APPENDIX E
Result 3 These results can be illustrated with a simple example. Consider the returns on a stock
4. A. If the Blume adjustment equation is fit and the appropriate equation is
what is your best forecast of beta for each of the stocks in Question 1?
B. If the parameters of the Vasicek technique are fit, and they are
what is your best forecast of beta for each of the stocks in Question 1?
5.
Security
A B C D
2 3 1 4
1.5 1.3 0.8 0.9
ei 3 1 2 4
Given the preceding data and the fact that R–
m 8 and m 5, calculate the following:
(a) The mean return for each security (b) The variance of each security’s return
(c) The covariance of returns between each security
6. Using the data in Problem 5 and assuming an equally weighted portfolio, calculate the following:
(a) p
(b) p
(c) 2p
(d) R–
p
σ σ
β σ σ
β β
β β
1 2
1 2 1
1 2
1 2
0 25 0 22
1 00
0 36 0 41
= =
= = =
. , . ,
.
. , .
A
B C
βit+1=0 41. +0 60. βi t,
7. Using Blume’s technique, where i2 0.343 0.677i1, calculate i2for the securi- ties in Problem 5.
8. Suppose –
1 1 and –1 0.25 A 0.21 B 0.32 C 0.18 D 0.20, fore- cast each security’s beta using the Vasicek technique.
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155
8
The Correlation Structure of Security Returns—Multi-Index Models and Grouping Techniques
In Chapter 7 we argued that because of both the huge number of forecasts required and the necessary restrictions on the organizational structure of security analysts, it was not feasi- ble for analysts to directly estimate correlation coefficients. Instead, some structural or behavioral model of how stocks move together should be developed. The parameters of this model can be estimated either from historical data or by attempting to get subjective estimates from security analysts. We have already examined one such model, the single- index model, which assumes that stocks move together only because of a common co- movement with the market. Two other approaches have been widely used to explain and estimate the correlation structure of security returns: multi-index models and averaging techniques.
Multi-index models are an attempt to capture some of the nonmarket influences that cause securities to move together. The search for nonmarket influences is a search for a set of economic factors or structural groups (industries) that account for common movement in stock prices beyond that accounted for by the market index itself. Although it is easy to find a set of indexes that is associated with nonmarket effects over any period of time, as we will see, it is quite another matter to find a set that is successful in predicting covariances that are not market related.
Averaging techniques are at the opposite end of the spectrum from multi-index models.
Multi-index models introduce extra indexes in the hope of capturing additional informa- tion. The cost of introducing additional indexes is the chance that they are picking up ran- dom noise rather than real influences. Averaging techniques smooth the entries in the historical correlation matrix in an attempt to “damp out” random noise and so produce bet- ter forecasts. The potential disadvantage of averaging models is that real information may be lost in the averaging process.
In this chapter we examine both multi-index models and averaging models. Several of the models put forth in the finance literature are discussed, as are some of the empirical evidence on their relative merits.
At this point, we should mention that there are other uses for multi-index models besides predicting correlation coefficients. Multi-index models can be used to form expectations about returns and study the impact of events, as a method for tailoring the return distribu- tion of a portfolio to the specific needs of an investor, and as a method for attributing the
cause of good or bad performance on a portfolio. These are subjects to which we will return later in the book. However, the reader should be alerted to these other possible uses.
We will close this chapter with a discussion of some multi-index models using fundamen- tal data that have recently been developed as a step toward building a general equilibrium model of security returns. We return to this class of model in Chapter 16.
MULTI-INDEX MODELS
The assumption underlying the single-index model is that stock prices move together only because of common movement with the market. Many researchers have found that there are influences beyond the market that cause stocks to move together. For example, as early as 1966, King (1966) presented evidence on the existence of industry influences. Two dif- ferent types of schemes have been put forth for handling additional influences. We have called them the general multi-index model and the industry index model.
General Multi-index Models
Any additional sources of covariance among securities can be introduced into the equa- tions for risk and return simply by adding these additional influences to the general return equation. Let us hypothesize that the return on any stock is a function of the return on the market, changes in the level of interest rates, and a set of industry indexes. If Riis the return on stock i, then the return on stock i can be related to the influences that affect its return in the following way:
In this equation I*jis the actual level of index j and b*ijis a measure of the responsive- ness of the return on stock i to changes in the index j. Thus b*ijhas the same meaning as i
in the case of the single-index model. A b*ij of 2 would mean that if the index increased (decreased) by 1%, the stock’s return is expected to increase (decrease) by 2%. As in the case of the single-index model, the return of the security not related to indexes is split into two parts: a*iand ciwhere a*iis the expected value of the unique return. This is the same meaning it had in the single-index model. Variable ci is the random component of the unique return; it has a mean of zero and a variance we will designate as 2ci.
Although a multi-index model of this type can be employed directly, the model would have some very convenient mathematical properties if the indexes were uncorrelated (orthog- onal). This would allow us to simplify both the computation of risk and the selection of opti- mal portfolios. Fortunately, this presents no theoretical problems because it is always possible to take any set of correlated indexes and convert them into a set of uncorrelated indexes. The method for doing so is outlined in Appendix A. Using this methodology, the equation can be rewritten as1
where all Ijare uncorrelated with each other. The new indexes still have an economic inter- pretation. Assume I*1was a stock market index and I*2an index of interest rates. I2is now an index of the difference between actual interest rates and the level of interest rates that
Ri = +ai b Ii1 1+b Ii2 2+b Ii3 3+ +L b IiL L+ci Ri =ai*+b Ii* *1 1 +b Ii*2 2*+ +L b IiL L* *+ci
1The asterisks have been removed to indicate that the indexes and coefficients are now different. Actually, if the procedure in Appendix A at the end of this chapter is followed, I1⫽I*1, but all others are different. In applica- tions it may be easier for analysts to estimate the model with correlated indexes. This model can then be trans- formed into one with uncorrelated indexes for purposes of portfolio selection.
CHAPTER 8 THE CORRELATION STRUCTURE OF SECURITY RETURNS 157
would be expected given the rate of return on the stock market (I1). Similarly, bi2becomes a measure of the sensitivity of the return on stock i to this difference. We can think of bi2 as the sensitivity of stock i’s return to a change in interest rates when the rate of return on the market is fixed.
Not only is it convenient to make the indexes uncorrelated, but it is also convenient to have the residual uncorrelated with each index. Formally, this implies that E[ci(Ij⫺I–
j)] ⫽ 0 for all j. The implication of this construction is that the ability of Equation (8.1) to describe the return on any security is independent of the value any index happens to assume. When the parameters of this model are estimated via regression analysis, as is usu- ally done, this will hold over the period of time to which the model is fitted.
The standard form of the multi-index model can be written as follows:
BASIC EQUATION:
for all stocks i⫽1, ..., N (8.1) BY DEFINITION
1. Residual variance of stock i equals ci2, where i⫽1, ..., N.
2. Variance of index j equals 2Ij,where j⫽1, ..., L.
BY CONSTRUCTION
1. Mean of ciequals E(ci) ⫽0 for all stocks, where i⫽1, ..., N.
2. Covariance between indexes j and k equals E[(Ij⫺I–
j) (Ik⫺I–
k)] ⫽0 for all indexes, where j⫽1, ..., L and k⫽1, ..., L (j⫽k).
3. Covariance between the residual for stock i and index j equals E[ci(Ij⫺I–
j)] ⫽0 for all stocks and indexes, where i⫽1, ..., N and j⫽1, ..., L.
BY ASSUMPTION
1. Covariance between ciand cjis zero (E(cicj) ⫽0) for all stocks where i⫽1, ..., N and j⫽1, ..., N ( j⫽i).
The assumption of the multi-index model is that E(cicj) ⫽ 0. This assumption implies that the only reason stocks vary together is because of common comovement with the set of indexes that have been specified in the model. There are no factors beyond these indexes that account for comovement between any two securities. There is nothing in the estima- tion of the model that forces this to be true. This is a simplification that represents an approximation to reality. The performance of the model will be determined by how good this approximation is. This, in turn, will be determined by how well the indexes that we have chosen to represent comovement really capture the pattern of comovement among securities.
The expected return, variance, and covariance among securities when the multi-index model describes the return structure are derived in Appendix B and are equal to the following:
1. Expected return is
(8.2) 2. Variance of return is
(8.3) σi2=bi21σI21+bi22σ2I2+ +L biL2σIL2 +σci2
Ri= +ai b Ii1 1+b Ii2 2+ +L b IiL L Ri = +ai b Ii1 1+b Ii2 2+b Ii3 3+ +L b IiL L+ci
3. Covariance between security i and j is
(8.4) From Equations (8.2), (8.3), and (8.4) it is clear that the expected return and risk can be estimated for any portfolio if we have estimates of aifor each stock, and estimates of bik for each stock with each index, an estimate of 2cifor each stock and, finally, an estimate of the mean (I–
j) and variance 2Ijof each index. This is a total of 2N⫹2L⫹LN estimates.
For an institution following between 150 and 250 stocks and employing 10 indexes, this calls for between 1,820 and 3,020 inputs. This is larger than the number of inputs required for the single-index model but considerably less than the inputs needed when no simplify- ing structure was assumed. Notice that now analysts must be able to estimate the respon- siveness of each stock they follow to several economic and industry influences.
This model can also be used if analysts supply estimates of the expected return for each stock, the variance of each stock’s returns, each index loading (bikbetween each stock i and each index k), and the means and variances of each index. This is the same number of inputs (2N⫹2L⫹LN). However, the inputs are in more familiar terms. As discussed at several points in this book, the inputs needed to perform portfolio analysis are expected returns, variances, and correlation coefficients. By having the analysts estimate means and variances directly, it is clear that the only input derived from the estimates of the multi- index models is correlation coefficients. We stress this point because later in this chapter, we evaluate the ability of a multi-index model to aid in the selection of securities by exam- ining its ability to forecast correlation coefficients.
There is a certain type of multi-index model that has received a large amount of atten- tion. This class of models restricts attention to market and industry influences. Alternative industry index models result from different assumptions about the behavior of returns and, hence, differ in the type and amount of input data needed. We now examine these models.
Industry Index Models
Several authors have dealt with multi-index models that start with the basic single-index model and add indexes to capture industry effects. The early precedent for this work can be found in King (1966), who measured effects of common movement between securities beyond market effects and found that this extra market covariance was associated with industries. For example, two steel stocks had positive correlation between their returns, even after the effects of the market had been removed.2
If we hypothesize that the correlation between securities is caused by a market effect and industry effects, our general multi-index model could be written as
where
Im is the market index
Ij are industry indexes that are constrained to be uncorrelated with the market and uncorrelated with each other
Ri = +ai b Iim m+b Ii1 1+b Ii2 2+ +L b IiL L+ci σij =b bi1 j1σ2I1+b bi2 j2σI22+ +L b biL jLσ2IL
2King (1966) found that over the entire period studied, 1927–1960, about half of the total variation in a stock’s price was accounted for by a market index, while an average of another 10% was accounted for by industry fac- tors. In the latter part of the period he studied, the importance of the market factor dropped to 30%, while the industry factors continued to explain 10% of price movement.