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bkmsol_ch06 - CHAPTER 6 EFFICIENT DIVERSIFICATION 1 E(rP...

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CHAPTER 6: EFFICIENT DIVERSIFICATION 1. E(r P ) = (0.5 × 15) + (0.4 × 10) + (0.10 × 6) = 12.1% 3. a. The mean return should be equal to the value computed in the spreadsheet. The fund's return is 3% lower in a recession, but 3% higher in a boom. However, the variance of returns should be higher, reflecting the greater dispersion of outcomes in the three scenarios. b. Calculation of mean return and variance for the stock fund: (A) (B) (C) (D) (E) (F) (G) Scenario Probability Rate of Return Col. B × Col. C Deviation from Expected Return Squared Deviation Col. B × Col. G Recession 0.3 -14 -4.2 -24.0 576 172.8 Normal 0.4 13 5.2 3.0 9 3.6 Boom 0.3 30 9.0 20.0 400 120.0 Expected Return = 10.0 Variance = 296.4 Standard Deviation = 17.22 c. Calculation of covariance: (A) (B) (C) (D) (E) (F) Deviation from Mean Return Scenario Probability Stock Fund Bond Fund Col. C × Col. D Col. B × Col. E Recession 0.3 -24 10 -240.0 -72 Normal 0.4 3 0 0.0 0 Boom 0.3 20 -10 -200.0 -60 Covariance = -132 Covariance has increased because the stock returns are more extreme in the recession and boom periods. This makes the tendency for stock returns to be poor when bond returns are good (and vice versa) even more dramatic. 6-1
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4. a. One would expect variance to increase because the probabilities of the extreme outcomes are now higher. b. Calculation of mean return and variance for the stock fund: (A) (B) (C) (D) (E) (F) (G) Scenario Probability Rate of Return Col. B × Col. C Deviation from Expected Return Squared Deviation Col. B × Col. G Recession 0.4 -11 -4.4 -20.0 400 160.0 Normal 0.2 13 2.6 4.0 16 3.2 Boom 0.4 27 10.8 18.0 324 129.6 Expected Return = 9.0 Variance = 292.8 Standard Deviation = 17.11 c. Calculation of covariance: (A) (B) (C) (D) (E) (F) Deviation from Mean Return Scenario Probability Stock Fund Bond Fund Col. C × Col. D Col. B × Col. E Recession 0.4 -20 10 -200.0 -80 Normal 0.2 4 0 0.0 0 Boom 0.4 18 -10 -180.0 -72 Covariance = -152 Covariance has increased because the probabilities of the more extreme returns in the recession and boom periods are now higher. This makes the tendency for stock returns to be poor when bond returns are good (and vice versa) more dramatic. 6. The parameters of the opportunity set are: E(r S ) = 15%, E(r B ) = 9%, σ S = 32%, σ B = 23%, ρ = 0.15, r f = 5.5% 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 529.0 110.4 Stocks 110.4 1024.0 The minimum-variance portfolio proportions are: ) r , r ( Cov 2 ) r , r ( Cov ) S ( w B S 2 B 2 S B S 2 B Min - σ + σ - σ = 6-2
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