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

Annuities on Joint Life and Last Survivor Statuses

Updated 11 October 2026 · Fact-checked

A joint life annuity pays while both lives survive. Its EPV is ä_xy = Σ v^t · tp_xy. A last survivor annuity pays while at least one life survives. Use ä_x̄ȳ = ä_x + ä_y − ä_xy. Premiums come from the equivalence principle: EPV of premiums equals EPV of benefits.

Understand Annuities on Joint Life and Last Survivor Statuses

An annuity on two lives pays a regular amount depending on who is alive. The EPV is the sum of each payment discounted for interest and weighted by the probability that it is paid.

The joint life status (xy) is alive only while both x and y are alive. It fails at the first death. So the probability that a payment at time t is made is t p_xy. With independent lives, t p_xy = t p_x × t p_y.

The last survivor status (x̄ȳ) is alive while at least one of x and y is alive. It fails at the second death. The probability is t p_x̄ȳ = t p_x + t p_y − t p_xy. This comes from P(A ∪ B) = P(A) + P(B) − P(A ∩ B).

Multiply that probability by v^t and sum over t. The same identity carries straight through to the annuity: ä_x̄ȳ = ä_x + ä_y − ä_xy. The identity holds for any joint distribution of the two lives. Independence is only needed to write t p_xy as a product.

For premiums, use the equivalence principle. Set the EPV of premiums equal to the EPV of benefits, then solve for the premium. Premiums are often payable only while the status is alive, for example while both lives survive. Check which status governs the premiums and which governs the benefits.

Key rules to remember

Joint life annuity-due: ä_xy = Σ (t = 0 to ∞) v^t · t p_xy; Payments at the start of each year while both lives are alive. With independent lives, t p_xy = t p_x · t p_y.
Last survivor survival probability: t p_x̄ȳ = t p_x + t p_y − t p_xy; Probability that at least one life survives t years.
Last survivor annuity-due: ä_x̄ȳ = ä_x + ä_y − ä_xy; Holds for any joint distribution of the two lives.
Annuity-immediate relation: a_xy = ä_xy − 1, and a_x̄ȳ = ä_x̄ȳ − 1; Applies for whole life annuities. The first payment is at time 0 for the due form.
Continuous joint life annuity: ā_xy = ∫ (0 to ∞) v^t · t p_xy dt; Payable continuously. The identity ā_x̄ȳ = ā_x + ā_y − ā_xy also holds.
Equivalence principle premium: P · (EPV of 1 per year premium annuity) = EPV of benefits; Use the premium annuity status specified by the contract.

How to solve Annuities on Joint Life and Last Survivor Statuses questions

Use this method for any two-life annuity or premium question.

1Identify the benefit status: joint life (both alive), last survivor (at least one alive), or only x alive, or another variant.
2Identify the payment timing: due (start of year), immediate (end of year), continuous or m-thly. Note any deferment or term.
3Write the EPV as Σ v^t · P(status survives t) for the correct timing. Decide if lives are independent.
4For last survivor, decompose with ä_x̄ȳ = ä_x + ä_y − ä_xy. Use given single and joint values directly if provided.
5Adjust timing if needed: a = ä − 1 for whole life annuities, or apply the correct m-thly or continuous approximation as stated in the question.
6For premiums, identify the premium status and the benefit EPV. Apply the equivalence principle with any expenses.
7Solve for the premium, then check the size is sensible: the last survivor annuity must be at least as large as either single life annuity, and the joint life annuity at most as large as either.

Quickest way: Decompose with the identity and use given values

When to use it: Use this when the question gives or lets you find ä_x, ä_y and ä_xy, which is common in exams.

Write ä_x̄ȳ = ä_x + ä_y − ä_xy immediately.
Substitute the given values and compute.
Convert to immediate with −1 if needed.
For premiums, write premium annuity and benefit annuity in terms of the same three values.
Check ordering: ä_xy ≤ ä_x, ä_y ≤ ä_x̄ȳ.

Common mistakes in Annuities on Joint Life and Last Survivor Statuses

Using ä_x + ä_y as the last survivor value
It feels natural to add two lives, but this double counts the period when both are alive.
Fix: Always subtract ä_xy: ä_x̄ȳ = ä_x + ä_y − ä_xy.
Writing t p_xy = t p_x + t p_y − something
Students mix up the formula for the joint life status with the one for last survivor.
Fix: Joint life is an intersection: product when independent. Last survivor is a union: add and subtract the intersection.
Converting due to immediate wrongly
Students subtract 1 from the wrong quantity or forget to do so for the last survivor annuity.
Fix: For whole life, a = ä − 1 applies to each status. Apply it to the final result, or to each term consistently.
Using the wrong status for premiums
The question may say premiums stop on the first death, but students use the last survivor annuity.
Fix: Underline the premium status and the benefit status separately before writing the equation.
Assuming independence without stating it
Students multiply survival probabilities by habit.
Fix: State the assumption of independent lives. The annuity identity itself does not need it, but the product form does.
Ignoring a sensible-answer check
Time pressure and arithmetic slips.
Fix: Confirm ä_xy is the smallest and ä_x̄ȳ the largest of the three annuity values.

Worked examples

Example 1

Given ä_x = 14.2, ä_y = 12.8 and ä_xy = 10.5, with interest at a fixed rate. Calculate the EPV of a whole life annuity of ₹50,000 a year, paid annually in advance, while at least one of (x) and (y) is alive.

Show the solution

The status is last survivor, payable in advance.
ä_x̄ȳ = ä_x + ä_y − ä_xy = 14.2 + 12.8 − 10.5.
14.2 + 12.8 = 27.0, and 27.0 − 10.5 = 16.5.
EPV = 50,000 × 16.5 = ₹8,25,000.
Check: 16.5 is larger than both 14.2 and 12.8, as expected.

Answer: ₹8,25,000

Example 2

A couple, (x) and (y), buy an annuity of ₹60,000 a year payable annually in advance while both are alive. Premiums are level, payable annually in advance while both are alive, for at most 10 years. Given ä_xy = 11.0 and the EPV of a 10-year temporary joint life annuity-due is ä_xy:10 = 7.0, ignoring expenses, find the annual premium.

Show the solution

Benefit EPV = 60,000 × ä_xy = 60,000 × 11.0 = ₹6,60,000.
Premium annuity EPV = P × ä_xy:10 = 7.0 P.
Equivalence principle: 7.0 P = 6,60,000.
P = 6,60,000 ÷ 7.0 = ₹94,285.71 (to the nearest paisa-level rounding).
Check: total premiums are about ₹9,42,857 over 10 years, above the benefit EPV, which is reasonable because of discounting and survival.

Answer: Annual premium ≈ ₹94,286

Exam tips

Write down the status and payment timing before any numbers. Most lost marks come from using the wrong status.
Use the identity ä_x̄ȳ = ä_x + ä_y − ä_xy, since questions usually give the single and joint values.
State your independence assumption clearly when you use a product of survival probabilities.
In computer-based Paper B, build t p_x and t p_y columns, compute t p_xy by product, and then t p_x̄ȳ, before discounting and summing.
Quote the sensible-answer check in written answers. It shows understanding and catches slips.

Practice questions from Assurance and annuity functions involving two lives

Annuities on Joint Life and Last Survivor Statuses: frequently asked questions

What is the formula for a joint life annuity?

For an annuity-due, ä_xy = Σ v^t · t p_xy, summed over t from 0. It pays while both lives survive. With independent lives, t p_xy = t p_x · t p_y.

How do you derive the last survivor annuity formula?

Start with t p_x̄ȳ = t p_x + t p_y − t p_xy, which is the union rule for probabilities. Multiply by v^t and sum over t. This gives ä_x̄ȳ = ä_x + ä_y − ä_xy.

Does the last survivor identity need independent lives?

No. The identity comes from the union rule and holds for any joint distribution. Independence is only needed to write t p_xy as t p_x times t p_y.

How do I calculate the premium for a joint life annuity?

Use the equivalence principle. Set the EPV of premiums equal to the EPV of benefits, including expenses if given, and solve for the premium. Check the premium status: it may differ from the benefit status.