Sums of compound Poisson models

Welcome to the World of Combined Risks!

In your actuarial journey, you’ve already met the Compound Poisson Model. It’s the "bread and butter" of aggregate modeling because it’s mathematically elegant and very flexible. But what happens when an insurance company manages multiple portfolios at once? What if you have one group of policies for auto insurance and another for homeowners insurance, and you want to know the total risk for the whole company?

In this chapter, we’ll learn a "mathematical magic trick." We will discover that when you add several independent compound Poisson risks together, the result is—drumroll, please—another single compound Poisson model! This makes our lives much easier because we don't have to learn a whole new set of rules to find the total risk. Let's dive in!

The Core Concept: The Summation Property

Imagine you have several different portfolios. Let's say you have \(n\) independent aggregate loss variables, \(S_1, S_2, ..., S_n\). Each one follows a compound Poisson distribution.

For each portfolio \(j\):
1. The number of claims follows a Poisson distribution with parameter \(\lambda_j\).
2. The individual claim amounts (severities) follow a distribution \(f_{X_j}(x)\).

The total aggregate loss for the entire company is \(S = S_1 + S_2 + ... + S_n\).

The Big Rule: If all \(S_j\) are independent, then their sum \(S\) is also a compound Poisson random variable! To treat the sum as one big compound Poisson model, we just need to find two things: the new total frequency (\(\lambda\)) and the new combined severity distribution (\(f_X\)).

1. Finding the New Frequency (\(\lambda\))

This is the easy part. To find the total frequency parameter, you simply add up the individual \(\lambda\) values from each portfolio.

\[ \lambda_{total} = \sum_{j=1}^{n} \lambda_j \]

Analogy: Think of this like rain gauges. If one gauge captures 3 drops per minute and another captures 5 drops per minute, combined they are capturing 8 drops per minute.

2. Finding the New Severity Distribution (\(f_X\))

This part is a bit more nuanced. The new severity distribution is a weighted average of the individual severity distributions. The "weights" are determined by how likely a claim is to come from a specific portfolio.

The probability that a claim comes from portfolio \(j\) is \(\frac{\lambda_j}{\lambda_{total}}\). Therefore, the combined severity distribution is:

\[ f_X(x) = \sum_{j=1}^{n} \frac{\lambda_j}{\lambda_{total}} f_{X_j}(x) \]

Why does this make sense? If Portfolio A has 90 claims and Portfolio B has only 10 claims, a "random" claim from the total pile is much more likely to look like a claim from Portfolio A. So, Portfolio A's severity distribution gets more "weight" in the average.

Summary Checklist for Combining Models:

1. Add the lambdas to get your new frequency.
2. Weight the severities by their respective \(\lambda / \lambda_{total}\).
3. Proceed as normal using your favorite aggregate method (like Panjer's recursion or MGFs) using these new parameters.

Step-by-Step Example

Let’s look at a quick example to make this concrete.

Suppose an insurer has two independent lines of business:
Line 1: Compound Poisson with \(\lambda_1 = 2\) and all claims are exactly 10.
Line 2: Compound Poisson with \(\lambda_2 = 3\) and all claims are exactly 20.

Step 1: Find the total \(\lambda\).
\( \lambda = 2 + 3 = 5 \).

Step 2: Find the combined severity distribution.
The probability that a claim is 10 is \( \frac{2}{5} = 0.4 \).
The probability that a claim is 20 is \( \frac{3}{5} = 0.6 \).

The Result: The total aggregate loss \(S\) is a compound Poisson model with \(\lambda = 5\) and a severity distribution where \(P(X=10) = 0.4\) and \(P(X=20) = 0.6\).

Common Pitfalls to Avoid

Mistake 1: Forgetting to Weight the Severities.
Students often just add the severity distributions or average them 50/50. Remember: the portfolio with the higher \(\lambda\) has a bigger influence on the final severity shape!

Mistake 2: Ignoring Independence.
This property only works if the portfolios are independent. If a hurricane hits and affects both your "Auto" and "Home" portfolios simultaneously, they aren't independent, and this simple addition won't work perfectly.

Mistake 3: Confusing Frequency and Severity.
Remember: \(\lambda\) is about how often things happen. \(X\) is about how much they cost. We add the frequencies, but we average the costs.

Quick Review: Why use this?

Did you know? This property is a lifesaver for computational efficiency. Instead of calculating the distribution for Portfolio A, then Portfolio B, and then doing a complex convolution to find \(S_A + S_B\), you can just combine the parameters at the start and do one calculation for the whole thing. It saves time and reduces the chance of math errors!

Key Takeaways for Exam Day

1. The Rule: Sum of independent Compound Poissons = A single Compound Poisson.
2. The Lambda: \(\lambda_{new} = \lambda_1 + \lambda_2 + ... + \lambda_n\).
3. The Severity: \(f_{X, new}(x) = w_1 f_{X_1}(x) + w_2 f_{X_2}(x) + ...\), where \(w_j = \lambda_j / \lambda_{total}\).
4. Moments: The mean of the sum is the sum of the means, and the variance of the sum is the sum of the variances (since they are independent).

Don't worry if the weighted average feels a bit strange at first. Just remember: "More claims (\(\lambda\)) means more weight." Once you have the new \(\lambda\) and the new severity distribution, you can solve for any probability or moment just like you did in the basic Compound Poisson chapter!