A 20-year-old purchases a 100,000 whole life policy with benefit payable at the moment of death. Given d = .05 and fix = .02 for all X, find each of the following:

(a) The net single premium for this policy.

(b) A number X such that the insurance company is 80% certain that the value at time of purchase of their eventual payout will be less than or equal to X. (Such a number X is sometimes called the 80^^ percentile of the present value of the benefit.)

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(a) Assuming uniform distribution of deaths throughout the year of age jc, show that ^ ^^ — i '^'tj-

(b) Extend the result of part

(a) to show that Ax = t A^^jf-