An insurer issues a 20-year term insurance policy to (35). The sum insured of $100 000 is payable at the end of the year of death, and premiums are paid annually throughout the term of the contract. The basis for calculating premiums and policy values is:

i. Survival model: Standard Select Survival Model

ii. Interest: 5% per year effective

iii. Expenses: Initial: $200 plus 15% of the first premium

Renewal: 4% of each premium after the first

Questions: a. (10 points) Show that the premium is $91.37 per year.

b. (5 points) Show that the policy value immediately after the first premium payment is 0+ V = $122.33

c. (5 points) Explain briefly why the policy value in (b) is negative.

d. (10 points) Calculate the policy values at each year end for the contract, just before and just after the premium and related expenses incurred at that time, and plot them on a graph. Explain the form of the graph. At what duration does the policy value first become strictly positive?

e. (10 points) Suppose now that the insurer issues a large number, N say, of identical contracts to independent lives, all aged 35 and all with sum insured $100 000. Show that if the experience exactly matches the premium/policy value basis, then the accumulated value at time 15 of all premiums less claims and expenses paid out up to time 15, expressed per surviving policyholder, is exactly equal to the policy value at time 15.