The single-index model relates the returns on a given asset, Ri,t, t= 1,2, . . . , T, to the returns on a market index, Rm,t, t= 1,2, . . . , T, through the model
Ri,t−Rf,t =αi+βi(Rm,t−Rf,t) +i,t, t= 1,2, . . . , T .
HereRf,t is the return on the risk-free asset at timet. The idea behind this model is that all assets are related to “the market” and volatility in the market induces volatility in the returns of individual assets. Furthermore, under this model, the correlation between the returns of two assets is a result of the fact that both assets are related to the market.
A general factor model extends the single-index model by including other risk factors, in addition to the market return, in the model. These factors may represent economic conditions that, like the return on a market index, affect all assets. Or the factors might reflect properties of the assets under consideration, such as the size of the company, in the case of a stock. This flexibility in the factors, which may be chosen to represent the analyst’s beliefs and goals, is one of the strengths of factor models. In this section, we consider the form and properties of a factor model, along with parameter estimation;
selection of the factors is considered in the following section.
Let F1,t, F2,t, . . . , FK,t denote the values of K factors at time t, t= 1,2, . . . , T. Fori= 1,2, . . . , N, letRi,t denote the return on assetiat timet.
Then a factor model that describesRi,t in terms ofF1,t, F2,t, . . . , FK,thas the form
Ri,t−Rf,t =αi+βi,1F1,t+βi,2F2,t+ã ã ã+βi,KFK,t+i,t, t= 1,2, . . . , T , where i,1,i,2, . . . ,i,T are unobserved mean-zero random variables that are uncorrelated with the factors. These terms represent the component of the asset’s excess return not explained by the factors.
Note that the values of the factors F1,t, F2,t, . . . , FK,t are the same for each asset and, hence, do not depend oni; in this sense, they may be viewed
as common factors. The parameters βi,1,βi,2, . . . ,βi,K, known as the factor sensitivities for asset i, measure how the factors affect a particular asset’s returns. Hence, these parameters depend oni; however, they are assumed to be constant over the observation period, so that they do not depend on t.
In the factor model, the factor sensitivities, likeβin the single-index model, are unknown parameters that must be estimated.
It is assumed that the same factor model applies to all assets under consideration so that, fort= 1,2, . . . , T,
R1,t−Rf,t =α1+β1,1F1,t+β1,2F2,t+ã ã ã+β1,KFK,t+1,t
R2,t−Rf,t =α2+β2,1F1,t+β2,2F2,t+ã ã ã+β2,KFK,t+2,t ...
RN,t−Rf,t=αN+βN,1F1,t+βN,2F2,t+ã ã ã+βN,KFK,t+N,t
or, in matrix notation,
Rt−Rf,t1=α+βFt+t, (10.1) whereα= (α1, . . . ,αN)T,βis theN×Kmatrix of factor sensitivities,
β=
⎛
⎜⎜
⎜⎝
β1,1 β1,2 ã ã ã β1,K
β2,1 β2,2 ã ã ã β2,K
... ... ã ã ã ... βN,1 βN,2 ã ã ã βN,K
⎞
⎟⎟
⎟⎠,
Ft= (F1,t, F2,t, . . . , FK,t)T is the K×1 vector of factor values at time t, andt= (1,t,2,t, . . . ,N,t)T is anN×1 random vector of unobserved model errors at timet.
We assume that the stochastic process {
(Rt−Rf,t1)T,FtT
T
:t= 1,2, . . .}
is a weakly stationary process, so any linear function of Rt−Rf,t1,Ft is a weakly stationary process; in particular, 1,2, . . . is a weakly stationary process. Furthermore, we assume that
Cov(t,Ft) =0, t= 1,2, . . . , T , so thatΣ, the covariance matrix ofRt, may be written as
Σ=βΣFβT+Σ, (10.2)
whereΣF denotes the covariance matrix ofFtandΣdenotes the covariance matrix oft; by the weak stationarity assumption, these parameters do not depend ont.
As with the single-index model, the errors for different assets are assumed to be uncorrelated so thatΣ is a diagonal matrix of the form
Σ=
⎛
⎜⎜
⎜⎜
⎝
σ2,1 0 . . . 0 0 σ2,2 . .. ... ... . .. . .. 0 0 . . . 0 σ2,N
⎞
⎟⎟
⎟⎟
⎠
whereσ2,i= Var(i,t),t= 1,2, . . . , N. Under this assumption, any correlation among asset returns for different assets is attributable to the common factors that affect all assets.
Thus, the factor model may be viewed as an extension of the single-index model in which the excess return on the market index,Rm,t−Rf,t, is replaced by the factorsF1,t, F2,t, . . . , FK,t and the vector β= (β1,β2, . . . ,βN)T of the single-index model, which gives the value of beta for each asset, is replaced by the matrixβ. Theith row ofβ, (βi,1,βi,2, . . . ,βi,K), gives the factor sensitivities for asseti; thejth column ofβ, (β1,j,β2,j, . . . ,βN,j)T gives the sensitivities to factorj for each of theN assets.
Portfolios
Consider a portfolio of the N assets under consideration based on a weight vector w = (w1, w2, . . . , wN)T. Then the return on the portfolio at time t, Rp,t, may be written as
Rp,t=wTRt, t= 1,2, . . . , T . It is straightforward to show that under the model (10.1)
Rp,t−Rf,t=αp+βTpFt+p,t, t= 1,2, . . . , T ,
whereαp=wTαandβp denotes theK×1 vector factors sensitivities for the portfolio,
βTp ≡(βp,1,βp,2, . . . ,βp,K) =wTβ, andp,t=wTt.
That is, the factor model applies to the portfolio as well, with the fac- tor sensitivities for the portfolio given by weighted sums of the asset factor sensitivities:
βp,j= N i=1
wiβi,j, j= 1,2, . . . , K.
The expected excess return on the portfolio at timetis given by E(Rp,t)−μf,t=wTα+wTβE(Ft)
=αp+βTpE(Ft) and the variance of the return at timetis given by
Var(Rp,t) =βTpΣFβp+wTΣw.
The first term in this expression, βTpΣFβp, represents a measure of the systematic risk of the portfolio, that is, the risk explained by the factors, while the second term, wTΣw, is a measure of the portfolio’s specific risk.
The systematic risk depends on the variation in the factors, as measured by ΣF, along with factor sensitivities of the portfolio,βp.
Estimation
Given a set of factors thought to be relevant to the asset returns under con- sideration, the factor sensitivities are estimated from the available data. Let F1,t, F2,t, . . . , FK,tdenote the values ofKfactors at timet,t= 1,2, . . . , T, and letRi,t, t= 1,2, . . . , T denote the returns on a given asset. Then, according to the factor model,
Ri,t−Rf,t =αi+βi,1F1,t+βi,2F2,t+ã ã ã+βi,KFK,t+i,t, t= 1,2, . . . , T , wherei,t is uncorrelated withF1,t, F2,t, . . . , FK,t; here, Rf,t is the return on the risk-free asset at timet.
Hence, as in the case of the single-index model, the parameter estimates for assetimay be obtained using least-squares regression based on the returns on asseti; that is, all parameter estimates may be obtained fromN regression analyses, one for each asset. Specifically, the parametersαi,βi,1,βi,2, . . . ,βi,K
may be estimated using least-squares regression with response variable Ri,t−Rf,t at timetand predictor variablesF1,t, F2,t, . . . , FK,t at timet. The error variance for asset i, σ2,i= Var(it), may be estimated using the usual estimator from the regression analysis.
Example 10.3 Consider the returns on JetBlue and EV Energy Partners stock, as discussed in Example 10.1. According to that example, oil prices apparently have an important effect on the returns on these stocks. Consider a factor model with two factors, the return on the S&P 500 index and the change in oil prices; recall that these data are stored in the variables sp500 andoil, respectively. Thus, using the general notation of this section,K= 2, withF1,tgiven by the excess returns on the S&P 500 index at timetandF2,t given by the change in West Texas Intermediate oil at timet.
The excess returns on JetBlue stock are stored in the variable jblu.
To estimate the parameters of the factor model for JetBlue stock, we use the lm function to fit a regression model with predictor variables sp500 andoil:
> summary(lm(jblu~sp500+oil)) Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) 0.00965 0.01282 0.75 0.4547 sp500 1.15544 0.34078 3.39 0.0013 **
oil -0.60598 0.19626 -3.09 0.0031 **
Residual standard error: 0.0948 on 57 degrees of freedom Multiple R-squared: 0.226, Adjusted R-squared: 0.199 F-statistic: 8.33 on 2 and 57 DF, p-value: 0.000672
Therefore, the parameter estimates for the factor model for JetBlue stock, which we take to be asset 1, are ˆα1= 0.00965, ˆβ1,1= 1.155, ˆβ1,2=−0.606, and ˆσ,1= 0.0948.
These results may be compared to the estimates from the market model for JetBlue stock:
> summary(lm(jblu~sp500)) Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) 0.0137 0.0137 1.00 0.321
sp500 0.8770 0.3520 2.49 0.016 *
Residual standard error: 0.102 on 58 degrees of freedom Multiple R-squared: 0.0967, Adjusted R-squared: 0.0811 F-statistic: 6.21 on 1 and 58 DF, p-value: 0.0156
Note that the estimated coefficient of the market index is different in the market model and the factor model. This is not unexpected; note that the coefficients have different interpretations. In the market model, the coefficient represents the change in the expected excess return on JetBlue stock corre- sponding to a change in the return on the market index, while in the factor model, it represents the change in the return on JetBlue stock corresponding to a change in the return on the market indexholding the change in oil prices constant.
Similarly, the negative coefficient ofoilin the factor model indicates that an increase in oil prices corresponds to a decrease in the expected excess return on JetBlue stock, holding the return on the S&P 500 constant.
Note that the value of R-squared in the factor model, 0.226, is larger than the value in the market model, 0.0967. Again, this is to be expected:
the factors sp500and oil explain more of the variation in the returns on JetBlue stock than does sp500 alone. In general, when adding predictor
variables to a regression model, the R-squared value increases or stays the same.
Therefore, when comparing R-squared values from regression models with different numbers of predictors, it is generally preferable to use adjusted R-squared. As the name suggests, adjusted R-squared includes an adjustment for the number of predictors in the model. Adding a predictor to a model can lead to a decrease in adjusted R-squared. In the present case, the adjusted R-squared value for the market model is 0.0811, while for the factor model it is 0.199, so the basic conclusion does not change: Including the change in oil prices in the model explains much more of the variation in the returns on JetBlue stock.
The estimates of the factor model for EV Energy Partners stock are given by
> summary(lm(evep~sp500+oil)) Coefficients:
Estimate Std. Error t value Pr(>|t|) (Intercept) 0.000828 0.011979 0.07 0.95 sp500 0.380479 0.318424 1.19 0.24 oil 0.782726 0.183384 4.27 7.5e-05 ***
Residual standard error: 0.0886 on 57 degrees of freedom Multiple R-squared: 0.297, Adjusted R-squared: 0.272 F-statistic: 12 on 2 and 57 DF, p-value: 4.43e-05
Here, the coefficient ofoilin the estimated regression model is negative, so that an increase in the change in oil prices corresponds to an increase in the expected excess return on EVEP stock, holding the return on the S&P 500 constant.
An estimate of the covariance matrix of the returns on JetBlue and EV Energy Partners stock based on the factor model may be obtained using the relationship given in (10.2). Note thatΣF, the covariance matrix of the fac- tors, may be estimated by the sample covariance matrix of observed factor values; the other parameters in (10.2) may be obtained from the factor model regressions. The relevant R commands are
> jblu.fm<-lm(jblu~sp500+oil)
> evep.fm<-lm(evep~sp500+oil)
> betamat<-rbind(jblu.fm$coef[2:3], evep.fm$coef[2:3])
> Sig.eps<-diag(c(summary(jblu.fm)$sigma, + summary(evep.fm)$sigma)^2)
> cov.fm<-betamat%*%cov(cbind(sp500, oil))%*%t(betamat) + + Sig.eps
Herejblu.fmandevep.fmcontain the results from the factor model regres- sions. The matrix betamat is the matrix of coefficient estimates; note that
$coef extracts the coefficient estimates from the result of lm. Thus, here betamatcontains the estimated factor sensitivities,
> betamat
sp500 oil [1,] 1.16 -0.606 [2,] 0.38 0.783
The matrixSig.epsis the estimate ofΣ from factor model regressions
> Sig.eps
[,1] [,2]
[1,] 0.00899 0.00000 [2,] 0.00000 0.00785
The matrixcov.fmis the estimate ofΣbased on the factor model
> cov.fm
[,1] [,2]
[1,] 0.01153 -0.00096 [2,] -0.00096 0.01104
and the corresponding correlation matrix is given by
> cov2cor(cov.fm) [,1] [,2]
[1,] 1.0000 -0.0851 [2,] -0.0851 1.0000
Thus, the factor model with two factors, the return on the S&P 500 index and the change in the price of WTI oil, captures the negative correlation between the returns on JetBlue and EV Energy Partners stock.