ProofOfMeanAndVarianceFormulas

ProofOfMeanAndVarianceFormulas - This proves the formula...

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Proof of Mean and Variance Formulas 1 In this handout we prove the following very useful relations: E ( R p ) = X 1 E ( R 1 ) + X 2 E ( R 2 ) (1) σ 2 p = X 2 1 σ 2 1 + X 2 2 σ 2 2 + 2 X 1 X 2 Cov( R 1 , R 2 ) (2) where E ( R p ) represents the mean of a portfolio, and σ 2 p represents the variance. We sometimes write the second formula in terms of correlation: σ 2 p = X 2 1 σ 2 1 + X 2 2 σ 2 2 + 2 X 1 X 2 σ 1 σ 2 ρ. First recall the rules of mean and covariance. For any random variables Z 1 , Z 2 , and Z 3 and constant a , 1. E ( Z 1 + Z 2 ) = E ( Z 1 ) + E ( Z 2 ) 2. E ( aZ 1 ) = aE ( Z 1 ) 3. Cov( Z 1 + Z 2 , Z 3 ) = Cov( Z 1 , Z 3 ) + Cov( Z 2 , Z 3 ) 4. Cov( aZ 1 , Z 2 ) = a Cov( Z 1 , Z 2 ) Our first goal is to prove the formula for the mean (1). From R p = X 1 R 1 + X 2 R 2 , it follows from the first rule that E ( R p ) = E ( X 1 R 1 ) + E ( X 2 R 2 ) . By the second rule: E ( R p ) = X 1 E ( R 1 ) + X 2 E ( R 2 ) . 1 Notes for Finance 100 (sections 301 and 302) prepared by Jessica A. Wachter.
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Unformatted text preview: This proves the formula for the mean. Now for the formula for the variance. First, recall that σ 2 p = Cov( R p , R p ) . By repeated applications of the third rule: Cov( R p , R p ) = Cov( X 1 R 1 + X 2 R 2 , X 1 R 1 + X 2 R 2 ) = Cov( X 1 R 1 , X 1 R 1 ) + Cov( X 2 R 2 , X 2 R 2 ) + 2Cov( X 1 R 1 , X 2 R 2 ) . Finally, applying the fourth rule: Cov( R p , R p ) = X 2 1 Cov( R 1 , R 1 ) + X 2 2 Cov( R 2 , R 2 ) + 2 X 1 X 2 Cov( R 1 , R 2 ) = X 2 1 σ 2 1 + X 2 2 σ 2 2 + 2 X 1 X 2 Cov( R 1 , R 2 ) . This completes the proof of the variance formula....
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ProofOfMeanAndVarianceFormulas - This proves the formula...

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