Actuary Certification Actuary Certification Loss Models 1 — Questions and Answers
Question 1: In loss models, which distribution is commonly used to model heavy-tailed claim severity due to its power-law tail?
- Normal distribution
- Pareto distribution (Correct answer)
- Binomial distribution
- Uniform distribution
Correct answer: Pareto distribution
The Pareto distribution has a power-law tail making it well-suited for modeling large, infrequent insurance claims with heavy tails.
Question 2: The excess loss variable (X − d | X > d) for a deductible d is called the:
- Limited expected value
- Left-censored variable
- Mean excess loss function (Correct answer)
- Stop-loss transform
Correct answer: Mean excess loss function
The mean excess loss function e(d) = E[X − d | X > d] gives the expected claim payment above the deductible d, given that the loss exceeds d.
Question 3: A compound Poisson frequency-severity model has aggregate loss S = X1 + X2 + ... + XN where N ~ Poisson(λ). The variance of S equals:
- λ · E[X]
- λ · E[X²] (Correct answer)
- λ · Var(X)
- λ² · E[X²]
Correct answer: λ · E[X²]
For a compound Poisson distribution, Var(S) = λ · E[X²], which combines both frequency and the second moment of severity.
Question 4: Which of the following is a member of the (a, b, 0) class of frequency distributions?
- Lognormal
- Negative Binomial (Correct answer)
- Weibull
- Pareto
Correct answer: Negative Binomial
The Negative Binomial distribution belongs to the (a, b, 0) class because its probability ratios pk/pk−1 = a + b/k for k ≥ 1, alongside Poisson and Binomial.
Question 5: The limited expected value E[X ∧ u] represents:
- The expected loss above the policy limit u
- The expected loss capped at u (Correct answer)
- The variance of X truncated at u
- The deductible at level u
Correct answer: The expected loss capped at u
E[X ∧ u] = E[min(X, u)] is the limited expected value, representing the expected payment when losses are capped at a policy limit u.
Question 6: Left-truncation of a loss variable at d (due to an ordinary deductible) means:
- Losses below d are set to zero in the data
- Losses below d are not observed at all (Correct answer)
- Losses above d are censored
- The policy pays only losses exactly equal to d
Correct answer: Losses below d are not observed at all
With left-truncation at d, losses below the deductible are never reported, so they are completely absent from the observed data.
