Chapter07 - Chapter 07 - Optimal Risky Portfolios CHAPTER...

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Chapter 07 - Optimal Risky Portfolios 7-1 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%, σ S = 30%, σ B = 15%, ρ = 0.10 From the standard deviations and the correlation coefficient we generate the covariance matrix [note that Cov(r S , r B ) = ρσ S σ B ]: Bonds Stocks Bonds 225 45 Stocks 45 900 The minimum-variance portfolio is computed as follows: w Min (S) = 1739 . 0 ) 45 2 ( 225 900 45 225 ) r , r ( Cov 2 ) r , r ( Cov B S 2 B 2 S B S 2 B = × + = σ + σ σ 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% σ Min = 2 / 1 B S B S 2 B 2 B 2 S 2 S )] r , r ( Cov w w 2 w w [ + σ + σ = [(0.1739 2 × 900) + (0.8261 2 × 225) + (2 × 0.1739 × 0.8261 × 45)] 1/2 = 13.92%
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Chapter 07 - Optimal Risky Portfolios 7-2 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. 6. 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%.
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Chapter 07 - Optimal Risky Portfolios 7-3 7. 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 + σ + σ σ = 4516 . 0 ] 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% σ p = [(0.4516 2 × 900) + (0.5484 2 × 225) + (2 × 0.4516 × 0.5484 × 45)] 1/2 = 16.54% 8. The reward-to-volatility ratio of the optimal CAL is: 4601 . 0 54 . 16 8 61 . 15 r ) r ( E p f p = = σ 9. 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 . 0 8 r ) r ( E r ) r ( E σ + = σ σ + = Setting E(r C ) equal to 14%, we find that the standard deviation of the optimal portfolio is 13.04%.
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Chapter07 - Chapter 07 - Optimal Risky Portfolios CHAPTER...

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