This question is complete.

Consider the Ramsey model max Illo , 1 {cumin-:0 g g(ct) ( ) subject to kt+1 = I]: Gt: (2) y: = Ah??- (3) The consumer maximises lifetime utility under a feasibility con- straint, given an initial value of capital k0. In each period t. the consumer has a stock of capital kt, must decide the level of consumption ct, and accumulates capital kt\" for the following period. Condition kt+1 2 0 is captured by the function Cmazuct) = Ak? (4) Condition kt\" = 1:; is captured by function 60!) = Arc: kt. (5) The key steady state of the model is given by [1633. 633] = [(Aa, 50955)]- (5) The policy functions for consumption and next period capital are em = (1 mama (7) k?+1(kt) = WM?- (3) It holds that A > 0. [3 E (0,1) and a E (0, 1). Figure 1 shows the setting of the model graphically. You can replicate this gure in your answer sheet and use it if it helps you with the questions. c (consumption) k (capital) (iv) Calculate the Kaplan-Meier estimates of the survival function for remaining clear of eczema for: (a) the group who continued to receive the steroid cream; and (b) the control group. [8] (v) (a) Recommend, without performing any calculations, a method of establishing whether the hazard of eczema returning is statistically lower for those continuing to receive the steroid cream. (b ) Comment on the chance of being able to conclude from the trial data that continuing to receive the steroid cream reduces the risk of recurrence of eczema. [3](i) Write down the formulae for the Kaplan-Meier estimator S() and Nelson-Aalen estimator S() 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 = e * over the range 0 Ex = 1. 1.00 0.90 0.80 0.70 0.60 -y=1-x Value of y 0.50 - - y = exp(-x) 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 (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, 18Mr and Mrs Jones both wish to buy stocks in Widgets Inc. They don't have enough money right now, so they are considering buying either forwards or options on the stocks, both with a term of 4 years. The stock price at time 0 is $10 with standard deviation of 12% per annum. The stock does not pay any dividend. The continuously compounded risk-free rate of interest is 5% per annum. (1) Calculate the 4 year forward price on one stock. [1] (ii) Calculate the price at time 0 of a 4 year call option on one stock with a strike price of 12.21. [3] Mrs Jones enters into one forward contract, while Mr Jones buys one call option. At time 4 the stock is worth $12. (iii) Calculate the accumulated profit or loss at time 4 for Mrs Jones. [1] (iv) Calculate the accumulated profit or loss at time 4 for Mr Jones. [2] (v) Explain why Mr Jones makes a loss despite having an option that does not force him to buy the stock. [2] (vi) Calculate the range of stock prices at time 4 which would leave Mr Jones better off than Mrs Jones. [3]