Overall average rates and rate changes: loss cost and loss ratio methods

Welcome to Ratemaking: How Much Should We Charge?

Hello, future actuary! Welcome to one of the most practical parts of the ASTAM syllabus. In this chapter, we are moving from "how much money do we owe for past claims?" (reserving) to "how much should we charge for future policies?" (pricing/ratemaking).

The goal of ratemaking is simple: set a price that is high enough to cover all claims and expenses while still allowing for a reasonable profit. If we charge too little, the company goes bankrupt. If we charge too much, customers leave for a competitor. It’s a delicate balancing act!

Don’t worry if this seems like a lot of math at first. Think of it like running a lemonade stand. You need to cover the cost of lemons (claims), the cost of the stand (fixed expenses), and the commission for the kid who helps you (variable expenses), all while making a little pocket money (profit).

Two Paths to the Same Goal

There are two primary methods we use to determine the correct rate. Both usually lead to the same answer, but they start from different angles:

1. The Pure Premium Method (Loss Cost Method): This method calculates an absolute dollar amount per exposure unit. It answers: "How many dollars should the rate be?"
2. The Loss Ratio Method: This method calculates a percentage change to the current rates. It answers: "By what percentage should we increase or decrease our current price?"

Key Takeaway

Pure Premium Method = "The new rate should be $500."
\nLoss Ratio Method = "The current rate should be increased by 5%."

1. The Pure Premium (Loss Cost) Method

We use this method when we have reliable data on the number of exposure units (like the number of cars insured). It is often used for new lines of business or when exposures are very stable.

The Formula

The Indicated Rate (the price we should charge) is calculated as:

\( \text{Indicated Rate} = \frac{\text{Pure Premium} + \text{Fixed Expense per unit}}{1 - \text{Variable Expense Ratio} - \text{Profit Margin}} \)

Breaking it Down

Pure Premium: This is the "Loss Cost." It is the expected loss and loss adjustment expenses (LAE) per unit of exposure. If you expect $200 in claims for every car you insure, your Pure Premium is $200.
\nFixed Expenses: These are costs that stay the same regardless of the premium (e.g., the cost of the building). We express this as a dollar amount per unit.
\nVariable Expenses: These are costs that change based on how much premium we collect (e.g., premium taxes or agent commissions). We express this as a percentage (decimal).
\nTarget Profit Margin: The percentage of every dollar we want to keep as profit.

Memory Aid: Think of the bottom of the fraction (the denominator) as the "money we get to keep" for our costs after the variable parts are taken out. This is often called the Variable Permissible Loss Ratio.

Step-by-Step Example

Imagine you insure 1,000 bikes. Total projected losses are $100,000. Your fixed expenses are $10 per bike. Your variable expenses are 15% and you want a 5% profit margin.

1. Calculate Pure Premium: \( \$100,000 / 1,000 = \$100 \)
\n2. Identify Fixed Expenses: \( \$10 \)
3. Identify Variable + Profit: \( 0.15 + 0.05 = 0.20 \)
4. Apply Formula: \( \frac{100 + 10}{1 - 0.20} = \frac{110}{0.80} = \$137.50 \)

2. The Loss Ratio Method

This method is more common for existing lines of business. It looks at how our current premiums compare to our losses to see if we need a raise or a discount.

The Formula

The Indicated Rate Change (the % we should move) is:

\( \text{Indicated Change} = \frac{\text{Experience Loss Ratio} + \text{Fixed Expense Ratio}}{1 - \text{Variable Expense Ratio} - \text{Profit Margin}} - 1 \)

What is the "Experience Loss Ratio"?

This is the most critical part! It is: \( \frac{\text{Projected Losses}}{\text{On-Level Earned Premium}} \)

Crucial Concept: On-Level Premium. Rates change over time. If you increased rates by 10% last year, the premiums you collected in the past aren't "at the current level." We have to adjust old premiums as if they had been charged at today's rates. This is called On-Leveling.

Quick Review: The Denominator

The denominator \( (1 - V - P) \) is often called the Target Loss Ratio or Permissible Loss Ratio (PLR). It represents the percentage of premium left over for losses and fixed expenses after accounting for variable expenses and profit.

Key Takeaway

If the result of the formula is 0.05, it means we need a 5% increase. If it’s -0.03, we need a 3% decrease.

Critical Adjustments: Preparing the Data

You can't just use raw historical numbers. Actuaries have to "clean" the data first. Did you know? Raw data is like raw vegetables—you have to prep them before they are ready for the final meal!

1. Loss Development

Old claims often grow in size as time goes on (IBNR). We use Loss Development Factors (LDFs) to "push" historical losses to their ultimate value. (Think back to your Reserving chapters!)

2. Trending

Inflation makes things more expensive every year. We apply Trend Factors to move losses from the past experience period into the future period when the new rates will be in effect. We usually trend both frequency (how often claims happen) and severity (how much each claim costs).

3. Catastrophes and Large Losses

A single massive hurricane can ruin your data. Actuaries often remove catastrophe losses from the data and replace them with a long-term "Catastrophe Loading" to keep the rates stable.

Comparison: Which Method to Use?

Use Pure Premium Method when:

• Exposures are easy to define (e.g., "per car").
• You are starting a new line of business.
• Current premiums are unavailable or unreliable.

Use Loss Ratio Method when:

• Exposures are hard to define (e.g., "complex commercial liability").
• You want to know the change from the current state.
• Data on historical premiums is readily available.

Common Pitfalls to Avoid

1. Mixing up Fixed and Variable Expenses: Always double-check if an expense is a dollar amount per unit (Fixed) or a percentage of premium (Variable). Putting them in the wrong place in the formula is a classic exam trap!
2. Forgetting to On-Level Premium: In the Loss Ratio method, you must use the current rate level for the premium. Using raw historical premium will give you a wrong answer.
3. Overlooking the "- 1": The Loss Ratio method gives you a "Ratio of New to Old." To get the change, you must subtract 1. If your formula gives 1.05, the change is 5%.

Summary Checklist for Success

✓ Pure Premium Method: Indicated Rate = (Loss + Fixed) / (1 - Variable - Profit)
✓ Loss Ratio Method: Indicated Change = (Loss Ratio + Fixed Ratio) / (PLR) - 1
✓ Permissible Loss Ratio: 1 - Variable Expenses - Profit Margin
✓ Adjustments: Always Develop and Trend your losses before plugging them into the formulas!

Keep practicing these formulas until they feel like second nature. You've got this!