Actuarial Mathematics for Modelling · Annuity and accumulation functions

Annuity-Immediate and Annuity-Due Functions: Formulas and Differences

Updated 11 October 2026 · Fact-checked

An annuity-immediate pays 1 at the end of each period for n periods. Its present value is a-angle-n = (1 − vⁿ) ÷ i and its accumulated value is s-angle-n = ((1 + i)ⁿ − 1) ÷ i. An annuity-due pays at the start of each period, so use d instead of i in the denominator.

Understand Annuity-Immediate and Annuity-Due Functions

A level annuity is a series of equal payments made at regular intervals. In CM1 you value it at a chosen date using a fixed effective interest rate i per payment period. Put simply, you add up discounted or accumulated single payments.

The annuity-immediate pays 1 at times 1, 2, ..., n, the end of each period. Its present value at time 0 is a-angle-n = v + v² + ... + vⁿ, where v = 1 ÷ (1 + i). This is a geometric series, so it sums to (1 − vⁿ) ÷ i. Its accumulated value at time n, just after the last payment, is s-angle-n = 1 + (1 + i) + ... + (1 + i)ⁿ⁻¹ = ((1 + i)ⁿ − 1) ÷ i.

The annuity-due pays 1 at times 0, 1, ..., n − 1, the start of each period. Its present value is ä-angle-n = 1 + v + ... + vⁿ⁻¹ = (1 − vⁿ) ÷ d, where d = i ÷ (1 + i) = 1 − v. Its accumulated value at time n is s̈-angle-n = (1 + i) + ... + (1 + i)ⁿ = ((1 + i)ⁿ − 1) ÷ d. Here the accumulation is to time n, one period after the last payment.

Each payment in the due annuity comes one period earlier than in the immediate annuity. So it is worth (1 + i) times as much at any fixed date: ä-angle-n = (1 + i) × a-angle-n and s̈-angle-n = (1 + i) × s-angle-n. Also, s-angle-n = (1 + i)ⁿ × a-angle-n, since accumulating from time 0 to time n multiplies by (1 + i)ⁿ.

The key skill is placing the payments on a timeline. Draw the timeline first, then decide which formula and which valuation date apply.

Key rules to remember

Present value, annuity-immediate: a-angle-n = v + v² + ... + vⁿ = (1 − vⁿ) ÷ i; Value one period before the first payment. v = 1 ÷ (1 + i).
Accumulated value, annuity-immediate: s-angle-n = ((1 + i)ⁿ − 1) ÷ i; Value at time n, the date of the last payment.
Present value, annuity-due: ä-angle-n = 1 + v + ... + vⁿ⁻¹ = (1 − vⁿ) ÷ d; Value at the date of the first payment. d = i ÷ (1 + i).
Accumulated value, annuity-due: s̈-angle-n = ((1 + i)ⁿ − 1) ÷ d; Value at time n, one period after the last payment.
Due versus immediate: ä-angle-n = (1 + i) a-angle-n; s̈-angle-n = (1 + i) s-angle-n; Valid for the same i and n.
Link between s and a: s-angle-n = (1 + i)ⁿ a-angle-n; Same relation holds for s̈ and ä.
Due and immediate by shifting: ä-angle-n = 1 + a-angle-(n−1); a-angle-n = ä-angle-n − 1 + vⁿ; Useful for payments counted from time 0 or time n.
Reciprocal relation: 1 ÷ a-angle-n = 1 ÷ s-angle-n + i; Holds for the immediate annuity with the same i and n.

How to solve Annuity-Immediate and Annuity-Due Functions questions

Use this method for any question on level annuities with payments at the end or start of each period.

1Draw a timeline and mark every payment with its amount and time.
2Check that the payment period matches the interest period. If not, convert the rate to an effective rate per payment period.
3Decide the type: payments at times 1 to n is immediate; payments at times 0 to n − 1 is due.
4Choose the valuation date. Immediate: PV at time 0, accumulation at time n. Due: PV at time 0, accumulation at time n, which is one period after the last payment.
5Write the equation of value using the correct symbol, for example R × a-angle-n, with R the payment size.
6Compute v, d or i from the given rate, then evaluate the annuity factor to enough decimals.
7Solve for the unknown, whether payment, present value or accumulated value, and state units in rupees.
8Check the answer: a due value must exceed the matching immediate value, and the PV must be less than the total payments if i > 0.

Quickest way: Immediate first, then multiply by (1 + i)

When to use it: When a question mixes immediate and due annuities, or gives only i.

Always compute a-angle-n or s-angle-n with the i-formula.
If the payments are in advance, multiply by (1 + i) once.
To move a value from time 0 to time n, multiply by (1 + i)ⁿ. To go back, multiply by vⁿ.
If payments start later or end earlier than standard, shift them: value a block with the standard formula, then discount it to the required date.
Use the reciprocal check 1 ÷ a-angle-n = 1 ÷ s-angle-n + i when a question gives one factor and asks for the other.

Common mistakes in Annuity-Immediate and Annuity-Due Functions

Using i in the denominator for an annuity-due.
Students memorise the immediate formula and apply it to every annuity.
Fix: Annuity-due uses d = i ÷ (1 + i). Or compute the immediate value and multiply by (1 + i).
Valuing an annuity-due at the wrong date.
The first payment is at time 0, so the PV formula gives the value at the first payment date, not one period before it.
Fix: Mark the timeline. ä-angle-n is the value at the time of the first payment, time 0.
Using the wrong number of periods n.
Students count years instead of payments, or confuse n with n − 1 for a due annuity.
Fix: Count payments. Then check that the interest rate matches the payment period.
Forgetting to convert the interest rate.
A question gives a nominal rate convertible monthly while payments are quarterly, and the formula is applied directly.
Fix: Convert to an effective rate per payment period first, then use i in the formulas.
Rounding v or i too early.
Short rounding feels convenient, but vⁿ amplifies the error for large n.
Fix: Keep at least six significant figures until the final answer, or use calculator memory.
Confusing accumulation of a due annuity with accumulation at the last payment date.
Students assume s̈-angle-n is valued at time n, the date of the last payment, as s-angle-n is. But the due payments are at times 0 to n − 1, so time n is one period after the last payment.
Fix: s̈-angle-n is valued at time n, one period after the last payment. s-angle-n is valued at time n, at the last payment.

Worked examples

Example 1

A person pays ₹10,000 at the end of each year for 8 years into a fund earning 6% per year effective. Find (a) the present value and (b) the accumulated value at the end of year 8.

Show the solution

Payments are at times 1 to 8, so this is an annuity-immediate with n = 8 and i = 0.06.
v = 1 ÷ 1.06 = 0.943396. v⁸ = 1 ÷ 1.06⁸. 1.06⁸ = 1.593848, so v⁸ = 0.627412.
a-angle-8 = (1 − 0.627412) ÷ 0.06 = 0.372588 ÷ 0.06 = 6.20979.
PV = 10,000 × 6.20979 = ₹62,098 (to the nearest rupee).
s-angle-8 = (1.593848 − 1) ÷ 0.06 = 0.593848 ÷ 0.06 = 9.89747.
Accumulated value = 10,000 × 9.89747 = ₹98,975.
Check: 62,098 × 1.593848 = 98,975, which matches.

Answer: Present value ≈ ₹62,098 and accumulated value at time 8 ≈ ₹98,975.

Example 2

An annuity-due pays ₹5,000 at the start of each year for 10 years. Interest is 5% per year effective. Find the present value and the accumulated value at the end of year 10.

Show the solution

Payments at times 0 to 9, so annuity-due with n = 10 and i = 0.05.
d = 0.05 ÷ 1.05 = 0.047619.
1.05¹⁰ = 1.628895, so v¹⁰ = 0.613913.
ä-angle-10 = (1 − 0.613913) ÷ 0.047619 = 0.386087 ÷ 0.047619 = 8.10782.
PV = 5,000 × 8.10782 = ₹40,539 (to the nearest rupee).
s̈-angle-10 = (1.628895 − 1) ÷ 0.047619 = 0.628895 ÷ 0.047619 = 13.2068.
Accumulated value at time 10 = 5,000 × 13.2068 = ₹66,034.
Check: 40,539 × 1.628895 = 66,034, which matches.

Answer: Present value ≈ ₹40,539 and accumulated value at time 10 ≈ ₹66,034.

Exam tips

Draw the timeline in every written answer. Examiners award marks for correct placement of payments and the valuation date.
Show the formula in standard notation, such as a-angle-n, before substituting numbers. Then show the factor, then the rupee value.
In MCQs, check whether the question says in advance, at the start, or in arrear. The wrong type often appears as a distractor option.
In Paper B (computer-based), build annuity factors from i in a single cell or line, then reuse it, so you can change n or i without retyping.
When an unknown n or i is asked, rearrange to isolate vⁿ or (1 + i)ⁿ first. Use logarithms for n, and trial-and-error or interpolation for i.

Practice questions from Annuity and accumulation functions

Annuity-Immediate and Annuity-Due Functions in other exams

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

Annuity-Immediate and Annuity-Due Functions: frequently asked questions

What is the difference between annuity-immediate and annuity-due?

An annuity-immediate pays at the end of each period. An annuity-due pays at the start. Each due payment is one period earlier, so the due value is (1 + i) times the immediate value at the same date.

How do I calculate s-angle-n?

Use s-angle-n = ((1 + i)ⁿ − 1) ÷ i. It is the accumulated value at time n of 1 paid at the end of each of n periods. You can also compute a-angle-n and multiply by (1 + i)ⁿ.

When do I use d instead of i?

Use d = i ÷ (1 + i) in the denominator for annuity-due functions ä-angle-n and s̈-angle-n. Alternatively, find the immediate value and multiply by (1 + i).

What does the angle symbol mean in a-angle-n?

The angle symbol shows the term n of the annuity in periods. It tells you the number of payments, not a rate. The rate is shown separately or stated in the question.