) = 2.207.
The estimated McDonalds return on that day, found by substituting .05 for the market
return into the WLS model would be .05761, while the estimated standard deviation of the
c 2011, Jeffrey S. Simonoff 23
error term for that day would be s/WT = .07457/
2.207 = .0502, where the s value also
comes from the WLS model. Note that this estimated standard deviation of the errors is
larger than that from the OLS model (which was .04171), which reflects that a day with a
market return of .05 will have higher than average variability. A rough prediction interval
for the McDonalds return on that day is thus .05761 (2)(.0502), or (.0428, .158).The exact prediction interval that comes out of Minitab requires more work. Here is
the output that comes out if confidence and prediction intervals are requested for a value
of market return equal to .05:
* WARNING * The prediction interval output assumes a weight of 1. An
adjustment must be made if a weight other than 1 is used.
Predicted Values for New Observations
New
Obs Fit SE Fit 95% CI 95% PI
1 0.05761 0.00928 (0.03917, 0.07605) (-0.09176, 0.20698)
Values of Predictors for New Observations
New Market
Obs return
1 0.0500
Note that Minitab provides a warning that the prediction interval is incorrect. The
problem is that the program assumes that the appropriate weight is equal to 1, even though
we just saw that it really should be 2.207. The correction for this must be made by hand.
The standard error of the fitted value (used for confidence intervals) is given correctly, but
we need to calculate the standard error of the predicted value. This equals
(Standard error of fitted value)2 + (Residual MS)/(Weight),
where the Residual MS comes from the WLS fit. Then, the prediction interval is
Predicted value tnp1/2 (Standard error of predicted value).
c 2011, Jeffrey S. Simonoff 24
The standard error of the predicted value in this case is.009282 + .00556/2.207 = .051.
For n = 89 and p = 1 the appropriate critical value for a 95% interval is 1.988, giving
prediction interval
.05761 (1.988)(.051) = (.0438, .159),
which is of course very similar to the rough prediction interval given earlier.
There is another mechanism by which variances of errors in a regression model can
be different for different observations, and related to a numerical variable. Say that the
response variable at the level of an individual follows the usual regression model,
yi = 0 + 1xi1 + + pxpi + i,
with i N(0, 2). Imagine, however, that the ith observed response is actually an averageyi for a sample of size ni with the observed predictor values {x1i, . . . , xpi}. The model isthus
yi = 0 + 1xi1 + + pxpi + i,
where
V (i) = V (yi|{x1i, . . . , xpi}) =2
ni.
An example of this kind of situation could be as follows. Say you were interested in
modeling the relationship between student test scores and (among other things) income.
While it might be possible to obtain test scores at the level of individual students, it would
be impossible to get incomes at that level because of privacy issues. On the other hand,
average incomes at the level of census tract or school district might be available, and could
be used to predict average test scores at that same level.
This is just a standard heteroscedasticity model, and WLS is used to fit it. In fact,
this is a particularly simple case, since the weights do not need to be estimated at all;
since V (i) = 2/ni, the weight for the ith observation is just ni. That is, quite naturally,
observations based on larger samples are weighted more heavily in estimating the regression
coefficients.
It should be noted, however, that this sort of aggregation is not without problems.
Inferences made from aggregated data about individuals are called ecological inferences,
c 2011, Jeffrey S. Simonoff 25
and they can be very misleading. As is always the case, we must be aware of confounding
effects of missing predictors; for example, if school districts with wealthier residents also
have lower proportions of non-native English speakers, a positive slope for income could be
reflecting an English speaker effect, rather than an income effect. In addition, ecological
inferences potentially suffer from aggregation bias, whereby the information lost when
aggregating (as it is clear that some information will be lost) is different for some individuals
than for others (for example, if there is more variability in incomes in some school districts
compared to others, more information is lost in those school districts), resulting in biased
inferences.
Minitab commands
To construct a scatter plot with a lowess curve superimposed on it, enter the appro-
priate variables under Y variables and X variables as usual. Click on Data View, then
Smoother, and click the button next to Lowess. Alternatively, a lowess curve can be su-
perimposed on an existing plot by right clicking on the plot, and clicking Add Smoother,and then OK.
c 2011, Jeffrey S. Simonoff 26