Introduction to Profit Testing

Welcome to one of the most practical parts of the CM1 syllabus! In previous chapters, you learned how to calculate premiums using the equivalence principle. However, in the real world, insurance companies don't just want to "break even"—they want to make a profit to reward their shareholders and ensure financial stability.

Profit testing is the process of projecting the year-by-year cashflows of an insurance policy to see how much profit it will generate. Think of it like a business plan for a single policy. We are going to look at four key metrics that tell us if a product is a "winner" or a "loser."

1. The Profit Vector \( (\underline{Pr}) \)

The profit vector is the starting point of our analysis. It is a list (a vector) of the expected profits arising at the end of each policy year, provided that the policy is still in force at the start of that year.

To calculate the profit in year \( t \) (denoted as \( Pr_t \)), we generally follow this logic:

\( Pr_t = \) (Income) \( - \) (Outgo) \( - \) (Increase in Reserves)

Specifically, for each year \( t \), we consider:

  • Income: Premiums received and investment income earned on the funds held.
  • Outgo: Benefits paid (death, maturity, etc.) and expenses (commission, administration).
  • Reserves: We subtract the amount we need to set aside to pay for future liabilities.

Key Point: The profit vector is "per policy in force at the start of the year." It doesn't yet account for the fact that some policyholders might die or cancel their policies before reaching year 10.

Analogy: Imagine you own a gym. The "Profit Vector" for a specific member is the \$50 they pay you each month minus the \$10 it costs to provide them with towels and water. It assumes the member is still showing up!

2. The Profit Signature \( (\underline{S}) \)

The profit signature takes the profit vector and adjusts it for the probability that the policy actually stays on the books. This is the profit the company expects to see from the perspective of Time 0.

To get the signature (\( S_t \)), we multiply the profit vector by the probability that the policyholder survived all the "dangers" of previous years.

\( S_t = Pr_t \times {}_{t-1}p_x \) (or \( {}_{t-1}p_x^{00} \) in a multiple-decrement model)

Where:

  • \( Pr_t \) is the profit for year \( t \).
  • \( {}_{t-1}p_x \) is the probability that a life aged \( x \) is still "active" and in the contract at the beginning of year \( t \).

Why do we do this? If a policy has a huge profit in year 20, but there is only a 10% chance the person will live that long, that "big profit" isn't worth as much to the company today.

Quick Review:
Year 1: \( S_1 = Pr_1 \times 1 \) (Because they must be alive at start of year 1 to buy it!)
Year 2: \( S_2 = Pr_2 \times p_x \)
Year 3: \( S_3 = Pr_3 \times {}_2p_x \)

3. Net Present Value (NPV)

Now that we have our "Signature" (the expected profits in each future year), we need to collapse them into a single number today. This is the Net Present Value (NPV).

We discount the profit signature using a special interest rate called the Risk Discount Rate (RDR), denoted as \( r \). The RDR is usually higher than the standard investment return because it accounts for the risk the company is taking.

\( NPV = \sum_{t=1}^{n} S_t \times (1+r)^{-t} \)

Interpretation:
If \( NPV > 0 \): The product is expected to be profitable.
If \( NPV < 0 \): The product is expected to lose money.

Did you know? Companies often set their RDR based on what their shareholders expect as a minimum return. If shareholders want 10%, the RDR will be 10%.

4. Profit Margin

The NPV tells us the amount of profit, but it doesn't tell us if that profit is "efficient." For example, a \$1 million profit is great for a small local product, but tiny for a global multi-billion dollar project.

The Profit Margin (often just called the "Margin") scales the NPV by the size of the policy. It is defined as:

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

Important: Both the NPV and the PV of Premiums must be discounted at the Risk Discount Rate (RDR) to keep the comparison fair.

Key Takeaway: The margin tells you "for every \$1 of premium I collect, how many cents of profit do I keep?" A 5% margin means you keep 5 cents of profit for every \$1 of premium received (on a present value basis).

Step-by-Step Summary: The "Profit Pipeline"

If you get a calculation question in Paper A or Paper B (Excel), follow this logical flow:

  1. Calculate Cashflows: Find the net cashflow at the end of each year.
  2. Create the Profit Vector \( (\underline{Pr}) \): Adjust cashflows for interest and changes in reserves.
  3. Calculate the Profit Signature \( (\underline{S}) \): Multiply \( Pr_t \) by the probability of the policy being in force at the start of the year (\( {}_{t-1}p_x \)).
  4. Find the NPV: Discount the Signature to Time 0 using the Risk Discount Rate \( (r) \).
  5. Find the Margin: Divide the NPV by the PV of premiums (discounted at \( r \)).

Common Pitfalls to Avoid

  • Wrong Probability: Students often use \( {}_tp_x \) for the signature instead of \( {}_{t-1}p_x \). Remember, the profit \( Pr_t \) is earned at the end of the year by those who were there at the start.
  • Wrong Discount Rate: Ensure you use the investment rate (\( i \)) for calculating the profit vector (investment income), but the risk discount rate (\( r \)) for calculating the NPV and the Margin.
  • Timing: Be careful with whether premiums and expenses occur at the start or end of the year. This affects your interest calculations!

Key Takeaway Box:
Vector: Potential profit per person.
Signature: Expected profit for the company.
NPV: Total value today.
Margin: Profitability efficiency.