(i) Write down the formulae for the Kaplan-Meier estimator (1) and Nelson-Aalen estimator S(t) of survival in the presence of a stated hazard, defining all terms used. [2] The following graph shows the functions: y=1-x and y=ex over the range 0x1. Value of y 1.00 0.90 0.80 0.70 0.60 0.50 0.40 0.30 0.20 0.10 0.00 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1.00 Value of x -y-1-x -- y= exp(-x) (ii) Demonstrate that the Nelson-Aalen estimator is never lower than the Kaplan-Meier estimator. [2] A trial is conducted amongst 20 patients who have suffered from eczema but are in remission (that is, they are clear of the condition). The trial is to assess whether continuing with periodic doses of a certain steroid cream in remission reduces the rate at which eczema recurs. Patients are invited to tests every 3 months for a period of up to 5 years from when first declared to be in remission. (iii) Describe THREE types of censoring present in the investigation. [3] The data for the trial are subdivided into a group who continued to receive the steroid cream, and a control group who did not receive the steroid cream. The data for the patients in the trial showing the quarterly test at which eczema recurred, or censoring occurred, are as follows (an indicates a patient who was censored): For group receiving steroid cream: 3, 5, 6, 7, 10, 10, 12, 14*, 18, 19* For control group: 6, 8, 8, 10, 11, 12, 14, 15*, 18, 18 3. Wolfson Inc. is considering to lease a new computer sever that costs 1,800. The machine will be depreciated on a five-year MACRS schedule and will be worth nothing at the end of six years. Assume that the administrative costs are 150 per year paid by the leasing company from Year 0 to 6. The leasing payments are also made in advance for six years. The corporate tax rate is 35 percent and the cost of capital is 12 percent. Depreciation Tax Shield based on Modified Accelerated Cost Recovery System (MACRS) Year 1 2 3 4 5 6 % 20 32 19.2 11.52 11.52 5.76 1) How much should be the minimum lease amount to break-even? (12 marks) 2) Explain the main differences between operating leases and financial leases. (4 marks) 3) Explain the main differences between direct leases and leveraged leases. (3 marks) 4) What is sale and lease-back and briefly explain the advantages of sale and lease-back. (3 marks) 5) Describe the main features of big-ticket leases. (3 marks) Consider a second-hand car market. There are two types of cars, "lemons" and "peaches". A seller values a lemon at $100 and a peach at $200. A buyer values a lemon at $120 and a "peach" at $300. 1. What are the gains-to-trade per car? If the probability of a car being a lemon is, what are the expected gains-to-trade? 2. Suppose that the information is perfect and the the gains-to-trade are split equally between the buyers and the sellers for each type of car. What are the prices and is the allocation pareto-efficient? Suppose that all gains-of- trade are assigned to the buyers. What are the corresponding prices? 3. Suppose that the buyers cannot observe the quality of the car. (a) What is the expected value of a car for a buyer if of the cars are lemons? What is the maximum price a buyer is willing to pay? (b) If of the cars are lemons, does there exist a pooling equilibrium in which both types of cars are sold? What is the minimal fraction of lemons such that an equilibrium does not involve any peaches sold? (c) Suppose that the fraction of lemons in the market is higher than the threshold obtained in (b). What cars are traded in equilibrium? Is the outcome pareto-efficient? (d) Suppose that the owners of peaches can offer a warranty on the car which the owners of lemons do not offer. What trade pattern emerges in equilibrium?