Question 2 - correlation and diversification

You have 3 assets A, B, C in a world with two future states, High and Low. The probability that the state High occurs is 0.5. These assets have the following net return structure:

Low High

A 0.02 0.02

B 0 0.1

C-0.2 0.2

1. Compute the covariance of the returns for each pair of these assets
2. Compute the correlation coefficient of the returns for each pair of these assets
3. Assume that correlation of returns of assets B and C is 1. Assume that you can only buy assets (shares of both assets must be non-negative).
4. Derive the formula for portfolio variance when the share of asset B is denoted by w.
5. Can you construct a portfolio of assets B and C that has a variance of zero? (Hint for b and c: recall how the efficient frontier for such assets looks likes)
6. How big should be the shares of both assets in your portfolio if you would like to achieve the highest possible variance?
7. Assume that correlation of returns of assets B and C is -1 (e.g., B pays now 0.1 in Low and 0 in High). Assume that you can only buy assets (shares of both assets must be non-negative).
8. Derive the formula for portfolio variance when the share of asset B is denoted by w.
9. Can you construct a portfolio of assets B and C that has a variance of zero? (Hint for b and c: recall how the efficient frontier for such assets looks likes)
10. How big should be the shares of both assets in your portfolio if you would like to achieve the highest possible variance?