Welcome to Premium Setting!

In our previous studies, we looked at how to calculate Net Premiums (which only cover benefits) and Gross Premiums (which cover benefits and expenses). But insurance companies aren't charities—they need to make a specific amount of profit to stay in business and satisfy their shareholders!

In this chapter, we are going to learn how to work backwards. Instead of just adding a small "load" to a premium, we will calculate the exact premium needed to hit a specific profit objective. Think of this like a chef pricing a meal: they don't just want to cover the cost of the ingredients; they want to make sure that at the end of the night, they have exactly 10% profit left over in the cash register.

Don't worry if the formulas look a bit long at first. We will break them down into simple steps that feel just like balancing a checkbook!

1. What is a Profit Objective?

A profit objective is a goal set by the actuary. The two most common objectives you will see on Exam ALTAM are:

  • Net Present Value (NPV) Objective: The company wants the total expected present value of all future profits to equal a specific dollar amount (e.g., \( \$100 \) per policy).
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  • Profit Margin Objective: The company wants the profit to be a specific percentage of the premiums collected. This is the most common approach in the real world!
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Quick Review: Remember that Profit in this context usually refers to the Emerging Profit (often denoted as \( Pr_t \)) which accounts for premiums, investment income, benefits, expenses, and the change in reserves.

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2. The Profit Margin Formula

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The Profit Margin (often denoted as \( k \)) is the ratio of the Expected Present Value of Profits to the Expected Present Value of Premiums.

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Mathematically, it looks like this:

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\[ \text{Profit Margin} (k) = \frac{NPV(0)}{E[PV(\text{Gross Premiums})]} \]

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Where:

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  • \( NPV(0) \) is the Net Present Value of profits at time 0, discounted at the hurdle rate (the company's required rate of return).
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  • \( E[PV(\text{Gross Premiums})] \) is the expected present value of the premiums the customer pays, also discounted at the hurdle rate.
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The Analogy: Imagine you sell lemonade. If it costs you \( \$0.80 \) to make a cup (including lemons, sugar, and your time) and you sell it for \( \$1.00 \), your profit is \( \$0.20 \). Your profit margin is \( 0.20 / 1.00 = 20\% \). In ALTAM, we are just doing this with "Time Value of Money" logic added in!

3. How to Calculate the Premium (\( G \))

To find the Gross Premium (\( G \)) that satisfies a specific profit margin \( k \), we use a "Components" approach. We break the total value of the policy into three parts:

  1. Benefits and Expenses: The "Outgo."
  2. Profit: The "Reward."
  3. Premium: The "Income."

The Core Equation:
The Present Value of Premiums must cover the Present Value of Benefits/Expenses plus the required Profit.

\[ E[PV(\text{Premiums})] = E[PV(\text{Benefits + Expenses})] + \text{Target Profit} \]

If our target profit is a margin \( k \), then:
\[ E[PV(\text{Premiums})] = E[PV(\text{Benefits + Expenses})] + k \times E[PV(\text{Premiums})] \]

Step-by-Step to solve for \( G \):
1. Group the premium terms together: \( E[PV(\text{Premiums})] \times (1 - k) = E[PV(\text{Benefits + Expenses})] \)
2. Since \( E[PV(\text{Premiums})] = G \times \ddot{a}_{x:n|} \), we can write:
\[ G = \frac{E[PV(\text{Benefits + Expenses})]}{(1 - k) \times \ddot{a}_{x:n|}} \]

Note: The annuity symbol \( \ddot{a}_{x:n|} \) must be calculated using the hurdle rate (the discount rate used for profit testing), not necessarily the pricing interest rate!

4. Important Nuances & Common Pitfalls

The Hurdle Rate vs. The Earned Rate

This is where many students get tripped up. In ALTAM profit testing:

  • The Earned Rate (\( i \)) is the interest the company actually earns on its assets.
  • The Hurdle Rate (\( r \)) is the interest rate used to discount the profits back to time zero.

Key Rule: When calculating the NPV for a profit margin objective, always use the Hurdle Rate (\( r \)) for your discounting factors (\( v \)).

Timing of Expenses

Remember that expenses usually come in two flavors:

  • Percentage of Premium: (e.g., commissions). These are handled by reducing the denominator in our premium formula.
  • Fixed Expenses: (e.g., \( \$50 \) per year). These are added to the numerator (the "Outgo" part).

Did you know? High-commission products (like many life insurance policies) often have negative profits in the first year because the expense of setting up the policy is higher than the first premium! This is why "NPV" is so important—it looks at the whole life of the policy, not just Year 1.

5. Summary and Key Takeaways

The "Big Secret" to solving these problems:

If you see a question asking for a premium based on a Profit Margin \( k \), follow this mental checklist:

  • Identify the target: \( \text{Profit} = k \times \text{Premium} \).
  • Set up the Balance: \( \text{Income} = \text{Outgo} + \text{Profit} \).
  • Check your Discount Rate: Ensure you are using the Hurdle Rate for the NPV and the annuity values.
  • Solve for \( G \): Treat \( G \) like \( x \) in an algebra equation and isolate it.

Common Mistake to Avoid: Don't forget that if the profit margin is \( 10\% \), you are effectively keeping \( 90\% \) of every dollar to pay for benefits and expenses. That's why we divide by \( (1 - 0.10) \) in the final formula!

Keep going! You've now mastered how to price a product to ensure a company stays profitable. This is one of the most practical skills an actuary can have.