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CalcMax

Time Value of Money Calculator

Range: 1 – 10,000,000

Range: 0 – 50

Range: 1 – 50

Result

16,288.95

Future value

Present value
6,139.13
Difference between future and present value
10,149.82

A dollar today and a dollar in ten years are not the same dollar, and this page shows you how far apart they are. Enter one amount, a rate and a number of years, and it reads that same sum at two points in time: forward, as what it grows into by the end of the term, and backward, as what a payment that far in the future is worth today. The third figure is the distance between those two — the amount of value that time itself adds or removes, which is the time value of money stated as a single number rather than as a rate. The rate can be compounded annually, twice a year, quarterly, monthly, daily or continuously, and the choice matters more than it looks: the same 5% over ten years is 16,288.95 compounded yearly and 16,487.21 compounded continuously, a difference of 198 that is entirely the frequency. What the page will not tell you is whether 5% is the right rate, or whether receiving the money later is actually a risk you want to take; it converts between dates, and that is all.

10,000 at 5% compounded yearly, read at six different horizons

YearsFuture valuePresent valueGap
1105009523.81976.19
2110259070.291954.71
311576.258638.382937.87
512762.827835.264927.56
1016288.956139.1310149.82
2026532.983768.8922764.09

One amount and one rate throughout — 10,000 at 5%, compounded annually — so the only thing moving is how many years the two readings are apart, and reading down the table is reading what time alone does. The columns behave completely differently, and that is the point. The future value climbs and accelerates: 10,500 at one year, 12,762.82 at five, 16,288.95 at ten and 26,532.98 at twenty, so the ten years from the ten-year row to the twenty-year row add more than the first five years did. The present value falls and decelerates, from 9,523.81 to 3,768.89 over the same span, because discounting divides by a factor that is growing. The last column is the one worth reading on its own: the gap is 976.19 at one year, 4,927.56 at five, 10,149.82 at ten and 22,764.09 at twenty. Doubling the horizon from one year to two more than doubles it, and going from ten years to twenty more than doubles it again — the effect is not linear in time and never will be, because the exponent is doing the work. Read the middle two columns as the same 10,000 seen from each end: at the twenty-year row, 26,532.98 is what today's money becomes and 3,768.89 is what a promise of 10,000 in twenty years is worth now.

Formula

Future value = Amount × (1 + Annual rate ÷ m)^(m × Years) Present value = Amount ÷ (1 + Annual rate ÷ m)^(m × Years)

Amount
The sum being read at two points in time; it is the amount you have today for the forward direction and the amount due at the end for the backward one
Annual rate
The nominal rate per year, before any compounding; enter 2 for 2 percent, and note that this is the quoted rate, not the effective one the compounding produces
m
Compounding periods per year — 1 for annual, 2 for semiannual, 4 for quarterly, 12 for monthly, 365 for daily; the higher it is, the more the same quoted rate earns
Years
How long the money is left, in years; the exponent is m × years, so it counts compounding periods rather than years
Future value
What today's amount becomes by the end of the term — the forward reading, and the page's main figure
Present value
What an amount that far in the future is worth today: the same formula run backwards, dividing instead of multiplying
Time value gap
Future value minus present value: the same sum read at two dates, and the difference between them is what the whole term of waiting is worth in money
Continuous compounding
The limiting case, where m grows without bound and the formula becomes Amount × e^(rate × years); it is a different expression from the one above rather than a frequency setting, and no finite m ever reaches it

Use the forward reading to answer what a sum becomes — what a deposit grows to, what a lump sum has to become to be worth saving, what a target number implies about the rate or the years. Use the backward reading to answer what a future payment is worth now, which is the operation behind discounting, and behind every discounted valuation: a lease, a settlement, a promise to pay in five years. Use the gap when you want to show someone the size of the effect rather than the rate that causes it, which is usually the more persuasive half of the argument. It is the wrong tool in three cases. It cannot price a stream of payments — for that the amount has to be repeated on a schedule, which is the annuity pages. It says nothing about risk: it converts between dates at a rate you supply, and if the money might not arrive at all, that is a separate question it does not touch. And it is agnostic about which direction is real: when you discount a payment a creditor has promised you, the rate you should use is the one you could actually earn on money of comparable risk, not the rate the creditor would like you to believe.

Worked examples

  1. 10,000 at 5% for ten years, compounded yearly

    1. Ten years at 5% compounded annually is the factor 1.05^10 = 1.628894627
    2. Future value: 10,000 × 1.628894627 = 16,288.95
    3. Present value: 10,000 ÷ 1.628894627 = 6,139.13
    4. The gap: 16,288.95 − 6,139.13 = 10,149.82

    The default, and the cleanest statement of the whole idea. Read the two amounts as the same 10,000 seen from two ends of a ten-year span: 16,288.95 is what today's money becomes, 6,139.13 is what a promise of 10,000 in ten years is worth right now. The 10,149.82 between them is not a fee or a loss — it is time, priced. Notice that the future value is larger than the amount by 6,288.95 and the present value smaller than it by 3,860.87, and that the gap is the sum of those two moves rather than either one of them; the page prints all three so that arithmetic is visible.

  2. A zero rate: 8,000 for fifteen years changes nothing

    1. At 0% the factor is 1.00^15 = 1
    2. Future value: 8,000 × 1 = 8,000
    3. Present value: 8,000 ÷ 1 = 8,000
    4. The gap: 8,000 − 8,000 = 0

    Not a degenerate case and not an error — a rate of zero is the honest way to say that in this particular comparison a dollar now and a dollar in fifteen years are being treated as the same thing. Everything the page does collapses: both directions return the amount unchanged and the gap is exactly zero. That is worth having as a reference point, because it isolates the whole effect: every other row of the reference table differs from this one only because the rate is not zero. If you ever see a gap of zero on a nonzero rate, one of the two amounts has not been applied.

  3. The same 10,000 and 5%, compounded continuously

    1. Continuous compounding uses e^(rate × years) instead of a power of (1 + rate ÷ m)
    2. The factor: e^(0.05 × 10) = e^0.5 = 1.648721271
    3. Future value: 10,000 × 1.648721271 = 16,487.21
    4. Present value: 10,000 ÷ 1.648721271 = 6,065.31, and the gap is 10,421.90

    The same inputs as the default example and 198.26 more, purely from the frequency. Continuous compounding is the ceiling the other frequencies approach: annual gives 16,288.95, daily gives 16,486.65, and letting the compounding interval shrink to nothing only adds another 0.56 on top of daily. That is the practical lesson — going from annual to monthly moves the number several hundred, and going from daily to continuous moves it by cents, so the choice between the low frequencies is the one that matters. Continuous compounding is also the one option here that is a different expression rather than a different setting of the same one.

  4. 5,000 at 6% for twenty years, compounded quarterly

    1. Quarterly means 1.5% per quarter over 80 quarters: 1.015^80 = 3.290662…
    2. Future value: 5,000 × 3.290662 = 16,453.31
    3. Present value: 5,000 ÷ 3.290662 = 1,519.45
    4. The gap: 16,453.31 − 1,519.45 = 14,933.86

    A longer term at a higher rate, and the gap is now nearly three times the original amount. That ratio is the thing to look at rather than the totals: over twenty years at 6%, the difference between the two readings of the same 5,000 is 14,933.86, which is almost exactly three times the sum itself. Doubling the horizon does more than double the effect, because the exponent is doing the work. Read the present value column here as the answer to a real question: a promise of 5,000 in twenty years, discounted at 6%, is worth about 1,519 today.

Limitations

Four things this page does not do. It converts a single amount between two dates; it cannot price a stream of payments, so rent, salaries, loan instalments and dividends need the annuity pages rather than this one. The rate is one you supply, and the page has no opinion about it — it does not know what inflation will be, what a comparable investment yields, or what the payment risk is, so a conversion at the wrong rate is a precise answer to the wrong question. Nothing is adjusted for inflation, so the figures are in whatever units you entered, which means a future value that looks large may buy less than the present value does. And the frequencies are not interchangeable conventions for the same thing: continuous compounding is a different expression, and any comparison between a figure produced here and a rate quoted elsewhere has to establish which convention the other side used before the two can be put side by side.

Frequently asked questions

What is the time value of money?
It is the fact that a sum of money is worth different amounts at different dates, which is why converting between dates is a calculation rather than a relabelling. This page measures it: 10,000 at 5% over ten years is 16,288.95 at the far end and 6,139.13 at the near one, and the 10,149.82 between those two figures is the value of the time itself. Nothing was earned or lost in making that statement — the same 10,000 was simply read at two dates.
Why are there two results, and which one do I want?
They are the same operation in opposite directions. Future value compounds forward and answers what today's money becomes; present value discounts backward and answers what a future amount is worth now. Take the future value if you are projecting a sum you hold today, and the present value if you are valuing a payment someone has promised you. The third figure, the gap between them, is for when you want to show the size of the effect rather than the rate behind it.
How much does the compounding frequency matter?
More than most people expect at the low end and less than they expect at the high end. On 10,000 at 5% for ten years: yearly gives 16,288.95, quarterly 16,436.19, monthly 16,470.09, daily 16,486.65, continuous 16,487.21. Moving from yearly to monthly adds about 181, and moving from daily to continuous adds less than a dollar. The quoted rate is unchanged in every case — what changes is how often the interest starts earning interest.
What does the gap between the two figures mean?
It is the value of the waiting, expressed in money rather than as a rate. It is simply the future value minus the present value, so on the default example it is 16,288.95 − 6,139.13 = 10,149.82. It is not a fee and it is not a loss; it is what the ten years are worth on this amount at this rate. It grows faster than the term does, because the exponent compounds: doubling the horizon more than doubles the gap.
Is continuous compounding just a very high frequency?
It is the limit the other frequencies approach but never reach, and it is computed by a different expression — Amount × e^(rate × years) rather than a power of (1 + rate ÷ m). The distinction is practical rather than philosophical: daily compounding is already within 56 cents of the continuous figure on 10,000 at 5% over ten years, so treating continuous as merely a sixth frequency would be accurate to under a dollar here. It becomes a genuinely different convention when rates are quoted continuously, which they often are in derivatives pricing, and then converting to an annually compounded rate before comparing is a required step rather than a nicety.
Does this account for inflation?
No. Every figure is in the units you entered, so a future value of 16,288.95 means 16,288.95 of nominal money at that date, and what it will buy is a separate question. If you want the real rather than the nominal reading, either enter a real rate — one already net of expected inflation — or deflate the result afterwards using an inflation assumption. The page will not stop you from entering a nominal rate and reading the output as if it were real, which is the single most common way this calculation is misused.

References

Related calculators

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