Problem 1.
1
An individual with a utility function U(c) = -exp (-Ac) (where A = 30 ln 4) and an initial wealth of
$50 must choose a portfolio of two assets. Each asset has a price of $50. The first asset is riskless and pays off $50 next period in each of the two possible states. The risky asset pays off zs in state s = 1, 2. Suppose also that the individual cares only about next period consumption (denoted by c1 or c2 depending on the state). The probability of state 1 is denoted by π.
(a) If the individual splits his wealth equally between the two assets, fill in the following table assuming that each of three scenarios is considered.
Scenario
1
2
3
(z1 , z2 )
(20,80)
(38,98)
(30,90)
π
1/5
1/2
1/3
(c1 , c2 )
E[c]
Var[c]
(b) How would this individual rank these three scenarios? Explain and give reasons to support your argument.
(c) Show that under each scenario the individual’s optimal decision is to invest an equal amount on each of the two assets.
Problem 2.
Consider an agent with a well-behaved utility function who must balance his portfolio between a riskless asset and a risky asset. The first asset, with price p1 has the certain payoff z1 while the second asset, with price p2 pays off z2 which is a random variable. The agent has an initial wealth
Y0 and he only cares about next period. Assuming that he holds x1 units of the riskless asset and x2 units of the risky asset;
(a) Write down his expected utility function in terms of Y0 , x1 , x2 , z1 and z2 . Write down his budget constraint as well. Find the equation (the FOC) that the optimal demand for the risky asset has to satisfy.
(b) Assume now that his utility function is of the form: U (Y ) = a − be−AY with a, b
