## By Definition

In the subsequent section we derive the expected return, standard deviation, and covariance when the single-index model is used to represent the joint movement of securities. The results are

2. The variance of a security's return, = P^cr2 + crv.

3. The covariance of returns between securities i and j, ar~ = p^cr^.

Note that the expected return has two components: a unique part oi; and a market-related part P;i?ra. Likewise, a security's variance has the same two parts, unique risk o^. and market-related risk 3?cr2. In contrast, the covariance depends only on market risk. This is what we meant earlier when we said that the single-index model implied that the only reason securities move together is a common response to market movements. In this section of the text delineated by the solid line we derive these results. The reader uninterested in the derivation can note the results and then skip to the end of the section. The expected return on a security is

Since the expected value of the sum of random variables is the sum of the expected values, we have a; and (3; are constants and by construction the expected value of ei is zero. Thus,

Result 1

The variance of the return on any security is lit

The variance of the return on any security is

Rearranging and noting that the a's cancel yields

Rearranging and noting that the a's cancel yields af=E[%(Rm-R,n) + ei]

Squaring the terms in the brackets yields a]=^E{Rm-Rm)2 +2\$lE\ei(Rm-Rm)] + E{ei)1 Recall that by assumption (or in some cases by construction) E[efRm - RtJ] = 0. Thus, o?=(i?E(Rm-Rm)2+E(e,)2

The covariance between any two securities can be written as

Substituting for Rp Rr Rjt and Rj yields a„ = £{[(«,. +PA +e,)-(a, + PA,)]

Simplifying by canceling the a's and combining the terms involving 0's yields ov=- *».)+- ■3.) •+■eD]

Carrying out the multiplication a, = - R,„f + p^/ft." *.)]

+ PiE[eJ{Rm-Rm)] + E(eieJ) Since the last three terms are zero, by assumption a„= Result 3

These results can be illustrated with a simple example. Consider the returns on a stock and a market index shown in the first two columns of Table 7.1. These returns are what an investor might have observed over the prior five months. Now consider the values for the single-index model shown in the remaining columns of the table. Column three just reproduced column one and is the return on the security. Accept for the moment that = 1.5. The fifth column is just the second column times 1.5 or the market return times a Beta of 1.5. Where does e. come from? Recall that the average value of e(- is zero. If the average value of e; is zero, then the sum of ei is also zero. The single-index model is an equality. The return over the 5 periods for the stock is 40; 30 of the 40 is market-related return, hence 10 must be non-market-related or unique. If ei sums to zero, then for the single-

index model to be an equality, a, must sum to 10. Since a- is a constant and there are 5 10

periods, a; is -y or 2 per period. Given the values for and for fi;Rm, and since the single-index model is an equality, e;

 Month

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