Banking and Insurance - Laws and Practice · Calculation of Interest and Annuities

Annuities: Types, Future Value and Present Value

Updated 11 October 2026 · Fact-checked

An annuity is a series of equal payments made at equal intervals. An ordinary annuity pays at the end of each period; an annuity due pays at the start. Find its future or present value by multiplying the payment by the right factor. A perpetuity never ends, so its present value is payment ÷ rate.

Understand Annuities: Types, Future and Present Value

An annuity is a stream of equal payments made at regular intervals for a fixed number of periods. Loan instalments, recurring deposits, rent and insurance pension payouts are all annuities. In banking and insurance, you use them to value a series of cash flows as one number.

The key question is when each payment falls. In an ordinary annuity (annuity immediate), payments are made at the end of each period. In an annuity due, payments are made at the start of each period. Because every payment in an annuity due is made one period earlier, it earns one more period of interest. So its value is higher than that of an ordinary annuity by a factor of (1 + i).

A perpetuity is an annuity that never ends, such as a fixed payment that continues forever. You cannot add infinite payments one by one. But the present value is finite because far-off payments are discounted to almost nothing. That gives a simple formula: payment ÷ rate.

The future value of an annuity is the total amount you hold at the end of the term, with each payment compounded up to that date. The present value is the single lump sum today that is equal to all the payments, each discounted back. A sinking fund is a fund built by equal periodic deposits so that it reaches a target sum, such as the amount needed to repay a debenture or replace an asset. It is the future value problem solved backwards 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 = P × [(1 + i)ⁿ − 1] ÷ i; P = payment per period, i = rate per period, n = number of periods. Payments at end of each period.
Present value of ordinary annuity: PV = P × [1 − (1 + i)⁻ⁿ] ÷ i; Gives today's value of n end-of-period payments.
Annuity due (future value): FV(due) = P × [(1 + i)ⁿ − 1] ÷ i × (1 + i); Ordinary annuity value multiplied by (1 + i).
Annuity due (present value): PV(due) = P × [1 − (1 + i)⁻ⁿ] ÷ i × (1 + i); Ordinary annuity present value multiplied by (1 + i).
Present value of perpetuity: PV = P ÷ i; Applies when payments are made at the end of each period forever. For a perpetuity due, add the first payment: PV = P + P ÷ i.
Sinking fund deposit: P = F × i ÷ [(1 + i)ⁿ − 1]; F = target sum needed at the end of n periods; deposits made at the end of each period.
Rate and period matching: i = annual rate ÷ m; n = years × m; m = number of payments per year.

How to solve Annuities: Types, Future and Present Value questions

Use this method for any annuity question, whether it asks for a future value, present value, instalment or sinking fund deposit.

1Read the question and mark what is asked: future value, present value, payment, or perpetuity value.
2Identify the type: ordinary annuity (payments at the end), annuity due (payments at the start) or perpetuity (no end date).
3Convert the rate and term to the payment period. For monthly payments, divide the annual rate by 12 and multiply the years by 12.
4Write the correct formula and substitute P, i and n. Compute (1 + i)ⁿ carefully first.
5Calculate the factor and multiply by the payment, or divide the target by the factor if you need the payment.
6For an annuity due, multiply the ordinary annuity result by (1 + i).
7Check that the answer is reasonable: future value exceeds total payments, and present value is less than total payments.
8State the answer in rupees with units and a one-line conclusion.

Quickest way: Factor-first shortcut

When to use it: Use it when the question gives or lets you compute the compound factor (1 + i)ⁿ, and time is short.

Compute A = (1 + i)ⁿ once.
Future value = P × (A − 1) ÷ i.
Present value = P × (1 − 1 ÷ A) ÷ i.
For annuity due, multiply by (1 + i) at the end.
For a sinking fund, divide the target by (A − 1) ÷ i. Use the given table value if one is supplied.

Common mistakes in Annuities: Types, Future and Present Value

Using the annual rate with monthly payments
Students copy the rate from the question without checking the payment frequency.
Fix: Divide the annual rate by the number of payments per year and multiply the years by the same number.
Treating an annuity due as an ordinary annuity
The words 'at the beginning of each year' are missed in a long question.
Fix: Underline the payment timing. If payments are at the start, multiply the ordinary annuity result by (1 + i).
Mixing up future value and present value formulas
Both use (1 + i)ⁿ and a division by i, so they look similar.
Fix: Remember that the future value has (1 + i)ⁿ − 1 and the present value has 1 − (1 + i)⁻ⁿ. The present value must be smaller than the sum of payments.
Using the perpetuity formula for a limited period
Students see 'regular payment' and apply P ÷ i directly.
Fix: Use P ÷ i only when payments continue forever. A fixed term needs the annuity formula.
Rounding (1 + i)ⁿ too early
Students round the factor to two decimals to save time.
Fix: Keep at least four decimal places in the factor and round only the final answer.
Using the sinking fund formula as if it gave the final sum
The future value and sinking fund formulas contain the same factor, so they get confused.
Fix: If the target sum is given, you are finding the deposit, so divide by the factor. If the deposit is given, you are finding the final sum, so multiply.

Worked examples

Example 1

A person 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. Also find the amount if the deposits are made at the start of each year.

Show the solution

P = ₹10,000, i = 0.10, n = 3.
(1.10)³ = 1.331.
Ordinary annuity: FV = 10,000 × (1.331 − 1) ÷ 0.10 = 10,000 × 3.31 = ₹33,100.
Annuity due: FV = 33,100 × 1.10 = ₹36,410.

Answer: The amount is ₹33,100 if deposits are made at the end of each year, and ₹36,410 if made at the start.

Example 2

A company wants to build a sinking fund to repay debentures of ₹5,00,000 after 2 years. Deposits are made at the end of each year and the fund earns 10% p.a. Find the annual deposit. Also, find the present value of a perpetual annual payment of ₹6,000 at 8%.

Show the solution

Sinking fund: F = ₹5,00,000, i = 0.10, n = 2.
(1.10)² = 1.21.
Factor = (1.21 − 1) ÷ 0.10 = 2.10.
Deposit P = 5,00,000 ÷ 2.10 = ₹2,38,095 (rounded).
Check: 2,38,095 × 1.10 + 2,38,095 = 2,61,904.5 + 2,38,095 = about ₹5,00,000.
Perpetuity: PV = 6,000 ÷ 0.08 = ₹75,000.

Answer: The annual sinking fund deposit is about ₹2,38,095. The present value of the perpetuity is ₹75,000.

Exam tips

Read the payment timing first. 'Beginning of the year' means annuity due and changes the answer.
Show the formula, the substitution and the factor calculation separately, as marks go for method even if arithmetic slips.
If the question gives table values for (1 + i)ⁿ or annuity factors, use them as given rather than recomputing.
Round only at the end and say clearly what rounding you used.
Add a short practical line, such as what the result means for the loan, deposit or policy payout in the question.

Practice questions from Calculation of Interest and Annuities

Annuities: Types, 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: Types, Future and Present Value: frequently asked questions

What is the difference between an ordinary annuity and an 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. An annuity due is worth more, by a factor of (1 + i), because each payment earns one extra period of interest.

How do you calculate the present value of a perpetuity?

Divide the periodic payment by the rate per period: PV = P ÷ i. For example, ₹6,000 a year forever at 8% has a present value of ₹75,000. This works for payments made at the end of each period.

What is the sinking fund formula?

The periodic deposit is P = F × i ÷ [(1 + i)ⁿ − 1], where F is the target sum. It is the future value annuity formula rearranged to find the payment. Deposits are assumed to be at the end of each period.

Are annuity numericals asked in CS Professional Banking and Insurance?

The syllabus covers calculation of interest and annuities, so you should prepare the formulas and numericals. The paper is written and descriptive, so show every step clearly.