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Unformatted text preview: Haas School of Business University of California at Berkeley UGBA 103 Avinash Verma C APITAL A SSET P RICING M ODEL Recall that in the last note, we discussed Capital Market Line (CML). CML gave us a relationship between risk (as measured by the standard deviation of returns) and reward (as measured by the expected return). However, CML holds only for efficient portfolios . In this note, we seek a relationship between risk and return that is true more generally rather than only for efficient portfolios. Capital Asset Pricing Model provides such a relationship between (a different measure of) risk and reward (as measured by expected return). 1. We start by asking the question: When we expand a portfolio by adding a new security to it, how much risk does the new security add to the portfolio risk? In other words, what part of the total risk of the expanded portfolio can be attributed to the security that has just been added? We know that the portfolio variance is given by: = = = = n i n j ij j i p p x x R V 1 1 2 ) ~ ( We also know that this double sum is an operation in which covariance between returns on any two securities in the unshaded matrix below is weighted with the fraction invested in each security in the pair (the fractions or the portfolio weights are in the yellow cells), and then a sum is taken over all possible pairs. S ECURITY 1 S ECURITY 2 S ECURITY n x 1 x 2 x n S ECURITY 1 x 1 2 1 11 = 21 12 = 1 1 n n = S ECURITY 2 x 2 12 21 = 2 2 22 = 2 2 n n = S ECURITY n x n n n 1 1 = n n 2 2 = 2 n nn = 2. Entries either in the first row or the first column in the matrix above can be attributed to our decision to add Security 1 to the portfolio. Therefore, the contribution that Security 1 made to the portfolio risk can be thought of as a weighted sum of all the entries in the first row [or, equivalently, the first column] where each entry is weighted with the fraction invested in every other security that Security 1 is paired with. Thus, we multiply (i) the portfolio weights, x 1 , x 2 , x n , in the white on black row with entries in the row for Security 1 , 11 , 12 , 1 n , and conclude that what Security 1 contributed to 2 p , the portfolio risk, is given by ( x 1 times): n n x x x 1 12 2 11 1 ... + + + . 3. Now, it is one of the properties of covariance that the covariance between a variable, say, Z ~ , and a weighted sum , such as return on a portfolio , is the same weighted sum of covariances between Z ~ and the constituents of the weighted sum. Thus: ( 29 Zn n Z Z p Zp x x x R Z COV ...... ) ~ , ~ 2 2 1 1 + + = ....
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- Spring '08