Compound distributions and their applications in risk modelling

CS2 – Risk Modelling and Survival Analysis · IFoA (Institute and Faculty of Actuaries) · 5 min read

Welcome to the World of Compound Distributions!

Hello there! Today, we are diving into one of the most practical and exciting chapters in the CS2 curriculum: Compound Distributions. If you’ve ever wondered how an insurance company calculates the total amount of money it might have to pay out in claims over a year, you’re in the right place.

At first glance, this topic might seem like a mountain of formulas, but don’t worry! We are going to break it down into bite-sized pieces. Think of it as a "two-step" process: first, we count how many events happen, and second, we look at how big each event is. Let’s get started!

1. What Exactly is a Compound Distribution?

In simple terms, a compound distribution represents the sum of a random number of random variables. In the world of risk modelling, we usually use this to represent the Aggregate Claim Amount (S).

Imagine you run a small insurance company. Two things are uncertain:
1. N (Frequency): How many people will make a claim this month?
2. X (Severity): For each person who claims, how much will it cost?

The total cost $ S $ is the sum of all those individual claims:

$ S = X_1 + X_2 + X_3 + ... + X_N $

Important Note: If $ N = 0 $, then $ S = 0 $. You can't have a total claim amount if nobody makes a claim!

Core Assumptions

For the math to work simply, we usually assume two things:
1. The individual claim amounts ($ X_1, X_2, ... $) are independent and identically distributed (i.i.d.). This means every claim is like a fresh roll of the same dice.
2. The number of claims ($ N $) is independent of the size of the claims ($ X $). The frequency doesn't change the severity.

Summary Takeaway: A compound distribution combines "How many?" ($ N $) and "How much?" ($ X $) to tell us "The total amount" ($ S $).

2. The Mean and Variance of S

Don't let the notation scare you! We can find the expected value (mean) and variance of the total claims $ S $ using the properties of $ N $ and $ X $.

The Mean (Expected Value)

This is quite intuitive. If you expect 10 claims and each claim averages \$500, you'd expect to pay \$5,000.

$ E[S] = E[N] \times E[X] $

The Variance

The variance is a bit more "spicy" because it has two components. It accounts for the uncertainty in how many claims occur and the uncertainty in how large each claim is.

$ Var(S) = E[N]Var(X) + Var(N)(E[X])^2 $

Memory Aid: Think of this as the "Law of Total Variance". The first part ($ E[N]Var(X) $) is the variance coming from the claim sizes, and the second part ($ Var(N)(E[X])^2 $) is the variance coming from the claim counts.

Quick Review Box:
- Total Mean: Mean Frequency $\times$ Mean Severity
- Total Variance: (Mean Frequency $\times$ Variance Severity) + (Variance Frequency $\times$ Mean Severity squared)

3. Generating Functions

In CS2, we often use Moment Generating Functions (MGFs) and Probability Generating Functions (PGFs) to describe distributions. For compound distributions, there is a very neat relationship.

If $ S $ is a compound distribution, its MGF, $ M_S(t) $, is found by "plugging" the MGF of $ X $ into the MGF of $ N $:

$ M_S(t) = M_N(\ln M_X(t)) $

Wait, why the natural log? Because the MGF of a sum involves exponentials! If you are working with PGFs (usually for the discrete claim counts $ N $), the formula looks even cleaner:

$ G_S(t) = G_N(G_X(t)) $

Don't worry if this seems tricky at first! Just remember that the generating function of the "total" is a composition (one function inside another) of the individual generating functions.

Did you know? This mathematical "nesting" is why compound distributions are sometimes called "stopped-sum" distributions!

4. The Star of the Show: The Compound Poisson Distribution

The most common compound distribution you will encounter is the Compound Poisson Distribution. This happens when the number of claims $ N $ follows a Poisson distribution with parameter $ \lambda $.

Why is it special?

Because for a Poisson distribution, the Mean and Variance are the same! ($ E[N] = Var(N) = \lambda $). This simplifies our formulas significantly:

$ E[S] = \lambda E[X] $

$ Var(S) = \lambda E[X^2] $

Common Mistake to Avoid: Students often forget that for Compound Poisson, $ Var(S) $ uses $ E[X^2] $ (the second raw moment), not just $ Var(X) $. Double-check this in your calculations!

The Additivity Property

If you have two independent Compound Poisson risks (e.g., one for Car Insurance and one for House Insurance), their sum is also a Compound Poisson distribution! You simply add their $ \lambda $ values together. This makes it very popular for modelling large portfolios.

Summary Takeaway: Compound Poisson is the "gold standard" in risk modelling because it's mathematically "well-behaved" and easy to combine.

5. Real-World Example: The Baker's Shop

Let's ground this in reality. Imagine a baker:
- The number of customers per hour ($ N $) follows a Poisson distribution with $ \lambda = 10 $.
- Each customer spends an amount ($ X $) that is normally distributed with mean \$15 and standard deviation \$3.

Step 1: Expected Total Sales
$ E[S] = E[N] \times E[X] = 10 \times 15 = \$150 $

Step 2: Variance of Total Sales
We know $ Var(X) = 3^2 = 9 $.
$ Var(S) = E[N]Var(X) + Var(N)(E[X])^2 $
$ Var(S) = (10 \times 9) + (10 \times 15^2) = 90 + 2250 = 2340 $

Key Takeaway: Even though individual spending is quite consistent (low variance), the total sales variance is high because the number of customers fluctuates.

6. Summary and Final Tips

We’ve covered the core of Compound Distributions. Here are the "Must-Know" points for your exam:

1. Definition: $ S $ is a sum of $ N $ random variables.

2. The Formulas: Memorize the Mean and Variance formulas; they are your best friends.

3. Independence: Always check if the frequency and severity are independent before applying these formulas.

4. Compound Poisson: Remember its special properties, especially the simplified variance ($ \lambda E[X^2] $) and the fact that you can add independent Poisson processes together.

Keep practicing! Compound distributions are just building blocks. Once you master these, the more complex models in CS2 will feel much more manageable. You've got this!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the general definition of a compound distribution for the aggregate claim amount $ S $?

$ S = X_1 + X_2 + ... + X_N $, which represents the sum of a random number ($ N $) of individual random claim amounts ($ X_i $).

What are the key independence assumptions required for standard compound distribution formulas?

The individual claim amounts $ X_i $ must be independent and identically distributed (i.i.d.), and the number of claims $ N $ must be independent of the claim sizes $ X $.

How is the expected value (mean) of the aggregate claim amount $ S $ calculated?

$ E[S] = E[N] \times E[X] $.

What is the general formula for the variance of the aggregate claim amount $ S $?

$ Var(S) = E[N]Var(X) + Var(N)(E[X])^2 $.

How is the Moment Generating Function (MGF) of a compound distribution $ S $ defined?

$ M_S(t) = M_N(\ln M_X(t)) $.

What is the Probability Generating Function (PGF) of the aggregate claim amount $ S $?

$ G_S(t) = G_N(G_X(t)) $.

How do the mean and variance formulas simplify for a Compound Poisson distribution?

$ E[S] = \lambda E[X] $ and $ Var(S) = \lambda E[X^2] $.

What is the additivity property of independent Compound Poisson risks?

The sum of independent Compound Poisson distributions is also a Compound Poisson distribution, with a rate parameter equal to the sum of the individual rates ($ \lambda_{total} = \sum \lambda_i $).

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