SOA Exam ALTAM Sample Question 21
Study guide: Exam ALTAM
A fully discrete, joint life 10-year endowment insurance policy, with sum insured 100,000, is issued to two independent lives, both aged 70. Premiums are payable annually until the earlier of the first death or the contract maturity.
You are given:
(i) Acquisition expenses are 400 plus 50% of the first year's premium.
(ii) Renewal expenses are 2% of each subsequent premium.
(iii) Mortality follows Standard Ultimate Life Table.
(iv) i = 0.05
(v) The gross premium is G = 10{,}658.
(vi) L_0 is the gross loss at issue random variable.
Part a (1 point)
Calculate the probability that both lives survive to the end of the contract.
Part b (2 points)
Show that the expected loss at issue for the policy is -4{,}470 to the nearest 10. You should calculate the value to the nearest 1.
Part c (3 points)
(i) Show that {}^2A_{70:70:\overline{10}|} = 0.44 to the nearest 0.01. You should calculate the value to the nearest 0.001.
(ii) Show that the standard deviation of L_0 is 30,000 to the nearest 10,000. You should calculate the value to the nearest 10.
Part d (2 points)
Now consider a portfolio of 100 identical, independent policies issued to couples both aged 70, with independent future lifetimes. Calculate the probability that the aggregate loss at issue is positive, using the Normal approximation to the aggregate loss distribution.
Part e (2 points)
In a sub-portfolio of 10 identical, independent policies issued to couples with lives both age 70, the age at death of the 10 couples' lives turned out to be the following:
| Policy Number | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| Age at death of first to die | 75.2 | 78.5 | 80.5 | 82.1 | 83.5 | 83.8 | 83.9 | 91.3 | 91.5 | 95.8 |
| Age at death of second to die | 77.2 | 93.1 | 81.7 | 90.1 | 92.3 | 95.0 | 96.3 | 91.7 | 94.2 | 100.8 |
Without further calculation, state whether the value of the total profit realized in this sub-portfolio will be greater or less than the expected value of the profit for the sub-portfolio. Justify your answer.
(This is Question 1 from the Spring 2020 Written Answer exam.)
Answer this question and see the worked solution