Welcome to Profit Analysis!

Hello future actuaries! Welcome to one of the most practical chapters in the ALTAM curriculum. Up until now, you’ve spent a lot of time calculating premiums and reserves based on "best estimates" or "pricing assumptions." But in the real world, things rarely go exactly as planned. Interest rates fluctuate, people live longer (or shorter) than expected, and business expenses can change.

In this chapter, we learn how to measure the difference between what we expected to happen and what actually happened. This is called Profit Analysis. It is the "post-game show" of actuarial science where we figure out exactly why the company made or lost money. Let's dive in!

1. The Foundation: Expected vs. Actual

Before we look at complex formulas, let’s use a simple analogy. Imagine you are running a lemonade stand.
Expected: You think you’ll sell 10 cups at $1.00 each, and the lemons will cost you $2.00.
Actual: You sell 12 cups, but the price of lemons went up to $3.00.
Your "Profit Analysis" would show a gain from higher sales but a loss from higher costs.

In life insurance, we do the same thing. We compare our assumptions (the math we used to set premiums and reserves) against the experience (what really happened during the year).

Key Terms to Know

Expected Profit: The profit we projected using our valuation assumptions. Often, if we are using "Net Premium" reserves, the expected profit is actually zero because we assume the premium perfectly covers the benefits and reserve changes. In "Gross Premium" valuation, the expected profit is the margin built into the premium.

Actual Profit: The surplus that remains at the end of the year after receiving actual premiums, paying actual expenses, earning actual interest, and paying actual death benefits.

Quick Review: If Actual > Expected, we have a Profit. If Actual < Expected, we have a Loss (or a smaller profit than planned).

2. The Recursive Nature of Profit

To calculate profit, we look at the change over one year (from time \( t \) to \( t+1 \)). We use the Recursive Reserve Formula as our baseline. Don't worry if this seems tricky; just think of it as a "balance sheet" for one policy.

The standard recursive formula for a reserve \( _{t}V \) is:
\( (_{t}V + P_t - e_t)(1+i) = q_{x+t} \cdot S_{t+1} + (1 - q_{x+t}) \cdot _{t+1}V \)

Where:
\( P_t \) = Gross Premium
\( e_t \) = Expenses
\( i \) = Assumed Interest Rate
\( q_{x+t} \) = Assumed Mortality Rate
\( S_{t+1} \) = Death Benefit (Sum Assured)

Actual Profit (\( Pr_{t+1} \)) is calculated by taking that same formula but using actual values (denoted with a "prime" symbol \( ' \)) and seeing what is left over at the end of the year:

\( Pr_{t+1} = (_{t}V + P_t - e'_t)(1+i') - [q'_{x+t} \cdot S_{t+1} + (1 - q'_{x+t}) \cdot _{t+1}V] \)

Why do we subtract the reserve at the end?

We must always set aside the reserve (\( _{t+1}V \)) for the survivors to ensure we can pay future claims. Profit is only what is "left over" after the reserve is fully funded.

3. Sources of Profit (The Decomposition)

This is the "meat" of the chapter. Management doesn't just want to know that they made a million dollars; they want to know how. We break the total profit down into three main "Gains":

A. Gain from Interest (\( G_i \))

We earned a different interest rate than we expected.
Formula: \( G_i = (_{t}V + P_t - e'_t) \cdot (i' - i) \)
Logic: We take the money we had at the start of the year (after expenses) and multiply it by the "extra" interest we earned.

B. Gain from Mortality (\( G_m \))

Fewer people died than we expected (for life insurance).
Formula: \( G_m = (q_{x+t} - q'_{x+t}) \cdot (S_{t+1} - _{t+1}V) \)
Key Term: \( (S_{t+1} - _{t+1}V) \) is called the Net Amount at Risk (NAR).
Logic: If a person dies, the company loses the death benefit but "saves" the reserve they were holding. So the net loss is the difference. If fewer people die (\( q > q' \)), we save that NAR!

C. Gain from Expenses (\( G_e \))

We spent less on administration than we thought.
Formula: \( G_e = (e_t - e'_t) \cdot (1+i) \)
Note: We usually multiply by the interest rate because expenses happen at the start of the year, so the "saving" earns interest all year long.

Important Note: The Total Profit is simply the sum of these gains:
\( Pr_{t+1} = G_i + G_m + G_e \)

4. Analysis of Surplus (Step-by-Step)

When solving Exam ALTAM problems, you might be asked to find the gain from one specific source. Always follow this order to avoid confusion:

  1. Identify the "Assumed" (Basis) values: These are usually the values used to calculate the reserve.
  2. Identify the "Actual" (Experience) values: These are given in the problem as "during the year..." or "the actual experience was...".
  3. Calculate the Gains one by one: Use the formulas above.
  4. Check your work: If the problem provides the total actual profit, make sure your individual gains add up to that total!

Common Mistake to Avoid: When calculating the Mortality Gain, many students forget to subtract the reserve (\( _{t+1}V \)) from the death benefit. Remember: the company doesn't lose the whole \( S_{t+1} \); they were already planning to eventually pay out the \( _{t+1}V \)!

5. Did You Know? (The Role of Prudence)

Did you know? Most insurance companies use "Prudent" assumptions rather than "Best Estimate" assumptions for reserves. This means they assume interest rates will be slightly lower and mortality will be slightly higher than they actually expect. This builds in a "cushion," meaning that in a normal year, the Expected Profit is positive. This cushion is often called the Provision for Adverse Deviation (PAD).

Summary / Key Takeaways:
- Profit Analysis identifies why actual results differ from expectations.
- Interest Gain: Based on the difference between actual and assumed interest rates applied to the starting funds.
- Mortality Gain: Based on the difference between assumed and actual death rates applied to the Net Amount at Risk.
- Expense Gain: Based on the savings in expenses, including the interest those savings earned.

6. Quick Review Box

Formula Cheat Sheet:
- NAR: \( S - V \)
- Gain Interest: \( (\text{Starting Funds}) \times (i_{act} - i_{exp}) \)
- Gain Mortality: \( (q_{exp} - q_{act}) \times (NAR) \)
- Gain Expense: \( (e_{exp} - e_{act}) \times (1+i) \)

Keep practicing! Profit analysis is like detective work—you are looking for where the money went. Once you master the "Gains" formulas, you'll find these points are some of the most reliable ones you can get on the exam!