Answer these questions clearly explaining your responses.

Prove each of the following results: (i) The probability that exactly x decrements will occur in a population consisting initially of n individuals subject to a single decrement with rate q per annum is: x 9' (1- 9 ) "- [3] (ii) The maximum likelihood estimator of the parameter q for the binomial model equals the number of decrements divided by the initial population. [4] (iii) The maximum likelihood estimator of the parameter / for the Poisson model is an unbiased estimator of the force of decrement. [3] [Total 10]The table below gives an extract of the data of a mortality investigation, which examines the mortality of lives aged between 70 and 71. For each life, you are given the age at which they were first observed, the age at which they ceased to be observed and their reason for leaving the investigation. Life Age at entry Age at exit Reason for leaving 1 70 years 0 months 71 years 0 months censored 2 70 years 0 months 70 years 1 1 months died W 70 years 0 months 70 years 8 months censored 70 years 1 month 70 years 3 months died 70 years 2 months 71 years 0 months censored 6 70 years 2 months 70 years 10 months censored 70 years 6 months 70 years 10 months died 70 years 6 months 70 years 8 months censored (i) Calculate the Kaplan-Meier estimate of the survival function ST (1) = P(770 > t) and sketch a graph of this function. [5] (ii) Construct a 90% confidence interval for 970 based on your estimate in part (i). [3] (iii) Calculate the total central exposed to risk and the total initial exposed to risk at age 70 last birthday for the group of lives given in the table above. [3] (iv) Calculate the actuarial estimate of 970 - [1] (v) Calculate the Poisson estimate of 970 and construct a 90% confidence interval for 970 based on this estimate. [5] (vi) Comment on the differences between the estimates of 970 calculated in (i), (iv) and (v). [4] [Total 21]16 Explain how the tax system might actively encourage taxpayers to make provision for their retirement. [5] 17 Three consultant actuaries have been in business, operating as a partnership, for several years. They are considering borrowing a substantial sum in order to expand and are considering incorporating their business as a limited company, with themselves as both shareholders and directors. Explain the advantages of a limited company compared with a partnership for the three partners. [5] 18 Agency theory suggests that there is a great deal of conflict between the interests of various stakeholders. This has led to a set of observable and significant costs associated with monitoring and protecting individual interests. Arguably, all stakeholders would benefit if they could agree to work together in the best interests of the entity, without incurring these agency-based costs. Explain why it would be difficult, if not impossible, to eliminate these agency costs. [5]