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5.why_works

# 5.why_works - University of California Los Angeles...

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University of California, Los Angeles Department of Statistics Statistics C183/C283 Instructor: Nicolas Christou Portfolio expected return and risk Suppose a portfolio consists of n stocks. Let ¯ R i and σ 2 i the expected return and variance of stock i , i = 1 , 2 , · · · , n . Also, let σ ij the covariance between stocks i and j . Let x 1 , x 2 , · · · , x n the fractions of the investors wealth invested in each one of the n stocks ( n i =1 x i = 1). The resulting portfolio is x 1 R 1 + x 2 R 2 + · · · + x n R n and at time t it has return: R pt = x 1 R 1 t + x 2 R 2 t + · · · + x n R nt The expected return of this portfolio is given by: ¯ R p = x 1 ¯ R 1 + x 2 ¯ R 2 + · · · + x n ¯ R n = n X i =1 x i ¯ R i = x 0 ¯ R where, x 0 = ( x 1 , x 2 , · · · , x n ) , and ¯ R = ( ¯ R 1 , ¯ R 2 , · · · , ¯ R n ) And its risk (variance) by: σ 2 p = var ( x 1 R 1 + x 2 R 2 + · · · + x n R n ) = n X i =1 x 2 i σ 2 i + n X i =1 n X j 6 = i x i x j σ ij = n X i =1 n X j =1 x i x j σ ij Or in matrix form: σ 2 p = x 0 Σx where, Σ is the symmetric, positive definite n × n

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