Actuary Certification Insurance Models 2 — Questions and Answers
Question 1: In the collective risk model, what does the aggregate loss S = X1 + X2 + ... + XN represent?
- Total number of claims in a period
- Sum of individual claim severities over a random number of claims (Correct answer)
- Average claim severity multiplied by exposure
- Net premium income minus expenses
Correct answer: Sum of individual claim severities over a random number of claims
S is the aggregate loss formed by summing N individual claim amounts, where N is a random claim count variable.
Question 2: Which distribution is used to model the time between successive claim arrivals in a Poisson process?
- Gamma
- Weibull
- Exponential (Correct answer)
- Lognormal
Correct answer: Exponential
Inter-arrival times in a Poisson process are exponentially distributed with the same rate parameter λ.
Question 3: A stop-loss reinsurance treaty pays losses exceeding a retention d. The net stop-loss premium equals:
- E[max(S − d, 0)] (Correct answer)
- E[S] − d
- E[S − d]
- P(S > d)
Correct answer: E[max(S − d, 0)]
The stop-loss premium is the expected value of the excess loss above the retention, E[max(S−d,0)].
Question 4: The coefficient of variation (CV) of aggregate losses S is lower than the CV of individual claim severity X when:
- The claim count N is deterministic
- The claim count variance exceeds its mean
- There are many independent claims (large n) (Correct answer)
- Claim severities are heavy-tailed
Correct answer: There are many independent claims (large n)
By the law of large numbers, pooling many independent claims reduces relative variability, lowering the CV of S compared to X.
Question 5: Under the individual risk model, the total loss for a group of n independent policies is modeled as:
- A compound Poisson distribution
- S = b1·I1 + b2·I2 + ... + bn·In where Ii are Bernoulli indicators (Correct answer)
- The convolution of n Poisson random variables
- A negative binomial aggregate
Correct answer: S = b1·I1 + b2·I2 + ... + bn·In where Ii are Bernoulli indicators
The individual model sums fixed benefit amounts bi scaled by Bernoulli claim indicators Ii for each policy.
Question 6: What is the primary advantage of using the recursive (Panjer) formula for aggregate loss distributions?
- It applies only to exponential severity distributions
- It avoids the need for moment generating functions
- It efficiently computes the aggregate distribution when (a,b,0) claim counts are combined with discrete severities (Correct answer)
- It provides closed-form solutions for all severity distributions
Correct answer: It efficiently computes the aggregate distribution when (a,b,0) claim counts are combined with discrete severities
Panjer recursion exploits the (a,b,0) property of N to compute aggregate probabilities iteratively without full convolution.
Question 7: In insurance pricing, the pure premium is calculated as:
- Expected losses divided by expected claim count
- Expected losses divided by earned exposure units (Correct answer)
- Gross premium minus expenses
- Loss ratio multiplied by earned premium
Correct answer: Expected losses divided by earned exposure units
The pure premium equals E[aggregate losses] / E[exposure], representing the expected loss cost per unit of exposure.
Actuary Certification Insurance Models 2
In the collective risk model, what does the aggregate loss S = X1 + X2 + ... + XN represent?