Annuity formulas: present and future value

An annuity, in the mathematical sense, is any series of regular payments: rent, a loan repayment, a pension, a savings plan. The annuity formulas value the whole series in one step. This page, part of our guides to how the maths works, covers payments for a fixed term, called an annuity-certain; payments that stop at death are life annuities.

The idea in plain English

The present value of a series of payments is the sum of the present values of each payment. With equal payments at equal intervals, that sum is a geometric series, which has a short closed form. The formulas below are nothing more than that sum, added up once and for all.

Two details change the answer:

Timing. Payments at the end of each period make an annuity-immediate (in finance, an “ordinary annuity”). Payments at the start make an annuity-due. Every payment of an annuity-due comes one period sooner, so it is worth more.
Growth. If each payment rises by a fixed percentage, as with an income linked to expected inflation, the series is a growing annuity.

The notation

Number of payments nHow many payments there are, or how many years a guarantee lasts.How it works: Annuity formulas: the number of payments; Interest rate iThe interest rate per period, used to move money between dates.How it works: Present value: the interest rate per period; Discount factor vWhat 1 paid one period from now is worth today: 1/(1 + i).How it works: Present value = 1/(1 + $i$ ).
Annuity-immediate aₙThe present value of 1 paid at the end of each period for n periods. The angle around n marks a fixed term.How it works: Annuity formulas: the present value of 1 paid at the end of each period for $n$ periods. The angle around $n$ marks a fixed term. Read it “a angle n”.
Annuity-due äₙThe present value of 1 paid at the start of each period for n periods. The two dots mark payments in advance.How it works: Annuity formulas: the same with payments at the start of each period. The two dots mark payments in advance.
Accumulated value sₙWhat 1 paid at the end of each period for n periods grows to at the end.How it works: Annuity formulas and Accumulated value s̈ₙWhat 1 paid at the start of each period for n periods grows to at the end.How it works: Annuity formulas: the accumulated (future) values, measured at the end of the $n$ th period.
Growing annuityThe present value of n payments that start at 1 and grow by g each period, paid at the end of each period.How it works: Annuity formulas: the present value of payments starting at 1 and growing by Growth rate gHow fast each payment grows over the one before it, per period.How it works: Annuity formulas each period.
PerpetuityThe present value of 1 paid at the end of every period forever: 1/i.How it works: Annuity formulas and Perpetuity-dueThe present value of 1 paid at the start of every period forever: (1 + i)/i.How it works: Annuity formulas: perpetuities, paying forever.
Payment PThe amount of each payment.How it works: Annuity formulas: the payment, and Present valueWhat a set of future payments is worth today, at a given interest rate.How it works: Present value and Future valueWhat a set of payments grows to by the end, at a given interest rate.How it works: Present value the present and future values.

The formulas

The present value of an annuity-immediate and an annuity-due:

\begin{aligned} a_{\angl{n}} &= v + v^{2} + \cdots + v^{n} \\ &= \frac{1 - v^{n}}{i} \\ \ddot{a}_{\angl{n}} &= 1 + v + \cdots + v^{n-1} \\ &= \frac{1 - v^{n}}{d} = (1+i)\,a_{\angl{n}} \end{aligned}

Here d = i/(1 + i) is the rate of discount. The annuity-due is the annuity-immediate moved one period earlier.

The future value of the same payments, at the end of the $n$ th period:

\begin{gathered} s_{\angl{n}} = \frac{(1+i)^{n} - 1}{i} = (1+i)^{n}\,a_{\angl{n}} \\ \ddot{s}_{\angl{n}} = (1+i)\,s_{\angl{n}} \end{gathered}

Accumulate the present value forward n periods.

A growing annuity, with payments 1, (1 + $g$ ), (1 + $g$ )², … at the end of each period, and a perpetuity:

\begin{gathered} (Ga)_{\angl{n}} = \frac{1 - \left(\frac{1+g}{1+i}\right)^{n}}{i - g} \;\;(g \ne i) \\ a_{\infty} = \frac{1}{i} \\ \ddot{a}_{\infty} = \frac{1}{d} = \frac{1+i}{i} \end{gathered}

When g = i the growing annuity is simply n/(1 + i). A perpetuity is the limit as n grows without end, for i above 0.

Each factor is the value of 1 a period. Multiply by the payment for the value, or divide a value by the factor for the payment: $PV = P\,a_{\angl{n}}$ and $P = PV / a_{\angl{n}}$ . For monthly payments, use the monthly rate and the number of months. The calculators convert a yearly rate to the equivalent monthly rate, $(1+i)^{1/12} - 1$ , so that twelve months of interest compound to exactly the yearly rate.

Worked example: $1,000 a year for 20 years at 5%

The discount factor for the whole term is $v^{20}$ = 0.376889, so $a_{\angl{20}}$ = (1 − 0.376889) ÷ 0.05 = 12.4622.
Paid at the end of each year, the payments are worth $1,000 × 12.4622 = $12,462.21 today.
Paid at the start, $\ddot{a}_{\angl{20}}$ = 13.0853, worth $13,085.32: $623.11 more, one year’s interest on the whole value.
Left to accumulate, the end-of-year payments grow to $1,000 × 33.0660 = $33,065.95 by the last payment date.
If the payments start at $1,000 and grow by 2% a year, the factor is 14.6654 and the value $14,665.40.
Paid forever, the factor is 1 ÷ 0.05 = 20.0000 and the value $20,000.00.

Turned around: a lump sum of $100,000 at 5% supports end-of-year payments of $100,000 ÷ 12.4622 = $8,024.26 a year for 20 years, after which nothing is left.

Present value of annuity table

Before calculators, these factors were read from printed tables. The table below gives $a_{\angl{n}}$ for yearly rates and terms, computed with the same engine that runs the calculators.

Present value of 1 a year paid at the end of each year for $n$ years, $a_{\angl{n}}$ , by interest rate. For payments at the start of each year, multiply by $1+i$ .
Years	1%	2%	3%	4%	5%	6%	7%	8%	10%
1	0.9901	0.9804	0.9709	0.9615	0.9524	0.9434	0.9346	0.9259	0.9091
2	1.9704	1.9416	1.9135	1.8861	1.8594	1.8334	1.8080	1.7833	1.7355
3	2.9410	2.8839	2.8286	2.7751	2.7232	2.6730	2.6243	2.5771	2.4869
4	3.9020	3.8077	3.7171	3.6299	3.5460	3.4651	3.3872	3.3121	3.1699
5	4.8534	4.7135	4.5797	4.4518	4.3295	4.2124	4.1002	3.9927	3.7908
6	5.7955	5.6014	5.4172	5.2421	5.0757	4.9173	4.7665	4.6229	4.3553
7	6.7282	6.4720	6.2303	6.0021	5.7864	5.5824	5.3893	5.2064	4.8684
8	7.6517	7.3255	7.0197	6.7327	6.4632	6.2098	5.9713	5.7466	5.3349
9	8.5660	8.1622	7.7861	7.4353	7.1078	6.8017	6.5152	6.2469	5.7590
10	9.4713	8.9826	8.5302	8.1109	7.7217	7.3601	7.0236	6.7101	6.1446
12	11.2551	10.5753	9.9540	9.3851	8.8633	8.3838	7.9427	7.5361	6.8137
15	13.8651	12.8493	11.9379	11.1184	10.3797	9.7122	9.1079	8.5595	7.6061
20	18.0456	16.3514	14.8775	13.5903	12.4622	11.4699	10.5940	9.8181	8.5136
25	22.0232	19.5235	17.4131	15.6221	14.0939	12.7834	11.6536	10.6748	9.0770
30	25.8077	22.3965	19.6004	17.2920	15.3725	13.7648	12.4090	11.2578	9.4269
40	32.8347	27.3555	23.1148	19.7928	17.1591	15.0463	13.3317	11.9246	9.7791

Try it: annuity factor calculator

Change the payment, rate, term and growth to see every factor side by side and what the payments are worth.

Annuity factor calculator

Payment a year

Interest rate, i

Years, n

Growth for the growing rows, g

Factor	Payments	Value of 1	Value of the payments
$a_{\angl{n}}$	Paid at the end of each year	12.4622	PV $12,462.21
$\ddot{a}_{\angl{n}}$	Paid at the start of each year	13.0853	PV $13,085.32
$(Ga)_{\angl{n}}$	Growing 2% a year, paid at the end	14.6654	PV $14,665.40
$(G\ddot{a})_{\angl{n}}$	Growing 2% a year, paid at the start	15.3987	PV $15,398.67
$s_{\angl{n}}$	Paid at the end, valued at the end	33.0660	FV $33,065.95
$\ddot{s}_{\angl{n}}$	Paid at the start, valued at the end	34.7193	FV $34,719.25
$a_{\infty}$	Forever, paid at the end	20.0000	PV $20,000.00
$\ddot{a}_{\infty}$	Forever, paid at the start	21.0000	PV $21,000.00

$1,000.00 a year for 20 years at 5% is worth $12,462.21 today if paid at the end of each year, or $13,085.32 if paid at the start: one year of interest more.

Assumptions to keep in mind

An annuity-certain assumes every payment is made and the interest rate is fixed for the whole term. Real income streams carry risk: a pension may stop at death, a bond issuer may default, rates change. The formulas still apply; the risk goes into the rate chosen or, for payments that depend on being alive, into survival probabilities, which turn $a_{\angl{n}}$ into the life annuity $a_{x}$ . The annuity calculator works out these values for your own numbers and shows the working.

Where this is used

Annuity calculator

Questions

What is the formula for the present value of an annuity?

For n equal payments of P at the end of each period, at interest rate i per period, the present value is P × (1 − (1 + i)^−n) ÷ i. The part after P is the annuity factor, written a with n in an angle. If payments are at the start of each period, multiply the result by (1 + i).

What is the difference between an ordinary annuity and an annuity due?

An ordinary annuity, or annuity-immediate, pays at the end of each period. An annuity due pays at the start. Each payment of an annuity due arrives one period earlier, so its present and future values are exactly (1 + i) times those of the ordinary annuity.

What is the future value of an annuity formula?

For n payments of P at the end of each period, the future value at the date of the last payment is P × ((1 + i)^n − 1) ÷ i. For payments at the start of each period, valued one period after the last payment, multiply by (1 + i).

How do you use a present value of annuity table?

Find the row for the number of payments and the column for the interest rate per period. The number there is the value of 1 paid at the end of each period. Multiply it by the payment to get the present value, or divide a present value by it to get the payment.

Annuity formulas: present value, future value and more