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

Actual Death Strain and Mortality Profit Explained

Updated 11 October 2026 · Fact-checked

Actual death strain (ADS) is the total death strain on the policies that really claimed in the year. Mortality profit is expected death strain (EDS) minus actual death strain. A positive result means fewer or lighter claims than expected. A negative result is a mortality loss. Compute both on the same policies and basis.

Understand Actual Death Strain and Mortality Profit

A life insurer holds a reserve for each policy. When a policyholder dies, the insurer pays the sum assured but no longer needs to hold that policy's reserve. The extra money it must find is the death strain at risk (DSAR) for that policy: DSAR = S − V, where S is the amount paid on death and V is the reserve that was held at the end of the year (the year-end policy value).

The expected death strain (EDS) is what the insurer expected to lose from deaths in the year. For each policy you multiply the probability of death in the year by its DSAR, then add across all policies. It uses the mortality in the valuation or pricing basis.

The actual death strain (ADS) is what really happened. You only count policies where the life died in the year. For each one, take its DSAR and add them up. A policy that did not claim adds nothing.

Mortality profit compares the two. If you expected a strain of ₹10 lakh but only ₹6 lakh arose, you gained ₹4 lakh from mortality. If ₹15 lakh arose, you lost ₹5 lakh. Mortality profit is one source in the analysis of surplus, along with interest, expenses and others.

The sign convention matters. The strain is a cost to the insurer. So profit equals expected cost minus actual cost. Think of it as: budget minus what you actually spent.

Key rules to remember

Death strain at risk: DSAR = S − V; S is the death benefit paid. V is the reserve at the end of the year (policy value at the time of death). Use the same reserve basis that the question states.
Expected death strain: EDS = Σ q × (S − V); Sum over all policies in force at the start of the year. q is the one-year death probability from the stated basis. Policies in force at the start of the year are the ones counted.
Actual death strain: ADS = Σ (S − V) over policies that died in the year; Only deaths count. Survivors contribute zero.
Mortality profit: Mortality profit = EDS − ADS; Positive means profit (fewer or lighter claims than expected). Negative means loss.
Expected number of deaths: Expected deaths = Σ q; Useful when all policies share the same DSAR: EDS = q × number of policies × DSAR.

How to solve Actual Death Strain and Mortality Profit questions

Use this method for any question on actual death strain and mortality profit. Keep the basis and the timing consistent throughout.

1Identify the group of policies in force at the start of the year and their sums assured or death benefits.
2Find the reserve V that applies at the end of the year for each group. Check whether the question gives V at the end of the year or at the start. Use the end-of-year value for DSAR.
3Calculate DSAR = S − V for each group. Check the units (rupees, lakh or crore).
4Find the death probability q for each group from the basis. Compute EDS = Σ (number of policies × q × DSAR).
5Find the number of actual deaths in each group. Compute ADS = Σ (number of deaths × DSAR).
6Compute mortality profit = EDS − ADS. State clearly whether it is a profit or a loss.
7Sense-check: if actual deaths are below expected deaths, the profit should be positive (when DSAR is positive). Comment briefly if the question asks.

Quickest way: Compare deaths by group, multiply by DSAR

When to use it: Use when policies fall into a few groups with the same DSAR. This is the common exam layout.

For each group, write DSAR = S − V.
Mortality profit for the group = (expected deaths − actual deaths) × DSAR.
Expected deaths = number in force × q.
Add the group results to get total mortality profit.
Check the sign: fewer deaths than expected gives a positive profit.

Common mistakes in Actual Death Strain and Mortality Profit

Computing mortality profit as ADS − EDS.
Students treat it as actual minus expected, as in most variance analyses.
Fix: Strain is a cost. Profit is expected cost minus actual cost, so use EDS − ADS. Write the sign check beside your answer.
Using the full sum assured S instead of S − V.
Students forget that the reserve is released on death.
Fix: Always compute DSAR = S − V first. Only then multiply by deaths or probabilities.
Using the reserve at the start of the year instead of the end of the year.
The start-of-year reserve is often the figure listed first in the question.
Fix: Death strain uses the reserve the insurer would have held at the end of the year. Read the table headings carefully.
Including survivors in the actual death strain.
Students apply the formula to every policy in force and multiply by q again.
Fix: ADS uses actual deaths only. Use q only for the expected figure.
Mixing a lakh figure with a rupee figure.
Sums assured and reserves are quoted in different units.
Fix: Convert everything to rupees before you start, or to lakh throughout.

Worked examples

Example 1

An insurer has 2,000 policies in force at the start of the year, each with sum assured ₹5,00,000 payable at the end of the year of death. The reserve at the end of the year is ₹1,00,000 per policy. The valuation mortality rate is q = 0.004. During the year 6 policyholders died. Calculate the mortality profit.

Show the solution

DSAR = S − V = 5,00,000 − 1,00,000 = ₹4,00,000.
Expected deaths = 2,000 × 0.004 = 8.
EDS = 8 × 4,00,000 = ₹32,00,000.
ADS = 6 × 4,00,000 = ₹24,00,000.
Mortality profit = EDS − ADS = 32,00,000 − 24,00,000 = ₹8,00,000.

Answer: Mortality profit is ₹8,00,000, a profit, because there were 6 deaths against 8 expected.

Example 2

A portfolio has two groups. Group A: 500 policies, S = ₹10,00,000, V = ₹2,00,000, q = 0.002, 2 deaths. Group B: 300 policies, S = ₹4,00,000, V = ₹1,00,000, q = 0.01, 4 deaths. Find the mortality profit or loss.

Show the solution

Group A DSAR = 10,00,000 − 2,00,000 = ₹8,00,000. Group B DSAR = 4,00,000 − 1,00,000 = ₹3,00,000.
Group A expected deaths = 500 × 0.002 = 1. Group B expected deaths = 300 × 0.01 = 3.
EDS = 1 × 8,00,000 + 3 × 3,00,000 = 8,00,000 + 9,00,000 = ₹17,00,000.
ADS = 2 × 8,00,000 + 4 × 3,00,000 = 16,00,000 + 12,00,000 = ₹28,00,000.
Mortality profit = 17,00,000 − 28,00,000 = −₹11,00,000.

Answer: There is a mortality loss of ₹11,00,000. Both groups had more deaths than expected, and Group A's deaths were costly.

Exam tips

Write DSAR for each group in a small table before doing anything else. It prevents most arithmetic slips.
State the sign in words, such as 'profit of ₹8,00,000'. Examiners give marks for interpreting the result.
Check which policies are in force at the start of the year. Expected deaths apply to those, not to the year-end count.
In written questions, give a short comment on why actual and expected may differ, for example random variation or a different mix of sums at risk.
For Paper B style work, keep a clear layout of inputs (S, V, q, deaths) so each formula is traceable.

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

Actual Death Strain and Mortality Profit: frequently asked questions

What is the formula for mortality profit?

Mortality profit = expected death strain − actual death strain. Each death strain is the sum of S − V over the relevant policies. A positive answer is a profit and a negative answer is a loss.

What is the difference between actual and expected death strain?

Expected death strain uses the mortality probability from the basis and applies to all policies in force at the start of the year. Actual death strain uses only the policies that really claimed. The difference between them is the mortality profit or loss.

Why do we subtract the reserve from the sum assured?

When a policyholder dies, the reserve held for that policy is released and can pay part of the claim. The extra amount the insurer must find is S − V. This is the death strain at risk.

Can mortality profit be negative?

Yes. If actual death strain exceeds expected death strain, the result is negative, which is a mortality loss. This happens when claims are more numerous or larger than the basis expected.