Time value of money calculator with worked steps

16,288.95FV

The value after 10 periods at 5% a year, payments at the end of each period.

Worked steps

Rate per periodi = I/Y ÷ 100 = 5 ÷ 100 = 0.05
Growth factor(1 + i)^N = 1.05¹⁰ = 1.628895
Future valueFV = −PV × 1.628895 = −(−10,000.00) × 1.628895 = 16,288.95

Timeline

t = 0PV −10,000.00
t = 1–
t = 2–
…
t = 9–
t = 10FV 16,288.95

Periods in between are left out; nothing is paid or received in them.

BA II Plus and HP 12C keystrokes for these numbers

Key facts

Future value: FV = PV × (1 + r)^n; present value: PV = FV ÷ (1 + r)^n, with r the rate per period and n the number of periods.
Payments at the start of each period (an annuity due) are worth (1 + r) times the same payments at the end of each period.
A level perpetuity with its first payment in one period is worth the payment divided by the rate: 840 a year at 6% is worth 14,000.
A nominal 7.2% a year compounded monthly is an effective annual rate of 7.4424%.
With continuous compounding the effective annual rate is e^j − 1: 10.5171% for a nominal 10%.

How is the result calculated?

All five time value of money values – N, I/Y, PV, PMT and FV – sit in one equation. Money received is positive and money paid out negative, as on a financial calculator, so the values balance to zero (NYU Stern, time value of money (secondary source)) (Texas Instruments, signs and payment timing):

PV\,(1+r)^n + PMT\,(1+r\,k)\,\frac{(1+r)^n-1}{r} + FV = 0

$r$ is the rate per period (I/Y ÷ 100 ÷ P/Y), $n$ the number of periods, and $k = 1$ for payments at the start of each period (an annuity due), $k = 0$ at the end (an ordinary annuity). Without payments the equation is the single-amount rule:

FV = PV(1+r)^n \qquad PV = \frac{FV}{(1+r)^n}

PV, FV and PMT follow from the equation directly, and N with a logarithm: $n = \ln\frac{c - FV}{PV + c} \div \ln(1+r)$ with $c = PMT\,(1+r\,k)/r$ . The rate has no formula once there are payments: the calculator tries rates between −99% and 1,000% a year until the equation is zero, then narrows the result down. At a rate of 0% the payment term is $PMT \times n$ .

A perpetuity pays without end. With the first payment one period from now and a constant rate, its present value is the payment divided by the rate; a payment that grows at $g$ per period is divided by $r - g$ , which needs $g < r$ (NYU Stern, time value of money (secondary source)). If the first payment is today, the value is $(1 + r)$ times as much.

PV = \frac{PMT}{r - g} \qquad EAR = \left(1+\frac{j}{m}\right)^m - 1 \qquad EAR_{\text{continuous}} = e^{j} - 1

The effective annual rate (EAR) turns a nominal rate per year $j$ , compounded $m$ times a year, into the rate that one year of compounding really earns; the reverse is $j = m\,\big((1+EAR)^{1/m} - 1\big)$ (NYU Stern, time value of money (secondary source)).

Worked example

1. Future value of a single amount

Invest 10,000 for 10 years at 5% a year, with interest added once a year: PV = −10,000 (paid out), N = 10, I/Y = 5, PMT = 0. The growth factor is 1.05 to the power of 10, 1.628895, so

FV = 10{,}000 \times 1.05^{10} \approx 16{,}288.95

The interest earned is 6,288.95. Discounting 16,288.95 back over the same 10 years at 5% gives the 10,000 again: present and future value are the same rule read in two directions. Likewise, 1,000 received at the end of each of those 10 years is worth 7,721.73 today.

2. An ordinary annuity and an annuity due

Five payments of 1,350 at 4.5% a year. Paid at the end of each year, they are worth 5,926.47 today and 7,385.46 at the last payment:

PV = 1{,}350 \times \frac{1 - 1.045^{-5}}{0.045} \approx 5{,}926.47 \qquad FV = 1{,}350 \times \frac{1.045^{5} - 1}{0.045} \approx 7{,}385.46

Paid at the start of each year (BGN), each payment has one period less of discounting and one more of compounding, so both values are 1.045 times as high: 6,193.16 today and 7,717.80 after five years. The calculator shows these values with a minus sign: they are what you would pay for receiving the five payments.

3. A perpetuity and an effective annual rate

840 a year without end, first payment in one year, at 6%: 840 ÷ 0.06 = 14,000.00. With the first payment today it is 14,840.00. A check: 6% of 14,000.00 is the 840 paid each year.

A nominal 7.2% a year compounded monthly is 0.60% a month. Over a year:

EAR = \left(1 + \frac{0.072}{12}\right)^{12} - 1 \approx 7.4424\%

Dividing an effective annual rate by 12 does not give the monthly rate; take the twelfth root of one plus the EAR and subtract one. Compounded continuously, a nominal 10% a year is $e^{0.1} - 1 \approx 10.5171\%$ effective.

What does each time value of money input mean?

The calculator inputs, as on a financial calculator
Input	Meaning
N	Number of periods: 10 for ten years paid yearly, 360 for 30 years paid monthly
I/Y	Interest rate per year in per cent (6 for 6%)
PV	Present value: the amount at the start, at time 0
PMT	The equal payment in each period
FV	Future value: the amount at the end of period N
P/Y	Payments per year; the rate per period is I/Y ÷ P/Y, compounded once per period
END / BGN	Payments at the end (ordinary annuity) or the start (annuity due) of each period

The same names are used on the BA II Plus (Texas Instruments, TVM keys). For the keys to press on a BA II Plus or an HP 12C, use the link under the result: it opens the online financial calculator with your numbers.

Which time value of money mistakes are common?

Rate and periods in different units. A monthly problem needs the monthly rate and the number of months; P/Y does this for you.
Dividing an effective annual rate by 12. That ignores compounding; only a nominal rate quoted with monthly compounding is divided by 12.
Payments at the start treated as at the end. An annuity due is worth (1 + r) times the ordinary annuity, so the wrong choice is off by one period of interest.
Every amount with the same sign. To find a rate or a number of periods, at least one amount must be paid out (negative) and one received (positive).
A growing perpetuity with g at or above r. The formula then has no finite value; the calculator says so instead of showing a number.

BA II Plus is a trade mark of Texas Instruments and HP 12C a trade mark of HP. Neither company is connected with this page.

What this tool covers

Last updated

30 October 2026

Based on

The time value of money equations (see the sources)
Your own numbers; the default values are illustrative

Includes

Any one of N, I/Y, PV, PMT and FV from the other four, with payments at the end or the start of each period and 1 to 365 payments a year
Level and growing perpetuities, with the first payment in one period or today
Nominal to effective annual rates and back, for 1 to 365 compounding periods a year or continuous compounding
Worked steps and a timeline of the cash flows

Not included

Uneven cash flows, NPV and IRR – the online financial calculator covers them
Growing annuities and payments that change over time
Compounding that differs from the payment frequency
Taxes, fees and inflation

Frequently asked questions

What is the time value of money formula?

For a single amount, FV = PV × (1 + r)^n and PV = FV ÷ (1 + r)^n, where r is the rate per period and n the number of periods. Equal payments at the end of each period add PMT × ((1 + r)^n − 1) ÷ r to the future value.

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

An ordinary annuity pays at the end of each period, an annuity due at the start. The annuity due is worth (1 + r) times as much: five payments of 1,350 at 4.5% are worth 5,926.47 today if paid at the end of each year and 6,193.16 if paid at the start.

How do I find the present value of a perpetuity?

Divide the payment by the rate per period, if the first payment is one period away: 840 a year at 6% is worth 14,000. For a payment that grows at g per period, divide by r − g; if the first payment is today, multiply by (1 + r).

How do I convert a nominal rate into an effective annual rate?

EAR = (1 + j ÷ m)^m − 1, with j the nominal rate per year and m the compounding periods per year. A nominal 7.2% compounded monthly is 7.4424% effective; with continuous compounding the EAR is e^j − 1.

How does the calculator find the interest rate?

Once there are payments, no formula gives the rate directly. The calculator tries rates between −99% and 1,000% a year until the amounts balance, then narrows the result down; the worked steps show the check.

Why are some amounts negative?

The calculator uses the convention of financial calculators: money you pay out is negative, money you receive positive. Investing 10,000 is PV = −10,000; the 16,288.95 you have after 10 years at 5% is FV = +16,288.95.

Does the calculator save my numbers?

No. The calculation runs in your browser. Your numbers appear only in the page address after the # sign, so you can share a result; they are not sent to us.

Sources

Aswath Damodaran, NYU Stern School of Business (secondary source): A Primer on the Time Value of Money. Accessed 29 October 2026.
Texas Instruments: Solution 11254: Time Value of Money Functions (TVM) on the BA II PLUS and BA II PLUS PROFESSIONAL. Accessed 29 October 2026.
Texas Instruments: Solution 21590: Unexpected Answer Using the BA II PLUS and BA II PLUS PROFESSIONAL. Accessed 29 October 2026.

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