(i) List four different situations in which pension rights can be converted to a

cash sum under a trust based defined benefit pension scheme. [2]

The trustees of the above scheme are reviewing the terms upon which a member can

convert their pension rights to a cash sum to ensure they are consistent in all

situations. In particular, they are considering whether to set the same conversion

terms for cash commutation at retirement and transfer value payments prior to

retirement.

(ii) Outline the underlying principles that the trustees should consider when

determining the approach to adopt for:

(a) cash commutation factors.

(b) transfer values. [16]

(iii) Discuss the implications for the scheme and the member if the trustees decide

to set the same conversion terms for cash commutation and transfer values.

[7]

2. Describe the empirical evidence relating to the continuous-time lognormal model for

security prices.

3

(i) State the expected utility theorem. [2] A risk averse investor makes decisions using a quadratic utility function: U( w) = w+ dw2. (ii) Derive an upper bound for d for this investor. [2] (iii) Explain why the investor can only use this utility function to make decisions over a limited range of wealth, w. Your answer should include a statement of this range. [2] The investor states that the upper limit of wealth where she can use this utility function is w = $1,000. (iv) Determine the value of d in the investor's utility function. [1] The investor wins a prize of $250 in a gameshow. She is then offered the opportunity to exchange this prize for a larger prize of $600 if she can answer one more question correctly. However, she will receive no prize at all if she gets the question wrong. She estimates her chances of answering the question correctly to be 50%. (v) Determine whether the investor should take this opportunity to exchange. [3]Claim amounts X, from a portfolio of insurance policies follow a gamma distribution with parameters & and 2. Each 2, also follows a gamma distribution with parameters a and B. (i) Show that the mixture distribution of losses is a generalised Pareto, with parameters o, B, k. [4] Claim amounts are now assumed to be exponentially distributed with parameter 2, (ii) Show, using your answer to part (i), that the mixture distribution of losses is now a standard Pareto distribution with parameters o, B. [2]The number of claims on a portfolio of insurance policies in a given year follows a Poisson distribution with unknown mean 2. Prior beliefs about 2 are specified by a gamma distribution with mean 60 and variance 360. Over a period of three and one- third years, the total number of claims is 200. (i) Calculate the Bayesian estimate of 2 under all-or-nothing loss. [7 (ii) Comment on your result for part (i). 1(i) Show that the following discrete distribution belongs to the exponential family of distributions. f()= (1-[)"-my y=0, [4] (ii) Derive expressions for the mean and variance of the distribution, E(y) and Var(y), using your answer to part (i). [4]