Welcome to the World of Ratemaking!

Hello there! Today, we are diving into one of the most practical parts of the FAM exam: Adjustments to Ratemaking Data. Think of an actuary as a chef who is trying to predict how many ingredients they need for a huge party next year. They look at what happened at last year's party, but things have changed—prices are up, more people might show up, and some bills from last year haven't even arrived yet!

In this chapter, we learn how to "clean up" historical data so it can accurately predict the future. We focus on three main tools: Premium On-Leveling, Loss Development, and Trending. Don't worry if these sound intimidating; we will break them down step-by-step!

1. Premium On-Leveling: Standardizing the Income

Insurance companies change their prices (rates) frequently. If you are looking at the total premium collected three years ago, it was based on old prices. To use that data to set future prices, we need to figure out what that premium would have been if we had charged today’s prices back then.

Why do we do this?

Imagine you sold 100 apples for $1 each last year ($100 total). This year, you charge $2. If you want to predict next year's income based on volume, you shouldn't look at the $100; you should "on-level" it to $200 (100 apples \(\times\) $2).

Methods of On-Leveling

A. Extension of Exposures Method
This is the most accurate way. You take every single policy written in the past and recalculate its premium using the current rate manual.
Pros: Extremely accurate.
Cons: Requires a lot of data and computer power.

B. The Parallelogram Method
We use this when we don't have detail for every policy and only have aggregate data (like "Total Premium for 2023"). We use a geometric approach to find the Average Relative Rate Level.

How to use the Parallelogram Method:

1. Draw a square representing the time period (usually a calendar year).
2. Draw a diagonal line representing when a rate change occurred.
3. Calculate the area of the square that falls under the "new" rate versus the "old" rate.
4. The On-Level Factor is: \( \frac{\text{Current Rate Level}}{\text{Average Rate Level in Period}} \).

Quick Tip: If a rate change happens at time \(k\) (where \(k\) is a decimal of the year, like 0.75 for October 1st), the area of the triangle "after" the change is \( \frac{1}{2}(1-k)^2 \).

Summary: On-leveling makes sure our historical premium reflects today's price levels.

2. Loss Development: Predicting the "Tail"

When an accident happens today, the insurance company might not know the final cost for years (think of a long lawsuit). Loss Development is the process of adjusting "immature" losses to their "ultimate" expected value.

Key Terms:

- Paid Losses: Money already sent to claimants.
- Incurred Losses: Paid losses + Case Reserves (money set aside for known claims).
- IBNR (Incurred But Not Reported): Claims that happened but the company doesn't know about them yet.

The Chain Ladder Method (Age-to-Age Factors)

We use a Loss Development Triangle to see how losses grow over time. We calculate Age-to-Age Factors (ATAF), also known as link ratios.

Example: If losses at 12 months are $100 and they grow to $120 at 24 months, the ATAF is \( 120 / 100 = 1.20 \). This means we expect losses to grow by 20% during that second year.

Step-by-Step Process:

1. Calculate link ratios for each year in your triangle.
2. Average those ratios (Simple average, Weighted average, or Medial average).
3. Multiply the averages together to get the Age-to-Ultimate Factor.
4. Multiply your latest "known" losses by the Age-to-Ultimate factor to get the Ultimate Losses.

Did you know? Incurred losses usually develop upward (get more expensive) for liability insurance, but they might stay flat for simple things like glass breakage!

Summary: Loss development ensures we don't underestimate the total cost of claims just because they haven't been fully paid yet.

On-leveling looked at the past. Development looked at the present. Trending looks at the future! Even if we know the ultimate cost of last year's claims, next year's claims will likely be more expensive because of inflation, safer cars, or changing legal environments.

Two Components of Trend:

1. Frequency Trend: How often do claims happen? (e.g., Are people driving more?)
2. Severity Trend: How much does each claim cost? (e.g., Are hospital bills rising?)
3. Pure Premium Trend: The combination of both! \( \text{Pure Premium} = \text{Frequency} \times \text{Severity} \).

The Math of Trending

We usually assume Exponential Trending (like compound interest):
\( \text{Trended Value} = \text{Current Value} \times (1 + r)^n \)
Where:
- \( r \) is the annual trend rate (e.g., 0.05 for 5%).
- \( n \) is the Trend Period in years.

Calculating the Trend Period (\(n\)):

This is where students often get confused. The trend period is the time from the average accident date of your historical data to the average accident date of the period when the new rates will be in effect.

Rule of Thumb: For a one-year historical period and a one-year effective period:
1. Find the midpoint of the historical experience period.
2. Find the midpoint of the period the rates will be "live" (the effective period).
3. Count the years between those two midpoints.

Common Mistake: Don't forget that if a policy is written on the last day of the effective period, it provides coverage for another full year. This shifts the "average accident date" of the future period!

Summary: Trending adjusts our data for inflation and changes in behavior between the past and the future.

4. Putting It All Together: The Big Picture

To calculate a new rate, we need to find the Adjusted Loss Ratio. Here is the general flow:

1. Take Historical Losses and multiply by the Loss Development Factor (to get Ultimate Losses).
2. Multiply those by the Trend Factor (to get Future Ultimate Losses).
3. Take Historical Premium and multiply by the On-Level Factor (to see what we'd earn today).
4. Divide the adjusted losses by the adjusted premium.

Quick Review Box:
- On-leveling = Adjusts Premium (Old rates \(\rightarrow\) New rates).
- Development = Adjusts Losses (Immature \(\rightarrow\) Ultimate).
- Trending = Adjusts Losses/Premium (Past/Present \(\rightarrow\) Future).

Final Encouragement

Ratemaking is all about consistency. As long as you remember to move all your "dollars" to the same point in time (the future effective period), the logic will hold up. Keep practicing those parallelogram areas and trend period timelines—they are the most common places to lose points, but they are easy to master with a bit of drawing! You've got this!