Expenses and the profit and contingencies loading in ratemaking

Exam FAM – Fundamentals of Actuarial Mathematics · SOA (Society of Actuaries) · 5 min read

Welcome to Ratemaking: Beyond Just Losses!

In your Exam FAM journey, you’ve spent a lot of time learning how to predict losses. But here is a secret: if an insurance company only charged enough to cover losses, they would go out of business by lunchtime! To survive and thrive, an insurer must cover its overhead costs and build in a little extra for a rainy day and profit. This is where Expenses and Profit & Contingencies come in.

In this chapter, we are going to learn how to bridge the gap between the "Pure Premium" (what covers claims) and the "Gross Premium" (what the customer actually pays).

1. Breaking Down Expenses

Not all costs are created equal. In ratemaking, we split expenses into two main buckets based on how they behave when we sell a policy.

Fixed Expenses

Fixed Expenses are costs that stay the same regardless of how much the premium is. Think of these as the "cost of keeping the lights on." Whether a policy costs \$500 or \$5,000, these expenses don't change much per policy.
• Examples: Building rent, salaries of the IT department, or the cost of the actuarial software.
• Analogy: Imagine you own a pizza shop. The rent for the building is the same whether you sell a small cheese pizza or a giant "everything" pizza. That’s a fixed expense.

Variable Expenses

Variable Expenses are costs that change in direct proportion to the premium. These are often expressed as a percentage.
• Examples: Premium taxes (the state takes a cut of every dollar you collect) and commissions (the agent gets a percentage of the sale).
• Analogy: In our pizza shop, the cost of the cardboard box is a variable expense. If you sell more pizzas, you spend more on boxes.

Quick Review:
Fixed: Stays the same per unit (expressed in dollars, $ F $).
Variable: Changes with the premium (expressed as a percentage, $ V $).

2. Profit and Contingencies (P&C) Loading

Insurance is a risky business. Even the best actuaries can’t predict the future perfectly. Because of this, companies add a Profit and Contingencies Loading (usually denoted as $ Q $).

• Profit: This is what the company keeps to satisfy shareholders and grow the business.
• Contingencies: This is an extra "safety cushion" in case the actual losses are much higher than the expected losses.

Did you know? This loading is almost always treated as a variable component—it’s calculated as a percentage of the final premium.

3. The Fundamental Formula: Putting it All Together

Don't worry if formulas look intimidating! Let’s build it logically. We want a Gross Premium ($ G $) that covers four things:
1. Expected Losses (Pure Premium, $ P $)
2. Fixed Expenses ($ F $)
3. Variable Expenses ($ V \times G $)
4. Profit & Contingencies ($ Q \times G $)

Mathematically, it looks like this:
$ G = P + F + (V \times G) + (Q \times G) $

If we use a little bit of algebra to move all the $ G $’s to one side, we get the Master Formula you need for the exam:

$ G = \frac{P + F}{1 - V - Q} $

Step-by-Step Calculation:

1. Identify your Pure Premium ($ P $): This is your expected loss.
2. Identify your Fixed Expenses ($ F $): Usually given as a dollar amount per exposure.
3. Sum your Variable items: Add your variable expense % ($ V $) and your profit % ($ Q $).
4. Find the "Permissible Loss Ratio": Subtract the variable percentages from 1 (this is the denominator).
5. Divide: Take your numerator ($ P + F $) and divide by your denominator.

4. Key Term: The Permissible Loss Ratio (PLR)

This is a favorite topic for exam writers. The Permissible Loss Ratio (also called the Target Loss Ratio) is the portion of the premium that is "available" to pay for losses and fixed expenses after the variable expenses and profit have been taken out.

Formula: $ PLR = 1 - V - Q $

If your actual loss ratio is higher than the PLR, the company is likely losing money. If it's lower, the company is performing better than expected!

5. Common Pitfalls to Avoid

1. Mixing Percentages and Decimals: If variable expenses are 15% and profit is 5%, make sure you use $ 0.15 $ and $ 0.05 $ in your formula. Using "15" and "5" will give you a very strange (and wrong!) answer.
2. Fixed vs. Variable Confusion: Always double-check if an expense is given as a dollar amount (Fixed) or a percentage (Variable). Taxes and Commissions are almost always variable.
3. Forgetting the Numerator: Remember that Fixed Expenses ($ F $) go in the numerator with the losses, while Variable Expenses ($ V $) go in the denominator.

6. Summary and Key Takeaways

• Gross Premium is the final price charged to the policyholder.
• Fixed Expenses are flat dollar amounts included in the numerator.
• Variable Expenses and Profit are percentages subtracted from 1 in the denominator.
• The Master Formula: $ G = \frac{P + F}{1 - V - Q} $
• PLR: The target ratio of (Losses + Fixed Expenses) to total Premium.

Memory Aid: "Fixed stays on top, Variables drop to the bottom!"

Think of the numerator as the "Total Costs" in dollars and the denominator as the "Keep Rate" (the percentage of each dollar the company gets to keep to pay those costs). You are simply dividing the costs by the keep rate to find the total premium!

Quick check

Can you answer these now?

Open each question to check the key ideas from this chapter.

What is the Gross Premium in insurance ratemaking?

The total price charged to the customer, covering expected losses, all expenses, and the target profit.

What are fixed expenses ($ F $) in the context of ratemaking?

Operating costs that remain constant as a flat dollar amount per policy, regardless of the premium size.

How are variable expenses ($ V $) typically expressed and calculated?

They are expressed as a percentage of the Gross Premium, changing in direct proportion to the sale price.

What is the purpose of the Profit and Contingencies ($ Q $) loading?

To provide a return for shareholders and a safety cushion in case actual losses exceed expected losses.

What is the Master Formula used to calculate the Gross Premium ($ G $)?

$ G = \frac{P + F}{1 - V - Q} $

How is the Permissible Loss Ratio (PLR) defined and calculated?

The portion of premium available to pay for losses and fixed expenses: $ PLR = 1 - V - Q $.

What is a common arithmetic pitfall when using the Gross Premium formula?

Mixing percentages and decimals (e.g., using 15 instead of 0.15 for a 15% expense ratio).

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