SOA Exam ALTAM Sample Question 3
Study guide: Exam ALTAM
For a special fully discrete two-year term insurance on an individual, age 60, you are given:
(i) The death benefit is 1000 plus the return of gross premiums paid with interest at 6%.
(ii) The following double decrement table, where decrement (d) is death and decrement (w) is withdrawal:
x |
q_x^{(d)} |
q_x^{(w)} |
|---|---|---|
| 60 | 0.06 | 0.04 |
| 61 | 0.12 | 0.00 |
(iii) There are no withdrawal benefits.
(iv) i = 0.06
(v) G denotes the annual gross premium.
(vi) L_0 denotes the insurer's loss at issue random variable for the individual's policy.
Part a (4 points)
Calculate the values in the following table. Express L_0 in terms of G where appropriate:
| Event | Value of L_0, Given that the Event Occurred |
Probability of Event |
|---|---|---|
| Death in year 1 | ||
| Withdrawal in year 1 | ||
| Death in year 2 | ||
| Neither death or withdrawal |
Part b (4 points)
(i) Show that E\left[L_0\right] = a - bG, where a is 150 to the nearest 10 and b is 1.58 to the nearest 0.01. You should calculate a to the nearest 1 and b to the nearest 0.001.
(ii) Show that Var\left(L_0\right) = cG^2 + dG + e, where c is 0.5 to the nearest 0.1, d is 480 to the nearest 10, and e is 116,000 to the nearest 1000. You should calculate c to the nearest 0.01, d to the nearest 1, and e to the nearest 100.
Part c (2 points)
The insurer expects to issue 200 such policies to insureds with independent future lifetimes. The premium for each policy is G = 130.
Let L_{agg} denote the insurer's aggregate future loss random variable at issue for these 200 policies.
Calculate \Pr\left(L_{agg} > 0\right) using the normal approximation without continuity correction.
(MLC, Fall 2015, Question 4.)
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