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# mid-sol - INVESTMENT ANALYSIS 70-492 Fall 2009 Suggested...

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INVESTMENT ANALYSIS 70-492 Fall 2009 Suggested Solution to Midterm Exam Professor Anisha Ghosh 1. CAPM (a) Let ω 1 denote the proportion of wealth invested in asset A and ω 2 the proportion in asset B. The tangency portfolio solves the following problem: max ω 1 2 E ( r T ) - r f σ T where E ( r T ) = ω 1 E ( r A ) + ω 2 E ( r B ) σ T = q ω 2 1 σ 2 A + ω 2 2 σ 2 B + 2 ω 1 ω 2 ρ AB σ A σ B ω 1 + ω 2 = 1 This gives the following equations with 2 unknowns z 1 and z 2 : z 1 z 2 = σ 2 A ρ AB σ A σ B ρ AB σ A σ B σ 2 B E ( r A ) - r f E ( r B ) - r f The composition of the tangency portfolio is obtained as ω * 1 = z 1 z 1 + z 2 ω * 2 = z 2 z 1 + z 2 For the given problem, we have ω * 1 = 0 . 605 , ω * 2 = 0 . 395. (b) Plugging the composition obtained in part (a) yields E ( r T ) = 0 . 605 × 0 . 02 + 0 . 395 × 0 . 09 = 0 . 048 . σ T = p (0 . 605) 2 × (0 . 02) 2 + (0 . 395) 2 × (0 . 07 2 ) = 0 . 030 (c) If the CAPM is true, the market portfolio is the tangency portfolio and, hence, has the highest Sharpe ratio. (d) The maximum Sharpe ratio is the Sharpe ratio of the tangency protfolio: 0 . 048 - 0 . 01 0 . 030 = 1 . 267 (e) An investor divides his wealth between the tangency portfolio and the risk-free rate as this gives him the best risk-return tradeoff. the expected return and the standard deviation of return on his portfolio is E ( r P ) = ωE ( r T ) + (1

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