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NotesSep27

# NotesSep27 - Diversication leads to smaller variance The...

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Diversification leads to smaller variance! The diversification works even better if one could pick two stocks for which the returns X and Y had negative covariance . In this case the risk is Var( X + Y ) = Var X + Var Y + 2Cov( X, Y ) < Var X + Var Y.

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A general rule for the variance of a sum : if X 1 , . . . , X n are random variables then Var n X i =1 X i = n X i =1 Var X i +2 n X i =1 n X j = i +1 Cov( X i , X j ) . If X 1 , . . . , X n are independent, then Var n X i =1 X i = n X i =1 Var X i .
Variance of a linear combination of random variables If X 1 , . . . , X n are random variables and a 1 , . . . , a n are numbers then Var n X i =1 a i X i = n X i =1 a 2 i Var X i + 2 n X i =1 n X j = i +1 a i a j Cov( X i , X j ) . If X 1 , . . . , X n are independent, then Var n X i =1 a i X i = n X i =1 a 2 i Var X i .

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Correlation The correlation or the correlation coefficient between random variables X and Y is ρ X,Y = Cov( X, Y ) Var X Var Y = Cov( X, Y ) σ X σ Y , where σ X and σ Y are the standard deviations of X and Y .
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NotesSep27 - Diversication leads to smaller variance The...

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