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Mean_Variance_Fall_722_2011

Mean_Variance_Fall_722_2011 - Mean Variance Analysis...

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Mean Variance Analysis Business Finance 722 Investment Management Fousseni D. Chabi-Yo 1 1 The Ohio State University, Fisher College of Business Mean Variance Analysis

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Mean Variance Analysis A Portfolio of Three Risky Assets We don’t live in a two risky asset world. We need to be able to deal with portfolios that have many assets. The return on a three-asset portfolio : r p = w a r a + w b r b + w c r c r a is the return and w a is the weight on asset A. r b is the return and w b is the weight on asset B. r c is the return and w c is the weight on asset C. w a + w b + w c = 1.
Mean Variance Analysis Three Asset Portfolio If A, B, and C are assets, w a , w b , and w c are the portfolio weights on each asset, and w a + w b + w c = 1, then the expected return of the portfolio is E ( r p ) = w a E ( r a ) + w b E ( r b ) + w c E ( r c ) the variance of the portfolio is σ 2 ( r p ) = w 2 a σ 2 ( r a ) + w 2 b σ 2 ( r b ) + w 2 c σ 2 ( r c ) + 2 w a w b ( r a , r b ) + 2 w a w c ( r a , r c ) + 2 w b w c ( r b , r c ) , and the standard deviation of the portfolio is σ ( r p ) = q σ 2 ( r p ) .

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Mean Variance Analysis A Portfolio of Three Risky Assets The expected return of a portfolio is just a weighted average of the individual security expected returns. The variance is a bit more complicated. Covariances are more influential than with a 2-asset portfolio. Three covariances now instead of one. As we add more assets, covariances become a more important determinate of the portfolio’s variance (diversification).
Mean Variance Analysis You hate Microsoft so you decide to invest in three competitors: Example: Three Risky Assets Covariance Matrix E ( r i ) Apple Sun Micro Red Hat Apple 0.20 0.09 0.045 0.05 Sun Micro 0.12 0.045 0.07 0.04 Red Hat 0.15 0.05 0.04 0.06 The diagonal of the covariance matrix holds variances. ( r apple , r apple ) = σ 2 ( r apple ) The covariance matrix is symmetric (row 1, column 2 is the same as row 2, column 1) because, ( r apple , r sun ) = ( r sun , r apple )

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Mean Variance Analysis Example: Three Risky Assets Q: What is E ( r p ) and σ ( r p ) if you invest equally in each stock? E ( r p ) = w a E ( r a ) + w s E ( r s ) + w h E ( r h ) = 1 3 ( 0 . 2 ) + 1 3 ( 0 . 12 ) + 1 3 ( 0 . 15 ) = 15 . 7 % σ 2 ( r p ) = w 2 a σ 2 ( r a ) + w 2 s σ 2 ( r s ) + w 2 h σ 2 ( r h ) + 2 w a w s ( r a , r s ) + 2 w a w h ( r a , r h ) + 2 w s w h ( r s , r h ) = 1 9 ( 0 . 09 ) + 1 9 ( 0 . 07 ) + 1 9 ( 0 . 06 ) + 2 ( 1 3 1 3 )( 0 . 045 ) + 2 ( 1 3 1 3 )( 0 . 05 ) + 2 ( 1 3 1 3 )( 0 . 04 ) = 0 . 0544 σ ( r p ) = 0 . 0544 = 23 . 3 %
Mean Variance Analysis Example: Three Risky Assets Q: What is E ( r p ) and σ ( r p ) if you invest equally in each stock? E ( r p ) = w a E ( r a ) + w s E ( r s ) + w h E ( r h ) = 1 3 ( 0 . 2 ) + 1 3 ( 0 . 12 ) + 1 3 ( 0 . 15 ) = 15 . 7 % σ 2 ( r p ) = w 2 a σ 2 ( r a ) + w 2 s σ 2 ( r s ) + w 2 h σ 2 ( r h ) + 2 w a w s ( r a , r s ) + 2 w a w h ( r a , r h ) + 2 w s w h ( r s , r h ) = 1 9 ( 0 . 09 ) + 1 9 ( 0 . 07 ) + 1 9 ( 0 . 06 ) + 2 ( 1 3 1 3 )( 0 . 045 ) + 2 ( 1 3 1 3 )( 0 . 05 ) + 2 ( 1 3 1 3 )( 0 . 04 ) = 0 . 0544 σ ( r p ) = 0 . 0544 = 23 . 3 %

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Mean Variance Analysis A Portfolio of Many Risky Assets Consider a portfolio that has n assets ( n can be a very large number like 1000, 7000, or even more than that). The return on the portfolio is, r p = w 1 r 1 + w 2 r 2 + · · · + w n r n = n X i = 1 w i r i r i is the return on the ith asset in the portfolio.
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Mean_Variance_Fall_722_2011 - Mean Variance Analysis...

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