Welcome to the World of Reinsurance!

Hello there! Today, we’re diving into a crucial topic for Exam FAM: Allocation of claim amounts between insurer and reinsurer. If you’ve ever wondered how insurance companies protect themselves from massive losses (like a hurricane or a giant pile-up on a highway), you’re in the right place. Think of reinsurance as "insurance for insurance companies." It’s all about sharing the risk so that one bad day doesn't put a company out of business.

Don't worry if this seems tricky at first. We’re going to break it down step-by-step, using simple analogies and clear formulas. By the end of these notes, you'll feel much more confident calculating exactly who pays what!


1. The Big Picture: Who are the Players?

Before we look at the math, let's get the terminology straight:

  • The Primary Insurer (The Cedant): This is the company that sells the policy to the customer. They "cede" (give away) part of the risk.
  • The Reinsurer: This is the company that accepts the risk from the primary insurer in exchange for a premium.
  • The Claim (\(X\)): This is the total amount of the loss before it gets split up.

Did you know? Reinsurance allows small insurance companies to take on much larger risks than they could handle alone. It’s like a group of friends splitting the cost of a giant pizza—it’s much easier for everyone to pay a small share than for one person to pay the whole bill!


2. Proportional Reinsurance (Quota Share)

This is the simplest way to share a claim. In Proportional Reinsurance, the insurer and reinsurer agree to share every single claim based on a fixed percentage.

How it Works

If the insurer keeps a proportion \(\alpha\) (where \(0 < \alpha < 1\)), then:

  • The Insurer's Share (\(Y_I\)) is: \( Y_I = \alpha X \)
  • The Reinsurer's Share (\(Y_R\)) is: \( Y_R = (1 - \alpha) X \)

Example: Imagine a "70/30 Quota Share" agreement. The insurer keeps 70% of the risk and the reinsurer takes 30%. If a claim of \$1,000 comes in, the insurer pays \$700 and the reinsurer pays \$300. Easy, right?

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Quick Review Box:
\nIn proportional reinsurance, the split is always a constant ratio, regardless of how big or small the claim is.

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3. Non-Proportional Reinsurance (Excess of Loss)

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This is where things get a bit more "actuarial." In Excess of Loss (XoL) reinsurance, the insurer pays for the small stuff, and the reinsurer steps in only when the claim exceeds a certain amount. This "certain amount" is called the Retention (often denoted as \(M\) or \(d\)).

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The Formulas

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If \(M\) is the retention level:

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  • Insurer's Share: \( Y_I = \min(X, M) \)
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  • Reinsurer's Share: \( Y_R = \max(0, X - M) \)
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Analogy: Think of the Retention (\(M\)) as a deductible for the insurance company. If you have a \$500 deductible on your car insurance, you pay the first \$500 (your retention), and the insurance company pays the rest.

Step-by-Step Logic:

  1. If the claim is smaller than \(M\), the insurer pays the whole thing. The reinsurer pays 0.
  2. If the claim is larger than \(M\), the insurer pays exactly \(M\). The reinsurer pays the "excess" (the leftover amount).

Common Mistake to Avoid: Don't confuse the Retention with a Policy Limit. A retention is what the insurer keeps, while a policy limit is the maximum the total policy will pay out. We will look at how they interact next!


4. Dealing with Policy Limits and Reinsurance

In the real world, most insurance policies have a Policy Limit (\(u\)). This means the total payment (\(X_{limit}\)) is \(\min(X, u)\). When we add reinsurance to this, we have to be careful about the order of operations.

Scenario A: Reinsurance on the "Net" Claim

Usually, the reinsurer only covers what the insurer is actually liable for. If the policy limit is \(u\) and the insurer's retention is \(M\), the allocation looks like this:

  • Total Payment: \( X_{paid} = \min(X, u) \)
  • Insurer Pays: \( Y_I = \min(X, M) \) (assuming \(M \le u\))
  • Reinsurer Pays: \( Y_R = \min(X, u) - \min(X, M) \)

Memory Trick: The reinsurer's share is the "slice" of the claim between \(M\) and \(u\). If the claim doesn't reach \(M\), the reinsurer pays nothing. If the claim goes past \(u\), the reinsurer's payment stops growing.


5. Calculating Expected Values (The Math Core)

On Exam FAM, you will often be asked to find the Expected Value of the insurer's or reinsurer's share. We use the Limited Expected Value notation: \( E[X \wedge L] \).

The Key Formulas:

For an Excess of Loss arrangement with retention \(M\) and policy limit \(u\):

  • Expected Cost to Insurer: \( E[Y_I] = E[X \wedge M] \)
  • Expected Cost to Reinsurer: \( E[Y_R] = E[X \wedge u] - E[X \wedge M] \)

Key Takeaway: The sum of the expected shares must equal the expected total payment:
\( E[Y_I] + E[Y_R] = E[X \wedge u] \).


6. Summary and Final Tips

We've covered a lot! Let's wrap up the most important points:

  • Quota Share = Multiplication (\( \alpha \times X \)). Everyone shares a percentage of every dollar.
  • Excess of Loss = Subtraction/Min/Max. The reinsurer only pays when the claim "overflows" the retention level.
  • Retention is the insurer's "deductible."
  • Expected Values are found by taking the difference between limited expected values at different layers.

Final Encouragement: Practice drawing "layer diagrams." Draw a vertical bar representing the claim. Mark the Retention (\(M\)) and the Limit (\(u\)). Coloring in the sections belonging to the Insurer vs. the Reinsurer can make these problems much easier to visualize!

Keep practicing those \( E[X \wedge d] \) calculations from your distribution tables, and you'll master this chapter in no time!