讲解 AMA533 Life Contingencies Assignment 1讲解 R程序

Department of Applied Mathematics

AMA533 Life Contingencies

Assignment 1

Due time: 23:00, March 7, 2025.

1. (20 pts) Verify by computations that the following recursive equations hold:

2. (10 pts) We know that q50 = 0.04, q51 = 0.06 and q52 = 0.07. Under the UDD assumption within each year, compute the value

3. (10 pts) The lifetime distribution is assumed to follow UDD within each year starting from birth. We also know that the force of mortality satisfies that µ60.5 = 0.032, µ61.5 = 0.054 and µ62.5 = 0.078. Compute 2 60 q .5 .

4. (10 pts) The force of mortality for a survival model is given by Compute the values 20|10q50 , 20q50.5 and ˚e50.

5. (10 pts) Under the UDD assumption within each year, verify that and hold.

6. (10 pts) For a given individual (x), let us consider a policy payable at the end of the year of death with a death benefit of 1 in the first year and an unspecified death benefit in the following years. Under qx = 0.06 and i = 0.1, and some given mortality probabilities at age x + 1 and beyond, the APV of the insurance policy is 0.42. If qx is in fact 0.03 and all other mortality probabilities at age x+ 1 and later remain the same, what is the new value of APV?

7. (10 pts) The insurance company has a group of policyholders all at the age of x, 70% of whom are non-smokers and 30% of whom are smokers. For the fixed age x, the insurance company’s model for mortality has (non-smoker) and (smoker).

(i) A policyholder is chosen at random from the group. Compute 1| q x and qx+1 for this policyholder.

(ii) Suppose that both non-smoker and smoker mortality follow UDD in each year of age.

Compute the value µx+0.2.

8. (10 pts) For a given individual (x), let Z1 be the PVRV for a policy issued to (x) with the death benefit of 1 that is payable at the end of 20 years if (x) dies within 20 years. Let Z2 be the PVRV for a policy issued to (x) with the death benefit of 1 that is payable at the end of 30 years if (x) dies between 10 and 30 years from the issue date. We know that Cov[Z1, Z2] = 0, 10qx = 0.12 and 20qx = 0.38. Compute the value of 30qx.