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as372w10a2soln

# as372w10a2soln - Solution 1(a Average Returns(assume...

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Solution 1. (a) Average Returns (assume arithmetic average): ¯ x small cap = 5 X i =1 R i = 0 . 1851 + · · · + 0 . 1255 5 = 17 . 384% ¯ x market = 0 . 1202 + · · · + 0 . 1453 5 = 11 . 5% (b) Variance and Standard deviation. First note that the sample (unbiased) variance estimates for a sample of N observations is given by σ 2 = 1 N - 1 N X i =1 ( x i - ¯ x ) 2 = 1 N - 1 N X i =1 x 2 i - N ¯ x 2 ! . Note, the sample estimate should be divided by N - 1, instead of N for the unbiasedness. Hence σ 2 small cap = 1 4 (0 . 34644504 - 5 × 0 . 17384 2 ) = 0 . 048835828 σ small cap = 0 . 048835828 = 22 . 10% σ 2 market = 1 4 (0 . 14178322 - 5 × 0 . 115 2 ) = 0 . 018914555 σ market = 0 . 018914555 = 13 . 75% (c) (unbiased) Sample covariance: σ small cap,market = 1 N - 1 N X i =1 ( x small cap,i - ¯ x small cap )( x market,i - ¯ x market ) = 1 N - 1 N X i =1 ( x small cap,i × x market,i ) - N ¯ x small cap × ¯ x market ! = 1 4 (0 . 1851(0 . 1202) + · · · + 0 . 1255(0 . 1453) - 5(0 . 17384)(0 . 115)) = 0 . 027085 Sample correlation ρ = σ small cap,market σ small cap × σ market = 0 . 891163 3

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2. μ A = 0 . 2( . 123) + . 5(0 . 065) + 0 . 3(0 . 03) = 6 . 61% μ B = 2 . 26% σ 2 A = 0 . 2( . 123 - μ A ) 2 + 0 . 5( . 065 - μ A ) 2 + 0 . 3( . 03 - μ A ) 2 = 0 . 00103909 σ A = 0 . 00103909 = 3 . 22349% σ B = 3 . 10329% σ A,B = . 2( . 123)( - 0 . 037) + . 5( . 065)(0 . 03) + . 3(0 . 03)(0 . 05) - μ A × μ B = - 0 . 0098 ρ A,B = σ A,B σ A σ B = - 0 . 097872 3. (a) x 1 μ 1 + (1 - x 1 ) μ 2 = 0 . 18 x
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as372w10a2soln - Solution 1(a Average Returns(assume...

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