Death strain at risk, expected and actual death strain, mortality profit: formula sheet

Key formulas

Death strain at risk (DSAR) for policy year t+1

DSAR = S_{t+1} + E_{t+1} − {t+1}V

S is the death benefit payable at the end of the year, E is any extra death-related expense or sum paid, and {t+1}V is the reserve held at the end of the year. Many questions take E = 0.

Sum at risk (simple form)

Sum at risk = Benefit on death − Reserve at end of year

Use the benefit actually payable on death in that year, which may include bonuses or an increasing sum assured.

Pure endowment with no death benefit

DSAR = 0 − {t+1}V = −{t+1}V

Negative whenever the reserve is positive. A death releases the reserve and pays nothing.

Sum assured with return of premiums

DSAR = S + premiums returned − {t+1}V, where S is the sum assured

Check the policy terms. If premiums are returned on death, add them to the benefit. If none are returned, the premiums term is zero.

Total expected death strain (link to next topic)

EDS = Σ over policies of q × DSAR

q is the probability of death in the year for each policy. Each policy has its own DSAR, so you sum q × DSAR over the policies. If all policies have the same DSAR, EDS = (expected number of deaths) × DSAR.

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.

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.

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.

Quick revision

• Death strain at risk (DSAR) = sum assured payable on death − reserve held at the end of the year for that policy (S − V).
• The insurer releases the reserve on death, which is why the cost is S − V and not S.
• Expected death strain = expected number of deaths × DSAR, or q × DSAR for one policy.
• Actual death strain = actual number of deaths × DSAR.
• Mortality profit = expected death strain − actual death strain.
• Fewer deaths than expected gives a positive mortality profit; more deaths gives a mortality loss.
• With several groups, compute DSAR for each group and sum the results; never use one average DSAR for mixed policies.
• Check whether the death benefit is paid at the end of the year or at the moment of death; the timing changes the formula and the figures.
• DSAR can be negative, for example on a policy whose reserve exceeds its sum assured, so then a death produces a gain.
• State your assumptions: the mortality rate q used for the expected figure and the reserve basis.
• Mortality is one source of surplus; analyse interest, expenses and other sources separately.
• Always finish by stating the sign and meaning of the result in words.

Common mistakes

• Using the reserve at the start of the year instead of the end of the year. Fix: For the death strain at risk, always use the reserve at time t+1 (the end of the year in which death occurs), with death assumed at the year end.
• Forgetting that the sum at risk can be negative. Fix: If the reserve is larger than the benefit, as in a pure endowment, the DSAR is negative. Report it with its sign.
• Using the sum assured S instead of S − V as the amount at risk 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 Fix: For benefits paid at year end, use the reserve at time t+1 for a survivor. Check the year labels.
• Computing mortality profit as ADS − EDS. 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. Fix: Always compute DSAR = S − V first. Only then multiply by deaths or probabilities.
• Using the reserve at the start of the year in DSAR instead of 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. Fix: Learn it as profit = expected − actual strain. A cost lower than expected is a gain.

Exam tips

• Write the time label next to every reserve you use. This avoids picking the wrong year in a table.
• State the assumption that death occurs at the end of the policy year, as it justifies the use of the year-end reserve.
• Show the sign. Examiners give marks for noticing that pure endowments have a negative DSAR.
• In written answers, define S, E and V before the formula. In MCQs, check the policy terms for bonuses or premium returns before subtracting.
• If a question asks for expected death strain, calculate DSAR first, then multiply by q, so you can earn method marks.
• 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.