ASTAM Aggregate Models
ASTAM aggregate models combine claim counts and claim sizes into a total loss distribution. The exam emphasis is convolution, Panjer recursion, severity discretization, and sums of compound Poisson models.
Quick Answer
The Spring 2026 ASTAM syllabus gives aggregate models a 12-22% weight. The outcomes name convolution, recursive formulas for (a,b,0) and (a,b,1) frequency classes, severity discretization, and sums of compound Poisson models.
The central random variable is S = X_1 + ... + X_N. Once you define N and the individual severity distribution, the rest is a distribution-building problem.
Panjer Recognition
Panjer recursion works when the frequency distribution belongs to the right recursive class and severity has been placed on a discrete grid. Poisson, binomial, and negative binomial are the key families to recognize quickly.
The exam can test the mechanics, but the setup matters more. Define the grid, define the mass at zero, identify a and b, and then start the recursion. Skipping the setup usually causes off-by-one errors.
Discretization Choices
Continuous severities must be discretized before recursive aggregate calculations. Rounding and local moment matching are both syllabus-level methods.
The actuarial issue is approximation error. A finer grid gives more accuracy but more computation. A coarser grid is faster but can distort layer premiums and tail probabilities.
Original Practice Drill
Let N be Poisson with mean 2.5 and let severity take values 1, 2, and 3 with probabilities 0.50, 0.30, and 0.20. Use Panjer recursion to calculate probabilities for S = 0 through 5, then check the mean by E[S] = E[N]E[X].
A complete written answer states the frequency family, the severity grid, the recursion seed, and the reason the mean check is a diagnostic rather than a second solution.
Common Traps
Trap 1: using Var(X) where compound Poisson needs E[X squared] in the variance of S.
Trap 2: forgetting that a zero severity mass changes the recursion seed and the interpretation of claim count.
Trap 3: treating sums of compound Poisson models as unrelated. Independent compound Poisson sums can often be recombined into one compound Poisson with a mixture severity.
Original Source-Backed Practice
3 questions built from syllabus outcomes and released-exam patterns. The prompts and answers are original, so they train the skill without copying official exam text.
ASTAM Aggregate and Panjer Drill
Practice for aggregate-loss setup, recursion seeds, zero severity mass, and compound Poisson mixtures.
- Question 1/Calculation
Compound Poisson seed
If N is Poisson with mean lambda and the discrete severity distribution has no mass at zero, what is P(S = 0)?
Solution And Grading Points
P(S = 0) = P(N = 0) = exp(-lambda). With no zero severity mass, total loss is zero only when there are no claims.
- Uses the Poisson zero-count probability.
- Explains why no zero severity mass matters.
- Treats the seed as a distribution probability, not a recursion coefficient.
- Question 2/Written Answer
Zero severity mass
Why does positive severity mass at zero change the aggregate-loss setup?
Solution And Grading Points
If individual severities can be zero, then S can be zero even when N is positive. The recursion seed must reflect both zero claims and claims with zero payment.
- Recognizes that positive claim counts may still produce zero total payment.
- Connects zero mass to the seed probability.
- Keeps claim count separate from payment count.
- Question 3/Written Answer
Sum of compound Poisson models
Two independent compound Poisson loss models have rates lambda_1 and lambda_2 and severity distributions F_1 and F_2. How can their sum be represented?
Solution And Grading Points
The sum is compound Poisson with rate lambda_1 + lambda_2 and mixture severity that chooses F_1 with weight lambda_1 / (lambda_1 + lambda_2) and F_2 with weight lambda_2 / (lambda_1 + lambda_2).
- Adds the Poisson rates.
- Uses rate-proportional severity mixture weights.
- States the independence assumption.