Financial Management and Business Data Analytics · Time Value of Money

Annuities: Future and Present Value, Sinking Fund and EMI

Updated 10 October 2026 · Fact-checked

An annuity is a series of equal cash flows at equal intervals. Ordinary annuity payments fall at the end of each period; annuity due payments fall at the start. Find future or present value with the annuity factor, then multiply by (1 + i) for annuity due. Sinking fund and EMI use the same factors in reverse.

Understand Annuities: Future and Present Value

An annuity is a series of equal payments made or received at equal intervals for a fixed number of periods. Rent, insurance premiums, recurring deposits and loan instalments are all annuities.

The only thing that separates the two types is timing. In an ordinary annuity (also called annuity immediate), each payment comes at the end of the period. In an annuity due, each payment comes at the start of the period. Because annuity due payments come one period earlier, each earns or loses one extra period of interest. So an annuity due is worth more than the matching ordinary annuity, both in future value and in present value.

The future value of an annuity is what all the payments grow to at the end of the last period. The present value is what all the payments are worth today, discounted at the given rate. You do not need to treat each payment separately. The annuity factor adds them all up in one step.

Two reverse problems come up often. A sinking fund asks: what equal amount must I save each period so that I reach a target sum on a future date? This uses the future value formula solved for the payment. A loan instalment (EMI) asks: what equal payment repays a loan of a given size over a given time at a given rate? The loan amount is the present value of the instalments, so you solve the present value formula for the payment.

Always match the rate to the period. If payments are monthly, use the monthly rate and the number of months.

Key rules to remember

Future value of ordinary annuity: FV = A × [(1 + i)ⁿ − 1] ÷ i; A = payment per period, i = rate per period, n = number of periods. The bracket is the FV annuity factor.
Present value of ordinary annuity: PV = A × [1 − (1 + i)⁻ⁿ] ÷ i; The bracket is the PV annuity factor. Payments are at the end of each period.
Annuity due conversion: FV (due) = FV (ordinary) × (1 + i); PV (due) = PV (ordinary) × (1 + i); Use it only when payments are at the start of each period.
Sinking fund payment: A = FV × i ÷ [(1 + i)ⁿ − 1]; i ÷ [(1 + i)ⁿ − 1] is the sinking fund factor, the reciprocal of the FV annuity factor.
Loan instalment (EMI): A = P × i ÷ [1 − (1 + i)⁻ⁿ]; P = loan amount. The instalment is P divided by the PV annuity factor. This is the capital recovery factor method.
Rate and period matching: i = annual rate ÷ periods per year; n = years × periods per year; Example: 12% p.a. paid monthly gives i = 1% and n = 12 × years.

How to solve Annuities: Future and Present Value questions

Use this order for any annuity question, whether the table factor is given or you must compute it.

1Read the question and decide what is unknown: future value, present value, the payment, or the number of periods.
2Check the timing of payments. 'End of each year' means ordinary annuity. 'Beginning of each year' or 'in advance' means annuity due.
3Match the rate and periods. If payments are monthly or quarterly, convert the annual rate and the number of years to the same period.
4Choose the formula: FV factor for a target or accumulated sum, PV factor for today's value or a loan.
5Get the factor from the table given in the question. If none is given, compute (1 + i)ⁿ first and then the factor.
6Calculate the amount. For annuity due, multiply the ordinary annuity result by (1 + i).
7For a sinking fund or EMI, divide the target or loan by the matching factor.
8Write the answer with the unit and a one-line check, such as whether the present value is lower than the total of payments.

Quickest way: Factor-first shortcut

When to use it: Use it when the question gives annuity factor tables or when the numbers are large and calculation time is short.

Underline the timing words and the period of payment first. This decides whether to apply the (1 + i) step.
Write only the factor you need: FVIFA for sums that grow, PVIFA for today's value and loans.
For payment problems, remember the pairing: sinking fund payment = target ÷ FVIFA, and EMI = loan ÷ PVIFA. No new formula is needed.
Multiply by (1 + i) once at the end for annuity due. Do not adjust the factor and the amount both.
Sense-check: FV must exceed the total payments, and PV must be below the total payments.

Common mistakes in Annuities: Future and Present Value

Treating an annuity due as an ordinary annuity
Students skip the words 'at the beginning of each year' and use the standard formula directly.
Fix: Circle the timing words first. For due, multiply the ordinary annuity answer by (1 + i).
Using the annual rate with monthly payments
The rate in the question is given per year, so it is used without conversion.
Fix: Divide the rate by the number of periods per year and multiply the years by the same number before using any factor.
Using the FV factor for a loan or the PV factor for a sinking fund
Both problems look like 'find the payment', so the factors get mixed up.
Fix: A loan is a value today, so use PVIFA. A sinking fund is a value in the future, so use FVIFA.
Multiplying instead of dividing when finding the payment
Students remember the formula with A on the right side and forget to rearrange it.
Fix: Write the equation first, for example 5,00,000 = A × factor, and then solve for A by division.
Adding (1 + i) to both the factor and the final amount for annuity due
Students apply the due adjustment twice while trying to be careful.
Fix: Apply it exactly once, on the final ordinary annuity value.
Rounding the factor too early
Students round intermediate values to two decimals, and the answer drifts.
Fix: Keep four to six decimals in the factor and round only the final rupee amount.

Worked examples

Example 1

Meera deposits ₹20,000 at the beginning of each year for 4 years in a scheme that pays 10% p.a. compound interest. Find the amount at the end of year 4.

Show the solution

Payments are at the beginning of each year, so this is an annuity due. A = ₹20,000, i = 10%, n = 4.
FV annuity factor (ordinary) = [(1.10)⁴ − 1] ÷ 0.10 = (1.4641 − 1) ÷ 0.10 = 4.641.
FV of ordinary annuity = 20,000 × 4.641 = ₹92,820.
Convert to annuity due: 92,820 × 1.10 = ₹1,02,102.

Answer: The amount at the end of year 4 is ₹1,02,102.

Example 2

Rohit takes a loan of ₹5,00,000 at 12% p.a., repayable in 36 equal monthly instalments at the end of each month. The PV annuity factor at 1% for 36 periods is 30.1075. Find the EMI.

Show the solution

Payments are at month-end, so this is an ordinary annuity. Monthly rate i = 12% ÷ 12 = 1%. n = 36.
The loan is the present value of the instalments: 5,00,000 = EMI × 30.1075.
EMI = 5,00,000 ÷ 30.1075 = ₹16,607 (approx.).
Check: total repaid = 16,607 × 36 = ₹5,97,852, which is more than the loan because of interest. The PV is lower than the total of payments, as expected.

Answer: The EMI is about ₹16,607 per month.

Exam tips

Look at the timing words and the payment period before touching a formula. Most marks lost on this topic come from timing and rate errors.
If the question supplies factor tables, use them as given. Show the factor, the multiplication and the final figure so you earn step marks.
Sinking fund and EMI questions often need an amortisation or fund schedule. Learn to show opening balance, interest, payment and closing balance in columns.
In MCQs, check the answer for reasonableness: annuity due value must exceed the ordinary annuity value, and PV must be below total payments. This can help you rule out options quickly.
Start the working with a one-line statement such as 'Ordinary annuity, i = 1% per month, n = 36'. It shows the examiner your approach even if the arithmetic slips.

Practice questions from Time Value of Money

Annuities: Future and Present Value in other exams

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

Annuities: Future and Present Value: frequently asked questions

What is the difference between ordinary annuity and annuity due?

In an 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 of an annuity due is one period earlier, its future value and present value are both the ordinary annuity value multiplied by (1 + i).

How do I calculate EMI using the present value of annuity?

Treat the loan amount as the present value of equal instalments. Convert the rate and tenure to the instalment period, find the PV annuity factor, and divide the loan by it. For monthly EMI, use the monthly rate and the number of months.

What is the sinking fund factor?

It is i ÷ [(1 + i)ⁿ − 1], the amount to set aside each period for every ₹1 you need at the end of n periods. Multiply it by the target sum to get the periodic deposit. It is the reciprocal of the FV annuity factor.

Why is the present value of an annuity less than the total of the payments?

Each future payment is discounted at a positive rate, so it is worth less today than its face value. The further away the payment, the larger the discount. The sum of these discounted values is therefore below the plain total.