Understand and use run-off triangles to estimate claims

Welcome to the World of Run-off Triangles!

Hello there! Welcome to one of the most practical and essential parts of the CM2 curriculum. If you have ever wondered how an insurance company knows how much money to set aside today for accidents that happened yesterday but won't be fully paid for until next year, you are in the right place.

In this chapter, we are going to learn about Run-off Triangles. Think of these as a "financial history book" that helps actuaries predict the future. Don't worry if the numbers look intimidating at first—we will break them down step-by-step until you're a pro at estimating claims!

1. The Basics: Why do we need Triangles?

In General Insurance (like car or home insurance), there is often a delay between an accident happening and the final claim being paid. For example, if you crash your car in December 2023, the insurance company might not finish paying for the repairs and legal fees until 2025.

Because of this delay, we need a way to track data. We use two time-scales:
1. Accident Year (i): The year the event actually happened.
2. Development Year (j): How many years have passed since the accident year.

Did you know?
The delay between an accident and the final payment is often called the "tail." Some insurance lines (like Motor) have short tails, while others (like Liability or Asbestos claims) have very "long tails" that can last decades!

Key Terms to Remember

IBNR (Incurred But Not Reported): These are "hidden" claims. The accident has happened, but the insurance company doesn't know about it yet.
Outstanding Liabilities: The total amount of money the insurer expects to pay in the future for accidents that have already occurred.
Ultimate Claims: The final, total cost of all claims for a specific year once every single penny has been paid.

Quick Review: We use triangles to organize data so we can see how claims "develop" over time. This helps us estimate the total amount we will eventually have to pay.

2. Understanding the Layout of a Triangle

A run-off triangle usually looks like a right-angled triangle.
The Rows represent the Origin Year (the year the policy started or the accident happened).
The Columns represent the Development Year (the age of the claim).
The Cells contain the claim amounts (either the amount paid in that specific year or the total amount paid up to that point).

The Golden Rule:
Always check if the triangle is Incremental or Cumulative.
Incremental: Just the money paid in that specific window of time.
Cumulative: The running total of all money paid for that accident year up to that point.

Analogy: Imagine a piggy bank for the year 2023.
Incremental = How many coins you dropped in each month.
Cumulative = The total weight of the piggy bank at the end of each month.

3. The Basic Chain Ladder Method

The Chain Ladder Method is the most common way to estimate future claims. It assumes that the pattern of claims seen in the past will continue in the future.

Step-by-Step Guide to the Chain Ladder

Step 1: Make it Cumulative
If your triangle is incremental, add the numbers up across the rows to make it cumulative. For example, if year 0 is \$100 and year 1 is \$50, the cumulative value for year 1 is \$150.

Step 2: Calculate Link Factors (Development Factors)
We want to know how much the claims "grow" from one year to the next. We calculate a factor \( f_j \) using this formula:
\( f_j = \frac{\sum \text{Claims in development year } j+1}{\sum \text{Claims in development year } j} \)
(Note: You only sum the rows where both years of data are available!)

Step 3: Project the Ultimate Claims
Multiply the latest known cumulative claim amount by the link factors for all the future years. This gives you the Ultimate Claim amount.

Step 4: Calculate the Reserve
Reserve = Ultimate Claim - Claims Paid to Date.

Common Mistake to Avoid:
When calculating the sum for link factors, make sure you don't include the "bottom" cell of a column if there is no corresponding data in the next column. You must use the same set of accident years for both the numerator and the denominator!

4. The Bornhuetter-Ferguson (BF) Method

Sometimes, the Chain Ladder method is a bit "jumpy." If one tiny claim in a very recent year happens to be unusually large, the Chain Ladder will multiply that "error" and give you a crazy result. The BF Method is more stable.

The BF method blends two things:
1. An initial estimate of ultimate claims (usually based on a Loss Ratio or expert judgment).
2. The actual data we have seen so far.

The BF Formula

The Reserve (the amount still to be paid) is calculated as:
\( \text{Reserve} = \text{Initial Estimate of Ultimate Claims} \times (1 - \frac{1}{\text{Cumulative Link Factor}}) \)

Memory Aid:
Think of the BF method as a weighted average. If we are early in the development of a year, we trust our initial guess more. As we get more data, we trust the actual experience more.

Key Takeaway: Use Chain Ladder when you have stable, mature data. Use BF when the data is "young" or volatile.

5. Dealing with Inflation

In the real world, prices go up! When using run-off triangles, we must consider two types of inflation:
1. Past Inflation: Already included in the numbers in our triangle.
2. Future Inflation: Not yet in the numbers, but it will affect future payments.

To handle this, we often use the Inflation-Adjusted Chain Ladder:
1. Adjust to current prices: Divide the historical data by an inflation index to put all numbers in "today's money."
2. Perform the Chain Ladder: Calculate link factors on these "constant price" claims.
3. Re-inflate: When projecting future payments, multiply them by the expected future inflation for those specific future years.

Important Point:
If inflation has been constant in the past and is expected to stay the same in the future, the basic Chain Ladder handles it automatically. You only need to do these extra steps if you expect inflation to change.

6. Limitations and Considerations

While these methods are powerful, they aren't perfect. Here are a few things that can "break" our models:

1. Changes in Law: A new law might suddenly make claims more expensive.
2. Changes in Policy: If the company starts selling to riskier drivers, the old patterns won't work.
3. Large One-Off Claims: A single massive claim (like a giant oil spill) can skew the averages.
4. Speed of Processing: If the claims department gets a new computer system and starts paying claims faster, the "development pattern" changes, making the link factors misleading.

Quick Review Box:
Chain Ladder: Relies purely on the development pattern of the triangle.
BF Method: Uses an external "initial guess" to stabilize the result.
Ultimate: The total cost once all claims are finished.
Reserve: Ultimate minus what we've already paid.

Summary of Key Concepts

Congratulations! You've covered the core of Run-off Triangles. Remember these three main takeaways:
- Triangles are tools to organize claims by the year they happened and how long they've been developing.
- The Chain Ladder is about finding "multipliers" (link factors) to grow current claims to their final size.
- The BF Method is a safer, hybrid approach that uses an initial estimate to avoid being fooled by weird data points in recent years.

Don't worry if the link factor calculations feel tedious—practice makes perfect! Try a few past paper questions, and you'll see the pattern quickly emerges. You've got this!