Quantitative Aptitude · Mathematics of Finance

Annuities: Ordinary, Due and Future Value

Updated 1 October 2026 · Fact-checked

An annuity is a series of equal payments made at equal intervals. In an ordinary annuity (annuity immediate) payments fall at the end of each period. In an annuity due they fall at the start. Find the FV or PV with the ordinary formula, then multiply by (1 + i) for an annuity due.

Understand Annuities: Ordinary, Due and Future Value

An annuity is a series of equal payments (or receipts) made at equal time gaps. Loan EMIs, recurring deposits and insurance premiums are everyday examples. The payment is usually called C (or A), the rate per period is i, and the number of payments is n.

The only thing that separates the types is when each payment is made. In an ordinary annuity, also called annuity immediate or annuity regular, each payment is made at the end of the period. In an annuity due, each payment is made at the start of the period.

The future value (FV) is the total worth of all payments at the end of the last period, with compound interest added. The present value (PV) is the single amount today that equals all the payments, after discounting each one back.

An annuity due pays every instalment one period earlier than an ordinary annuity. So each payment earns interest for one extra period. That is why the annuity due value is always the ordinary value multiplied by (1 + i). This one link lets you remember only two core formulas.

Always match i and n. If payments are monthly, i is the monthly rate and n is the number of months. If the question gives an annual rate with quarterly payments, divide the rate by 4 and multiply the years by 4.

Key formulas to remember

FV of ordinary annuity: FV = C × [(1 + i)ⁿ − 1] ÷ i; Payments at the end of each period. FV is measured at the end of period n, right when the last payment is made.
PV of ordinary annuity: PV = C × [1 − (1 + i)⁻ⁿ] ÷ i; Payments at the end of each period. PV is measured one period before the first payment.
FV of annuity due: FV = C × (1 + i) × [(1 + i)ⁿ − 1] ÷ i; Payments at the start of each period. FV is measured at the end of period n.
PV of annuity due: PV = C × (1 + i) × [1 − (1 + i)⁻ⁿ] ÷ i; Payments at the start of each period. The first payment is made today, so it is not discounted.
Link between due and ordinary: Value of annuity due = Value of ordinary annuity × (1 + i); Holds for both PV and FV, for the same C, i and n.
Finding the instalment: C = FV × i ÷ [(1 + i)ⁿ − 1] or C = PV × i ÷ [1 − (1 + i)⁻ⁿ]; Rearranged ordinary formulas. For an annuity due, also divide the answer by (1 + i).

How to solve Annuities: Ordinary, Due and Future Value questions

Use this method for any annuity question. It stops you from picking the wrong formula or the wrong rate.

1Read the timing words. 'At the end of each year' or 'annuity immediate' means ordinary. 'At the beginning of each year' or 'annuity due' means due. If nothing is said, assume ordinary.
2Decide what is asked: future value, present value, or the instalment C.
3Convert the rate and time to match the payment gap. Use i = annual rate ÷ number of payments per year, and n = years × payments per year.
4Write the ordinary formula for FV or PV and put in C, i and n.
5Calculate (1 + i)ⁿ carefully, step by step. Use the factor given in the question if there is one.
6If it is an annuity due, multiply the ordinary result by (1 + i). If you are finding C for a due annuity, divide by (1 + i) instead.
7Check the answer is sensible. FV should be more than n × C. PV should be less than n × C. Then pick the matching option.

Quickest way: Ordinary first, then adjust, and use bounds to eliminate

When to use it: Use it in the MCQ paper when options include both the ordinary and due values, which is the usual trap.

Always compute the ordinary value first. Then multiply by (1 + i) once if the annuity is due.
Use bounds to cut options. FV must be above n × C and PV must be below n × C. This often removes two options in seconds.
Note that the due value is exactly (1 + i) times the ordinary value. If two options differ by that ratio, the question is testing timing. Choose by the wording.
Use small cases to check. For n = 2, the ordinary FV is C(2 + i). It is quicker than expanding powers.
If (1 + i)ⁿ needs heavy calculation and four-digit accuracy is not needed, compute approximately and match the nearest option. Check that the options are far enough apart first.
Skip a question if it needs a long power with no given factor and the options are very close. Each wrong answer costs 0.25 marks.

Common mistakes in Annuities: Ordinary, Due and Future Value

Using the ordinary formula for an annuity due, or the reverse.
Students skip the timing words such as 'beginning of each year' and apply the first formula they remember.
Fix: Underline the timing phrase first. Due means multiply the ordinary value by (1 + i).
Dividing the instalment by (1 + i) when the question is ordinary, or forgetting to divide when finding C for an annuity due.
The (1 + i) adjustment is applied by habit, without thinking about which direction it should go.
Fix: Remember that due values are higher. To get C from a given due value, divide by (1 + i).
Using the annual rate with monthly or quarterly payments.
Students copy the rate as stated and forget that i and n must match the payment period.
Fix: Convert first. Quarterly payments at 12% a year mean i = 3% and n = 4 × years.
Getting a negative power wrong in PV, such as treating (1 + i)⁻ⁿ as −(1 + i)ⁿ.
The negative index is confused with a negative number.
Fix: Write (1 + i)⁻ⁿ as 1 ÷ (1 + i)ⁿ. Compute the power first, then take the reciprocal.
Mixing up the date at which FV and PV are measured.
Students forget that the ordinary PV sits one period before the first payment, while the FV sits at the last payment.
Fix: Draw a quick timeline. Mark payments and the valuation point, then see if any extra (1 + i) is needed.
Writing i as 10 instead of 0.10 in the formula.
The rate is quoted as a percentage and is not converted to a decimal.
Fix: Divide the percentage by 100 before using it. 10% means i = 0.10.

Worked examples

Example 1

You deposit ₹1,000 at the end of each year for 3 years in an account paying 10% p.a. compounded annually. What is the amount at the end of 3 years? Options: (A) ₹3,000 (B) ₹3,300 (C) ₹3,310 (D) ₹3,641

Show the solution

The deposits are at the end of each year, so this is an ordinary annuity. C = 1,000, i = 0.10, n = 3.
FV = C × [(1 + i)ⁿ − 1] ÷ i.
(1.10)³ = 1.331, so (1.10)³ − 1 = 0.331.
FV = 1,000 × 0.331 ÷ 0.10 = 1,000 × 3.31 = ₹3,310.
Check with bounds: FV must be above 3 × 1,000 = ₹3,000. Option (D) ₹3,641 is the annuity due value, which is 3,310 × 1.10, so it is the trap.

Answer: (C) ₹3,310

Example 2

A person receives ₹1,000 at the beginning of each year for 3 years. Find the present value of these receipts at 10% p.a. compound interest. Options: (A) ₹2,487 (B) ₹2,736 (C) ₹3,000 (D) ₹3,310

Show the solution

The receipts are at the beginning of each year, so this is an annuity due. C = 1,000, i = 0.10, n = 3.
Ordinary PV = C × [1 − (1 + i)⁻ⁿ] ÷ i.
(1.10)⁻³ = 1 ÷ 1.331 = 0.751315 (approx).
Ordinary PV = 1,000 × (1 − 0.751315) ÷ 0.10 = 1,000 × 2.48685 = ₹2,486.85.
Annuity due PV = 2,486.85 × 1.10 = ₹2,735.54, which is about ₹2,736.
Cross-check by discounting each receipt: 1,000 + 1,000 ÷ 1.10 + 1,000 ÷ 1.21 = 1,000 + 909.09 + 826.45 = ₹2,735.54.
Option (A) is the ordinary PV, and (C) is above the bound because PV must be below ₹3,000.

Answer: (B) ₹2,736

Example 3

A firm wants ₹3,31,000 at the end of 3 years. It will deposit an equal amount at the end of each year at 10% p.a. compounded annually. What yearly deposit is needed? Options: (A) ₹90,909 (B) ₹1,00,000 (C) ₹1,10,000 (D) ₹1,21,000

Show the solution

Deposits at the end of each year make this an ordinary annuity. FV = 3,31,000, i = 0.10, n = 3.
C = FV × i ÷ [(1 + i)ⁿ − 1].
(1.10)³ − 1 = 0.331.
C = 3,31,000 × 0.10 ÷ 0.331 = 33,100 ÷ 0.331 = ₹1,00,000.
Check: 1,00,000 × 3.31 = ₹3,31,000, which matches.
Option (A) ₹90,909 is the deposit if payments were at the start of each year (1,00,000 ÷ 1.10), so it is a trap.

Answer: (B) ₹1,00,000

Exam tips

Look at the timing words in the question before touching a formula. Most wrong answers come from ordinary versus due confusion.
Expect the options to include both the ordinary and the due value. Do not choose the first number that matches your calculation. Check the timing again.
Practise powers of (1 + i) for common rates such as 5%, 10% and 12% for 2 to 5 years. It saves time when no factor is given.
Use bounds: FV is more than n × C and PV is less than n × C. They help eliminate options quickly.
If the question asks for the instalment, check whether you should divide by (1 + i) for an annuity due. Do not skip this last step.

Practice questions from Mathematics of Finance

Annuities: Ordinary, Due and Future Value: frequently asked questions

What is the difference between annuity immediate and annuity due?

In an annuity immediate (ordinary annuity), payments are made at the end of each period. In an annuity due, they are made at the start of each period. Because each payment in an annuity due is one period earlier, its value is (1 + i) times the ordinary value.

How do I find the future value of an annuity in CA Foundation?

For an ordinary annuity, use FV = C × [(1 + i)ⁿ − 1] ÷ i. Make sure i and n match the payment period. If the annuity is due, multiply the result by (1 + i).

If the question does not say whether payments are at the start or end, which should I assume?

Assume an ordinary annuity, with payments at the end of each period. Most questions state the timing clearly when they mean an annuity due, using phrases such as 'at the beginning of each year'.

Is the PV of an annuity due always higher than the PV of an ordinary annuity?

Yes, for the same C, positive i and n. Each payment is received one period earlier, so it is discounted for one less period. The due value is exactly (1 + i) times the ordinary value.