bkmsol_ch08_09 - CHAPTER 8: OPTIMAL RISKY PORTFOLIOS 1. The...

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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 minimum-variance 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. 9-1 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 reward-to-variability ratio of the optimal CAL is: 9-2 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 T-bill fund, remember that the mean of the complete portfolio (i.e., 14%) is an average of the T-bill 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.

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bkmsol_ch08_09 - CHAPTER 8: OPTIMAL RISKY PORTFOLIOS 1. The...

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