...Assignment needed soon 1. The advisor to a very large defined benefit pension scheme is about to analyse the scheme's post retirement mortality experience. (i)
...Assignment needed soon
1. The advisor to a very large defined benefit pension scheme is about to analyse the
scheme's post retirement mortality experience.
(i) Describe, with reference to the Actuarial Control Cycle, how this analysis
would be performed. [7]
(ii) Discuss the limitations and any adjustments that should be considered when
using the results of the analysis. [5]
2.
Consider a binomial tree model for the stock price S,. Let So = 50 and let the price rise by 10% or fall by 5% each month for the next three months. Assume also that the risk-free rate is 5% per annum continuously compounded. (i) State the conditions under which the market is arbitrage free. [2] (ii) Calculate the price at time / = 0 of a European call option on this stock, which expires in three months and is struck at-the-money (i.e. strike price K = 50). [4] A special option, called a knock-out barrier option, goes out of existence (i.e. expires without any payoff or value) if the underlying asset reaches a pre-specified barrier b > 0 either from above (down-and-out) or from below (up-and-out). The down-and-out call has the following payoff at time T: max(ST - K,0) if min S, 2b, OSIST 0 otherwise. Assume this special option is written on the given stock, has the same strike price and maturity as the European call option described in part (ii) and the barrier b is fixed at 48. (iii) Calculate the price of this contract using the binomial tree model and risk- neutral valuation. [3] (iv) Determine the price of the down-and-out contract when b = 40, without performing any further calculations. [21In order to model the seasonality of a particular data set an actuary is asked to consider the following model: (1-B 2)(1-(a +B)B +aBB? )X, = &, where B is the backshift operator and &, is a white noise process with variance o-. The actuary intends to apply a seasonal difference V, X, = Y,- (i) Explain why s should be 12 in this case (i.e. Y, = X - X1-12). [1] (ii) Determine the range of values for o and B for which the process will be stationary after applying this seasonal difference. [3] Assume that after the appropriate seasonal differencing the following sample autocorrelation values for observations of Y, are p, =0 and p2 =0.09. (iii) Estimate the parameters o and B. [5] The actuary observes a sequence of observations X1, X2, ..., Xy of X,, with > 12. (iv) Derive the next two forecasted values for next two observations x7+, and Xia, as a function of the existing observations. [4]
