Solve the following questions. Explain them fully.

A life insurance company sells 5-year-term, single-premium, unit-linked bonds each for a premium of $10,000. There is no bid/offer spread and the allocation percentage is 100%. (i) Assuming that the only charge is a 2% annual management charge and assuming unit growth of 9% pa, calculate the unit reserve at the start and end of each year and the management charge each year. [3] (ii) Calculate the net present value of the contract assuming: Commission of 5% of the premium Initial expenses of $50 Annual renewal expenses of $20 in the Ist year, inflating at 5% pa. Independent rate of mortality is a constant 0.5% Independent rate of surrender is 5% pa Non unit fund interest rate is 9% pa Risk discount rate 12% pa The company holds unit reserves equal to the full value of the units and zero non-unit reserves. You may assume that expenses are incurred at the start of the year and that death and surrender payments are made at the end of the year. [4] [Total 7]Claims on a portfolio of policies occur according to a Poisson process with a mean rate of 5 claims per day. Claim amounts are 10, 20 or 30. 20% of claims are of amount 10, 70% are of amount 20 and 10% are of amount 30. (i) Calculate the expected waiting time until the first claim of amount 30. [1] (ii) Calculate the probability that there are at least 10 claims during the first 2 days, given that there were exactly 6 claims during the first day. [2] (iii) Calculate the probability that there are at least 2 claims of amount 20 during the first day and at least 3 claims of amount 20 during the first 2 days. [3] (iv) Calculate the conditional variance of the number of claims during the first day, given that there are 2 claims of amount 10 during the first day. [3] [Total 9]