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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling

Death Strain at Risk, Expected and Actual Death Strain, Mortality Profit

Death strain at risk (DSAR) is the amount the insurer must find if a policyholder dies in the year: the sum assured paid minus the reserve released, DSAR = S − V at year end. Expected death strain is q × DSAR. Actual death strain uses actual deaths. Mortality profit = expected minus actual death strain.

What this chapter covers

This chapter looks at one source of profit or loss in a life insurance portfolio: mortality. You compare the deaths you expected in a year with the deaths that actually happened, and measure the money effect of the difference.

The key idea is that a death does not cost the full sum assured. The insurer pays the sum assured, but it no longer needs to hold the reserve for that policy. So the real cost is the death strain at risk, also called the sum at risk: DSAR = S − V, where S is the sum assured (plus any death benefit payable at the end of the year) and V is the reserve held at the end of the year for a surviving policy. You then multiply by the expected number of deaths, or by the actual number, to get expected death strain (EDS) and actual death strain (ADS).

This chapter builds on the pricing and reserving work in CM1: policy values, reserves, and recursions over a single year. It leads straight into the analysis of surplus, where mortality profit sits next to the profit from investment, expenses and other sources. In CM1 you must be able to compute each piece and explain what the sign of the result means.

Mortality profit is a standard calculation in the pricing and reserving part of CM1, which carries the largest syllabus weighting. It is short, numerical and follows a fixed pattern, so well-prepared students can collect marks reliably. It also tests whether you understand reserves, since you cannot get the sum at risk right without them. The same logic reappears in the analysis of surplus and in Paper B style spreadsheet work, so time spent here pays off more than once.

Death strain at risk, expected and actual death strain, mortality profit: topics in the order to study them

  1. 1Sum at Risk and Death Strain at RiskEverything else uses DSAR, so learn why a death costs S − V and not S before you do any other calculation.
  2. 2Expected Death StrainOnce you have DSAR per policy, expected death strain is a simple extension: multiply by the expected number of deaths, q × DSAR per policy.
  3. 3Actual Death Strain and Mortality ProfitActual death strain uses the real number of deaths, and the difference from expected gives the mortality profit, so you need the expected figure first.
  4. 4Analysis of Surplus by Source: Mortality ExperienceThis puts mortality profit in the wider surplus analysis, so study it last when you can see where the single-source calculation fits.

How to prepare Death strain at risk, expected and actual death strain, mortality profit

This chapter rewards a clear method more than long theory. Practise the same short sequence until it becomes automatic, then test it on varied questions.

  1. Write down the definition DSAR = S − V and explain in one sentence why the reserve is released on death. Check which reserve and which benefit timing the question uses.
  2. Work simple one-policy examples first. For a policy with S = ₹10,00,000 and an end-year reserve V = ₹2,00,000, DSAR = ₹8,00,000. If q = 0.004, expected death strain per policy is 0.004 × ₹8,00,000 = ₹3,200.
  3. Move to groups of policies with different sums at risk. Compute expected death strain as Σ q × DSAR across the groups, and actual death strain as Σ (actual deaths × DSAR).
  4. Calculate mortality profit as expected death strain minus actual death strain. Then state in words whether it is a profit or a loss and what it says about the mortality assumption.
  5. Practise tying the result into a full surplus analysis, where you also deal with interest, expenses and other sources. Keep each source separate and in the same order.
  6. Finish with timed past-paper style questions, written and MCQ. Show the formula, the working and a one-line conclusion each time. Repeat a spreadsheet version for Paper B practice.

Common mistakes in Death strain at risk, expected and actual death strain, mortality profit

  • Using the full sum assured as the cost of a death.

    Fix: Write DSAR = S − V at the top of every answer and check that V is the reserve at the end of the year for that policy.

  • Using the wrong reserve, such as the opening reserve instead of the year-end reserve.

    Fix: Label times clearly (t and t+1) and use the reserve at the time the death benefit is paid.

  • Reversing the sign of the mortality profit.

    Fix: Remember profit = expected strain − actual strain. If actual strain is lower than expected, the answer is positive and is a profit.

  • Applying one average DSAR to policies with different sums at risk.

    Fix: Compute DSAR separately for each group, then multiply by that group's expected or actual deaths and add up.

  • Ignoring policies where DSAR is negative.

    Fix: Keep the sign. A negative DSAR means a death releases more reserve than the benefit paid, so it adds to surplus. Say so in your explanation.

  • Giving a number without explaining it.

    Fix: Add one line saying whether it is a profit or a loss, and what it implies about the mortality assumption.

Last-day revision: Death strain at risk, expected and actual death strain, mortality profit

  • 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.

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

Death strain at risk, expected and actual death strain, mortality profit: frequently asked questions

What is the difference between sum at risk and death strain at risk?

In most CM1 treatments they mean the same thing: the sum assured payable on death minus the reserve held for that policy. Use the definition given in the question, and state it clearly in your answer.

How do I calculate mortality profit?

Work out DSAR for each policy or group. Expected death strain is expected deaths × DSAR and actual death strain is actual deaths × DSAR. Mortality profit is expected minus actual death strain.

Can mortality profit be negative?

Yes. If more people die than expected, or the deaths come from policies with a high DSAR, actual death strain exceeds expected death strain. The result is then a mortality loss.

Does this chapter appear in Paper B as well as Paper A?

It can be examined in either, since the calculation is easy to put in a spreadsheet. In Paper A show the formula and working clearly. In Paper B build the calculation with clear inputs and cell formulas that can be checked.