Financial economics

Please see attached file..

You have 1 dollar of initial wealth and would like to invest in some assets to consume tomorrow. In addition, you have the following utility function from consumption tomorrow, U(Y₁)=ln(Y₁).

1. Assume that you have only two assets to choose from, a risky one and risk-free one. The probability of the High state is π, the net return on the risk-free asset is r_f and the risky asset has a net return of r_H in the High state and r_L in the Low state. Find the optimal amount of dollars a you would invest in the risky asset. Provide derivations (8 points)
2. How does the sign of the optimal amount of dollars invested in the risky asset depend on the expected returns of the risky and risk free assets? Why? (6 points)
3. Does the optimal amount of dollars invested in the risky asset depend on the initial wealth? (3 points)
4. Does the optimal share of the risky asset in your portfolio allocation depend on the wealth? Can you justify your finding? (3 points)
5. Using the result in 1, plot the optimal amount of dollars invested in the risky asset as a function of the probability of High state for two cases: when the investor chooses only between asset A and B, and when she chooses between A and C. You can use Excel. (6 points)
6. Compare the optimal allocation towards asset B and C from question 5 and explain the differences. (4 points)