(10 points) CTI is a life insurance company that plans to issue a new, high volume, low sum insured term insurance policy through direct marketing. Its current business is predominantly traditionally-brokered term insurance. CTI assumes the following for calculating premiums and reserves.

(i) For brokered policies, mortality follows the Standard Ultimate Life Table, and expenses are 8% of each premium.

(ii) For direct-marketed policies, mortality follows the Standard Ultimate Life Table with a 5-year addition to the policyholder's age. Expenses are 3% of each premium.

(iii) i = 0.05 (iv) Premiums are calculated using the equivalence principle.

(a) (2 points) Explain why the mortality and expense assumptions are different for the direct-marketed policies compared to the brokered policies.

Consider a fully discrete direct-marketed 10-year term insurance policy of 50,000 issued to (40), with premium P.

(b) (1 point) Let K* denote the curtate future lifetime of (40). Write down the gross loss-at-issue random variable in terms of K*, P, and interest rate functions.

(c) (1 point) Show that P = 57 to the nearest 1. You should calculate your answer to the nearest 0.01

(d) (3 points) (i) Show that the standard deviation of the present value of the benefit is 4088 to the nearest 1. You should calculate the value to the nearest 0.01.

(ii) The standard deviation of the gross loss-at-issue is 4088.5. Without further calculation, explain why the standard deviation of the gross loss-at-issue is very close to the standard deviation of the benefit.

(e) (2 points) CTI would like to offer a revised, direct-marketed 10-year term policy, where the annual premium in the first five years is of the annual premium in the last five years. Show that the initial annual premium for a 50,000 policy issued to (40) is 40 to the nearest 10. You should calculate the value to the nearest 0

(f) (1 point) CTI issues 5-year term insurance policies of 50,000 by direct marketing. At most ages the premium for this policy is greater than the initial premium for the policy described in part (e). Therefore, the new policy is expected to be very popular. Explain why CTI might not launch this product despite its anticipated popularity.

I only need the solution to part e (in bold and underlined)