## Appendix B

mean return, variance, and covariance of a MULTI-INDEX model

In this appendix we derive the mean return variance and covariance of return when the multi-index model is assumed to describe the return structure in the market.

^he index could also be written as dt + 70 Each is appropriate. The way we have defined it the mean is zero.

Expected Return

The expected return on a security with the multi-index model is

E(Rt ) = % + V, + b,2I2 + • • • + b,JL + C; )

Since the expected value of the sum of random variables is the sum of the expected values, we have

E{R,) = E{0i) + £( VO + E{bnh ) + - + ) + %) Recognizing that a and b are constants and that by construction E(c) = 0, we have

E(R, ) = a,+ ba /, + bnl2+- + bJL This is the result stated in the text.

Variance of Return

The variance of the return on a security is o- = E{R,~R)2

Substituting for R{ and R{, we have af = E[{a, + Vi + hi h + •'' + Kh + c,)

The next step is to square the terms in the brackets. The results of this can be seen if we examine all terms involving the first index. The first index times itself and each of the other terms is

The expected value of the sum of random variables is the sum of the expected values and, since the b.'s are constants, we have

Canceling the <j.'s and rearranging yields

<7,2 = 46n( J, -l) + bn(l2-l2) + - + biL(lL - 7t) + c,f

bÎE(ll-ïlf+bnb,2E[(ll-ït)(l2-ï2j\ + -+ - /,)(/, -/,)] + *„£[(/, -l){c,)}

By construction and

4(w,)(c,)]=o thus, the only nonzero term involving index one is

When we examine terms involving the c;, we get the c; with each index that has an xpected value of zero. We also get £(c.)2 = a-2.; thus,

The Covariance

[ he covariance between securities i and j is Substituting in the expressions for and Rj yields aiJ=E{[bn(l1-Ii)+bi2(l2-I2) + - + biL(. ■[bjl(li-Il) + bJ2{l2-I2) + - + bjL(l

Noting that the a's cancel, and combining the terms involving the same b's yields

The next step is to multiply out the terms. The results of this multiplication can be seen by onsidering the terms involving bn. They are

fhe expected value of all terms involving different indexes, for example, (/j - Ix)(Jk — lk) is zero by construction. Furthermore, the expected value of bn(Ij — /¡)c^ is zero by con-•truction. Thus, the only nonzero term is bnbnE^-l)1 ^b^a),

There are two types of terms involving the cs. First, there are terms like bjk{Ik — Ik)Cj, which is zero by construction. Second, there is the term cf~ This is zero by assumption. Thus

= bnbna2n + bnbj2<t22 + b,3bp a2, + • ■ • + biLbjLa2IL QUESTIONS AND PROBLEMS

1. Given that the correlation coefficient between all securities is the same, call it p*, and the assumption of the single-index model is accepted, derive an expression for the Beta on any stock in terms of p*. - Complete the procedure in Appendix A for reducing a general three-index model to a three-index model with orthogonal indexes.

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