Question 1 microeconomics

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.

(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.

(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.

(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.

Calculate all2) `70/60 On the following basis: Mortality: 70-year-old: PMA92C20 60-year-old: PFA92C20 Interest: 4% pa Two lives, aged 40 and 44, purchase a policy from an insurance company that pays 50,000 in 20 years' time, if at least one of them is still alive at that time. Assuming that the mortality of each life follows the AM92 Select table, and the annual effective interest rate is 6%, calculate the expected present value of the benefits from this policy. Two lives aged x and y take out a policy that will pay f15,000 immediately on the death of (x) provided that (y) has died at least 5 years earlier and no more than 15 years earlier. (i) Express the present value of this benefit in terms of the random variables denoting the future lifetimes of (x) and (y). [2] (ii) Write down an integral expression (in terms of single integrals only) for the expected present value of the benefit. [3] (iii) Prove that the expected present value is equal to: 15,000 5 5Px Au+sy -v15 15Px At+15:] [3] (iv) Describe the appropriate premium payment term for this policy, assuming premiums are to be paid annually in advance. [2] [Total 10] A 65-year-old male and a 62-year-old female take out a joint whole life policy with a sum assured of f10,000 that is payable immediately on the first death. Premiums are payable monthly in advance while the policy is in force for at most 5 years. (i) Show that the monthly premium is f100, using the basis given below. [6] (ii) Calculate the prospective reserve at the end of the third policy year, using the basis given below. [4] Basis: Mortality: PMA92C20 for the male life, PFA92C20 for the female life Interest: 4% pa effective Expenses: None