Question 1:

Sally is an investor who faces the possibility of investing in two projects, A and B. Her utility function is  U(x) = 1 − e− , where x is measured in dollars and her current wealth is 200 dollars. The expected profit (in dollars) of the two projects depends on the outcome of 2 independent uncertain events. If she invests in Project A she is expected to make a profit of

300 with a probability of 0.5, a profit of 500 with a probability of 0.25 and a profit of 800 with a probability of 0.25. Investing in Product B she is expected to make a profit of 200 with a probability of 0.4 and a profit of 1000 with a probability of 0.6.

a.   Draw a decision tree describing her final wealth level.

b.   What is the optimal policy if Sally’s goal is to maximise her utility. Calculate her Certainty equivalent and Risk Premium for this policy.

c.   Calculate the value of perfect information with respect to project B. Give a value in terms of utility and then translate this to an increase in her dollar value.

Question 2:

The NSW health authority is considering purchasing vaccines for swine flu. A local firm is offering to sell a newly developed vaccine, but the price is dependent on a court ruling. The price is expected to be 10 dollars or 20 dollars with equal probabilities. A Mexican firm is offering their vaccine for a fixed price, but due to exchange rate uncertainty, the expected price will be14 dollars with a probability of 0.6 and 17 dollars with a probability of 0.4.

a.   Describe the decision faced by the NSW health authority using a decision tree. What is the optimal policy based on EMV?

b.   What is the expected value of perfect information with respect to the Mexican vaccine price?

c.   An expert offers his consulting services with respect to the court ruling. His track record  is very impressive; he has predicted the  correct  outcome  of 80%  of court

hearings  he  has  attended  (both  successful  and  unsuccessful  ones).  What  is  the maximum amount you would be willing to pay the expert?

Question 3:

Jack is a parking ranger in the Sydney CBD. Based on his experience 20% of drivers don’t purchase a parking ticket and should be fined. Jack has just started his shift and many        interesting questions are arousing his curiosity.

a.   After inspecting 10 random cars, what is the probability that he will give at most one fine?

b.   What is the probability that he will hand out his third fine to the 9th car he inspects?

c.   Jack has observed that on average there are 2 car accidents on average every hour        during the morning. The rate of occurrences seems to follow a random process and the accidents are independent. What is the probability of no accidents occurring between 9 am and 10 am on a given day? What is the probability that exactly 3 accidents occur    between 10 am and 12 pm on a given day?

d.   What is the average time interval between two consecutive accidents on a given day?   What is the probability that the time between two consecutive accidents is smaller than this average?