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Unformatted text preview: CHAPTER 8: OPTIMAL RISKY PORTFOLIOS 1. The parameters of the opportunity set are: E(r S ) = 20%, E(r B ) = 12%, S = 30%, B = 15%, =0.10 From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(r S , r B ) = S B ]: Bonds Stocks Bonds 225 45 Stocks 45 900 The minimumvariance portfolio is computed as follows: w Min (S) = 1739 . ) 45 2 ( 225 900 45 225 ) r , r ( Cov 2 ) r , r ( Cov B S 2 B 2 S B S 2 B =  + = +  w Min (B) = 1  0.1739 = 0.8261 The minimum variance portfolio mean and standard deviation are: E(r Min ) = (0.1739 20) + (0.8261 12) = 13.39% Min = 2 / 1 B S B S 2 B 2 B 2 S 2 S )] r , r ( Cov w w 2 w w [ + + = [(0.1739 2 900) + (0.8261 2 225) + (2 0.1739 0.8261 45)] 1/2 = 13.92% 2. Proportion in stock fund Proportion in bond fund Expected return Standard Deviation 0.00% 100.00% 12.00% 15.00% 17.39% 82.61% 13.39% 13.92% minimum variance 20.00% 80.00% 13.60% 13.94% 40.00% 60.00% 15.20% 15.70% 45.16% 54.84% 15.61% 16.54% tangency portfolio 60.00% 40.00% 16.80% 19.53% 80.00% 20.00% 18.40% 24.48% 100.00% 0.00% 20.00% 30.00% Graph shown on next page. 91 3. 0.00 5.00 10.00 15.00 20.00 25.00 0.00 5.00 10.00 15.00 20.00 25.00 30.00 Tangency Portfolio Minimum Variance Portfolio Efficient frontier of risky assets CML INVESTMENT OPPORTUNITY SET The graph indicates that the optimal portfolio is the tangency portfolio with expected return approximately 15.6% and standard deviation approximately 16.5%. 4. The proportion of the optimal risky portfolio invested in the stock fund is given by: ) r , r ( Cov ] r ) r ( E r ) r ( E [ ] r ) r ( E [ ] r ) r ( E [ ) r , r ( Cov ] r ) r ( E [ ] r ) r ( E [ w B S f B f S 2 S f B 2 B f S B S f B 2 B f S S +  +   = 4516 . ] 45 ) 8 12 8 20 [( ] 900 ) 8 12 [( ] 225 ) 8 20 [( ] 45 ) 8 12 [( ] 225 ) 8 20 [( =  +  +    = w B = 1  0.4516 = 0.5484 The mean and standard deviation of the optimal risky portfolio are: E(r P ) = (0.4516 20) + (0.5484 12) = 15.61% p = [(0.4516 2 900) + (0.5484 2 225) + (2 0.4516 0.5484 45)] 1/2 = 16.54% 5. The rewardtovariability ratio of the optimal CAL is: 92 4601 . 54 . 16 8 61 . 15 r ) r ( E p f p = =  6. a.If you require that your portfolio yield an expected return of 14%, then you can find the corresponding standard deviation from the optimal CAL. The equation for this CAL is: C C P f p f C 4601 . 8 r ) r ( E r ) r ( E + =  + = Setting E(r C ) equal to 14%, we find that the standard deviation of the optimal portfolio is 13.04%. b. To find the proportion invested in the Tbill fund, remember that the mean of the complete portfolio (i.e., 14%) is an average of the Tbill rate and the optimal combination of stocks and bonds (P). Let y be the proportion invested in the portfolio P. The mean of any portfolio along the optimal CAL is: E(r C ) = (l  y)r f + yE(r P ) = r f + y[E(r P )  r f ] = 8 + y(15.61  8) Setting E(r...
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This note was uploaded on 03/10/2011 for the course FMIS 3601 taught by Professor Vizanko during the Spring '08 term at University of Minnesota Duluth.
 Spring '08
 Vizanko

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