Question 1

Davina has $1 she can invest in two assets, A and B. A dollar invested in A has a 50-50 chance of returning 16 or 0 and a dollar invested in B has a 50-50 chance of returning 9 or 0. Davina's utility over wealth is given by the function U(w) = ln(w + 1).

(1) Suppose the assets returns are independent.

(a) Despite the fact that A has a much higher expected return than B, show that Davina would prefer to invest half of her money in B rather than investing everything in A.

(b) Let a be the fraction of the dollar she invests in A: What is the optimal value of a? (At least get the FOC).

(2) Suppose the assets returns are perfectly negatively correlated. That is, when A has a positive return, B returns nothing, and vice versa.

(a) Show that Maria is better off investing half her money in each asset now than when the assets returns were independent.

(b) If she can choose how much to invest in each, how much does she invest in A? (At least get the FOC).