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IAI Actuarial Core Principles · Actuarial Mathematics for Modelling

Assurance and Annuity Functions Involving Two Lives

Two-life functions price benefits that depend on whether two people are alive or dead. A joint life status ends at the first death. A last survivor status ends at the second death. Define the status, find its survival probability, then sum or integrate discounted payments.

What this chapter covers

This chapter extends single-life assurance and annuity work to two lives, (x) and (y). You define a status, a condition that is either alive or failed at each time. The joint life status xy lasts while both are alive. The last survivor status x̄ȳ lasts while at least one is alive.

Once the status is defined, the maths is familiar. You find the probability that the status survives to time t, then value the payments. For an annuity you use Σ v^t × tpxy. For an assurance you use the distribution of the time the status fails. Most results follow from one identity: the time of the first death plus the time of the second death equals the sum of the two individual lifetimes. So you can write a last survivor function as the sum of the two single-life functions minus the joint life function.

This chapter builds on survival models and the single-life assurance and annuity work earlier in CM1, and on interest rate theory. It also feeds into pricing and reserving, where policies on two lives (joint life annuities, survivor pensions, joint life term assurances) need premiums and reserves. In the exam, it is usually tested with independence assumed unless stated otherwise.

Pricing and reserving is the heaviest part of the CM1 syllabus, and two-life benefits are a standard way to test whether you really understand status definitions, survival probabilities and valuation. Questions can appear in the multiple-choice section and in longer written questions, and the same ideas can be examined in the computer-based paper through a spreadsheet or R calculation. The chapter is also compact. The same few identities solve most problems, so careful practice earns marks that students often lose through careless status definitions.

Assurance and annuity functions involving two lives: topics in the order to study them

  1. 1Joint Life and Last Survivor StatusEverything else depends on defining the statuses correctly and seeing how they relate.
  2. 2Future Lifetime Distributions for Two LivesYou need survival probabilities, such as tpxy and tpx̄ȳ, before you can value any benefit.
  3. 3Assurances on Joint Life and Last Survivor StatusesAssurances apply the status distributions to a single payment at failure, which is the simpler valuation.
  4. 4Annuities on Joint Life and Last Survivor StatusesAnnuities sum payments over time and use the same survival probabilities and the relationship between assurances and annuities.
  5. 5Reversionary Annuities and Contingent AssurancesThese combine the earlier ideas with a condition on who dies first, so study them last.

How to prepare Assurance and annuity functions involving two lives

Work from definitions to valuation, and practise each step with a clear written setup before you calculate.

  1. Write down, in your own words, what each status means. For each one, say when it fails and what must happen for it to survive to time t.
  2. Derive the survival probabilities. Under independence, tpxy = tpx × tpy, and tp x̄ȳ = tpx + tpy − tpxy. Practise until you can write these without hesitation.
  3. Learn the identity linking last survivor to single and joint functions, and use it for assurances and annuities alike. Check it on a small numerical example.
  4. Value assurances and annuities from first principles with a timeline of payments, then use the recursive and relationship formulas to check your answer.
  5. Treat reversionary annuities and contingent assurances as a payment that starts only when a specific life dies first. Express each as a difference of known functions, for example ax|y = ay − axy.
  6. Do past-paper questions in timed blocks. Write the status, the assumption of independence, the formula and the working, then the result with units.
  7. For the computer-based paper, build a spreadsheet or R function that calculates tpxy from a life table and sums the annuity. Compare it with your hand answer.

Common mistakes in Assurance and annuity functions involving two lives

  • Mixing up joint life and last survivor statuses

    Fix: Write the status in words first: joint life ends at first death, last survivor ends at second death. Then pick the formula.

  • Using tpxy = tpx × tpy when the question gives dependence

    Fix: Read the question for any dependence, and state your assumption in the working.

  • Using the wrong relationship for last survivor annuities

    Fix: Remember ax̄ȳ = ax + ay − axy and test it by imagining both lives certain to survive.

  • Getting reversionary annuities backwards

    Fix: Draw a timeline marking who is alive, who is paid and when payment starts. Then write the value as ay − axy or ax − axy as appropriate.

  • Wrong payment timing in annuities and assurances

    Fix: Underline the timing in the question, and write the first payment time and the payment formula before any numbers.

  • Skipping a sense check

    Fix: Check that last survivor values are larger than single life values and that joint life values are smaller.

Last-day revision: Assurance and annuity functions involving two lives

  • Joint life status xy fails at the first death. Last survivor status x̄ȳ fails at the second death.
  • Under independence, tpxy = tpx × tpy.
  • Under independence, tp x̄ȳ = tpx + tpy − tpx × tpy.
  • Time to first death plus time to last death equals Tx + Ty, so last survivor functions equal the sum of single-life functions minus the joint life function.
  • So ax̄ȳ = ax + ay − axy, and the same pattern holds for assurances.
  • Annuity-due on a status: ä = Σ v^t × tp(status) for t = 0, 1, 2, ...
  • A reversionary annuity paid to (y) after the death of (x) has value ay − axy.
  • Contingent assurances pay only if a named life dies first. Check the order of death carefully.
  • State independence explicitly unless the question gives dependence.
  • Write each status before writing a formula, and check the payment timing is right.
  • Check answers are sensible: last survivor annuity ≥ either single life annuity ≥ joint life annuity.

Assurance and annuity functions involving two lives practice questions

Assurance and annuity functions involving two lives in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Assurance and annuity functions involving two lives: frequently asked questions

What is the difference between joint life and last survivor status?

A joint life status continues only while both lives are alive, so it fails at the first death. A last survivor status continues while at least one life is alive, so it fails at the second death.

Do I always assume the two lives are independent?

Independence is the usual assumption in this chapter, but you should state it in your working. If a question gives any other dependence, use that instead.

How do I get last survivor values from single life values?

Use the identity that a last survivor function equals the sum of the two single-life functions minus the joint life function. For annuities, ax̄ȳ = ax + ay − axy.

How is a reversionary annuity valued?

A reversionary annuity pays to one life only after the other has died. If it pays to (y) after the death of (x), its value is ay − axy, because the annuity to (y) is reduced by the part where (x) is still alive.

Can I solve these questions without a formula sheet?

Many can be solved from definitions of the status and survival probabilities. Learn the main relationships, but check them against the status definitions, which helps you avoid memory slips.