Introduction to Pricing via Profit Testing

In your earlier studies, you learned about the equivalence principle. This is the "break-even" approach where the expected present value of premiums equals the expected present value of benefits and expenses. While that’s a great starting point, real-world insurance companies aren't just looking to break even—they need to make a profit to satisfy shareholders and provide a cushion against risks.

Using a profit test to price a contract is a more dynamic approach. Instead of solving a single equation, we project the cashflows year by year, see how much profit we make, and adjust the premium until we hit a specific profit target. This method is the "gold standard" for products like unit-linked contracts or complex conventional policies where the timing of cashflows really matters.

The Core Concept: Moving Beyond Equivalence

When we price using a profit test, we are usually looking to satisfy a specific profit criterion. Common criteria include:

  • Profit Margin: The Net Present Value (NPV) of profits divided by the Net Present Value of premiums should equal a target percentage (e.g., \(5\%\)).
  • NPV: The total discounted value of profits should be a specific monetary amount.
  • Internal Rate of Return (IRR): The interest rate at which the NPV of profits is zero (the yield to the company) should exceed a "hurdle rate."

Think of it like this: If the equivalence principle is finding a price that pays for the ingredients of a cake, profit testing is finding a price that pays for the ingredients, the baker's time, the electricity, AND leaves the bakery with enough money to grow the business!

The Step-by-Step Pricing Process

Pricing via profit testing is often an iterative process (trial and error), especially in Paper B (Excel). Here is the logical flow:

1. Select an Initial Premium

You have to start somewhere! Often, you might calculate a "starting" premium using the equivalence principle as a baseline, or the question might provide a trial premium to test.

2. Set Up the Profit Test Model

You project the expected cashflows for a single policy (or a model office) based on assumptions for:

  • Decrements: Mortality, surrenders, and lapses (using a multiple decrement model).
  • Investment Return: The interest the company expects to earn on its funds.
  • Expenses: Initial, renewal, and termination expenses.
  • Reserves: Any money that must be set aside to meet future liabilities.

3. Calculate the Profit Vector and Signature

The profit vector gives the expected profit at the end of each year for a policy that is still in force. The profit signature (\(S_t\)) adjusts these values for the probability that the policy actually survives to that year. (Note: These are covered in detail in the "Profit vector, profit signature..." chapter.)

4. Calculate the Profit Criterion

Discount the profit signature at the risk discount rate to find the \(NPV\). Then, check if your target is met. For example, if you want a \(10\%\) profit margin:

\(Profit Margin = \frac{NPV}{PV(Premiums)}\)

5. Adjust and Repeat

If the profit margin is too low, increase the premium. If it's too high, decrease it. In an exam (especially CM1B), you can use Excel's Goal Seek tool to find the exact premium that makes the \(NPV\) or Profit Margin hit the target perfectly.

Key Variables: What Changes the Price?

When pricing, the actuary must decide on the assumptions. These are critical because a small change can make a product look very profitable or very risky:

  • Risk Discount Rate (RDR): This is the interest rate the company uses to discount its profits. A higher \(RDR\) means the company values future profits less, which usually means they need a higher premium today to meet their profit targets.
  • Assumed Interest Rate: This is what the company earns on its assets. If we assume we will earn more interest, we can afford to charge a lower premium.
  • Expenses: Higher expected expenses lead to higher premiums.
Quick Review: The Goal Seek Logic

In the exam, you might be asked: "Calculate the premium required to achieve a profit margin of \(7.5\%\)."
The logic:
1. Set the Profit Margin cell as the "Set Cell".
2. Enter \(0.075\) as the "To Value".
3. Set the Premium cell as the "By Changing Cell".

Common Pitfalls to Avoid

1. Confusion of Interest Rates: Students often mix up the investment return (earned on the fund) and the risk discount rate (used to discount the profits). Remember: The RDR is the company's "hurdle rate"—the return shareholders demand for the risk they are taking.

2. Forgetting Expenses: In pricing, you must include all expenses (initial, renewal, and claim-related). Missing the initial commission will vastly overstate the profit!

3. Timing of Cashflows: Pay close attention to whether premiums are paid at the start of the year (in advance) and whether claims/expenses are paid at the start, middle, or end of the year. This affects the discounting by \(v^{1/2}\) or \(v\).

Summary of Pricing via Profit Testing

  • It is an iterative approach to find a premium that meets a specific profit goal.
  • It accounts for the timing of cashflows and reserves more accurately than simple formulas.
  • The Profit Margin is the most common target: \(Margin = \frac{NPV \text{ of profits}}{PV \text{ of premiums}}\).
  • In Paper B, use Goal Seek. In Paper A, you might be asked to "show that" a premium works or perform one step of the iteration.

Pro Tip: Don't worry if your first guess for a premium is way off. The profit test is designed to be adjusted. Focus on setting up your cashflow columns correctly first!