Consider the following 3-state model for critical illness insurance:

A. Calculate the premium rate for the death benefit for the policy.

B. Calculate the additional premium rate for the rider.

C.

Compute the reserve, for the death benefit, for the policy after ten years, assuming

(i) the policyholder is still healthy;

(ii) the policyholder is already critically ill.

D. Let tV (0) be the time-t reserve for the death benefit for the policy. Write down the

differential equation for tV (0) and compute d/dt tV (0) for t = 10.

0. Healthy 1. Critically ill 2. Dead A 20-year term insurance with an optional critical illness rider to a life age 53 is valued using the model. The contractual details are as follows: Death benefit is 100,000 immediately on death The rider offers a benefit of 30,000 immediately on diagnosis of critical illness Premiums are payable continuously while the policyholder is healthy. The premium and the net premium reserve basis are as follows: Mortality and illness: Mm1 = 4 x 10-4 +3.5 x 10-6 x 1.15, M 2 = 5 x 10-4 = %+8 x 10-5 x 1.054, 12 ,01 , = = 2 uit = 0.05 Interest: i = 6% An actuary has computed the following probabilities and actuarial functions: 01 02 ,00 tP53 tP53 053+t ASI A: t 0 10 20 0 0.099359 0.337623 0 0.038544 0.159902 10.942138 7.780714 4.379097 53+t 53+t 0.338538 0.177501 0.524398 0.262689 0.727951 0.350550 It is further given that 10P63 = 0.582853 and 10P63 = 0.321726. 0. Healthy 1. Critically ill 2. Dead A 20-year term insurance with an optional critical illness rider to a life age 53 is valued using the model. The contractual details are as follows: Death benefit is 100,000 immediately on death The rider offers a benefit of 30,000 immediately on diagnosis of critical illness Premiums are payable continuously while the policyholder is healthy. The premium and the net premium reserve basis are as follows: Mortality and illness: Mm1 = 4 x 10-4 +3.5 x 10-6 x 1.15, M 2 = 5 x 10-4 = %+8 x 10-5 x 1.054, 12 ,01 , = = 2 uit = 0.05 Interest: i = 6% An actuary has computed the following probabilities and actuarial functions: 01 02 ,00 tP53 tP53 053+t ASI A: t 0 10 20 0 0.099359 0.337623 0 0.038544 0.159902 10.942138 7.780714 4.379097 53+t 53+t 0.338538 0.177501 0.524398 0.262689 0.727951 0.350550 It is further given that 10P63 = 0.582853 and 10P63 = 0.321726