SOA Exam ALTAM Sample Question 4
Study guide: Exam ALTAM
A life insurer issues a 5-year term insurance, with premiums payable half-yearly, to a person who is age 60. The death benefit of 100,000 is paid at the end of the half-year of death.
You are given:
(i) Expenses are 10% of each semi-annual premium.
(ii) i = 0.05
(iii) Mortality follows the Standard Ultimate Life Table; the force of mortality of the Standard Ultimate Life Table is:
\mu_{x+t} = A + Bc^{x+t}, \text{ where } A = 2.2\times 10^{-4},\ B = 2.7\times 10^{-6},\ c = 1.124
(iv) The gross premium is 270 per half-year.
Part a (3 points)
(i) Show that 100{,}000\ {}_{0.5}q_{64} = 260 to the nearest 10. You should calculate the value to the nearest 1.
(ii) Show that the gross premium policy value at time 4 is 30 to the nearest 10. You should calculate the policy value to the nearest 0.1.
Part b (2 points)
In the final year of the policy, the insurer decides to re-evaluate the gross premium policy value due to a virus that is spreading through the community. The insurer uses the following multiple state model.
(as printed)
A Healthy person will move to State 1, At Risk, if exposed to the virus. Every person moving into the At Risk state will be tested for the virus. If the result of the test is positive, the person moves to State 2, Sick; otherwise, the person will move back to the Healthy state.
The transition intensities for all x are:
\mu_x^{01} = 0.06
\mu_x^{03} is the force of mortality of the Standard Ultimate Life Table, as in (iii) above.
\mu_x^{10} = 20.0; \quad \mu_x^{12} = 80.0
\mu_x^{20} = 4.0; \quad \mu_x^{23} = 5\mu_x^{03}
You are also given that the insured is Healthy at time 4, and that premiums are payable conditional on being in State 0, 1, or 2.
Calculate the probability that the insured will remain in the Healthy state throughout the final year of the policy, {}_1p_{64}^{\overline{00}}.
Part c (3 points)
The insurer re-evaluates the gross premium policy value at time 4 using the model above, with the same interest rate and expense assumptions as in part (a).
You are given the following probabilities:
x |
{}_{0.5}p_x^{00} |
{}_{0.5}p_x^{01} |
{}_{0.5}p_x^{02} |
{}_{0.5}p_x^{03} |
{}_{0.5}p_x^{13} |
{}_{0.5}p_x^{23} |
|---|---|---|---|---|---|---|
| 64.0 | 0.98661 | 0.00059 | 0.01016 | 0.00264 | 0.00605 | 0.00700 |
| 64.5 | 0.98646 | 0.00059 | 0.01016 | 0.00279 | 0.00640 | 0.00740 |
Calculate the revised gross premium policy value for the insured's policy at time 4.
Part d (2 points)
Calculate the expected time, in days, of each sojourn in the At Risk state.
(LTAM, Spring 2021, Question 1)
(Part (a) is covered by the FAM-L exam.)
Answer this question and see the worked solution