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Unformatted text preview: ACTSC 372 Assignment 1 Solutions 1. [3 points] Suppose you can invest in two stocks, A and B. The returns R A , R B , on these stocks depend on the state of the economy and are modelled as follows (where returns are expressed in percentages): state probability R A R B Recession 0.258 Normal 0.5 10 14 Boom 0.3 20 21 (a) Compute the expected return for each stock. Solution: E( R A ) = 0 . 2 ( . 05) + 0 . 5 . 1 + 0 . 3 . 2 = 0 . 10 = 10% and E( R B ) = 0 . 2  . 08 + . 5 . 14 + 0 . 3 . 21 = 0 . 117 = 11 . 7%. (b) Compute the standard deviation of the return for each stock. Solution: A = (0 . 2( . 05 . 1) 2 +0 . 5(0 . 1 . 1) 2 +0 . 3(0 . 2 . 1) 2 ) 1 / 2 = 8 . 66% and B = (0 . 2( . 08 . 117) 2 + 0 . 5(0 . 14 . 117) 2 + 0 . 3(0 . 21 . 117) 2 ) 1 / 2 = 10 . 31%. (c) Compute the covariance and correlation between the two stocks. Solution: Cov( R A , R B ) = 0 . 2( . 05 . 1)( . 08 . 117) + 0 . 5(0 . 1 . 1)(0 . 14 . 117) + 0 . 3(0 . 2 . 1)(0 . 21 . 117) = 0 . 0087 and AB = Cov( R A , R B ) / ( A B ) = 0 . 974778. 2. [3 points] Suppose there are N stocks in the economy, each with an expected return of 9% and a standard deviation of the return equal to 12%. The covariance between the returns is the same for all pairs of (distinct) stocks. (a) Suppose N = 100. If an equally weighted portfolio composed of these N = 100 assets has a standard deviation of 9%, what is the covariance between the returns (for distinct pairs of stocks)? Solution: Let c be the common covariance between two stocks. We have that 2 R = 0 . 09 2 = 100 X i =1 1 100 2 . 12 2 + 2 99 X i =1 100 X j = i +1 1 100 1 100 c Therefore . 09 2 = 0 . 01 . 12 2 + 2 c . 01 2 99 100 2 and so c = 0 . 008036....
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This note was uploaded on 09/18/2011 for the course ACTSC 372 taught by Professor Maryhardy during the Winter '09 term at Waterloo.
 Winter '09
 MARYHARDY

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