Proportional and excess of loss reinsurance

Welcome to the World of Reinsurance!

Hello! If you’ve ever wondered how insurance companies handle massive claims (like a hurricane or a giant multi-car pileup) without going bankrupt, you’re in the right place. In this chapter, we explore Reinsurance—which is essentially "insurance for insurance companies."

Think of it like this: If you and a friend agree to split the cost of a giant pizza, you’re sharing the "risk" of being too full or spending too much money. Reinsurance works the same way. The primary insurance company (the Ceding Company) passes some of its risk to another company (the Reinsurer) so that one big disaster doesn't wipe them out. Let’s dive into the two main ways they split the bill!

1. Proportional Reinsurance: Sharing the Pie

In Proportional Reinsurance, the ceding company and the reinsurer share everything—premiums and losses—based on a fixed percentage. It is the simplest form of sharing.

The Quota Share Treaty

This is the most common type of proportional reinsurance. The two parties agree on a percentage (let's call it \( \alpha \)) that the ceding company will keep (this is the retention). The rest (\( 1 - \alpha \)) goes to the reinsurer.

How the math works:

If \( X \) is the total loss from a claim:

The Ceding Company pays: \( Y = \alpha \cdot X \)
The Reinsurer pays: \( Z = (1 - \alpha) \cdot X \)

Example: If a company has a 70% quota share treaty (meaning they keep 70%), and a claim of \$1,000 comes in:
\nThe Ceding Company pays: \( 0.70 \cdot 1,000 = \$700 \)
The Reinsurer pays: \( 0.30 \cdot 1,000 = \$300 \)

Step-by-Step Explanation:
\n1. Identify the retention percentage (\( \alpha \)).
\n2. Multiply the total claim amount by \( \alpha \) to find the ceding company's share.
\n3. The leftover amount is what the reinsurer pays.

Quick Review: Proportional Pros & Cons

• Pro: Very simple to administer.
\n• Pro: The reinsurer also gets that same percentage of the premiums!
\n• Con: The ceding company still pays a portion of every single claim, even the tiny ones.

2. Excess of Loss (XL) Reinsurance: The Safety Net

Excess of Loss is non-proportional. It’s like a deductible for the insurance company. The ceding company agrees to pay for all losses up to a certain dollar amount (called the retention level or attachment point, denoted as \( d \)). Anything over that amount is paid by the reinsurer.

How the math works:

If \( X \) is the total loss and \( d \) is the retention level:
\nThe Ceding Company pays: \( Y = \min(X, d) \)
\nThe Reinsurer pays: \( Z = \max(0, X - d) \)

Analogy: Think of a "Safety Net." If you fall, the net only catches you if you fall further than a certain distance. If you just trip, you handle it yourself. If you fall off a ladder, the net catches the "excess" part of the fall.

Example: An insurer has an XL treaty with a retention of \$50,000.
• Case A: A claim of \$10,000. The insurer pays the whole \$10,000. The reinsurer pays \$0.
\n• Case B: A claim of \$150,000. The insurer pays \$50,000. The reinsurer pays the "excess" \$100,000.

Did you know? This type of reinsurance is great for protecting against "catastrophic" losses while letting the insurer keep all the profit from smaller, everyday claims.

3. Two Types of Excess of Loss

Don't worry if this seems tricky at first; just remember that the difference is simply what we are totaling up before we apply the deductible.

A. Per Risk Excess of Loss

The retention \( d \) applies to each individual claim. If ten different houses burn down, the deductible applies to each house separately.

B. Aggregate Excess of Loss (Stop-Loss)

The retention \( d \) applies to the total sum of all claims over a period (usually a year). Once the ceding company has paid out a total of \( d \) across all its customers, the reinsurer steps in to pay everything else for the rest of the year.

Key Takeaway: Per-risk protects against one "giant" claim. Aggregate/Stop-loss protects against a "bad year" where many small claims add up.

4. The Impact of Inflation (Common Exam Trap!)

In Exam FAM, they love to ask how inflation affects these two types of reinsurance. Pay close attention here!

Proportional: Inflation hits both parties equally. If all claims go up by 10%, the ceding company’s share goes up 10% and the reinsurer’s share goes up 10%. Easy!

Excess of Loss: Inflation hits the Reinsurer much harder!
Why? Because the ceding company's payment is capped at \( d \). Once the claim hits that cap, every extra dollar of inflation is paid entirely by the reinsurer.

Example: Retention \( d = 100 \).
Original claim: 110. Ceding pays 100, Reinsurer pays 10.
If 10% inflation makes the claim 121: Ceding still pays 100, but Reinsurer now pays 21.
The claim went up 10%, but the Reinsurer’s cost went up 110%!

5. Summary and Common Mistakes

Memory Aid: The "P" Rule

• Proportional = Percentage split.
• Primary pays a Part of every claim.

Common Mistakes to Avoid:

1. Mixing up the percentage: Always double-check if the problem says "the amount ceded" (given away) or "the amount retained" (kept).
2. Applying XL formulas to Proportional: Remember, in Quota Share, there is no "deductible." They share from the very first dollar.
3. Forgetting the Cap: In Excess of Loss, the Ceding Company never pays more than \( d \). If your calculation shows them paying more than the retention, something is wrong!

Quick Review Box

Quota Share: \( Y = \alpha X \) | \( Z = (1-\alpha)X \)
Excess of Loss: \( Y = \min(X, d) \) | \( Z = \max(0, X-d) \)
Inflation: Affects XL reinsurers disproportionately because the ceding company's loss is capped.

Keep practicing these formulas with different claim amounts, and you'll find that reinsurance is one of the most logical parts of the FAM curriculum. You've got this!