For t= 1,2, . . . , T, letRi,t denote the return on asseti at time t. Let Rm,t denote the return on a market index at timetand letRf,tdenote the return on the risk-free asset at time t. Recall that, although the return on the risk-free asset has zero variance, the rate of return itself varies over time.
Assume that the stochastic process given by {(Ri,t−Rf,t, Rm,t−Rf,t)T: t= 1,2, . . .}is weakly stationary; weak stationarity of a pair of random vari- ables holds if any real-valued linear function of the random variables is weakly stationary in the usual sense. Hence, under weak stationarity, the means, variances, and covariances ofRi,t andRm,t do not depend ont.
The market model states that
Ri,t−Rf,t=αi+βi(Rm,t−Rf,t) +i,t, t= 1,2, . . . , T (8.1) where αi,βi are unknown parameters and i,1,i,2, . . . ,i,T are unobserved random variables each with mean zero and variance σ2,i. Furthermore, we assume that
Cov(i,t, Rm,t) = 0, t= 1,2, . . . , T . (8.2) The random variables i,t, t= 1,2, . . . , T, are known as the residual returns. They may be interpreted as the component of the excess return on the asset that is uncorrelated with the market returns; see Corollary 7.1 for a similar random variable in the context of the CAPM.
Note that some analysts define the residual returns to be αi+i,t, t= 1,2, . . . , T so that αi represents the mean residual return, while here we define the residual returns to have mean zero; however, the basic idea is the same—residual returns represent that part of an asset’s returns that remains after accounting for a linear relationship with the market return.
Note that assumption (8.2) is equivalent to the assumption that the parameter βi in (8.1) can be expressed in terms of Cov(Ri,t, Rm,t) and Var(Rm,t):
βi= Cov(Ri,t, Rm,t)
Var(Rm,t) . (8.3)
To see this, first note that, if Cov(i,t, Rm,t) = 0, then, by (8.2), Cov(Ri,t, Rm,t) =βiVar(Rm,t),
which yields the expression (8.3) forβi.
Conversely, according to (8.2),
Cov(Ri,t, Rm,t) =βiVar(Rm,t) + Cov(i,t, Rm,t); (8.4) recall that Cov(Rm,t, Rf,t) = 0. Therefore, if (8.3) holds, then
Cov(Ri,t, Rm,t) = Cov(Ri,t, Rm,t) + Cov(i,t, Rm,t) so that Cov(i,t, Rm,t) = 0.
We also assume that the errors are uncorrelated,
Cov(i,t,i,s) = 0 for all t, s= 1,2, . . . , T , t=s and that
Cov(i,t, Rm,s) = 0 for all t, s= 1,2, . . . , T .
Therefore, the market model is a regression model with response variable Yt=Ri,t−Rf,t, t= 1,2, . . . , T ,
the excess returns of asseti, and predictor variable Xt=Rm,t−Rf,t, t= 1,2, . . . , T , the excess returns on a market index
Yt=α+βXt+t, t= 1,2, . . . , T
whereα=αi,β=βi,andt=i,t. Under the assumptions described here, the errors1,2, . . . ,T have mean zero, constant variance, and are uncorrelated;
furthermore, Cov(t, Xt) = 0. A model of this type for (Yt, Xt),t= 1,2, . . . , T is often described as a “simple linear regression model.”
Relationship to the CAPM
The market model is similar to the CAPM, but there are important differences.
The CAPM is a model for the relationship between the expected excess return on an asset and the expected excess return on a hypothetical market portfolio, which is assumed to achieve the maximum possible Sharpe ratio, while the market model is a statistical model for observed excess returns on an asset and the observed excess returns on a market index. Note that, even though the CAPM and the market model use the same basic notation for the returns on the market portfolio and the returns on a market index, these two sets of returns are not identical.
Taking expectations in (8.1), the market model implies that μi−μf =αi+βi(μm−μf),
where μi−μf = E(Ri,t−Rf,t) and μm−μf = E(Rm,t−Rf,t). Thus, if the return on the market index used in the market model may be viewed as the return on the market portfolio, the market model and the CAPM describe similar relationships. The most important difference between these models is that, under the CAPM, the efficiency of the market portfolio implies that the intercept parameter satisfies αi= 0; conversely, if αi= 0, the market model implies a form of the CAPM using the returns on a market index in place of the returns on the market portfolio.
Interpretation of βi
As in the CAPM, the parameterβi in the market model is a measure of the relationship between the excess returns on an asset and the excess returns on the market index and, in many respects, the interpretation of βi follows from the interpretation of beta in the CAPM, as discussed in Chapter 7.
For instance, it may be used to decompose the variance of an asset’s returns into market and nonmarket components; this will be discussed in detail in Section 8.5.
An alternative interpretation ofβi is as a measure of the sensitivity of an asset’s excess returns to the excess return on the market index. However, such an interpretation does not follow directly from the assumptions of the market model as given in this chapter. In particular, the interpretation of βi as a measure of sensitivity is valid only if the relationship betweenRi,t−Rf,t and Rm,t−Rf,t is a linear one.
That is, suppose that the condition thati,t andRm,tare uncorrelated is strengthened to E(i,t|Rm,t−Rf,t) = 0; then
E(Ri,t−Rf,t|Rm,t−Rf,t=r) =αi+βir.
It follows that
βi= d
drE(Ri,t−Rf,t|Rm,t−Rf,t=r)
andβi may be interpreted as the measure of sensitivity described previously.
However, if E(i,t|Rm,t−Rf,t) is a nonzero function ofRm,t−Rf,t, then d
drE(Ri,t−Rf,t|Rm,t−Rf,t=r)
might not be equal to βi. Fortunately, it is generally reasonable to assume that E(i,t|Rm,t−Rf,t) = 0 does hold and, hence, the interpretation ofβi as a measure of sensitivity is typically appropriate.
Estimation
We now consider estimation of the parameters of the market model. As discussed previously, the market model may be viewed as a simple linear
regression model with response variable Yt=Ri,t−Rf,t, where Ri,t is the return on a specific asset in periodt, andRf,t is the risk-free rate in periodt, and predictor variableXt=Rm,t−Rf,t, whereRm,tis the return of a market index in periodt.
Therefore, the parametersαiandβimay be estimated using ordinary least squares. The formulas for the estimators are
ˆβi= T
t=1(Yt−Y¯)(Xt−X¯) T
t=1(Xt−X)¯ 2 and
αˆi= ¯Y−ˆβiX¯
where ¯Y and ¯X are the sample means of the Yt and Xt, respectively; these expressions are sometimes useful for studying the properties of the estimators, but they are not needed for numerical work.
Thus, the remaining issue is selection of the data to be used in the analysis:
the market index, the risk-free asset, the return interval, and the observation period.
As discussed in Section 8.2, the “market portfolio” is a hypothetical con- cept; hence, in estimating the parameters of the market model, we use a market index chosen to measure the general behavior of the equity market.
The most commonly used index in this context is the S&P 500 index. Although it includes only 500 stocks, the return on the S&P 500 is generally believed to reflect the return on the entire market. There are a number of broader indices that can be used such as the Russell 3000 index and the Wilshire 5000 index. As shown in Example 8.1, the S&P 500 index, the Russell 1000 index, the Russell 3000 index, and the Wilshire 5000 index are generally highly cor- related with each other; hence, the choice from among these indices has a relatively small impact on the estimates. Here we will use the return on the S&P 500 index as the return on the market portfolio.
For the risk-free rate to use in the analysis, we will use the return on a 3-month Treasury Bill, as discussed in Example 6.1. These are generally reported as annual percentage rates, which must be converted to proportional monthly rates. LetRf a be an annual percentage rate; recall that this may be converted to a monthly rate by
Rf = (1 +Rf a/100)1/12−1.
For the return interval, we could use daily, weekly, monthly, quarterly, or yearly returns. The return interval should reflect the investment horizon of interest. For instance, if investment decisions are made on a monthly basis, it generally makes sense to use monthly returns. Here we will use monthly returns.
The observation period refers to the number of return intervals to use in the analysis; for example, for monthly data, we need to choose how many
months of data to include. For a given return interval, a longer observation period clearly yields more data and smaller standard errors. However, in using a longer observation period, we are implicitly assuming thatβis constant over that time. Over a short observation period, this may be reasonable, but such an assumption becomes questionable as the observation period increases due to changes in the firms under consideration or changes in economic conditions.
Three to five years is commonly used. Here we will use five years of monthly data.
Example 8.3 Consider the monthly excess returns on IBM stock, which we assume have been calculated and stored in the variable ibm; as discussed in Example 8.1, the excess returns on the S&P 500 index are stored in the vari- ablesp500. As with any statistical analysis, before estimating the parameters of the linear regression model relatingibmto sp500, it is a good idea to plot the data, such a plot is given in Figure 8.1. The plot indicates an approximate linear relationship among the variables that would be accurately described by the market model.
To estimate the parameters of the market model in R, we use the function lm. The syntax of the command to estimate the market model relating returns on IBM stock to returns on the S&P 500 index, as contained in the variables ibmandsp500, respectively, is
> lm(ibm~sp500)
The expressionibm~sp500is known as amodel formula and may be read as “ibmis described bysp500.” The screen output from the command contains
−0.05 0 0.05 0.10
−0.10
−0.05 0 0.05 0.10
Return on S&P 500 index
Return on IBM stock
FIGURE 8.1
Plot of IBM monthly returns versus the returns on the S&P 500 index.
ˆα and ˆβ, the least-squares estimates of the parametersαand β, respectively, in the market model for the returns on IBM stock:
> lm(ibm~sp500) Coefficients:
(Intercept) sp500 -0.000707 0.618789
Therefore, for the data under consideration, ˆβ= 0.619 and ˆα=−0.000707.
However, much more information is available from the command, and it may be accessed by using certain extractor functions. Therefore, it is often useful to save the results of thelmfunction in a variable, which can be accessed as necessary:
> ibm.mm<-lm(ibm~sp500)
The variableibm.mmnow contains the results of the linear regression anal- ysis relating the returns on IBM stock to the returns on the S&P 500 index.
Thesummarycommand may be used to display a summary of the results:
> summary(ibm.mm) Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) -0.000707 0.005358 -0.13 0.9 sp500 0.618789 0.138073 4.48 3.5e-05 ***
---
Signif. codes: 0 *** 0.001 ** 0.01 * 0.05 . 0.1 1 Residual standard error: 0.0398 on 58 degrees of freedom Multiple R-squared: 0.257, Adjusted R-squared: 0.244 F-statistic: 20.1 on 1 and 58 DF, p-value: 3.54e-05
This output contains much useful information. For instance, the standard error of ˆβis 0.138, so that an approximate 95% confidence interval for β is given by
0.619±(1.96)(0.138) = (0.349,0.889).
An estimate of the error standard deviation in the market model for returns on IBM stock, σ, is given by the “Residual standard error” on the output.
Hence, usingσto denote the residual standard deviation in the market model for IBM stock, and using ˆσto denote the estimate ofσbased on least-squares regression, ˆσ= 0.0398.
It is also useful to know that this information may be accessed directly through the components of the result of the summaryfunction. For instance, in the IBM example,summary(ibm.mm)$coefficientsis a 2×4 matrix:
> summary(ibm.mm)$coefficients
Estimate Std. Error t value Pr(>|t|) (Intercept) -0.000707 0.00536 -0.132 8.96e-01 sp500 0.618789 0.13807 4.482 3.54e-05
For example, the estimate ofβis given bysummary(ibm.mm)$coefficients [2,1]:
> summary(ibm.mm)$coefficients[2,1]
[1] 0.619
Other useful components are $sigma, which contains ˆσ, and $r.squared, which containsR-squared for the regression:
> summary(ibm.mm)$sigma [1] 0.0398
> summary(ibm.mm)$r.squared [1] 0.257
The lm command may be used with a matrix argument as the response variable in order to provide results for several stocks at one time. Recall that in Chapter 6 we analyzed the variablebig8, a matrix containing the monthly returns for the stocks of eight large companies; see Example 6.6 for details.
To obtain the market model estimates ofαi,βi for all the stocks represented in the data matrixbig8, we use the command
> big8.mm<-lm(big8~sp500)
> big8.mm Coefficients:
AAPL BAX KO CVS
(Intercept) 0.015361 -0.000353 0.004547 0.009559 sp500 0.920307 0.716442 0.485664 1.071935
XOM IBM JNJ DIS
(Intercept) -0.001335 -0.000707 0.005637 0.007799 sp500 0.878737 0.618789 0.540500 1.193193
Thus, the values of ˆβfor these stocks range from 0.486 for Coca-Cola to 1.193
for Disney.