SOA Exam ALTAM Fall 2024, Question 5
Study guide: Exam ALTAM
For a Type B universal life policy with annual premiums, you are given the following information:
(i) The Additional Death Benefit is 100,000.
(ii) The minimum annual premium is 2,000.
(iii) The death benefit is paid immediately on death.
(iv) The expense charge is 100 plus 2% of the premium.
(v) The CoI is calculated based on 120% of the Standard Ultimate Mortality Model.
(vi) CoIs are not discounted (that is, i_q = 0).
(vii) The annual effective credited interest rate of i_c = 0.06 is credited continuously to account values.
(viii) There is no corridor factor.
A life age 50 purchases this policy. She pays annual premiums of 2,000 throughout the term of the policy.
Part a (3 points)
(i) Show that the Account Value at the end of the second year is 3730 to the nearest 10. You should calculate the value to the nearest 0.1.
(ii) Roughly sketch a graph of the account values, AV_t, as a function of t, for 0 \le t < 2. You should identify key values on your sketch.
Part b (2 points)
Show that the youngest age at which the deductions (cost of insurance and expenses) are greater than the premium for this policyholder is age 74.
Part c (3 points)
(i) Assume that the policyholder holds the policy until her death at age 91.6.
You are given that AV_{41} = 109{,}467.
Calculate the total amount of death benefit paid.
(ii) Roughly sketch a graph of the account values, AV_t, as a function of t, for 41 \le t < 41.6.
Part d (3 points)
The insurer offers a special joint life and last survivor version of the policy, under which a death benefit of 50,000 is paid out on the first death of a couple. On the second death, the death benefit is 50,000 plus the account value. The CoI is calculated assuming that the lives have independent future lifetimes and that the mortality of each life is 120% of the Standard Ultimate Mortality Model. All other policy details and assumptions are as given above.
A couple, Linda and Xiaobai, who are both age 50, purchases this policy. Linda also buys the single life policy.
(i) Show that in the first year, the CoI for this policy is the same as the CoI for the single life policy.
(ii) Describe how the account values of the single life policy and joint life policy would compare at the end of the t-th year if both Linda and Xiaobai survive to time t-1.
(iii) Describe how the account values of the single life policy and joint life policy would compare at the end of the t-th year if only Linda survives to time t-1.
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