3. (AMLCR) 6.16. A life insurance company sells annuities to men aged exactly 60. Each policyholder pays a single net premium, P, and then receives an annuity of $30,000 a year in arrear (so that the...

Extracted text: 3. (AMLCR) 6.16. A life insurance company sells annuities to men aged exactly 60. Each policyholder pays a single net premium, P, and then receives an annuity of $30,000 a year in arrear (so that the first annuity payment is on the 61st birthday). Assume that mortality follows the Standard Select Life Table, and that the interest rate is 5% per year. (a) Calculate P. (b) Calculate the probability that the present value of profit on a single policy is positive. (c) Calculate the standard deviation of the present value of profit on a single policy. (d) Now suppose that the office sells 1,000 such annuities simultaneously to independent lives. Calculate the value of P such that the probability that the present value of the profit to the insurance company is positive is 95%.