The single-index model yields a simple expression for the covariance matrix ofRt, as described in the following proposition.
Proposition 9.1. For eacht= 1,2, . . . , T, letRt denote anN×1vector of asset returns and suppose that the single-index model holds forRt.
LetΣdenote the covariance matrix ofRtand letΣ denote the covariance matrix oft. Then
Σ=σ2mββT+Σ (9.3)
whereσ2m=Var(Rm,t).
Proof. Let a and b be elements of N and consider Cov(aTRt,bTRt).
According to the single-index model,
Cov(aTRt,bTRt) = Cov(aTβRm,t+aTt,bTβRm,t+bTt)
= Cov(aTβRm,t,bTβRm,t) + Cov(aTβRm,t,bTt) + Cov(aTt,bTβRm,t) + Cov(aTt,bTt).
Note thataTβRm,t= (aTβ)Rm,t where (aTβ) is a scalar; it follows that Cov(aTβRm,t,bTβRm,t) = (aTβ)(bTβ)σ2m= (aTβ)(βTb)σ2m=aTββTbσ2m. By the properties of covariance matrices,
Cov(aTt,bTt) =aTΣb
and, by the assumptions of the single-index model,
Cov(aTβRm,t,bTt) = Cov(aTt,bTβRm,t) = 0.
It follows that
Cov(aTRt,bTRt) = (aTβ)(bTβ)σ2m+aTΣb
= (aTβ)(βTb)σ2m+aTΣb
=aT
ββTσ2m+Σ b.
Since this holds for any vectorsa,b, it follows that ββTσ2m+Σ must be the
covariance matrix ofRt.
Under the assumption thatΣis a diagonal matrix, the correlation between the returns of any two assets is attributable entirely to the fact that the returns are related to the market index. That is, the correlation structure of Rt is attributable to a “single index,” the return on the portfolio corresponding toRm,t.
The primary assumption underlying the single-index model, that the covariance matrix of t is a diagonal matrix, is a strong one, but it greatly simplifies the structure of the covariance matrix ofRt. In general, the covari- ance matrix ofRt hasN(N+ 1)/2 parameters,N variances plus (N2−N)/2 covariances. Under the single-index model, (9.3) has 2N+ 1 parameters, N elements ofβ,N diagonal elements of Σ, andσ2m. For example, forN = 50 assets, in general, the covariance matrix ofRthas 1225 unknown parameters, while under the single-index model it has only 101.
Furthermore, the parameters in (9.3) can be estimated using the regression output from applying the market model to each of theN assets together with the sample variance of the excess returns on the market index.
Correlation of Asset Returns under the Single-Index Model
To better understand the implications of the single-index model, consider the properties of two assets,i and j. Letσ2i and σ2j denote the variances ofRi,t andRj,t, respectively. Then, under either the market model or the single-index model,σ2i andσ2j can be decomposed into market and nonmarket components:
σ2i =β2iσ2m+σ2,i and σ2j =β2jσ2m+σ2,j.
Let ρi and ρj denote the correlations of Ri,t and Rj,t, respectively, with Rm,t. Under either the market model or the single-index model,
Cov(Ri,tRm,t) =βiσ2m and Cov(Rj,t, Rm,t) =βjσ2m; it follows that
ρi= βiσ2m σm
β2iσ2m+σ2e,i = βiσm
β2iσ2m+σ2,i
and
ρj= βjσm
β2jσ2m+σ2,j.
Now considerρij, the correlation ofRi,t andRj,t. Under the single-index model, the covariance ofRi,t andRj,t is the (i, j)th element of
σ2mββT+Σ; hence,
Cov(Ri,t, Rj,t) =βiβjσ2m. It follows that
ρij= βiβjσ2m β2iσ2m+σ2,i
β2jσ2m+σ2,j =ρiρj. Thus, we have proven the following result.
Corollary 9.1. For each t= 1,2, . . . , T, let Rt denote an N×1 vector of asset returns and suppose that Σ, the covariance matrix of Rt is of the form
Σ=σ2mββT+Σ (9.4)
whereΣ is a diagonal matrix.
Letρij denote the correlation ofRi,t andRj,t, letρi denote the correlation ofRi,t andRm,t, and letρj denote the correlation of Rj,t andRm,t. Then for i, j= 1,2, . . . , N,i=j,
ρij =ρiρj.
Thus, under the single-index model, the correlation of the returns on any two assets is equal to the product of the correlations of each assets’ returns with the returns on the market index.
Example 9.1 Suppose that for a set of three assets, the single-index model holds with β= (0.8,0.5,1.1)T,σ,1= 0.20,σ,2= 0.25,σ,3= 0.10, and σm= 0.05. Then the covariance matrix of the assets is given by
> beta<-c(0.8, 0.5, 1.1)
> Sig1<-(0.05^2)*(beta%*%t(beta)) + diag(c(0.2,0.25, 0.1)^2)
> Sig1
[,1] [,2] [,3]
[1,] 0.0416 0.00100 0.00220 [2,] 0.0010 0.06313 0.00138 [3,] 0.0022 0.00138 0.01303
The corresponding correlation matrix can be obtained using the function cov2cor, which converts a covariance matrix into a correlation matrix.
> cov2cor(Sig1)
[,1] [,2] [,3]
[1,] 1.0000 0.0195 0.0945 [2,] 0.0195 1.0000 0.0480 [3,] 0.0945 0.0480 1.0000
Recall that, for a given asset, βi=ρi(σi/σm) where σ2i = Var(Ri,t).
Therefore, the vector of correlations (ρ1,ρ2,ρ3)T of each asset’s returns with the returns on the market index is given by
> cor_vec<-beta*(0.05/(diag(Sig1)^.5))
> cor_vec
[1] 0.1961 0.0995 0.4819
Products of the form ρiρj for i=j may be easily obtained from the off- diagonal elements ofcor_vec%*%t(cor_vec):
> cor_vec%*%t(cor_vec) [,1] [,2] [,3]
[1,] 0.0385 0.0195 0.0945 [2,] 0.0195 0.0099 0.0480 [3,] 0.0945 0.0480 0.2322
Note that the off-diagonal elements of the matrixcor_vec%*%t(cor_vec)are identical to the off-diagonal elements ofcov2cor(Sig1).
Partial Correlation
This property of the correlation between the returns of any two assets in a given time period may be described in terms of theirpartial correlation. If, as in the single-index model, the correlation betweenRi,t andRj,tis attributable entirely to the fact that both assets’ returns are linearly related to the market return, thenρij =ρiρj. It follows that
ρij−ρiρj
represents the correlation ofRi,t andRj,t relative to the value of the correla- tion that would be obtained if the correlation betweenRi,tandRj,tis because of the relationships of the assets’ returns to the market return.
The partial correlation coefficient of Ri,t and Rj,t given Rm,t is a scaled version of this difference:
ρijãm= ρij−ρiρj
(1−ρ2i)(1−ρ2j)
. (9.5)
This partial correlation coefficient describes the extent to which Ri,t and Rj,tare linearly related, after removing the effect of the market return on this relationship; this is often expressed by saying thatρijãmgives the correlation between Ri,t and Rj,t “controlling for” the market return. Like the usual correlation coefficient, it is based on the assumption that the relationships amongRi,t, Rj,t, andRm,tare all linear ones. If the single-index model holds, thenρijãm= 0 for alli, j= 1,2, . . . , N, i=j.
Example 9.2 In this chapter, we apply the single-index model to the returns on the stocks of five companies, Cablevision Systems Corp. (symbol CVC), Edison International (EIX), Expedia, Inc. (EXPE), Humana, Inc. (HUM), and Wal-Mart Stores, Inc. (WMT). Five years of monthly returns for the period ending December 31, 2014, were calculated for each stock and stored in variables with the same name as the stock symbol; for example,cvccontains the excess returns for Cablevision.
The matrix of excess returns for all five stocks is stored in the variable stks, which is analogous to the matrix stored in the variablebig8 used in Example 6.6, as well as other examples in Chapters 6 through 8. The vari- ablesp500 contains similar excess returns on the Standard & Poors (S&P) 500 index.
An estimate of the partial correlation coefficient is given by replacing the correlation coefficients in (9.5) by the sample correlation coefficients. Although such an estimate is easily calculated using results from the corfunction, it is convenient to use the pcor.test function in package ppcor, which also includes a test ofρijãm= 0.
For instance, consider the estimated partial correlation of the returns on Edison stock and Wal-Mart stock given the returns on the S&P 500 index.
> library(ppcor)
> cor(eix, wmt) V1 V1 0.2769
> pcor.test(eix, wmt, sp500)
estimate p.value statistic n gp Method 1 0.1574 0.229 1.203 60 1 pearson
Therefore, the sample correlation coefficient for the returns on Edison and Wal-Mart stock is 0.277 and the estimated partial correlation coefficient is 0.157. The test of the null hypothesis that the partial correlation coefficient for Edison and Wal-Mart returns, controlling for the returns on the S&P 500 index, is zero hasp-value 0.229. Hence, there is no evidence to reject the null hypothesis, and it appears that the correlation between Edison and Wal-Mart stock returns is attributable entirely to their relationships with the market return, as measured by the return on the S&P 500 index. That is, there is no evidence to reject the hypothesis that the single-index model holds for Edison and Wal-Mart stock.
To calculate the partial correlation coefficients for all pairs of assets, we can use nested loops that calculate the partial correlation and the corresponding p-value for the returns on each pair of stocks.
> pcor<-pvalue<-matrix(0, 5, 5)
> for (i in 1:4){
+ for (j in (i+1):5){
+ res<-pcor.test(stks[,i], stks[, j], sp500) + pcor[i, j]<-res[1, 1]
+ pvalue[i, j]<-res[1, 2]
+ }
+ }
The matrixpcorwill contain the estimated partial correlation coefficients and the matrixpvaluewill contain the associatedp-values. Note, although there are five columns in the return matrix stks, i runs from 1 to 4 and j runs from i+ 1 to 5 to avoid computing each estimate and p-value twice and to avoid computing the partial correlation coefficient of a vector of returns with itself.
It is easier to read the results if we add column names and row names to the result matrices:
> rownames(pcor)<-colnames(stks)
> colnames(pcor)<-colnames(stks)
> rownames(pvalue)<-colnames(stks)
> colnames(pvalue)<-colnames(stks)
> pcor
CVC EIX EXPE HUM WMT
CVC 0 -0.0856 -0.0354 -0.0318 -0.1018 EIX 0 0.0000 -0.0396 -0.0285 0.1574 EXPE 0 0.0000 0.0000 -0.2486 0.1526 HUM 0 0.0000 0.0000 0.0000 -0.0667 WMT 0 0.0000 0.0000 0.0000 0.0000
> pvalue
CVC EIX EXPE HUM WMT CVC 0 0.517 0.789 0.8101 0.440 EIX 0 0.000 0.765 0.8297 0.229 EXPE 0 0.000 0.000 0.0527 0.244 HUM 0 0.000 0.000 0.0000 0.614 WMT 0 0.000 0.000 0.0000 0.000
Because allp-values are relatively large, there is no evidence that the single- index model is inappropriate.
When there are many zeros in a table, it is often convenient to replace them by a different symbol; this is particularly true when, as in the present case, the zeros do not provide any information—they are simply placeholders.
This can be achieved by converting the matrix to a “table” and then using
the functionprint, which includes an argument for the symbol used for zeros.
For example,
> print(as.table(pvalue), zero.print=".")
CVC EIX EXPE HUM WMT
CVC . 0.5167 0.7892 0.8101 0.4398 EIX . . 0.7649 0.8297 0.2290
EXPE . . . 0.0527 0.2436
HUM . . . . 0.6138
WMT . . . . .
Using nested loops in this way generally works well when analyzing a small or moderate number of assets. However, when analyzing a large number of assets, it may be preferable to use one of the vector-based functions available in R, such asouter, which is often more efficient in such cases.
Note that when testing a large number of hypotheses, as in the previous example, it is important to be aware of the multiple testing issue, as discussed in Section 8.4. That is, when interpreting thep-values for tests ofρijãm= 0 for a large number of stocks, we expect a few smallp-values even if the single-index model holds for all stocks. As discussed in Section 8.4, the Bonferroni method may be used in such cases to calculate an adjustedp-value that is valid for test- ing the hypothesis that the single-index model holds for all stocks considered.