One role of the single-index model is to model the covariance between the returns on two assets in terms of the relationship between each asset’s returns on the returns on a market index. However, as noted in the introduction, in some cases, there may be important economic variables, in addition to the return on the market, that have important effects on the relationship between the returns on the assets. This is illustrated in the following example.
Example 10.1 Consider the returns on two stocks, JetBlue Airways Corp.
(symbol JBLU) and EV Energy Partners, L.P. (EVEP), an oil and natu- ral gas company. The variables jblu and evep contain 5 years of monthly excess returns on JBLU and EVEP stock, respectively, for the period end- ing December 31, 2014, and suppose that sp500contains the corresponding excess returns on the Standard & Poors (S&P) 500 index. Then the estimated correlation of the returns on these stocks is given by
> cor(jblu, evep) [1] -0.150
The estimated correlations of each return with the return on the S&P 500 index are given by
> cor(jblu, sp500) [1] 0.311
> cor(evep, sp500) [1] 0.268
Therefore, each stock’s returns arepositively correlated with the return on the market index, but the returns arenegatively correlated with each other.
Note that relationships of this type are not possible under the single-index model. The estimates of beta for the two stocks are given by
> lm(jblu~sp500)$coef (Intercept) sp500
0.0137 0.8770
> lm(evep~sp500)$coef (Intercept) sp500
-0.00437 0.74013
The estimated return variance for the S&P 500 index is
> var(sp500) [1] 0.00141
According to the single-index model, the estimated covariance of the returns on JBLU and EVEP stock is
(0.877)(0.740)(0.00141) = 0.000915, corresponding to an estimated correlation of
> 0.000915/(sd(jblu)*sd(evep)) [1] 0.0832
Hence, although the sample correlation of returns on JBLU and EVEP stock is negative (−0.150), the estimated correlation based on the single-index model is positive (0.0832).
One reason for this behavior may be the presence of other economic vari- ables that are affecting the returns on JBLU and EVEP stock. For example, JBLU, as an airline stock, is likely to be negatively affected by increasing oil prices; EVEP, on the other hand, as a gas and oil stock, is likely to be positively affected by increasing oil prices. Hence, oil prices might have an important effect on the relationship between the returns on these two stocks.
One commonly used benchmark for crude oil prices is the price of West Texas Intermediate (WTI) oil, which is generally refined in the United States.
Historical prices for WTI oil are available on the Federal Reserve Eco- nomic Data (FRED) website at https://research.stlouisfed.org/fred2/series/
DCOILWTICO/downloaddata; like stock prices, these data are available for different sampling frequencies, such as daily or monthly prices. Let the vari- ableoildenote the proportional change in monthly prices of WTI oil for the 5-year period ending December 31, 2014; thus, oil is calculated the same way that asset returns are calculated, except that oil prices, rather than stock prices, are used.
Note that, as expected, the returns on JBLU stock are negatively cor- related with the change in oil prices, while the returns on EVEP stock are positively correlated with the change in oil prices:
> cor(jblu, oil) [1] -0.265
> cor(evep, oil) [1] 0.528
Therefore, in modeling the relationship between JBLU and EVEP stock, it may be important to take into account changes in oil prices. This is likely to be true when analyzing the returns on other stocks thought to be related
to oil prices.
A second use of the single-index model is in understanding the role of an asset’s relationship with the market index in the expected return on the asset.
According to the single-index model or, equivalently, the market model, the expected excess return on asseti, μi−μf, is related to the expected excess return on the market index,μm−μf, by
μi−μf =αi+βi(μm−μf),
whereαi andβi are the parameters in the market model for asseti.
If the asset is priced correctly, in the sense described in Section 8.4, then αi= 0 and the expected excess return on an asset is proportional to its value of beta; see Section 7.7 for further details. This fact suggests that assets with greater values of beta will tend to have higher expected excess returns. The following example shows that this interpretation of beta is not always useful in practice.
Example 10.2 Consider stocks for firms represented in the S&P 500 index.
Five years of monthly returns for the period ending December 31, 2014, were analyzed; 474 of the stocks had returns for that entire period.
For each stock, the parameters of the market model were estimated, along with the sample mean excess return. These results suggest that all 474 stocks are priced correctly; for instance, the minimump-value for testing αi= 0 is 0.00236 so that, using the Bonferroni method, we fail to reject the hypothesis that allαi are equal to zero at any level.
The estimates of beta for the 474 stocks are stored in the variable sp474.mmbeta. Note that there is considerable variation in the estimates of beta:
> summary(sp474.mmbeta)
Min. 1st Qu. Median Mean 3rd Qu. Max.
0.004 0.732 1.060 1.080 1.390 2.870
Hence, according to the CAPM, we expect that stocks with large estimates of beta will tend to have higher sample mean excess returns.
Figure 10.1 contains a plot of the sample mean excess returns versus the estimated value of beta for the 474 stocks. Note that there is, at most, a very weak relationship between a stock’s sample mean excess return and its estimate of beta. Furthermore, this plot does not support the idea that stocks with larger values of beta tend to have large mean excess returns. For instance, the sample correlation of the estimates of beta and the sample mean excess returns based on these data is only 0.0359. The standard error of this estimate when the true correlation is zero is 1/√
N, whereN is the number of observa- tions; hereN= 474. Hence, the standard error of the estimate is 0.0459 and the correlation is not significantly different than zero. The squared sample
0 0.5 1.0 1.5 2.0 2.5
−0.01 0 0.01 0.02 0.03 0.04 0.05
Estimates of beta
Sample mean excess returns
FIGURE 10.1
Plot of sample mean excess returns versus estimates of beta for stocks in the S&P 500 index.
correlation is about 0.0013, so only about 0.13% of the variation in the sam- ple mean excess returns on the stocks may be explained by their estimates of beta. Thus, the theoretical relationship expressed in Figure 7.1 does not hold
for these data.
The results in the previous example suggest that, at least in some cases, the relationship between the returns on an asset and the returns on a market index is not sufficient to effectively describe the mean excess returns on an asset.
That is, these results are consistent with the idea that it may be important to include factors other than the return on a market index in a model for asset returns.