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

Expected Death Strain Formula and How to Calculate It

Updated 11 October 2026 · Fact-checked

Expected death strain is the amount of money the insurer expects to lose, or gain, from deaths in a year. For each policy, multiply the death strain at risk (DSAR) by the expected mortality rate q. Then add the results across all policies in the portfolio.

Understand Expected Death Strain

A life insurer holds a reserve for each policy. If the policyholder dies, the insurer pays the sum assured but no longer needs to hold the reserve. The extra money the insurer must find from its own funds is the death strain at risk (DSAR).

For a policy in force at the start of the year, with death benefit S paid at the end of the year of death, the DSAR is S minus the reserve held at the end of the year. The reserve here is the one for a surviving policy at the year end, written as V at time t+1. The benefit is S, so the DSAR is S − V. The reserve is released on death, so only the shortfall is a strain.

The expected death strain (EDS) is the average strain you would expect from a policy in the year. A policy has probability q of dying in the year. If it dies, the strain is the DSAR. If it survives, the death strain is zero. So the expected value is q × DSAR.

For a portfolio you add up the expected strain of each policy. If many policies share the same DSAR and q, you group them and multiply by the number of policies. The q used is the mortality assumption of the valuation basis, unless the question says otherwise.

The EDS is the benchmark. You later compare it with the actual death strain (ADS), which uses the real deaths. The difference gives the mortality profit or loss. Here we only build the expected figure.

Key rules to remember

Death strain at risk (benefit paid at end of year of death): DSAR = S − V, where S = death benefit and V = reserve at the end of the year for a surviving policy; Use the reserve at the end of the year, with the same basis as the question. If the benefit is paid at the end of the year, this is the standard form. A negative DSAR is possible and must keep its sign.
Expected death strain, one policy: EDS = q × DSAR; q is the probability of death during the year, taken from the stated mortality basis for the policyholder's age.
Expected death strain, portfolio: EDS = Σ (q_i × DSAR_i); Sum over all policies in force at the start of the year. For identical policies, use number of policies × q × DSAR.
Group of identical policies: EDS = N × q × (S − V); N is the number of policies in force at the start of the year in that group.

How to solve Expected Death Strain questions

Use this method for any question that asks for expected death strain, whether the data is one policy or a table of groups.

1Read the timing of the death benefit. The standard case is the benefit paid at the end of the year of death, so DSAR = S − V with V the end-of-year reserve.
2For each policy or group, find the sum assured S and the end-of-year reserve V. Use the reserve for a survivor, not the opening reserve.
3Calculate DSAR = S − V for each group. Keep any negative sign.
4Find the mortality rate q for each group from the basis given. Check that it is a one-year rate for the right age.
5Multiply q × DSAR for each group. Then multiply by the number of policies N if the group has several.
6Add the group results to get the portfolio EDS. Check the answer is in rupees and the scale is right.
7State the answer with units. Write a short comment if the result is negative or if the question asks for a view.

Quickest way: Table method: DSAR then q then N

When to use it: Use it when the question gives several policy groups with numbers of policies, sums assured, reserves and mortality rates.

Draw a table with columns: N, S, V, DSAR, q, and N × q × DSAR.
Fill DSAR = S − V first for every row.
Compute N × q first, which is the expected number of deaths, then multiply by DSAR.
Add the last column. Keep the working visible for method marks.

Common mistakes in Expected Death Strain

Using the sum assured S instead of S − V as the amount at risk
Students link death with the full benefit and forget the reserve is released.
Fix: Always write DSAR = S − V first. The insurer only needs to find the difference.
Using the reserve at the start of the year instead of the end of the year
Both reserves are in the table and the opening one is easy to pick.
Fix: For benefits paid at year end, use the reserve at time t+1 for a survivor. Check the year labels.
Dropping the negative sign when V is larger than S
Students think a strain cannot be negative.
Fix: A negative DSAR is possible. Its expected value is negative and reduces the portfolio total.
Forgetting to multiply by the number of policies
The calculation is done for one policy and the group size is overlooked.
Fix: Put N in the table. Check the final figure is on the whole portfolio.
Using the wrong q, such as a multi-year rate or a rate for the wrong age
Mortality tables have many columns and ages shift during the year.
Fix: Use the one-year rate for the age at the start of the year, from the stated basis.

Worked examples

Example 1

A life office has 200 identical policies in force at the start of the year. Each has a death benefit of ₹5,00,000 paid 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 mortality rate for the year is q = 0.004. Calculate the expected death strain.

Show the solution

DSAR = S − V = 5,00,000 − 1,20,000 = ₹3,80,000.
Expected death strain per policy = 0.004 × 3,80,000 = ₹1,520.
Portfolio EDS = 200 × 1,520 = ₹3,04,000.

Answer: ₹3,04,000

Example 2

A portfolio has two groups, both with death benefit paid at the end of the year of death. Group A: 100 policies, S = ₹10,00,000, end-year reserve V = ₹2,50,000, q = 0.002. Group B: 50 policies, S = ₹4,00,000, end-year reserve V = ₹4,20,000, q = 0.010. Calculate the total expected death strain.

Show the solution

Group A DSAR = 10,00,000 − 2,50,000 = ₹7,50,000.
Group A EDS = 100 × 0.002 × 7,50,000 = 0.2 × 7,50,000 = ₹1,50,000.
Group B DSAR = 4,00,000 − 4,20,000 = −₹20,000.
Group B EDS = 50 × 0.010 × (−20,000) = 0.5 × (−20,000) = −₹10,000.
Total EDS = 1,50,000 − 10,000 = ₹1,40,000.

Answer: ₹1,40,000

Exam tips

Write DSAR = S − V as your first line. It earns method marks even if a number slips later.
Keep negative DSARs. Examiners often include a group with a reserve above the sum assured to test this.
Check the timing of the benefit in the question before choosing which reserve to use.
Show expected deaths N × q as an intermediate step. It helps you link to actual death strain in later parts.

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

Expected Death Strain: frequently asked questions

What is the formula for expected death strain?

For each policy, EDS = q × DSAR, where DSAR = S − V for a benefit paid at the end of the year of death. For a portfolio you sum this over all policies. For groups of identical policies, multiply by the number in the group.

Is expected death strain the same as expected claims?

No. Expected claims would use q × S. Expected death strain uses q × (S − V), because the reserve for the dead policy is released and does not need to be found from other funds.

Can expected death strain be negative?

Yes. If the end-year reserve is larger than the sum assured, as can happen with annuities or some savings contracts, the DSAR is negative. Deaths then release money, so the expected strain is negative.

Which mortality rate q should I use?

Use the one-year mortality rate for the policyholder's age at the start of the year from the basis given in the question. Do not use a rate from a different table unless the question tells you to.