CFA Level I Exam · The Time Value of Money in Finance

Annuities and Perpetuities: Present and Future Value

Updated 7 October 2026 · Fact-checked

An annuity is a fixed payment made at equal intervals for a set number of periods. A perpetuity pays forever. Find the present value by discounting the payments: PV of an ordinary annuity = PMT × [1 − (1 + r)^−N] ÷ r, and PV of a perpetuity = PMT ÷ r. For an annuity due, multiply by (1 + r).

Understand Annuities and Perpetuities

A single cash flow is easy to value. Many real products pay a series of equal amounts: a bond's coupons, a loan's instalments, a pension. An annuity is a finite series of equal cash flows at equal intervals. Instead of discounting each payment one by one, you use a formula that sums them.

The key choice is timing. In an ordinary annuity the first payment comes at the end of the first period (time 1). In an annuity due the first payment comes at the start of the first period (time 0). Because every payment in an annuity due arrives one period earlier, its PV and FV are both larger than the ordinary annuity's by a factor of (1 + r).

A perpetuity is an annuity with no end. Its PV is finite because far-off payments are discounted to almost nothing. The PV is simply PMT ÷ r, valued one period before the first payment. A perpetuity has no FV, because it never ends.

A deferred annuity starts later than the usual first date. Value it in two steps: find the annuity's PV at the date one period before its first payment, then discount that value back to today as a single cash flow. The same logic works for a deferred perpetuity.

You can also solve for the unknown. Given PV or FV, the rate and the number of periods, you can solve for the payment. This is how loan instalments and savings plans are sized. Always match the rate to the payment frequency: monthly payments need a monthly rate and N in months.

Key formulas to remember

PV of an ordinary annuity: PV = PMT × [1 − (1 + r)^−N] ÷ r; Value sits one period before the first payment. r is the rate per payment period, N is the number of payments.
FV of an ordinary annuity: FV = PMT × [(1 + r)^N − 1] ÷ r; Value sits at the date of the last payment.
Annuity due conversion: PV(due) = PV(ordinary) × (1 + r); FV(due) = FV(ordinary) × (1 + r); Each payment is one period earlier, so value is higher. Valid for the same PMT, r and N.
PV of a perpetuity: PV = PMT ÷ r; Value sits one period before the first payment. Requires r > 0.
Deferred annuity: PV today = PV of annuity at date (k) ÷ (1 + r)^k; k is the number of periods from today to the date one period before the first payment.
Solving for payment: PMT = PV × r ÷ [1 − (1 + r)^−N] (ordinary annuity); PMT = FV × r ÷ [(1 + r)^N − 1]; For an annuity due, divide the resulting PMT by (1 + r).

How to solve Annuities and Perpetuities questions

Use this routine for any annuity or perpetuity question. Draw a timeline first; most errors come from timing.

1Write the timeline. Mark today as time 0 and mark each payment date.
2Identify the type: ordinary (first payment at time 1), due (first payment at time 0), perpetuity or deferred.
3Convert the rate and N to the payment frequency, for example monthly rate = annual rate ÷ 12 when compounding is monthly.
4Decide what you need: PV, FV, PMT, r or N.
5Compute using the formula or calculator. For an annuity due, set the calculator to BGN mode or multiply by (1 + r).
6For a deferred case, value the annuity at the date one period before its first payment, then discount that amount back to time 0.
7Check reasonableness: PV should be less than the sum of payments, FV more, and the due value above the ordinary value.

Quickest way: Calculator time-value keys (TI BA II Plus)

When to use it: Any ordinary annuity, annuity due, or solve-for-payment question where you have N, I/Y, PV or FV and PMT.

Press 2ND, CLR TVM to clear old inputs before each question.
Enter N, I/Y (percent per period), PMT and PV or FV. Use a negative sign for cash outflows and positive for inflows.
Press CPT, then the key you want (PV, FV, PMT, N or I/Y).
For an annuity due, press 2ND, BGN, then 2ND, SET so the display shows BGN, then 2ND, QUIT. Compute as usual.
Reset to END mode afterwards (2ND, BGN, 2ND, SET) so the next question is not wrong.
On the HP 12C, press g, BEG for annuity due and g, END for ordinary. Enter n, i, PMT, then press PV or FV.
For a perpetuity, skip the keys and divide PMT by r.

Common mistakes in Annuities and Perpetuities

Treating an annuity due as an ordinary annuity
The wording 'payments at the start of each year' is easy to skim past.
Fix: Draw the timeline. If a payment sits at time 0, it is a due. Multiply by (1 + r) or use BGN mode.
Forgetting to switch the calculator back to END mode
BGN mode stays on between questions.
Fix: Check that BGN is not shown on the display at the start of every question.
Using the annual rate with monthly or quarterly payments
Candidates plug in the stated rate without matching it to the payment period.
Fix: Divide the annual rate by the compounding periods per year and multiply N by the same number.
Discounting a deferred annuity by the wrong number of periods
The annuity formula gives a value one period before the first payment, which is not time 0.
Fix: If the first payment is at time t, the annuity value is at t − 1. Discount it back (t − 1) periods.
Valuing a perpetuity at the wrong date
PMT ÷ r is applied as if it were value at the first payment date.
Fix: PMT ÷ r is the value one period before the first payment. Adjust if the first payment is not at time 1.
Wrong signs on the calculator
PV and PMT are entered with the same sign, giving an error or a wrong answer.
Fix: Use opposite signs for money paid out and money received.

Worked examples

Example 1

A bond-like contract pays €5,000 at the end of each year for 8 years. The discount rate is 6% a year. What is its present value? Options: A) €29,490 B) €31,049 C) €33,000

Show the solution

Ordinary annuity: first payment at year 1.
(1.06)^8 = 1.593848, so (1.06)^−8 = 0.627412.
Annuity factor = (1 − 0.627412) ÷ 0.06 = 6.20979.
PV = 5,000 × 6.20979 = 31,048.95.
Calculator: N = 8, I/Y = 6, PMT = −5,000, CPT PV gives about 31,049.

Answer: B) €31,049

Example 2

A company will pay $2,000 a year for 5 years, with the first payment 3 years from today (at time 3). The discount rate is 8%. What is the present value today? Options: A) $6,247 B) $6,846 C) $8,001

Show the solution

First payment at time 3, so the annuity value is at time 2 (one period before).
Annuity factor for 5 years at 8% = (1 − 1.08^−5) ÷ 0.08. 1.08^5 = 1.469328, so 1.08^−5 = 0.680583.
Factor = 0.319417 ÷ 0.08 = 3.99271.
Value at time 2 = 2,000 × 3.99271 = 7,985.42.
Discount 2 periods: 7,985.42 ÷ 1.08^2 = 7,985.42 ÷ 1.1664 = 6,846.2.
This rounds to $6,846, which is option B. Option C ($8,001) is close to the time 2 value and is the trap for forgetting to discount.

Answer: B) $6,846

Exam tips

Read the timing words closely: 'at the end of each year' means ordinary, 'at the beginning' means due. Examiners use this as the trap.
Use the elimination logic: the due value must exceed the ordinary value, and a perpetuity PV must exceed the PV of any finite annuity with the same PMT and r.
Options are in ascending order and one is often the ordinary-annuity answer for a due question. Compute with the right mode before choosing.
For deferred annuities, do a quick timeline sketch on your scratch paper. It takes ten seconds and prevents a wrong discounting period.
Clear TVM and check BGN/END every question. With about 90 seconds per question, build this into your routine.

Practice questions from The Time Value of Money in Finance

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?

An ordinary annuity pays at the end of each period, and an annuity due pays at the start. Because each payment comes one period earlier, an annuity due is worth (1 + r) times as much as the matching ordinary annuity, in both PV and FV.

How do you calculate the present value of a perpetuity?

Divide the constant payment by the discount rate per period: PV = PMT ÷ r. The result is the value one period before the first payment. If the first payment is later, discount the result further.

How do you find the present value of a deferred annuity?

Find the annuity's PV at the date one period before its first payment, then discount that single amount back to today. The number of periods used in the second step is the number of periods from today to that date.

Does a perpetuity have a future value?

No. A perpetuity has no final payment, so there is no date at which to measure a future value. You only calculate its present value.