Actuarial Mathematics for Modelling · Key assurance and annuity contracts

Annuity Contracts: Immediate, Due, Deferred and Temporary

Updated 11 October 2026 · Fact-checked

A life annuity pays a regular amount while a person is alive. Its value is the expected present value of the payments. Immediate annuities pay at the end of each period, due annuities at the start. Temporary annuities stop after n years. Deferred annuities start after n years. Add survival probabilities to discount factors.

Understand Annuity Contracts: Immediate, Due, Deferred and Temporary

A life annuity is a series of payments made only while the annuitant is alive. Because payments stop at death, each payment is worth its discount factor multiplied by the probability that the person survives to the payment date. The value of the annuity is the sum of these terms. This is its expected present value (EPV).

The first choice is timing. An annuity-due pays at the start of each year, beginning now, so the first payment is certain for a life aged x. Its EPV is ä_x. An annuity-immediate pays at the end of each year, so the first payment needs survival for one year. Its EPV is a_x. The two differ by exactly one certain payment of 1 at time 0, so ä_x = 1 + a_x.

The second choice is the term. A whole life annuity pays until death. A temporary (term) annuity pays for at most n years, so it stops at the earlier of death and time n. A deferred annuity pays nothing for the first n years and then pays whole life, provided the person survives to time n. Whole life = temporary + deferred, which is a useful check.

The third choice is frequency. An m-thly annuity pays 1/m of the annual amount m times a year. A continuous annuity pays at a constant rate each instant, with EPV ā_x. Approximations link these to the annual-due value, using the UDD assumption or a simple adjustment. Use the one your question asks for.

The notation n|ä_x uses a vertical bar for deferment. The symbol ä_x:n̅| means a temporary annuity-due over n years. Always state whether the life is aged x and whether payments are in advance or arrear.

Key rules to remember

Whole life annuity-due: ä_x = Σ (t = 0 to ∞) v^t × tp_x; First payment at time 0 with certainty. Use v = 1/(1 + i).
Whole life annuity-immediate: a_x = Σ (t = 1 to ∞) v^t × tp_x = ä_x − 1; First payment at time 1, needs survival to age x + 1.
Temporary annuity-due: ä_x:n̅| = Σ (t = 0 to n − 1) v^t × tp_x; n payments at times 0, 1, …, n − 1.
Temporary annuity-immediate: a_x:n̅| = Σ (t = 1 to n) v^t × tp_x = ä_x:n̅| − 1 + v^n × np_x; n payments at times 1, …, n. The last term of the sum is at time n.
Deferred annuity-due: n|ä_x = v^n × np_x × ä_x+n = ä_x − ä_x:n̅|; Payments start at time n, only if the life survives to age x + n.
Deferred annuity-immediate: n|a_x = v^(n+1) × (n+1)p_x + v^(n+2) × (n+2)p_x + … = a_x − a_x:n̅|; First payment at time n + 1.
Pure endowment factor: nE_x = v^n × np_x = v^n × l_(x+n) ÷ l_x; Use it to move values back n years: n|ä_x = nE_x × ä_x+n.
Recursion: ä_x = 1 + v × p_x × ä_x+1; Works backwards from a known value at an older age.
Annuity-due from assurance: ä_x = (1 − A_x) ÷ d, where d = i ÷ (1 + i); Whole life, annual payments in advance, with A_x paid at the end of the year of death.
m-thly annuity approximation: ä_x^(m) ≈ ä_x − (m − 1) ÷ (2m); Approximation for whole life annuity-due with payments of 1 per year split into m instalments. For a temporary annuity, also adjust for the term end as the question demands.
Continuous annuity approximation: ā_x ≈ ä_x − 1/2; Approximation; exact form is ∫ v^t × tp_x dt from 0 to ∞.

How to solve Annuity Contracts: Immediate, Due, Deferred and Temporary questions

Use this method for any question on life annuity values. It turns the wording into a sum of discounted survival-weighted payments.

1Identify the age x, the payment amount, the payment frequency and whether payments are in advance or arrear.
2Decide the term. Whole life, temporary for n years, or deferred by n years. Mark the first and last payment times on a timeline.
3Write the EPV as payment × Σ v^t × tp_x over those times. Do not compute anything yet.
4Convert to the nearest standard symbol, such as ä_x, a_x:n̅| or n|ä_x. Use the identities ä = 1 + a, deferred = whole life − temporary, and nE_x to relate values.
5Get the inputs from the life table or given values: l_x, v = 1/(1 + i), and the annuity values at older ages if given.
6Compute carefully. Keep four or more decimal places in v and the survival ratios until the final step.
7If payments are m-thly or continuous, apply the approximation or exact method asked, then multiply by the annual payment.
8Check the answer. Whole life ≥ temporary, due ≥ immediate, and a deferred value is less than the whole life value.

Quickest way: Anchor on ä and use identities

When to use it: When you are given ä_x values or a life table and the question asks for a different annuity type.

Reduce the contract to ä_x, ä_x+n and nE_x.
Use n|ä_x = nE_x × ä_x+n for deferred annuities.
Use ä_x:n̅| = ä_x − nE_x × ä_x+n for temporary annuities.
Use a = ä − 1 for whole life immediate. For temporary immediate, subtract 1 and add nE_x.
Apply the m-thly or continuous adjustment last, to the annual-due value.

Common mistakes in Annuity Contracts: Immediate, Due, Deferred and Temporary

Using a_x = ä_x − 1 for temporary annuities.
The identity is learned for whole life and applied everywhere.
Fix: For temporary annuities use a_x:n̅| = ä_x:n̅| − 1 + nE_x, because the last payment moves from time n − 1 to time n.
Starting a deferred annuity-due at time n + 1.
Confusing n|ä_x with n|a_x.
Fix: For due, the first payment is at time n. For immediate, it is at time n + 1. Draw the timeline.
Forgetting survival to the deferment date.
Treating the deferred period like a certain-annuity delay.
Fix: Multiply by nE_x = v^n × l_x+n ÷ l_x, not just v^n.
Applying the m-thly adjustment to the wrong base.
The approximation is for the whole life annuity-due and students apply it to an immediate value.
Fix: First convert to the annuity-due value, adjust, then convert back if needed.
Using the wrong age in the survival ratio.
Mixing up l_x and l_x+n, or using x as the age at the deferred start.
Fix: Survival is measured from the valuation age x. The annuity after deferment uses ä at age x + n.
Mixing annual amounts with instalments.
For m-thly annuities, students forget each payment is 1/m of the annual amount.
Fix: Work out the EPV of 1 per year, then multiply by the annual amount.

Worked examples

Example 1

A life aged 60 buys an annuity of ₹40,000 a year, paid annually in advance for life but for at most 3 years. Given l_60 = 1000, l_61 = 980, l_62 = 955 and i = 5%, find the EPV.

Show the solution

Payments are at times 0, 1 and 2, so this is a temporary annuity-due ä_60:3̅|.
v = 1/1.05 = 0.952381 and v² = 0.907029.
1p_60 = 980 ÷ 1000 = 0.98 and 2p_60 = 955 ÷ 1000 = 0.955.
ä_60:3̅| = 1 + 0.952381 × 0.98 + 0.907029 × 0.955.
= 1 + 0.933333 + 0.866213 = 2.799546.
EPV = 40,000 × 2.799546 = ₹1,11,981.84.

Answer: About ₹1,11,982.

Example 2

Given ä_65 = 12.0, ä_70 = 10.0, 5E_65 = 0.80 and v = 0.95, find the EPV of a deferred annuity-due of 1 a year to a life aged 65, payable for life from age 70, and the EPV of a whole life annuity-immediate to the same life.

Show the solution

Deferred annuity-due: 5|ä_65 = 5E_65 × ä_70.
= 0.80 × 10.0 = 8.0.
Whole life annuity-immediate: a_65 = ä_65 − 1.
= 12.0 − 1 = 11.0.
Check: 8.0 < 12.0, so the deferred value is less than the whole life value.

Answer: Deferred annuity-due = 8.0; whole life annuity-immediate = 11.0 per unit of annual payment.

Exam tips

Write the timeline first. Most lost marks come from off-by-one errors in payment times.
Show the formula in standard notation before substituting numbers, so you earn method marks even if the arithmetic slips.
Keep v and survival ratios to at least four decimal places and round only at the end.
In MCQs, check ä > a and whole life > temporary or deferred to remove wrong options quickly.
In the computer-based paper, build the sum Σ v^t × tp_x in a column and check it against the closed-form identity.

Practice questions from Key assurance and annuity contracts

Annuity Contracts: Immediate, Due, Deferred and Temporary in other exams

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

CMA FoundationAnnuity: Immediate, Due and Perpetuity

Annuity Contracts: Immediate, Due, Deferred and Temporary: frequently asked questions

What is the difference between annuity due and annuity immediate?

An annuity-due pays at the start of each period, so the first payment is made at once. An annuity-immediate pays at the end of each period. For a whole life annuity, ä_x = 1 + a_x.

How do I calculate the present value of a deferred life annuity?

Multiply the pure endowment factor nE_x = v^n × l_x+n ÷ l_x by the annuity value at age x + n. For a due annuity, n|ä_x = nE_x × ä_x+n. You can also use whole life minus temporary.

What is a temporary life annuity?

It pays while the person is alive but for no more than n years. Payments stop at death or at the end of the term, whichever comes first. Its EPV is the sum of v^t × tp_x over the payment times.

How are m-thly and continuous annuities handled?

Work out the annual-due value ä_x and then adjust. A common approximation is ä_x^(m) ≈ ä_x − (m − 1) ÷ (2m) for whole life, and ā_x ≈ ä_x − 1/2 for continuous payments.