Analysis of gains by source

Welcome to the Analysis of Gains!

Hello there! If you’ve made it to the Analysis of Gains by Source, you are already deep into the world of Advanced Long-Term Actuarial Mathematics. Don't worry if this chapter feels a bit like detective work at first—that's exactly what it is!

In the previous chapters, we learned how to calculate premiums and reserves based on certain assumptions (like how many people will die or how much interest we will earn). But in the real world, things never go exactly as planned. This chapter teaches us how to look back at the end of the year and figure out exactly why our actual profit was different from what we expected. Was it because we spent too much on administration? Or because the stock market performed better than thought? Let's dive in and find out!

1. The Big Picture: What is a "Gain"?

In actuarial terms, a gain (also called a profit or surplus) occurs when the actual experience of the insurance company is "better" than what was assumed in the valuation of reserves.

Think of it like a personal budget. If you expected your electric bill to be \$100 but it was only \$80, you have a "gain" of \$20. In ALTAM, we do this for three main categories: Expenses, Interest, and Mortality.

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Quick Review: The Recursive Reserve Formula

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Before we calculate gains, we must remember our "Home Base" formula. The reserve at the end of the year is built from the reserve at the start, plus premiums, minus expenses, plus interest, minus the cost of insurance (death benefits).

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\( (V_t + P_t - E_t)(1 + i) - q_{x+t}(S_{t+1} - V_{t+1}) = V_{t+1} \)

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Where:
\n\(V_t\) = Reserve at time \(t\)
\n\(P_t\) = Premium received at time \(t\)
\n\(E_t\) = Expenses paid at time \(t\)
\n\(i\) = Interest rate
\n\(q_{x+t}\) = Probability of death
\n\(S_{t+1}\) = Death benefit paid at \(t+1\)

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2. Breaking Down the Gain: The Three Main Sources

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We typically break the total gain into three specific buckets. It is important to calculate them in a specific order to ensure that the sum of the individual gains equals the total gain.

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

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This is the difference between what we thought we would spend on running the policy and what we actually spent.

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The Concept: If actual expenses (\(E_t^{act}\)) are lower than expected expenses (\(E_t^{exp}\)), the company saves money. Since expenses are paid at the start of the year, we also account for the interest that "saved" money would have earned.

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The Formula:
\n\( G_e = (E_t^{exp} - E_t^{act})(1 + i) \)

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Common Mistake: Forgetting to include the interest! Because expenses usually happen at the beginning of the year, the gain must be rolled forward to the end of the year to be comparable with other gains.

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

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This is the difference between the interest rate we earned (\(i^{act}\)) and the interest rate we assumed (\(i^{exp}\)).

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The Concept: We apply the difference in interest rates to all the money we had sitting in the "pot" throughout the year. This includes the starting reserve, the premium, and the actual expenses paid.

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The Formula:
\n\( G_i = (V_t + P_t - E_t^{act})(i^{act} - i^{exp}) \)

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Analogy: Imagine you put \$100 in a savings account expecting 2% interest, but the bank gave you 5%. Your "gain" is that extra 3% on your \$100.

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

This is the difference between the number of people we expected to die and the number who actually died.

The Concept: This is often the most confusing part. When someone dies, the company pays the death benefit (\(S\)) but "saves" the reserve (\(V\)) they were holding for that person. The net cost to the company is the Death Strain at Risk (DSAR), which is \((S - V)\).

The Formula:
\( G_m = (q_{x+t}^{exp} - q_{x+t}^{act})(S_{t+1} - V_{t+1}) \)

Did you know? If actual mortality is higher than expected, the gain will be negative (a loss) for life insurance, but it might be positive for a pension or annuity! This is because, in an annuity, the company stops paying when someone dies.

Key Takeaway Summary

Total Gain = \(G_e + G_i + G_m\)
Always use Actual values for components already calculated in the sequence. For example, when calculating the Gain from Interest, use the actual expenses.

3. The Step-by-Step Process: How to Solve Problems

When you see a problem asking for the "Analysis of Gains," follow these steps to stay organized:

Step 1: Calculate the Total Gain.
Find the difference between the actual year-end reserve and what the reserve would have been using actual experience. Alternatively, use the actual surplus formula.

Step 2: Calculate Gain from Expenses first.
Use the expected interest rate.
\( G_e = (E^{exp} - E^{act})(1 + i^{exp}) \)

Step 3: Calculate Gain from Interest second.
Use the actual expenses you just dealt with.
\( G_i = (V_t + P_t - E^{act})(i^{act} - i^{exp}) \)

Step 4: Calculate Gain from Mortality last.
Use the expected mortality rate vs. the actual mortality rate multiplied by the Death Strain at Risk.
\( G_m = (q^{exp} - q^{act})(S - V_{t+1}) \)

Step 5: Check your work!
Add \(G_e + G_i + G_m\). If they don't equal your Total Gain, go back and check your interest rate timing or your signs!

4. Common Pitfalls to Avoid

1. The Sign Error: Remember that for expenses, (Expected - Actual) is a gain. For interest, (Actual - Expected) is a gain. If you mix these up, your whole analysis will be backwards!

2. The "Whoops, wrong Reserve": Always use the valuation reserve (the one calculated with expected assumptions) when calculating the Death Strain at Risk, unless the problem specifically tells you otherwise.

3. Timing of Cash Flows: In ALTAM, premiums and expenses are usually at the start of the year, while death benefits and interest are at the end of the year. Pay close attention to when the money moves!

5. Why does the order matter?

"Don't worry if this seems tricky at first..." Many students ask why we have to go in a specific order.

If we calculated mortality gain first using expected interest, and then interest gain using actual mortality, we might miss the "interaction" between the two. By following a consistent order (usually Expenses -> Interest -> Mortality), we ensure that every dollar of profit is assigned to exactly one source, leaving no "leftover" unexplained profit.

Quick Review Box

• Gain from Expenses: (Expected Expenses - Actual Expenses) accumulated with expected interest.
• Gain from Interest: (Actual Interest Rate - Expected Interest Rate) applied to the initial funds (Reserve + Premium - Actual Expenses).
• Gain from Mortality: (Expected Mortality Rate - Actual Mortality Rate) multiplied by the Death Strain at Risk.
• Goal: The sum of these three equals the total deviation from the expected year-end position.

You've got this! Practice a few problems using this step-by-step approach, and you'll be the Sherlock Holmes of actuarial gains in no time!