Actuarial Mathematics for Modelling · Death strain at risk, expected and actual death strain, mortality profit

Analysis of Surplus by Source: Mortality Experience and Profit

Updated 11 October 2026 · Fact-checked

Analysis of surplus splits a year's total profit into sources such as interest, expenses and mortality. Mortality profit = expected death strain − actual death strain, where each death strain is the number of deaths × the death strain at risk (sum assured payable − end-year reserve). Positive means profit, negative means loss.

Understand Analysis of Surplus by Source: Mortality Experience

An insurer's total profit in a year comes from several places. Investment return differs from what the reserve basis assumed. Expenses differ from the loading. Deaths differ from the number expected. Analysis of surplus pulls these apart so you can see which source made or lost money.

The mortality source asks one question: did more or fewer policyholders die than the valuation basis expected, and what did that cost? A death costs the insurer the death strain at risk (DSAR). This is the amount paid on death minus the reserve the insurer no longer needs to hold for that policy. So DSAR = sum assured payable on death − end-of-year reserve.

The expected death strain (EDS) uses the expected number of deaths, which is the mortality rate on the valuation basis × policies in force at the start of the year. The actual death strain (ADS) uses the actual number of deaths. Mortality profit = EDS − ADS. Fewer deaths than expected give a profit when DSAR is positive, as on most assurances.

Do not assume that more deaths always means a loss. For annuities, DSAR is usually negative, because a death releases the reserve and pays no benefit. Then more deaths than expected give a profit, and fewer deaths give a loss. Always work out the sign of DSAR first.

The reserve basis matters because DSAR contains the end-of-year reserve. A higher reserve lowers DSAR on an assurance, so the same extra deaths cost less. The valuation basis also sets the expected number of deaths. The total profit does not depend on how you split it, but the size of each source can depend on the order you calculate them in. State your order and basis clearly in an exam answer.

Key rules to remember

Death strain at risk: DSAR = S − V; S is the amount payable on death in the year. V is the reserve at the end of the year for a surviving policy. DSAR can be negative, for example for annuities.
Expected death strain: EDS = (number in force at start) × q × DSAR; q is the mortality rate for the year on the valuation basis. Number × q is the expected number of deaths.
Actual death strain: ADS = (actual number of deaths) × DSAR; Use the same DSAR as in EDS, so the difference depends only on the number of deaths.
Mortality profit: Mortality profit = EDS − ADS = (expected deaths − actual deaths) × DSAR; Positive is profit, negative is loss. Check the sign of DSAR before interpreting the result.
Total surplus: Total profit = interest profit + expense profit + mortality profit + other sources; The sources add up to the total. The split between them depends on the order and basis used.

How to solve Analysis of Surplus by Source: Mortality Experience questions

Use this method for any question on mortality profit within an analysis of surplus.

1Write down the data: number in force at the start, mortality rate q, sum assured or benefit on death, end-of-year reserve, and actual deaths.
2Calculate DSAR = amount payable on death − end-of-year reserve. Note whether it is positive or negative.
3Calculate the expected number of deaths = number in force × q.
4Calculate EDS = expected deaths × DSAR and ADS = actual deaths × DSAR.
5Calculate mortality profit = EDS − ADS. Keep the sign.
6Interpret in words: compare actual deaths with expected deaths, then use the sign of DSAR to say profit or loss.
7If the question asks about reserve basis or ordering, recompute DSAR with the changed reserve, or explain that the source split depends on the order used while the total stays the same.
8Check that the mortality profit fits with the other sources if you are asked for the total.

Quickest way: Deaths difference times DSAR

When to use it: Use when the question gives expected and actual deaths and the same DSAR applies to every policy in the group.

Find DSAR = benefit on death − end-of-year reserve.
Find expected deaths = number in force × q.
Take (expected deaths − actual deaths) × DSAR.
Sanity check: fewer deaths than expected and positive DSAR must give a profit. If DSAR is negative, the sign flips.

Common mistakes in Analysis of Surplus by Source: Mortality Experience

Using the reserve at the start of the year in DSAR instead of the end of the year.
Students link reserve with the opening position and forget that a death releases the reserve at the end of the year.
Fix: DSAR uses the reserve at time t+1 for the policy, because the death benefit and the release of reserve both fall at the year end.
Writing mortality profit as ADS − EDS.
Students copy the order from other variance formulas where actual comes first.
Fix: Learn it as profit = expected − actual strain. A cost lower than expected is a gain.
Calling more deaths a loss without checking the sign of DSAR.
It is true for most assurances, so students apply it to annuities too.
Fix: For annuities DSAR is negative because a death releases reserve with no benefit. More deaths than expected then give a profit.
Using different DSAR values for EDS and ADS.
Students re-derive the reserve on an actual basis for the actual strain.
Fix: Use the same DSAR from the valuation basis for both. Only the number of deaths changes.
Saying the order of analysis changes the total profit.
Students confuse the split by source with the total.
Fix: The total is fixed by actual experience. Order changes how it is shared between sources, so state your order.
Forgetting to scale by the number of policies, giving a per-policy figure.
Students compute DSAR and q × DSAR and stop there.
Fix: Multiply by the number in force for EDS, and by actual deaths for ADS.

Worked examples

Example 1

A portfolio has 1,000 identical term assurance policies in force at the start of the year. Each pays a sum assured of ₹5,00,000 at the end of the year of death. The reserve at the end of the year for each surviving policy is ₹1,20,000. The valuation mortality rate is 0.004 and 6 policyholders die. (a) Find the mortality profit. (b) Find it if the end-of-year reserve were ₹1,50,000 instead.

Show the solution

DSAR = 5,00,000 − 1,20,000 = ₹3,80,000.
Expected deaths = 1,000 × 0.004 = 4.
EDS = 4 × 3,80,000 = ₹15,20,000.
ADS = 6 × 3,80,000 = ₹22,80,000.
Mortality profit = 15,20,000 − 22,80,000 = −₹7,60,000, a loss. Check: 2 extra deaths × 3,80,000 = 7,60,000.
(b) DSAR = 5,00,000 − 1,50,000 = ₹3,50,000.
EDS = 4 × 3,50,000 = ₹14,00,000 and ADS = 6 × 3,50,000 = ₹21,00,000.
Mortality profit = 14,00,000 − 21,00,000 = −₹7,00,000.

Answer: (a) A mortality loss of ₹7,60,000. (b) A mortality loss of ₹7,00,000. The higher reserve lowers DSAR, so the loss from the extra deaths is smaller.

Example 2

An insurer has 200 annuitants alive at the start of the year. There is no death benefit. The reserve at the end of the year for each surviving annuitant is ₹8,00,000. The valuation mortality rate is 0.03. During the year 4 annuitants die. Find the mortality profit and explain why the result is a loss even though fewer people died than expected.

Show the solution

DSAR = 0 − 8,00,000 = −₹8,00,000.
Expected deaths = 200 × 0.03 = 6.
EDS = 6 × (−8,00,000) = −₹48,00,000.
ADS = 4 × (−8,00,000) = −₹32,00,000.
Mortality profit = EDS − ADS = −48,00,000 − (−32,00,000) = −₹16,00,000.
Check: (6 − 4) × (−8,00,000) = −₹16,00,000.
Explanation: each death releases a reserve of ₹8,00,000. Two fewer deaths than expected means two reserves were not released, so the insurer keeps paying annuities for longer than assumed.

Answer: Mortality loss of ₹16,00,000. DSAR is negative for annuities, so fewer deaths than expected give a loss.

Exam tips

Write DSAR first and state its sign. Examiners give marks for the correct DSAR even if later arithmetic slips.
Give a one-line interpretation after the number: profit or loss, and whether deaths were above or below expected.
For multiple-choice items, check the sign logic before calculating. Most wrong options come from the sign or from using the start-of-year reserve.
In written answers on reserve basis, say that the reserve changes DSAR, and say what happens to the expected number of deaths if the mortality assumption changes.
When asked about ordering, say that the total profit is unchanged and only the split between sources moves. State the order you used.

Practice questions from Death strain at risk, expected and actual death strain, mortality profit

Analysis of Surplus by Source: Mortality Experience: frequently asked questions

What is mortality profit in life insurance?

It is the part of the year's surplus that comes from deaths differing from those expected on the valuation basis. It equals expected death strain minus actual death strain. Positive means profit and negative means loss.

What is the difference between expected and actual death strain?

Both use the same death strain at risk. Expected death strain multiplies it by the expected number of deaths from the valuation mortality rate. Actual death strain multiplies it by the deaths that really occurred.

Why can fewer deaths than expected be a loss?

This happens when DSAR is negative, as on annuities. A death releases reserve and pays no benefit, so a death is a gain. Fewer deaths than expected then reduce the release and give a loss.

Does the reserve basis affect mortality profit?

Yes. The end-of-year reserve is part of DSAR, and the mortality rate sets the expected deaths. A higher reserve lowers DSAR on an assurance, so the same deaths cause a smaller profit or loss.