The time value of money formula, step by step
The core time value of money formula is FV = PV × (1 + r)^n: money today (PV) grows at a rate r for n periods into a future value (FV). Add a regular payment (PMT) and one equation ties five numbers together, so if you know any four you can solve for the fifth.
Updated October 20, 2026 · 9 min read · Plain math, no product pitches
On this page
- The five variables and the sign convention
- Time value of money calculator
- Future value of a lump sum
- Compounding periods and the effective rate
- Future value of an annuity
- Present value of a lump sum and an annuity
- Perpetuities and growing annuities
- Solving for the payment, years or rate
- FV, PV, PMT, RATE and NPER in Excel
- Loans use the same equation
What are the five time value of money variables?
Every time value of money problem uses the same five numbers, and if you know four you can find the fifth. (For the idea itself, why a dollar today beats a dollar later, start with the time value of money explained. This page is the math.)
- PV (present value): money at the start, such as a deposit today or a loan amount.
- FV (future value): money at the end, such as a savings goal or a balance left on a loan.
- PMT (payment): the same amount added or paid every period.
- r (written I/Y on financial calculators): the interest rate per period, as a decimal.
- n (written N): the number of periods.
Put together, they make one equation. This is the form spreadsheets and financial calculators solve:
- i
- rate per period (yearly rate ÷ periods per year)
- n
- number of periods (years × periods per year)
- = 0
- every dollar in and out balances once moved to the same date
The sign convention
Why does it equal zero? Because of signs. Spreadsheets and financial calculators treat money you pay out as negative and money you receive as positive. Put $10,000 into savings (−10,000), add $200 a month (−200), and the balance you take out at the end comes back positive. On a loan it flips: the bank hands you the loan (+), and your payments go out (−).
A negative answer is not a wrong answer; it is telling you money flows out. If you get an error instead, check that PV, PMT and FV do not all have the same sign. Something has to come in for something to go out.
Time value of money calculator
Choose what to solve for and fill in the rest. This calculator uses plain positive numbers (no sign convention) and shows the formula with your numbers plugged in, so you can check every step below against it.
Time value of money formula calculator
Solve for future value, present value, the deposit, the rate or the years.
Solve for
Future value
$144,572.72
Starting with $10,000 and adding $200 a month at 7% for 20 years grows to $144,573.
You put in
$58,000
Growth
$86,573
60% of the total
FV = PV(1 + i)n + PMT × [((1 + i)n − 1) ÷ i]
i = 7% ÷ 12 = 0.5833% a month; n = 20 × 12 = 240 months
$144,572.72 = $10,000(1 + 0.005833)240 + $200.00 × [((1 + 0.005833)240 − 1) ÷ 0.005833]
How do you calculate the future value of a lump sum?
Multiply the starting amount by (1 + r) once for every period. Each period’s growth earns growth in the next one.
- FV
- future value
- PV
- present value, the amount today
- r
- rate per period, as a decimal
- n
- number of periods
How do compounding periods change the formula?
Divide the yearly rate by the number of compounding periods per year (m), and multiply the years by m. Interest is then added more often, in smaller pieces.
- r
- yearly (nominal) rate
- m
- compounding periods per year (12 = monthly)
- t
- years
The effective annual rate
The effective annual rate (EAR, called APY on savings accounts) is the yearly rate you actually get once compounding is counted. It lets you compare rates that compound on different schedules.
6% compounded monthly: (1.005)12 − 1 = 6.17%. In a spreadsheet, =EFFECT(6%, 12) returns 0.061678.
Continuous compounding
As m grows without limit, the formula approaches a ceiling that uses the number e (about 2.71828). This is continuous compounding:
$5,000 at 6% for 10 years: $5,000 × e0.6 = $5,000 × 1.822119 = $9,110.59. The effective rate is e0.06 − 1 = 6.18%.
| $10,000 at 6% for 10 years | Ends at | Effective yearly rate |
|---|---|---|
| Yearly | $17,908.48 | 6.000% |
| Quarterly | $18,140.18 | 6.136% |
| Monthly | $18,193.97 | 6.168% |
| Daily | $18,220.29 | 6.183% |
| Continuous | $18,221.19 | 6.184% |
Going from yearly to monthly matters a little; going from daily to continuous barely matters at all.
How do you calculate the future value of an annuity?
An annuity is a series of equal payments at equal intervals. Its future value is the payment times the annuity factor, which adds up every payment’s growth in one step.
- PMT
- payment each period
- i
- rate per period
- n
- number of payments
An ordinary annuity pays at the end of each period. An annuity due pays at the start, so every payment gets one extra period of growth: multiply the result by (1 + i).
With a starting balance as well, add the two pieces: FV = PV × (1 + i)n + PMT × [((1 + i)n − 1) ÷ i].
Where the annuity factor comes from
It is just every payment’s own future value, added up. Take three yearly payments of $100 at 10%, each paid at the end of the year. The first grows for two years, the second for one, and the last not at all:
$100 × (1.10)2 + $100 × 1.10 + $100 = $121 + $110 + $100 = $331
That sum is a geometric series, and it always collapses to the same short form: $100 × [((1.10)3 − 1) ÷ 0.10] = $100 × (0.331 ÷ 0.10) = $331. With 360 payments, the formula saves you 359 additions.
How do you calculate present value?
Run the future value formulas backwards: divide by the growth instead of multiplying by it. This is called discounting.
$20,000 needed in 6 years, at 5%: PV = $20,000 ÷ (1.05)6 = $20,000 ÷ 1.340096 = $14,924.31.
What is the formula for a perpetuity or a growing annuity?
A perpetuity pays the same amount forever. Its present value is the payment divided by the rate, because as n gets huge the (1 + i)−n term in the annuity formula shrinks to zero.
$1,000 a year forever, at 5%: $1,000 ÷ 0.05 = $20,000. Put $20,000 in at 5% and you can take $1,000 a year without ever touching the principal.
A growing perpetuity starts at PMT one period from now and rises by a rate g each period. It only works when r is greater than g.
- PMT
- the first payment, one period from now
- g
- growth rate of the payment per period
$1,000 next year, growing 2% a year forever, at 5%: $1,000 ÷ (0.05 − 0.02) = $33,333.33.
A growing annuity does the same for a fixed number of payments, such as an income that rises with inflation:
How do you solve for the payment, the years or the rate?
Solving for the payment (PMT)
Rearrange the annuity formulas. To reach a savings goal from zero:
$50,000 in 10 years at 5%, monthly: i = 0.0041667 and (1.0041667)120 = 1.647009, so PMT = $50,000 × 0.0041667 ÷ 0.647009 = $321.99 a month. To pay off a loan, use PMT = PV × i ÷ [1 − (1 + i)−n], worked in the loan section below.
Solving for the number of periods (n)
The unknown sits in the exponent, so take logarithms. For a lump sum, (1 + r)n = FV ÷ PV. Taking the natural log (ln) of both sides gives n × ln(1 + r) = ln(FV ÷ PV), so:
$10,000 to $50,000 at 7% a year: n = ln(5) ÷ ln(1.07) = 1.609438 ÷ 0.067659 = 23.8 years.
With payments, the algebra takes three steps. Start from FV = PV × (1 + i)n + PMT × [((1 + i)n − 1) ÷ i]:
- Multiply both sides by i: FV × i = PV × i × (1 + i)n + PMT × (1 + i)n − PMT
- Move PMT across and factor: FV × i + PMT = (1 + i)n × (PV × i + PMT)
- Divide, then take logs of both sides:
Solving for the rate (r)
For a lump sum with no payments, there is a closed form. Take the nth root of the growth:
$10,000 that became $25,000 in 12 years: r = (2.5)1/12 − 1 = 1.079348 − 1 = 7.93% a year. This is the compound annual growth rate covered in the rate of return guide.
Once payments are involved, no formula isolates r. You (or the software) guess, check, and narrow in. Say you can invest $300 a month for 30 years and want $400,000. What rate does that take?
| Try a rate | $300 a month for 30 years grows to | Next step |
|---|---|---|
| 6% | $301,355 | Too low, go higher |
| 7.5% | $404,234 | Too high, go lower |
| 7% | $365,991 | Too low, try between 7% and 7.5% |
| 7.45% | $400,000 | Found it |
Spreadsheets do the same thing faster: Excel documents that RATE is calculated by iteration and returns a #NUM! error if it cannot settle on an answer. Notice that 7.45% is above the 7% a year (before inflation) we use as a cautious planning figure, so this plan depends on a return it may not get.
How do you use FV, PV, PMT, RATE and NPER in Excel?
Excel and Google Sheets share these functions with the same argument order: rate per period, number of periods, payment, present value, future value, and an optional type (0 = payments at the end of each period, the default; 1 = at the start).
| Formula | Result | What it says |
|---|---|---|
| =FV(7%/12, 360, -200, -10000) | 325,159.17 | $10,000 plus $200 a month at 7% for 30 years |
| =PV(5%, 20, 12000) | −149,546.52 | Pay this today for $12,000 a year for 20 years |
| =PMT(5%/12, 120, 0, 50000) | −321.99 | Save this monthly to reach $50,000 in 10 years |
| =RATE(12, 0, -10000, 25000) | 0.0793 | 7.93% a year turns $10,000 into $25,000 in 12 years |
| =RATE(360, -300, 0, 400000)*12 | 0.0745 | The yearly rate for the $400,000 goal above |
| =NPER(7%, 0, -10000, 50000) | 23.79 | Years for $10,000 to grow to $50,000 at 7% |
| =NPER(7%/12, -200, -10000, 100000) | 190.72 | Months to $100,000 with $200 a month |
How is a loan payment calculated?
A loan is the same equation with the signs flipped: the lender gives you PV, you pay PMT each month, and FV is zero when it is paid off.
- PV
- the amount borrowed
- i
- yearly rate ÷ 12
- n
- number of monthly payments
The first payment is $1,733.33 interest ($320,000 × 0.00541667) and only $289.28 principal. Each month the balance falls a little, so less of the next payment goes to interest. After five years you have paid about $121,357, but only about $20,445 of it reduced the balance, which still stands near $299,555. That is the time value of money working for the lender, and it is why extra principal payments early in a loan save so much interest.
The same equation works in your favor when you invest. The Freedom calculator’s expert mode runs it on your own savings, lump sums and income goal.
How compound interest works →Growth with monthly deposits, the cost of waiting, and how much compounding frequency really matters.Questions people ask
What is the formula for time value of money?
The basic formula is FV = PV × (1 + r)^n, where PV is the money today, r the rate per period and n the number of periods. With a regular payment it becomes FV = PV × (1 + r)^n + PMT × [((1 + r)^n − 1) ÷ r].
How do you solve time value of money problems on a financial calculator?
Enter four of the five keys (N, I/Y, PV, PMT, FV) and compute the fifth. Enter money you pay out as a negative number, check how many payments per year the calculator is set to, and clear the time value registers before each new problem.
Why does Excel give a negative answer for PV or PMT?
Spreadsheets treat money you pay out as negative and money you receive as positive. A negative PMT means you pay that amount each period; the size of the answer is the same, and the sign only shows the direction.
Do I divide the interest rate by 12 for monthly payments?
Yes. Use the yearly rate ÷ 12 as the rate and the number of months as the periods. If you solve for a monthly rate with RATE, multiply by 12 to get the quoted yearly rate, or use (1 + monthly rate)^12 − 1 for the effective yearly rate.
What is the difference between nominal and effective interest rates?
The nominal rate is the quoted yearly rate before compounding. The effective rate is what you actually earn or pay in a year once compounding is counted: 6% compounded monthly is an effective 6.17%.
When do I use an annuity due instead of an ordinary annuity?
Use an annuity due when each payment happens at the start of the period, as with rent, many leases and deposits made on the first of the month. Loan payments and most savings plans are treated as ordinary annuities, paid at the end of each period.
What does the time value of money formula assume?
One constant rate, equal payments and equally spaced periods. When the amounts or dates vary, value each cash flow on its own, or use spreadsheet functions built for that, such as NPV for uneven amounts and XNPV or XIRR for uneven dates.
What is the difference between an annuity and a perpetuity?
An annuity pays for a fixed number of periods; a perpetuity pays forever. A perpetuity's present value is simply the payment divided by the rate, which is also the limit an annuity's value approaches as it gets longer.