...Solve

1. The manager of a life insurance company wishes to revise the premiums for termassurance policies. He has asked a trainee to compare the latest mortality estimatesfrom the Continuous Mortality Investigation (CMI) for ages 40-64 years inclusivewith the estimates the company has been using in its premium calculations, using a95% significance level.The trainee says: "I've done the Signs Test and we just pass - one more positive sign

and we would have failed!".

(i) Calculate the number of ages for which the company's mortality estimate washigher than that produced by the CMI. [3]

Ten minutes later the trainee says: " I tried the Grouping of Signs test and we just fail. We needed one more positive run!".

(ii) Determine the number of runs of positive signs in the company's data. [3]

2. (i) Derive the median of a Pareto distribution with parametersand . [3]

Let gamma = 2 andsigma = 3.

(ii) Comment on the skewness of this Pareto distribution

3.

Consider a three-period binomial tree model for the stock price process S, Let So = 100 and let the price rise by 10% or fall by 5% at each time step. Assume also that the risk-free rate is 4% per time period, continuously compounded. (i) (a) State the conditions under which the market is arbitrage free. (b) Verify that there is no arbitrage in the given market. [2] (ii) Calculate the price of a European call option on this stock, with maturity at the end of the third period and a strike price of 103. [4] A special option, called a European "Paylater" call option, has the following payoff at maturity T: (ST - K-c) if S, > K and zero otherwise. K is the strike price and c is the premium paid for the option. The premium is paid at maturity, and is only paid if the option expires in-the-money. Further, the option premium is set such that the value of the option at time / = 0 is zero. Assume that K = 103 and the maturity of the contract is at time / = 3. (iii) Determine the premium c of this contract. [31(i) Define the following types of stochastic process: (a) a Poisson process (b) a compound Poisson process [3] Consider the modelling of the following situations: A the number of claims for motorcycle accidents received by an insurer's telephone claim line B the number of breakfast bagels sold by a New York bagel bar C the number of breakdowns of freezers in a large supermarket D the cost of wasted food caused by breakdowns of freezers in a large supermarket. (ii) Comment on which of the following stochastic processes will be most suitable for modelling each of the four situations above: time-homogeneous Poisson process time-inhomogeneous Poisson process time-homogeneous compound Poisson process time-inhomogeneous compound Poisson process