Lecture2011 - Lecture 11: Risk and Return I Continued I....

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Lecture 11: Risk and Return I Continued I. Risk and Return A) We will de…ne the risk of any asset (or portfolio of assets) as its variance. B) Expected Return of Portfolios Given N assets with respective expected returns r 1 ; r 2 ;:::;r N , the expected return of a portfolio with respective weights w 1 ; w 2 ; ..., w N is ER p ( w 1 ; w 2 ;:::; w N ) = N X n =1 w n r n C) Variance of Portfolios: Given N assets with respective variances ¾ 2 1 ; ¾ 2 2 ;:::; ¾ 2 N , let ¾ j;n denote the covariance between the n-th and j-th asset. Then the variance of a portfolio with respective weights w 1 ; w 2 ; ..., w N is V p ( w 1 ; w 2 ;:::; w N ) = N X n =1 w 2 n ¾ 2 n + N X n =1 N X j =1 For n 6 = j w n w j ¾ j;n For example is N = 3 V p ( w 1 ; w 2 ;:::; w N ) = w 2 1 ¾ 2 1 + w 2 2 ¾ 2 2 + w 2 3 ¾ 2 3 +2 w 1 w 2 ¾ 1 ; 2 + 2 w 1 w 3 ¾ 1 ; 3 +2 w 2 w 3 ¾ 2 ; 3 II. Diversi…cation A) Return is a linear function of the relative portfolio weights, risk, however, is often a non-linear function of these same weights. In fact, when the correlation between any two assets is less than one there exist diversi…cation bene…ts to investing money in both assets. When correlation between any two assets is less than one, it is possible to decrease risk without decreasing return by mixing the relative portfolio weights optimally. B) To compute the e¢cient frontier one must …nd the weights that minimize variance for any given return, K. w ¤ 1 ;:::; w ¤ N to solve Min w 1 ;:::;w N V p ( w 1 ; w 2 ;:::; w N ) subject to ER p ( w 1 ; w 2 ;:::; w N ) = K 1
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As we alter K, we can compute new minimum variance portfolios to trace the e¢cient frontier. One can solve this problem by substituting the constraints
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This note was uploaded on 04/12/2008 for the course ECON 435 taught by Professor Chabot during the Winter '08 term at University of Michigan.

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Lecture2011 - Lecture 11: Risk and Return I Continued I....

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