Consider the following two assets. Returns on asset 1 has a mean of µ1 and standard deviation of σ1. Returns on asset 2 has a mean of µ2 and standard deviation of σ2. The correlation coefficient ρ1,2 measures how the two assets’ returns are correlated, and it takes on values between -1 and +1. An investor puts W1 fraction of her wealth into stock 1, and W2 = 1 − W1 fraction of her wealth into stock 2.
1. Using the equation on the standard deviation of portfolio returns, argue that the portfolio risk is increasing in ρ1,2.
2. When ρ1,2 = 1, argue using mathematical formulas that the portfolio standard deviation is equal to the weighted average of the standard deviation of the individual stocks in the portfolio. (Hint, Square of Summation).
3. Now suppose ρ1,2 = −1. If the investor wants to minimize her risk in investing in this portfolio, how should she choose W1 and W2 = 1 − W1? (Hint, Square of Difference).