Free SOA Exam FAM (Fundamentals of Actuarial Mathematics) Severity, Frequency, and Aggregate Models Practice Questions

Severity, frequency, and aggregate loss models on SOA Exam FAM test parametric distributions (exponential, Pareto, lognormal), compound Poisson models, and the Panjer recursive method for computing aggregate loss probabilities.

152 Questions
76 Easy
51 Medium
25 Hard
2026 Syllabus

Sample Questions

Question 1 Easy
Which of the following is a property of a coherent risk measure?
Solution
A is correct.

Subadditivity is one of the four axioms of coherent risk measures (monotonicity, positive homogeneity, translation invariance, subadditivity).
Question 2 Medium
Which of the following statements about TVaR is FALSE?
Solution
B is correct.

TVaR0=E[X]\text{TVaR}_0 = E[X], while VaR0\text{VaR}_0 is the minimum possible value (often 0 for loss distributions). Since E[X]E[X] generally does not equal the minimum, the statement that TVaR at level p=0p = 0 equals VaR at level p=0p = 0 is false.
Question 3 Hard
Continuing from the Panjer recursion with Poisson λ=3\lambda = 3, f1=0.6f_1 = 0.6, f2=0.4f_2 = 0.4, and g0=0.04979g_0 = 0.04979, g1=0.08962g_1 = 0.08962. Calculate g2=P(S=2)g_2 = P(S = 2).
Solution
E is correct.

For compound Poisson (a=0a = 0, b=λ=3b = \lambda = 3), Panjer's recursion simplifies to: gs=λs∑j=1sj⋅fj⋅gs−jg_s = \frac{\lambda}{s}\sum_{j=1}^{s} j \cdot f_j \cdot g_{s-j} For s=2s = 2: g2=32[1⋅f1⋅g1+2⋅f2⋅g0]g_2 = \frac{3}{2}[1 \cdot f_1 \cdot g_1 + 2 \cdot f_2 \cdot g_0] =1.5[0.6×0.08962+0.8×0.04979]= 1.5[0.6 \times 0.08962 + 0.8 \times 0.04979] =1.5[0.05377+0.03983]=1.5×0.09360=0.14041= 1.5[0.05377 + 0.03983] = 1.5 \times 0.09360 = 0.14041. Verification by direct calculation: P(S=2)P(S=2) requires either one claim of size 2 or two claims each of size 1. P(N=1)P(X=2)+P(N=2)[P(X=1)]2=3e−3(0.4)+9e−32(0.36)=0.04979×2.82=0.14041P(N=1)P(X=2) + P(N=2)[P(X=1)]^2 = 3e^{-3}(0.4) + \frac{9e^{-3}}{2}(0.36) = 0.04979 \times 2.82 = 0.14041. Common errors: dropping the λ/s=1.5\lambda/s = 1.5 leading factor and reporting the bracket sum 0.093600.09360; using j=1j = 1 instead of j=2j = 2 on the size-2 term to get 0.110530.11053; omitting the 2⋅f2⋅g02 \cdot f_2 \cdot g_0 term entirely for 0.080660.08066; or applying the non-simplified formula incorrectly to get 0.070200.07020, which is g2/2g_2/2 rather than g2g_2.
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