Free SOA Exam FAM (Fundamentals of Actuarial Mathematics) Short-Term Insurance and Reinsurance Coverages Practice Questions

Short-term insurance coverages on SOA Exam FAM cover property and casualty policy forms, deductibles, coinsurance, and reinsurance structures. This topic introduces how insurers transfer and manage underwriting risk.

72 questions 27 easy 28 medium 17 hard 2026 syllabus

Sample Questions

Question 1 Easy
Losses follow an exponential distribution with mean 800. Calculate the mean excess loss at d=300d = 300.
Solution
D is correct.

The exponential distribution is memoryless: e(d)=E[X−d∣X>d]=θe(d) = E[X - d \mid X > d] = \theta for all d≥0d \ge 0. With θ=800\theta = 800, e(300)=800e(300) = 800.
Question 2 Medium
An excess-of-loss reinsurance treaty has a retention of 50,000. A loss of 80,000 occurs. How much does the reinsurer pay?
Solution
B is correct.

Under excess-of-loss reinsurance with retention M=50,000M = 50{,}000, the reinsurer pays max⁡(X−M,0)=max⁡(80,000−50,000,0)=30,000\max(X - M, 0) = \max(80{,}000 - 50{,}000, 0) = 30{,}000.
Question 3 Hard
Losses follow a Pareto distribution with α=3\alpha = 3 and θ=2,000\theta = 2{,}000. Calculate the loss elimination ratio (LER) for an ordinary deductible of d=500d = 500.
Solution
D is correct.

For a Pareto distribution with α=3\alpha = 3 and θ=2,000\theta = 2{,}000:

E[X]=θα−1=2,0002=1,000E[X] = \frac{\theta}{\alpha - 1} = \frac{2{,}000}{2} = 1{,}000.

The limited expected value at d=500d = 500:
E[X∧d]=θα−1[1−(θθ+d)α−1]=1,000[1−(0.8)2]=1,000×0.36=360E[X \wedge d] = \frac{\theta}{\alpha - 1}\left[1 - \left(\frac{\theta}{\theta + d}\right)^{\alpha - 1}\right] = 1{,}000\left[1 - (0.8)^{2}\right] = 1{,}000 \times 0.36 = 360
LER=E[X∧d]E[X]=3601,000=0.3600\text{LER} = \frac{E[X \wedge d]}{E[X]} = \frac{360}{1{,}000} = 0.3600
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