You can invest in three stocks that A, B and C where the standard deviation of returns are: A = 0.15, B .20, C .25 and A,B = A,C = B,C 1.

You can invest in three stocks that A, B and C where the standard deviation of returns are: σA = 0.15, σB .20, σC .25 and σA,B=σ A,C=σ B,C

1. What’s the minimum variance portfolio and what’s its variance and covariance with each of the three assets?

2. Redo part a) for arbitrary standard deviations σA , σB , σC

3. Prove the covariance between the minimum variance portfolio and any other portfolio is the variance of the minimum variance portfolio.  Show this by contradiction. (Assume there is an asset for which the above is not true and consider a portfolio that consists of x in the minimum variance portfolio and 1-x in the other portfolio)