IAI Actuarial Core Principles · Risk Modelling and Survival Analysis

Compound Distributions and Their Applications in Risk Modelling

A compound distribution models total claims S = X₁ + X₂ + … + X_N, where N is a random number of claims and each X is a random claim size. To solve questions, find E[S] and Var(S) by conditioning on N, then use a named distribution, recursion or approximation for probabilities.

What this chapter covers

This chapter deals with aggregate claims. An insurer does not only care how big one claim is. It cares about the total paid over a period. You model that total as S = X₁ + X₂ + … + X_N, where N is the number of claims and the X's are individual claim amounts. Usually the X's are assumed independent and identically distributed, and independent of N. State these assumptions every time.

The chapter moves in a clear line. You first set up the model. Then you get the mean, variance and moment generating function by conditioning on N. Next you meet the standard choices for N: Poisson, binomial and negative binomial. Then you learn how to get actual probabilities, by recursion or by approximation. Finally you apply the model to reinsurance and ruin.

It connects to the rest of CS2 in several ways. It builds on the claim frequency and claim severity distributions from the risk modelling distributions part of the paper. It uses ideas from conditional expectation and moment generating functions. Ruin theory links to the stochastic processes part, and reinsurance links to deductibles and limits. Paper A tests the theory and written working. Paper B can ask you to simulate or compute aggregate claims in R.

Risk modelling distributions carry a 20% syllabus weighting in CS2, and compound distributions are one of the most formula-driven, predictable areas inside it. Questions follow a pattern: set up S, state assumptions, compute moments, then price or test a reinsurance arrangement. If you practise the conditioning steps and the standard results, you can collect most of the marks in a written question. The same skills help in Paper B, where simulation of aggregate claims is a natural R task, so the effort pays in both papers.

Compound distributions and their applications in risk modelling: topics in the order to study them

1Compound Distributions and Aggregate Claims BasicsYou need the model S = X₁ + … + X_N and its assumptions before any formula makes sense.
2Moments of Compound DistributionsThe conditioning results for E[S], Var(S) and the MGF are used in every later topic.
3Compound Poisson, Binomial and Negative BinomialThese are the standard frequency models, and you apply the moment results to each of them.
4Recursive and Numerical Methods for Aggregate ClaimsOnce you know the frequency families, you can see which ones allow recursion and how to approximate the rest.
5Applications: Reinsurance and Ruin in Risk ModellingThis topic applies everything before it, so it is best done last, when the tools are secure.

How to prepare Compound distributions and their applications in risk modelling

Treat this chapter as a short chain of tools. Each tool feeds the next, so secure the early ones first and then practise full exam-style questions.

Write the model in your own words: what N is, what X is, what S is, and the independence assumptions. Do this until you can write it from memory.
Derive the mean and variance by conditioning: E[S] = E[N]·E[X] and Var(S) = E[N]·Var(X) + Var(N)·(E[X])². Learn the derivation, not only the result, since written questions may ask for it.
Build a table for the three frequency families with their mean, variance and PGF. For the compound Poisson case with rate λ, remember that E[S] = λ·E[X] and Var(S) = λ·E[X²].
Practise getting probabilities three ways: direct convolution for small cases, recursion where the frequency family allows it, and a normal or other approximation with correct parameters.
Work reinsurance questions. Split S into the insurer's and reinsurer's parts, for example under proportional, excess of loss and stop loss, and compute expected amounts and variances for each side.
Do a few ruin questions. Define the surplus process and premium loading, and be clear about which result you are using and its conditions.
In R, simulate N, then simulate the claims, sum them, and repeat many times. Check the simulated mean and variance against your formulas.

Common mistakes in Compound distributions and their applications in risk modelling

Using Var(S) = E[N]·Var(X) and dropping the second term.
Fix: Write the full formula Var(S) = E[N]·Var(X) + Var(N)·(E[X])² first. Then simplify for the specific frequency model.
Mixing up Var(X) and E[X²] in the compound Poisson variance.
Fix: For compound Poisson, Var(S) = λ·E[X²]. Compute E[X²] = Var(X) + (E[X])² explicitly before substituting.
Forgetting to state independence and identical distribution assumptions.
Fix: Open every written answer with one line listing the assumptions. Examiners give marks for them.
Applying a recursion to a claim size distribution or frequency model it does not fit.
Fix: Check the frequency family and that claim sizes are discrete (or have been discretised) before you use it.
Mixing up per-claim and aggregate reinsurance.
Fix: Write down what the retention is applied to before computing. Define the insurer's and reinsurer's payments as separate random variables.
Treating a normal approximation as exact in the tail.
Fix: Use it only when the question allows, say it is an approximation, and note that skewness can make tail probabilities inaccurate.

Last-day revision: Compound distributions and their applications in risk modelling

S = X₁ + … + X_N, with the X's independent and identically distributed and independent of N.
E[S] = E[N]·E[X].
Var(S) = E[N]·Var(X) + Var(N)·(E[X])².
MGF of S: M_S(t) = M_N(ln M_X(t)), where it exists.
Compound Poisson with rate λ: E[S] = λ·E[X] and Var(S) = λ·E[X²].
For Poisson N, the mean and variance of N are equal; binomial has variance below the mean; negative binomial has variance above the mean.
Panjer recursion applies when the claim number distribution is in the (a, b, 0) class, with discrete claim sizes.
A normal approximation to S needs the exact mean and variance, and may be poor when claims are few or skewed.
Under excess of loss with retention M, the insurer pays min(X, M) per claim and the reinsurer pays max(X − M, 0).
Under stop loss with retention d, the reinsurer pays max(S − d, 0).
Always state your assumptions before you start a calculation.

Compound distributions and their applications in risk modelling practice questions

Compound distributions and their applications in risk modelling in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Compound distributions and their applications in risk modelling: frequently asked questions

What is the difference between individual and collective risk models?

The individual risk model sums the claims from each separate policy. The collective risk model treats the portfolio as a random number of claims, each with a random size. This chapter is about the collective model.

Do I need to memorise the Panjer recursion?

You should know the conditions under which it applies and be able to use it for a short number of steps. Check whether the exam formula sheet gives it. Even then, practise setting it up so you can apply it quickly.

Why is compound Poisson so common in exam questions?

Its moments are simple, and the sum of independent compound Poisson variables is again compound Poisson. This makes it easy to examine and also useful for modelling portfolios.

How does this chapter appear in Paper B?

Paper B is computer-based, and aggregate claims are a natural simulation task in R. You simulate the number of claims, simulate each claim size, sum them, and repeat. You then compare the results with the theoretical mean and variance.