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Unformatted text preview: CHAPTER 7: OPTIMAL RISKY PORTFOLIOS PROBLEM SETS 1. (a) and (e). 2. (a) and (c). After real estate is added to the portfolio, there are four asset classes in the portfolio: stocks, bonds, cash and real estate. Portfolio variance now includes a variance term for real estate returns and a covariance term for real estate returns with returns for each of the other three asset classes. Therefore, portfolio risk is affected by the variance (or standard deviation) of real estate returns and the correlation between real estate returns and returns for each of the other asset classes. (Note that the correlation between real estate returns and returns for cash is most likely zero.) 3. (a) Answer (a) is valid because it provides the definition of the minimum variance portfolio. 4. 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 ( , ) S B S B Cov r r = ]: 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) = .1339 = 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% 71 5. 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 below. 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 r f = 8.00 6. The above graph indicates that the optimal portfolio is the tangency portfolio with expected return approximately 15.6% and standard deviation approximately 16.5%. 72 7. The proportion of the optimal risky portfolio invested in the stock fund is given by: 2 2 2 [ ( ) ] [ ( ) ] ( , ) [ ( ) ] [ ( ) ] [ ( ) ( ) ] ( , ) S f B B f S B S S f B B f S S f B f S B E r r E r r Cov r r w E r r E r r E r r E r r Cov r r   = +  + [(.20 .08) 225] [(.12 .08) 45] 0.4516 [(.20 .08) 225] [(.12 .08) 900] [(.20 .08 .12 .08) 45]  = = +  + 1 0.4516 0.5484 B w = = The mean and standard deviation of the optimal risky portfolio are: E(r P ) = (0.4516 .20) + (0.5484 .12) = .1561 = 15.61% p = [(0.4516 2 900) + (0.5484 2 225) + (2 0.4516 0.5484 45)] 1/2 = 16.54% 8. The rewardtovolatility ratio of the optimal CAL is:...
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