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(i) Express fully in words xval [3] (ii) Express a- as the expected value of random variables and hence show that 1- A [4] [Total 7] A 20-year special endowment assurance policy is issued to a group of lives aged 45 exact Each policy provides a sum assured of f.10,000 payable at the end of the year of death or [20,000 payable if the life survives until the maturity date. Premiums on the policy are payable annually in advance for 15 years or until earlier death. You are given the following information: Number of deaths during the 13" policy year Number of policies in force at the end of the 13" policy year 195 (i) Calculate the profit or loss arising from mortality in the 13" policy year. [7] (ii) Comment on your results. [2] Basis: Mortality AM92 Ultimate Interest 4% per annum Expenses none [Total 9] An employer wishes to introduce a lump-sum retirement benefit payable immediately on retirement at 65 or carlier other than on the grounds of ill-health. The amount of the benefit is E1,000 for each year of an employee's service, with proportionate parts of a year counting. (i) Give a formula to value this benefit for an employee currently aged x with n years of past service, defining all terms used. [5] (ii) Using the Pension Scheme Tables from the Actuarial Formulae and Tables, calculate the value for an employee cunendy aged 30 exact with exactly 10 years past service. [2] (mi) Calculate the level annual contribution payable continuously throughout this employee's service to fund the future retirement benefit. [3] [Total 10](i) Define the following terms without giving detailed formulae: (@) Crude Mortality Rate (b) Directly Standardised Mortality Rate (c) Indirectly Standardised Mortality Rate [3] (ii) The data in the following table are taken from data published by the Office of National Statistics in 2001. England and Wales Tyne and Wear Population Number of Population Number of births births Under 25 3.149.000 153.000 71.000 4.000 25-35 3.769.000 339.000 74.000 6.000 35+ 3.927.000 103,000 82.000 1.000 (a) Using the population for England and Wales as the standard population calculate crude birth rates and the directly and indirectly standardised birth rates for Tyne and Wear. (b) State an advantage of using the Indirectly Standardised Birth Rate and comment briefly on the answers you have obtained. [8] [Total 1 1]