Fundamentals of Business Mathematics and Statistics · Time Value of Money and Annuity - Simple and Compound Interest

Annuity: Immediate, Due and Perpetuity Explained

Updated 10 October 2026 · Fact-checked

An annuity is a series of equal payments made at equal intervals. In an immediate (ordinary) annuity, payments fall at the end of each period; in an annuity due, at the start. A perpetuity never ends. Find the type, then apply the future value, present value or P ÷ i formula.

Understand Annuity: Immediate, Due and Perpetuity

An annuity is a series of equal payments made at equal time gaps. Rent, loan instalments, insurance premiums and recurring deposits are all annuities. The question is always: what is the whole series worth at one point in time?

In an annuity immediate (also called ordinary annuity), each payment is made at the end of the period. In an annuity due, each payment is made at the start of the period. Because every payment in an annuity due is made one period earlier, it earns one extra period of interest. So its value is the ordinary value multiplied by (1 + i).

The future value is what all payments add up to at the end of the last period, with interest. The present value is what the whole series is worth today, found by discounting each payment back. Present value is the lump sum you would need today to be equal to the series.

A perpetuity is an annuity that continues forever. Its present value is a simple fraction: the payment divided by the rate of interest. Future value of a perpetuity has no meaning because it never ends.

Two common applications are built on these formulas. A sinking fund is money set aside in equal instalments to reach a target sum later, such as replacing a machine. An EMI (equated monthly instalment) is the equal payment that repays a loan with interest. Sinking fund is a future value problem worked backwards; EMI is a present value problem worked backwards.

Key formulas to remember

Future value of annuity immediate: A = P × [(1 + i)ⁿ − 1] ÷ i; P is the payment per period, i the rate per period (as a decimal), n the number of periods. Payments at the end of each period.
Present value of annuity immediate: PV = P × [1 − (1 + i)⁻ⁿ] ÷ i; Gives today's value of n end-of-period payments.
Annuity due (future or present value): Due value = Immediate value × (1 + i); Applies to both future and present value, because each payment is one period earlier.
Present value of perpetuity (immediate): PV = P ÷ i; First payment is one period from now.
Present value of perpetuity due: PV = P + P ÷ i = P × (1 + i) ÷ i; First payment is made today, so add one payment to the immediate value.
Sinking fund instalment: P = F × i ÷ [(1 + i)ⁿ − 1]; F is the target amount. Uses the future value formula solved for P.
EMI (loan instalment): EMI = Loan × i ÷ [1 − (1 + i)⁻ⁿ]; Uses the present value formula solved for P. Loan is the present value.

How to solve Annuity: Immediate, Due and Perpetuity questions

Use this method for any annuity question, whether it asks for a value, a payment or a loan instalment.

1Read the wording and decide the type: immediate (end of period), due (start of period) or perpetuity (forever).
2Decide the direction: is the target a future value (sinking fund, accumulation) or a present value (loan, purchase price today)?
3Convert the rate and time to match the payment gap. For half-yearly payments, use i = annual rate ÷ 2 and n = years × 2.
4Write the correct formula and substitute P, i and n. Compute (1 + i)ⁿ carefully first.
5If the question says due, multiply the immediate result by (1 + i). For a perpetuity due, add one payment to P ÷ i.
6If the payment is the unknown, rearrange: divide the known value by the annuity factor.
7Check that the answer is sensible: future value should exceed total payments; present value should be less than total payments.

Quickest way: Factor shortcut with option checking

When to use it: Use in the 1-hour MCQ paper when the numbers are clean and the options are far apart.

Compute the annuity factor once: for future value, [(1 + i)ⁿ − 1] ÷ i; for present value, [1 − (1 + i)⁻ⁿ] ÷ i.
Multiply by P for a value, or divide by the factor for an unknown payment.
Check bounds before calculating fully: future value must be above n × P, present value must be below n × P. This often removes two options.
For due questions, find the immediate answer first, then multiply by (1 + i). Check whether one option equals the immediate answer; that is usually the trap.
For perpetuity, just do P ÷ i. Do not spend time on anything else.

Common mistakes in Annuity: Immediate, Due and Perpetuity

Using the immediate formula when the payments are made at the start of each period.
Students skip words like 'at the beginning' or 'in advance' while reading.
Fix: Underline the timing phrase first. If payments are in advance, multiply the immediate result by (1 + i).
Not adjusting the rate and number of periods for half-yearly or monthly payments.
The annual rate is given and students plug it in directly.
Fix: Always make i and n match the payment gap: i = annual rate ÷ periods per year, n = years × periods per year.
Applying P ÷ i to a perpetuity due without adding the first payment.
Students remember only one perpetuity formula.
Fix: For a perpetuity due, the answer is P + P ÷ i, because the first payment is made today.
Confusing sinking fund with EMI and using the wrong formula.
Both ask for an equal payment, so they look alike.
Fix: Sinking fund builds a future target, so use the future value factor. EMI repays a loan today, so use the present value factor.
Writing the rate as 10 instead of 0.10 in the formula.
Rates are given in percent and students forget to convert.
Fix: Divide the percentage by 100 before substituting. Write i = 0.10 on your rough sheet.
Multiplying the factor instead of dividing when the payment is unknown.
Students memorise the value formula and copy it without rearranging.
Fix: Ask: am I given the payment or the total? If the total is given, divide it by the annuity factor.

Worked examples

Example 1

Mr. Sharma deposits ₹10,000 at the end of each year for 3 years at 10% p.a. compound interest. Find the amount at the end of 3 years. What would the amount be if he deposited at the start of each year?

Show the solution

Payment P = ₹10,000, i = 0.10, n = 3. Payments at year-end, so this is an annuity immediate.
(1.10)³ = 1.331.
Future value factor = (1.331 − 1) ÷ 0.10 = 0.331 ÷ 0.10 = 3.31.
Future value = 10,000 × 3.31 = ₹33,100.
For the annuity due, multiply by (1 + i): 33,100 × 1.10 = ₹36,410.

Answer: ₹33,100 for end-of-year deposits; ₹36,410 for start-of-year deposits.

Example 2

A loan of ₹55,000 is to be repaid in 2 equal annual instalments, the first due at the end of year 1, with interest at 20% p.a. compounded annually. Find the instalment.

Show the solution

This is a present value problem: Loan = ₹55,000, i = 0.20, n = 2, payments at year-end.
(1.20)² = 1.44, so (1 + i)⁻ⁿ = 1 ÷ 1.44.
1 − 1 ÷ 1.44 = 0.44 ÷ 1.44.
EMI = 55,000 × 0.20 ÷ (0.44 ÷ 1.44) = 11,000 × 1.44 ÷ 0.44.
11,000 ÷ 0.44 = 25,000, and 25,000 × 1.44 = 36,000.
Check: 36,000 ÷ 1.2 = 30,000 and 36,000 ÷ 1.44 = 25,000. Total = ₹55,000, which matches the loan.

Answer: The annual instalment is ₹36,000.

Exam tips

Look for timing words first: 'end of each year' means immediate, 'at the beginning' or 'in advance' means due. Examiners often put the immediate answer as a wrong option.
Perpetuity questions are usually one-step: P ÷ i. Check whether the question says the first payment is today (due) or after one period.
For sinking fund and EMI, work out which one it is from the story: saving for a future target or repaying a loan today.
Use the bounds check: future value above total payments, present value below total payments. With no negative marking, eliminate and still answer every question.
Practise the factor for small n, such as 2, 3 and 4 years at 10% and 20%, so you can compute quickly without tables.

Practice questions from Time Value of Money and Annuity - Simple and Compound Interest

Annuity: Immediate, Due and Perpetuity 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, Due and Perpetuity: frequently asked questions

What is the difference between annuity immediate and annuity due?

In an annuity immediate, payments are made at the end of each period. In an annuity due, they are made at the start. The due value equals the immediate value multiplied by (1 + i), for both future and present value.

How do I calculate the present value of an annuity?

Use PV = P × [1 − (1 + i)⁻ⁿ] ÷ i, with i and n matched to the payment gap. This gives today's value of all the payments. For an annuity due, multiply the result by (1 + i).

What is the perpetuity formula in Business Mathematics?

The present value of a perpetuity is P ÷ i, where P is the fixed payment per period and i is the rate per period. For example, ₹5,000 a year forever at 10% is worth ₹50,000 today. If the first payment is made today, add one payment to the result.

How is a sinking fund different from an EMI?

A sinking fund is a series of equal deposits that grows to a target sum in the future, so it uses the future value formula. An EMI is the equal payment that repays a loan taken today, so it uses the present value formula.