DEPARTMENT OF ECONOMICS
ECON0001
ECONOMICS OF FINANCIAL MARKETS
Formative assessment n.1 on the first part of the unit
Section A: investment under risk or uncertainty (30 marks)
1. [Total 9 points] Consider the decision problem of investing an amount of wealth W = $1000000 into a risky asset with return
and into a risk-less asset with risk-free interest rate r = 5%. You are a risk averse investor with a CRRA utility function
where ρ = 0.5, and WT is the amount of wealth at the end of the investment.
(a) Find the optimal allocation in risky and risk-less assets as a function of the probability p and the initial amount of wealth W invested. How does it depend on p? Comment on your result.
(b) Compute the optimal allocation for a probability p = 50% and for p = 51%. What is the optimal allocation for p = 1? Comment.
(c) What is the lower bound of the probability p for investing a positive share into the risky asset? Answer with and without using the utility function, and comment. What is this lower bound of probability for a risk neutral investor? (d) Find the lower bound of the probability p for starting to borrow money at the risk-less interest rate and invest an amount larger than W in the risky asset.
2. [Total 7 points] Consider an asset with a risky rate of return over a time period T expressed by
The investment of an amount of wealth W returns WT at the end of the period T, and investors have utility function U(WT ) = ln(WT ).
(a) Why this asset and this utility imply a positive risk premium?
(b) Find the Certainty Equivalent of the investment for such utility as a function of the probability p of the higher return value and the invested wealth W. How does it depend on p? Comment on your results.
(c) Compute Certainty Equivalent and Risk Premium for a probability p = 50%.
(d) What is the risk premium for p = 0 and for p = 1? Comment on your results.
(e) How does the risk premium depend on the probability p? Proof that the risk premium is anon-monotonic function of the probability p.
3. [Total 8 points] An investor with utility function
faces the decision problem of allocating her wealth x in the following assets: a risky asset that gives a return of 20% or a loss of -5% with equal probability, and a risk-less asset with interest rate equal to 5%.
(a) How does her relative risk aversion depend on wealth x, if σ = 10-6 $-1 ? If her wealth increases from £1 Mln to £10 Mln, how does relative risk aversion changes? And absolute risk aversion?
(b) Find the optimal investment in risky and risk-less asset as a function of σ and as a function of wealth x, and comment.
(c) Study the optimal allocation for σ ! 1 and σ ! 0. What is the condi- tion on σ for investing 100% in the risk-less asset? Justify your answer both conceptually and mathematically.
(d) If she invests a total wealth of £1 Mln, what is the condition on risk aversion σ for the optimal allocation to be 100% invested in the risky asset? Comment on your answer.
(e) Evaluate the marginal effect of σ on the optimal share of investment in the risky asset.
4. [Total 6 points] Consider this situation of choice under uncertainty: an urn contains 1500 balls. 500 balls are green, while the remaining balls are either blue or yellow. Which gamble would you choose between (1) and (2)?
(1) If the ball is green you get £500.
(2) If the ball is blue you get £500.
Which gamble would you choose between (3) and (4)?
(3) If the ball is yellow or green you get £500.
(4) If the ball is blue or yellow you get £500.
Experimental evidence shows people tend to prefer gamble (1) over gamble (2), and gamble (4) over gamble (3). Explain the rational behind such choices, and why they are not consistent with expected utility theory.