NISM-Series-X-A: Investment Adviser (Level 1) · Time Value of Money

Annuities and Perpetuities: Formulas and Solved Examples

Updated 11 October 2026 · Fact-checked

An annuity is a series of equal cash flows at equal intervals for a fixed time. A perpetuity continues forever. To solve, find rate per period and number of periods, check whether payments fall at the end (ordinary) or start (due), then apply the correct present or future value formula.

Understand Annuities and Perpetuities

An annuity is a stream of equal payments made at regular intervals for a fixed number of periods. A monthly SIP of ₹10,000, a loan EMI and a pension of fixed amount are all annuity-type flows.

The key question is timing. In an 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. A SIP debited on the first day of each month is closer to an annuity due. Rent paid in advance is an annuity due. Loan EMIs paid at month-end are an ordinary annuity.

Because an annuity due payment starts earning one period earlier, its value is higher than that of an ordinary annuity with the same payment, rate and periods. The annuity due value equals the ordinary annuity value multiplied by (1 + r). This holds for both present value and future value.

A perpetuity is an annuity that never ends. Its present value is finite because far-off payments are worth very little today. You use it for things like a fixed payout that continues indefinitely, or a rough value of a preference share paying a fixed dividend. A perpetuity has no future value, because it never ends.

Always match the rate to the period. If payments are monthly, use the monthly rate (annual rate ÷ 12) and the number of months. This is the most common place where marks are lost.

Key formulas to remember

Future value of ordinary annuity: FV = PMT × [((1 + r)ⁿ − 1) ÷ r]; PMT is the payment per period, r the rate per period, n the number of periods. Payments at end of each period.
Present value of ordinary annuity: PV = PMT × [1 − (1 + r)⁻ⁿ] ÷ r; Gives value today of n equal end-of-period payments.
Annuity due conversion: FV (due) = FV (ordinary) × (1 + r); PV (due) = PV (ordinary) × (1 + r); Use the same PMT, r and n. Due value is always higher for r > 0.
Present value of perpetuity: PV = PMT ÷ r; First payment is one period from now. r must be the rate per payment period.
Growing perpetuity: PV = PMT₁ ÷ (r − g); PMT₁ is the first payment, one period away. Valid only if r > g.
Periodic rate: r = annual rate ÷ number of periods per year; For monthly payments, divide by 12 and use n in months.

How to solve Annuities and Perpetuities questions

Use this method for any annuity or perpetuity question. It keeps timing and rate errors out.

1Decide what is asked: value today (PV) or value at the end (FV). If the payments never end, it is a perpetuity and only PV applies.
2Write down PMT, the annual rate and the payment frequency.
3Convert to the periodic rate r and the total number of periods n so they match the payment frequency.
4Check timing. Payments at the end of each period mean ordinary annuity. Payments at the start mean annuity due.
5Apply the ordinary annuity formula (or PMT ÷ r for a perpetuity).
6If it is an annuity due, multiply the result by (1 + r).
7Sanity check: FV should exceed total payments made; PV should be less than total payments received; annuity due should exceed ordinary.

Quickest way: Ordinary first, then adjust

When to use it: Use when the question gives options and you have limited time on a calculator-based question.

Always compute the ordinary annuity value first, then multiply by (1 + r) only if payments are in advance.
For perpetuity questions, just divide PMT by r. Check that r is a decimal (8% = 0.08).
Eliminate options that break logic: an FV below the sum of payments, or a PV above the sum of payments, is wrong.
If an option equals your ordinary answer times (1 + r), that is the likely trap for the due version. Choose based on the timing words in the question.

Common mistakes in Annuities and Perpetuities

Using the annual rate with monthly payments.
The rate is quoted per year, so students plug it in directly.
Fix: Divide the annual rate by 12 and use n as the number of months.
Treating an annuity due as ordinary.
Students miss words like 'at the beginning of each month' or 'in advance'.
Fix: Underline the timing words. For due, multiply the ordinary result by (1 + r).
Multiplying by (1 + r) twice, or using it for a perpetuity without being told payments are in advance.
Students memorise the adjustment as a habit.
Fix: Apply (1 + r) once, and only when payments are at the start of the period.
Entering the rate as 8 instead of 0.08.
Rushing under time pressure.
Fix: Convert the percentage to a decimal before substituting.
Calling a perpetuity's value at time zero when the first payment is today.
The formula PMT ÷ r assumes the first payment is one period away.
Fix: If the first payment is today, add that payment to PMT ÷ r.
Looking for a future value of a perpetuity.
Students apply FV logic to every cash flow stream.
Fix: Remember a perpetuity never ends, so only present value is defined.

Worked examples

Example 1

You invest ₹10,000 at the end of every year for 3 years in a fund earning 10% per annum. What is the future value at the end of year 3?

Show the solution

PMT = ₹10,000, r = 0.10, n = 3. Payments are at year-end, so it is an ordinary annuity.
(1.10)³ = 1.331.
FV = 10,000 × (1.331 − 1) ÷ 0.10 = 10,000 × 3.31.
FV = ₹33,100.

Answer: ₹33,100. If payments were made at the start of each year, the value would be ₹33,100 × 1.10 = ₹36,410.

Example 2

A preference share is expected to pay a fixed dividend of ₹6 per year forever. If your required rate of return is 8%, what is its present value? What is the present value of the same ₹6 for 2 years only as an ordinary annuity at 8%?

Show the solution

Perpetuity: PV = PMT ÷ r = 6 ÷ 0.08 = ₹75.
Two-year annuity: (1.08)² = 1.1664, so (1.08)⁻² = 1 ÷ 1.1664 = 0.857339.
PV = 6 × (1 − 0.857339) ÷ 0.08 = 6 × 0.142661 ÷ 0.08.
6 × 1.783265 = ₹10.70 (approx).

Answer: Perpetuity value is ₹75. The 2-year ordinary annuity value is about ₹10.70.

Exam tips

Read the timing words first. 'At the end of each year' means ordinary; 'in advance' or 'at the beginning' means due.
Check the compounding and payment frequency before touching the formula. Monthly SIP means monthly r and n in months.
Expect the due version of an ordinary answer to appear as a trap option. Confirm which one the question needs.
On 2-mark caselet questions, a wrong answer costs more under negative marking. Use the sanity checks to eliminate options before guessing.
For perpetuities, if r and the payment are given, the answer is a single division. Do not overcomplicate it.

Practice questions from Time Value of Money

Annuities and Perpetuities in other exams

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

Annuities and Perpetuities: 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 + r), because each payment earns interest for one extra period.

How do you calculate the present value of an annuity?

Use PV = PMT × [1 − (1 + r)⁻ⁿ] ÷ r, with r and n matched to the payment frequency. For an annuity due, multiply the result by (1 + r).

What is the perpetuity formula?

PV = PMT ÷ r, where r is the rate per period. For example, ₹5,000 a year forever at 10% has a present value of ₹50,000. The first payment is assumed one period from now.

How is the SIP future value calculated?

A SIP is treated as an annuity. Use the future value formula with the monthly rate and number of months. If the instalment is made at the start of each month, multiply by (1 + r) for the annuity due value.