Microeconomics

In a medical study of survival rates for patients who have received a new type of

treatment, observations have only been recorded weekly. The resulting data show that 5

lives died and 2 lives were censored in week 10. State the convention that is usually

applied when calculating the product-limit estimates when such coincident events are

present

William, aged 75, and Laura, aged 80, are the guardians of a child. They take out a life assurance policy that provides a payment of f25,000 immediately when the second of them dies. Level annual premiums are payable in advance whilst the policy is in force. (i) Calculate the annual gross premium, using the basis given below. [4] (ii) Calculate the gross premium prospective reserve just before the sixth premium is paid, using the basis given below, assuming that both William and Laura are still alive at that time. [3] [Total 7] Basis: Mortality: PMA92C20 for William, PFA92C20 for Laura Interest: 4% pa effective Expenses: Initial: E250 Renewal: 5% of each premium, excluding the first A life insurance company issues an annuity to a male, aged 68, and a female, aged 65. The annuity of f10,000 pa is payable annually in arrears and continues until both lives have died. The insurance company values this benefit using PMA92C20 mortality for the male life, PFA92 C20 mortality for the female life and 4% pa interest. (i) Calculate the expected present value of this annuity. [2] (ii) Derive an expression for the variance of the present value of this annuity in terms of appropriate single life and joint life assurance functions. [4] Let X be the present value of the insurer's profit from this policy. (iii ) If the insurance company charges a premium of $150,000 for this policy, calculate P(X > 0) . [4] [Total 10] A man aged 60 exact and a woman aged 65 exact wish to purchase an annuity that provides: 625,000 pa payable while they are both alive, E20,000 pa payable for the remainder of the woman's life, if the man dies first, E15,000 pa payable for the remainder of the man's life, if the woman dies first. All the annuity payments are made annually in arrears. (i) Write down an expression for the present value random variable of this benefit. [2] (ii) Calculate the expected present value of this annuity benefit, using the following basis: Mortality: PMA92C20 for the male life, PFA92C20 for the female life Interest: 4% pa effective [2] [Total 4]