capm - do you want fries with that?

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


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


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:


-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


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


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


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.


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.


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):


Regression Analysis


Absolute residuals = 0.691 - 1.35 Market return + 119 Markretsquared


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%


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


lgsressq = - 1.51 + 1.28 Market return + 262 Markretsquared


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%


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:


-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


McDonalds return = 0.00956 + 0.961 Market return


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%


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


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


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


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:


absres = 0.816 - 0.61 Market return - 8.3 Marketsquared


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%


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


-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


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

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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:


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),


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


McDonalds excess rate = 0.00773 + 1.09 Market excess rate


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%


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


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


lgsressq = - 1.51 + 3.75 Market excess rate + 266 Markexsq


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%


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


McDonalds excess rate = 0.00936 + 0.945 Market excess rate


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%



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


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


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


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).


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,


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.


capm - do you want fries with that?

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