PS 4 Solutions.pdf

# PS 4 Solutions.pdf - Solutions to Problem Set 4 Corporate...

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Solutions to Problem Set 4 Corporate Finance, Prof. William Diamond 1. Because the proceeds from Year 1 are reinvested in Year 2 etc., the proper summary measure for returns is the geometric average of the annual returns. Thus, the geometric average returns are: Fund A = (1 . 16 × 1 . 10 × 1 . 14 × 1 . 02 × 1 . 04) 1 / 5 - 1 = . 090631 Fund B = (1 . 3 × . 9 × 1 . 28 × 1 . 17 × . 98) 1 / 5 - 1 = . 114196 The amount of money at the end can be calculated in either of 2 ways for each fund: Fund A = \$100(1 . 16 × 1 . 10 × 1 . 14 × 1 . 02 × 1 . 04) = \$154 . 308 or \$100(1 . 090631) 5 = \$154 . 308. Fund B = \$100(1 . 3 × . 9 × 1 . 28 × 1 . 17 × . 98) = \$171 . 715 or \$100(1 . 114196) 5 = \$171 . 715. 2. (a) The general formula for expected return for a two-security portfolio is: ¯ R p = X 1 ¯ R 1 + X 2 ¯ R 2 Also, the variance of return for a two-security portfolio is: σ 2 p = X 2 1 σ 2 1 + X 2 2 σ 2 2 + 2 X 1 X 2 σ 1 σ 2 ρ. Recall that X 2 = 1 - X 1 . Substituting the numerical values given in the question into these equations pro- duces the following results (remember to square terms or take the square root as necessary). Expected Standard Return Deviation i) .15 .200 ii) .20 .200 iii) .25 .245 iv) .30 .316 v) .35 .400 1

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3. Using the formulas ¯ R p = X 1 ¯ R 1 + X 2 ¯ R 2 and σ 2 p = X 2 1 σ 2 1 + X 2 2 σ 2 2 + 2 X 1 X 2 σ 1 σ 2 ρ we calculate ρ = - 1 ρ = 0 ρ = . 2 ρ = . 5 ρ = 1 ¯ R p . 142 . 142 . 142 . 142 . 142 σ p . 102 . 158 . 166 . 179 . 198 Note that the mean does not depend on the correlation. On the other hand, the higher is the correlation, the higher is the portfolio standard deviation. For stocks that move closely together, there are fewer gains from diversification from combining them into a portfolio. The standard deviation when ρ = 1 represents the highest standard deviation for this portfolio. Notice from the formula that the portfolio variance is increasing in ρ . Therefore σ p = ( X 2 1 σ 2 1 + X 2 2 σ 2 2 + 2 X 1 X 2 σ 1 σ 2 ρ ) 1 2 ( X 2 1 σ 2 1 + X 2 2 σ 2 2 + 2 X 1 X 2 σ 1 σ 2 ) 1 2 = X 1 σ 1 + X 2 σ 2 4. (a) i. The investor should choose security 2 because security 2 has a higher expected return than security 1 ( ¯ R 2 = . 16 while ¯ R 1 = . 10).
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