Dani has to decide how to combine three assets whose returns are denoted X1, X2 and X3. Each of these assets yields 24 with probability 0.5 and 12 with probability 0.5. The random returns are independent. Dani is risk-averse and his utility function is u(x)=150 (×-10)-1/3, where x is the monetary payoff.

a. Dani thinks first about combining X1 and X2. Let Y=0.5 X1 + 0.5 X2.

Compute the expected returns and expected utility of X and Y.

Does Dani have a preference between X and Y? Explain.

b. Dani considers another combination. Let Z = 0.25 X1 + 0.75 X2.

Before any computation, explain why Dani would prefer Y over Z. Then, show the computation.

c. X1 and X2 are uncorrelated. If Dani had the choice, would he prefer X1 and X2 to be correlated or not?

Explain.

d. Dani now thinks about combining X1, X2 and X3.

What would be the weights of each asset in an optimal portfolio?

Explain the intuition without computing.

e. Compute the expected utility of such an optimal portfolio with X1, X2 and X3. Conclude.

How would the answers to the previous questions be different if Dani were risk-neutral?