Sum at Risk and Death Strain at Risk Explained

Understand Sum at Risk and Death Strain at Risk

Key rules to remember

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.

How to solve Sum at Risk and Death Strain at Risk questions

1. 1Identify the policy year and the time points. Find the end-of-year time, t+1, at which death is assumed to occur.
2. 2Write down the death benefit payable in that year. Include bonuses, return of premiums or an increasing sum assured if the contract has them.
3. 3Add any extra death-related costs, such as a claim expense, if the question gives them.
4. 4Find the reserve at the end of the year, {t+1}V. If only the start-of-year reserve is given, roll it forward using the premium, expenses, interest and mortality, or use the stated table.
5. 5Subtract the reserve from the total benefit: DSAR = S + E − {t+1}V.
6. 6Check the sign. A positive value is a strain. A negative value is a release or gain on death.
7. 7For a group, repeat for each policy group and multiply by the number of policies only if the question asks for totals. State units and assumptions clearly.

Quickest way: Benefit minus year-end reserve

Common mistakes in Sum at Risk and Death Strain at Risk

• Using the reserve at the start of the year instead of the end of the year.

  The policy value at the start is the one students see first and use for the interest roll-forward.

  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.

  Students expect a claim to always cost money.

  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 only and ignoring bonuses or return of premiums.

  Students read the headline benefit and skip the policy terms.

  Fix: Read what is payable on death in that year. Add attached bonuses, accumulated premiums or increasing benefits.

• Treating DSAR as the expected cost.

  The words strain and risk suggest a probability.

  Fix: DSAR is the amount if the death happens. The expected death strain multiplies it by the probability of death, q.

• Mixing up the year-end reserve in the DSAR with the reserve at the end of a different year when several years are given.

  Tables list reserves by duration and students pick the wrong row.

  Fix: Label each time point. If the policy is in year 3, death occurs at time 3 and you need the reserve at time 3.

Worked examples