Actuarial Mathematics for Modelling · Assurance and annuity functions involving two lives

Reversionary Annuities and Contingent Assurances Explained

Updated 11 October 2026 · Fact-checked

A reversionary annuity pays (y) a benefit only after (x) has died, while (y) is still alive. Its value is a_x|y = a_y − a_xy. A contingent assurance pays on one death only if the deaths occur in a stated order. Split the joint outcomes by order, then use the identities that link them.

Understand Reversionary Annuities and Contingent Assurances

Two-life problems depend on the order of deaths. Write (x) and (y) for the two lives. Assume their future lifetimes are independent unless the question says otherwise.

A reversionary annuity pays to (y) once (x) has died, for as long as (y) lives. Nothing is paid while both are alive. Nothing is paid once (y) has died. So the payments run only in the period when (x) is dead and (y) is alive. The notation is a_x|y. The vertical bar means 'after (x) dies, to (y)'.

The easiest way to value it is by subtraction. An annuity to (y) for life pays whenever (y) is alive. Take away the payments made while both are alive, which is the joint life annuity a_xy. What remains is the reversionary annuity. This gives a_x|y = a_y − a_xy.

A contingent assurance pays a sum on a death only if the deaths happen in a given order. A^1_xy pays on the death of (x) if (x) dies before (y). The '1' above x marks the life whose death triggers the payment. This life must fail first. A^2_xy pays on the death of (y) if (x) has already died. The '2' above y means y must be the second to die.

The key link is that the first death is either (x) or (y). So A^1_xy + A^1_yx = A_xy, where A_xy is the assurance paid on the first death. Also (y) dies either first or second, so A_y = A^1_yx + A^2_xy. Most exam questions come down to these identities, plus the order of death.

Key rules to remember

Reversionary annuity (annual, in arrears): a_x|y = a_y − a_xy; Paid to (y) while y is alive and x is dead. Joint-life annuity a_xy is subtracted from the single-life annuity on (y).
Reversionary annuity (annuity-due and continuous forms): ä_x|y = ä_y − ä_xy and ā_x|y = ā_y − ā_xy; Use the same type of annuity on both sides. Do not mix due and immediate values.
Joint life and last survivor annuities: a_xy = value while both alive; a_(x̄ȳ) = a_x + a_y − a_xy; The last survivor annuity pays while at least one life is alive. It is not the same as a reversionary annuity.
Decomposition of the last survivor annuity: a_(x̄ȳ) = a_xy + a_x|y + a_y|x; Both alive, only y alive, only x alive. Useful as a check on your numbers.
Contingent assurance, death of x first (continuous): A^1_xy = ∫₀^∞ v^t · t p_xy · μ_(x+t) dt; Pays 1 at the moment of death of (x) if (y) is still alive. Under independence, t p_xy = t p_x × t p_y.
Contingent assurance, death of y second (continuous): A^2_xy = ∫₀^∞ v^t · t p_y · μ_(y+t) · (1 − t p_x) dt; Pays 1 on the death of (y) if (x) has died earlier. This form assumes independent lives.
Order-of-death identities: A^1_xy + A^1_yx = A_xy and A_y = A^1_yx + A^2_xy, so A^2_xy = A_y − A^1_yx; The first identity splits the first-death assurance by who dies first. The second splits the assurance on (y) by whether y dies first or second.

How to solve Reversionary Annuities and Contingent Assurances questions

Use this method for any reversionary annuity or contingent assurance question. It works for numerical and algebraic questions.

1Identify who is paid and who must die. Write down the payee, the life whose death triggers a payment, and the order of deaths required.
2Choose the notation. Reversionary annuity: a_x|y. Death of x first: A^1_xy. Death of y after x: A^2_xy. Check which life is the payee.
3Convert the benefit into known building blocks. For annuities use a_x|y = a_y − a_xy. For assurances use the identities A^1_xy + A^1_yx = A_xy and A_y = A^1_yx + A^2_xy.
4Check the payment timing. Decide whether the annuity is in advance or in arrears, and whether the assurance is paid at the moment of death or at the end of the year of death. Use the matching function throughout.
5Substitute the given values. Keep the same interest rate and mortality basis for all terms.
6Multiply by the annual payment or the sum assured. Include any deferral, reduction or other contract feature the question describes.
7Check the answer. It must be positive, and a_x|y must be smaller than a_y. Check that a_xy + a_x|y + a_y|x equals the last survivor annuity if you have all three. Check that a contingent assurance is smaller than the related single-life assurance.

Quickest way: Subtract and split

When to use it: Use this when the question gives you single-life and joint-life values and asks for a reversionary annuity or a contingent assurance. It saves you from integrating.

For a reversionary annuity to (y): compute a_y − a_xy immediately.
For A^1_xy: look for A^1_xy = A_xy − A^1_yx, or the other way round, if one of them is given.
For A^2_xy: compute A_y − A^1_yx. If A^1_yx is not given, get it from A_xy − A^1_xy.
Scale by the annuity payment or the sum assured.
Do a size check: the reversionary annuity is below a_y, and A^2_xy is below A_y.

Common mistakes in Reversionary Annuities and Contingent Assurances

Treating a_x|y as paying to x
The order of letters feels like 'x then y' and students read the benefit as going to the first life.
Fix: Read a_x|y as 'to (y) after (x) dies'. The payee is the life after the bar. The life before the bar must die first.
Calling a reversionary annuity a last survivor annuity
Both involve two lives and both pay after a death.
Fix: A last survivor annuity pays while at least one life is alive, including when both are alive. A reversionary annuity pays only after the named life dies, to the other life. Nothing is paid while both are alive.
Using a_x − a_xy instead of a_y − a_xy
Students subtract the joint life from the wrong single-life annuity.
Fix: Subtract the joint life annuity from the annuity on the payee. For a_x|y the payee is (y), so use a_y.
Confusing A^1_xy with A^2_xy
The superscript position is easy to misread.
Fix: A superscript 1 over a letter means that life dies first and triggers the payment. A superscript 2 over a letter means that life dies second. Write the event in words before you calculate.
Mixing annuity-due and annuity-immediate values
Given values may be a_y and ä_xy, or students forget that the subtraction needs the same payment timing.
Fix: Check the notation of every input. Convert using ä = 1 + a for single and joint life annuities before subtracting.
Forgetting that the identities need the right pairing
Students write A_y = A^1_xy + A^2_xy by pattern matching.
Fix: When y dies first the term is A^1_yx, not A^1_xy. So A_y = A^1_yx + A^2_xy. Always put the life that dies first as the first letter.

Worked examples

Example 1

Two lives (x) and (y) are valued on a given basis. a_x = 12.0, a_y = 14.0 and a_xy = 10.5 (annuity-immediate, annual in arrears). Assume the lives are independent. (a) Find the value of a reversionary annuity of ₹40,000 a year payable to (y) after the death of (x). (b) Find a_y|x, and check that the three parts add up to the last survivor annuity.

Show the solution

(a) The reversionary annuity to (y) is a_x|y = a_y − a_xy = 14.0 − 10.5 = 3.5.
Multiply by the annual payment: 40,000 × 3.5 = ₹1,40,000.
(b) The annuity to (x) after the death of (y) is a_y|x = a_x − a_xy = 12.0 − 10.5 = 1.5.
Last survivor annuity: a_(x̄ȳ) = a_x + a_y − a_xy = 12.0 + 14.0 − 10.5 = 15.5.
Check the decomposition: a_xy + a_x|y + a_y|x = 10.5 + 3.5 + 1.5 = 15.5. This matches.

Answer: (a) a_x|y = 3.5, so the value is ₹1,40,000. (b) a_y|x = 1.5, and the check gives 15.5, equal to the last survivor annuity.

Example 2

For two independent lives (x) and (y), the following assurance values are given on a certain basis, each for a sum assured of 1 paid at the moment of death: A_xy = 0.50 (first death), A^1_xy = 0.22 and A_y = 0.45. Find A^2_xy, and hence the value of a benefit of ₹10,00,000 payable on the death of (y) if (x) has died earlier.

Show the solution

First find the assurance for the case where (y) dies first. Since A^1_xy + A^1_yx = A_xy, we have A^1_yx = 0.50 − 0.22 = 0.28.
Now split the assurance on (y) by whether (y) dies first or second: A_y = A^1_yx + A^2_xy.
So A^2_xy = A_y − A^1_yx = 0.45 − 0.28 = 0.17.
Check the size: 0.17 is below A_y = 0.45, as it must be.
Multiply by the sum assured: 10,00,000 × 0.17 = ₹1,70,000.

Answer: A^2_xy = 0.17, so the benefit is worth ₹1,70,000.

Exam tips

Write the event in words before using any formula, for example 'pay to y, only if x has died first'. This catches most notation errors.
Show the subtraction a_y − a_xy clearly in written answers. Examiners award method marks even if the arithmetic slips.
In Paper B or R-based tasks, compute a_xy from survival probabilities t p_x × t p_y when the lives are independent, then subtract. State the independence assumption.
Always state whether the annuity is in advance or in arrears, and whether the assurance is paid at death or at year end.
Use the sum check a_xy + a_x|y + a_y|x = a_(x̄ȳ) as a quick verification when you have time.

Practice questions from Assurance and annuity functions involving two lives

Reversionary Annuities and Contingent Assurances: frequently asked questions

What is the formula for a reversionary annuity a_x|y?

a_x|y = a_y − a_xy. It values payments to (y) that start after (x) dies and continue while (y) lives. The same form holds for annuity-due and continuous annuities, provided you use the same type on both terms.

What is the difference between a reversionary annuity and a last survivor annuity?

A last survivor annuity pays while at least one of the two lives is alive. A reversionary annuity pays only to (y), and only after (x) has died. A last survivor annuity is the sum of the joint life annuity and both reversionary annuities.

What do A^1_xy and A^2_xy mean?

A^1_xy is the assurance paid on the death of (x) if (x) dies before (y). A^2_xy is the assurance paid on the death of (y) if (x) has already died, so (y) dies second. The number marks whether the life with that number above it dies first or second.

How do I value a contingent assurance if I am not given the integrals?

Use the identities. A^1_xy + A^1_yx = A_xy gives the first-death split. A_y = A^1_yx + A^2_xy gives the second-death value. Rearrange to find the one you need, then multiply by the sum assured.