{[ promptMessage ]}

Bookmark it

{[ promptMessage ]}

Lecture2011

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

This preview shows pages 1–3. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
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 N P n =1 w n r n = K and N P n =1 w n = 1 into the objective function V p ( w 1 ;w 2 ;:::;w N ) ; to get an unconstrained minimization problem. When there are many assets, however, it is often di¢cult to solve this problem by substitution. We need a way to compute the
This is the end of the preview. Sign up to access the rest of the document.

{[ snackBarMessage ]}

Page1 / 5

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

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online