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# ch_07_sol - CHAPTER 7 EFFICIENT DIVERSIFICATION 1 2 3 4...

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CHAPTER 7: EFFICIENT DIVERSIFICATION 1. E(r P ) = .5 x 15 + .4 x 10 + .10 x 6 = 12.1% 2. a. –1.0 3. a. only i 4. a. The mean return should be the same. The fund's return is 2% lower in a recession, but 2% 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 variance for the stock fund. Deviation from Rate of Expected Squared Scenario Return Return Deviation Recession 9% –20% 400 Normal +12 + 1 1 Boom +30 +19 361 Expected return = 11% Variance* = 254 Standard deviation = 15.94% *Variance = average of squared deviations from the expected value. c. Calculation of covariance Stock fund Bond fund Deviation Deviation from from Rate of Expected Rate of Expected Product of Scenario Return Return Return Return Deviations Recession 9% –20% +17% +10% –200 Normal +12 + 1 + 7 + 0 0 Boom +30 +19 3 –10 –190 7-1

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Covariance = average of product of deviations = 1/3 x (–200 + 0 –190) = –130 Covariance is higher 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. 5. a. One would expect variance to increase since the probabilities of the extreme outcomes are now higher. b. Calculation of new stock variance. Deviation from Rate of Expected Squared Scenario Return Return Deviation Recession 9% –19.8% 392.04 Normal +12 + 1.2 1.44 Boom +30 +19.2 368.64 Expected return .4 x (–9) + .2 x 12 + .4 x 30 = 10.8% Variance* .4 x 392.04 + .2 x 1.44 + .4 x 368.64 = 304.56 Standard deviation = = 17.45% *Variance = expected value of squared deviations from the mean return. c. Calculation of new covariance Stock fund Bond fund Deviation Deviation from from Rate of Expected Rate of Expected Product of Scenario Return Return Return Return Deviations Recession 9% –19.8% +17% +10% –198 Normal +12 + 1.2 + 7 + 0 0 Boom +30 +19.2 3 –10 +192 Covariance = average of product of deviations = .4 x (–198) + .2 x 0 + .4 x (–192) = –156 7-2
Covariance is higher 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. 7-3

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6. The parameters of the opportunity set are: E(r S ) = 22%, E(r B ) = 13%, σ S = 32%, σ B = 23%, ρ=.15 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 is found by applying the formula: w Min (S) = = = .3142
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