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Present value calculator

Present value is what a future cash flow is worth today at a stated rate. $10,000 due in 10 years at 6 percent, credited yearly, is $5,583.95 today. A $500 monthly payment for 20 years at 5 percent is worth $75,762.66 today.

Present value

$5,583.95

What $10,000.00 in 10 years is worth today at 6 percent.

Present value of the lump
$5,583.95
Present value of the payments
$0.00
Number of periods
10
Present value
$5,583.95
$

An amount due at the end of the term. Set 0 to value only the payment stream.

$

Level amount paid at the end of each period. Set 0 to discount only the lump.

%
yr

How often the rate is applied, and how often the payment arrives.

The formula

PV=F(1+i)n+PMT×1−(1+i)−niPV = \frac{F}{(1+i)^n} + PMT \times \frac{1 - (1+i)^{-n}}{i}

FF is a lump due after nn periods, PMTPMT a level end-of-period payment, and ii the rate per period. At a zero rate the present value is just the undiscounted sum, because nothing is being given up by waiting.

What discounting is doing

A dollar later is not a dollar now whenever money can earn a rate, because the dollar now could have been earning that rate in the meantime. Present value undoes that growth. At 6 percent a year, $10,000 in 10 years is divided by 1.0610=1.7908471.06^{10} = 1.790847, which is $5,583.95. Put $5,583.95 to work at 6 percent for 10 years and it becomes the $10,000; that is the check that the discounting was done right.

The rate in the formula is a nominal rate split across the compounding frequency, the same convention the rest of this site uses. Six percent compounded monthly is 0.5 percent a month for 12t12t months, not 6 percent applied once a year. Mixing those two is the usual way a present-value sum comes out wrong by a few percent, which on a long stream is real money.

Time value of money is the long-form version of this idea. This page is the working tool: a lump, a payment stream, or both.

A payment stream is a stack of lumps

The annuity term is not a different theory. It is the lump-sum formula applied to every payment and added up. A $500 payment at the end of each month for 20 years is 240 separate present values. At 5 percent a year, i=0.05/12i = 0.05/12, the closed form PMT×(1−(1+i)−n)/iPMT \times (1 - (1+i)^{-n}) / i adds them in one step and gives $75,762.66.

Payments at the start of each period (an annuity due) are each worth one extra period of growth, which multiplies the annuity present value by 1+i1+i. Rent paid in advance is the usual case. Bond coupons and loan payments are paid in arrears, which is the ordinary-annuity default on this calculator.

A bond is both terms at once: coupons as the payment stream, face value as the lump at maturity. The bond price calculator is that application with the coupons derived from a coupon rate, rather than typed in as a payment.

The rate is doing all of the work

Raise the rate and every future cash flow shrinks in today's money, because more growth is being undone. Cut the rate and present value rises. At a zero rate nothing is being undone and present value equals the cash, which is why a $10,000 lump in 10 years at 0 percent is still $10,000 today.

That sensitivity is duration's starting point: how far the present value moves when the rate moves. The bond duration calculator measures it for a coupon bond. On a single lump the whole present value sits at one date, so a rate move has nowhere to average, and the sensitivity is just the wait itself.

The rate is also the thing a reader has to choose. A safe cash flow discounted at a risky rate is understated; a risky cash flow discounted at a safe rate is overstated. This calculator will not pick the rate. It will only apply the one you give it, which is the honest scope of a present-value identity.

Net present value is the next step, not a synonym

Present value discounts inflows. Net present value subtracts what you pay today to buy them. A project that costs more today than the $5,583.95 present value of $10,000 in 10 years at 6 percent has a negative NPV, so it does not earn 6 percent. The NPV calculator is that subtraction on a whole cash-flow series.

The two pages share a formula and answer different questions. Use this one to put a price on a future amount or a payment stream. Use NPV when an outlay sits at time zero and you want to know whether the discounted inflows cover it.

Worked examples

A lump of \$10,000 in 10 years at 6 percent

What is $10,000 due in 10 years worth today at 6 percent compounded once a year?

  1. Period rate is 0.06 and there are 10 periods.
  2. Divide by the growth factor: 10000/1.061010000 / 1.06^{10}.
  3. 1.0610=1.79084771.06^{10} = 1.7908477, so 10000/1.7908477=5583.9510000 / 1.7908477 = 5583.95.

The present value is $5,583.95. Left to grow at 6 percent for 10 years, that sum becomes the $10,000, which is the check that the discounting matches the compounding.

\$500 a month for 20 years at 5 percent

A level $500 is paid at the end of each month for 20 years. The annual rate is 5 percent. What is the stream worth today?

  1. Period rate 0.05/12=0.00416670.05/12 = 0.0041667, and n=240n = 240.
  2. Annuity factor: (1−1.0041667−240)/0.0041667=151.525313(1 - 1.0041667^{-240}) / 0.0041667 = 151.525313.
  3. Multiply by the payment: 500×151.525313=75762.66500 \times 151.525313 = 75762.66.

The present value is $75,762.66. That is 240 discounted payments added together, not a lump, which is why a 20-year wait does not shrink 240 payments of $500 all the way down to a 20-year lump of the same size.

A small bond: coupons plus face

Face $1,000 in 5 years, plus $30 at the end of each half-year, discounted at 5 percent compounded semiannually. What is it worth today?

  1. Period rate 0.05/2=0.0250.05/2 = 0.025, and n=10n = 10.
  2. Present value of the face: 1000/1.02510=781.201000 / 1.025^{10} = 781.20.
  3. Present value of the ten coupons: 30×(1−1.025−10)/0.025=262.5630 \times (1 - 1.025^{-10}) / 0.025 = 262.56.
  4. Add them: 781.20+262.56=1043.76781.20 + 262.56 = 1043.76.

The package is worth $1,043.76 today: $781.20 for the face and $262.56 for the coupons. It prices above face because the coupon rate, 6 percent a year, sits above the 5 percent yield.

The mistake that costs the most

Discounting a monthly stream at an annual rate, or a yearly lump at a monthly rate, without converting the rate to the period.

$500 a month for 20 years at '5 percent' is not 500×(1−1.05−20)/0.05500 \times (1 - 1.05^{-20}) / 0.05. That formula would be 20 annual payments of $500, a different stream, and it gives a present value near six thousand rather than $75,762.66. The ii and the nn have to describe the same period. Monthly payments need a monthly rate and a month count. Annual lumps need an annual rate and a year count. Mixing them is not a rounding issue. It is a different cash-flow series.

Common questions

Is present value the same as net present value?

No. Present value discounts future cash flows to today. Net present value subtracts the amount you pay today to buy them. A positive NPV means the discounted inflows more than cover the outlay at that rate. Present value alone has not yet asked what the inflows cost.

What rate should I use?

A rate that matches the risk of the cash flows, and that is expressed with the same compounding frequency the formula will split it across. This page will not pick one. A wrong rate is a wrong present value, however carefully the rest of the arithmetic is done.

Does inflation belong in the rate?

Only if the cash flows are nominal. Real cash flows want a real rate. Mixing a real rate with nominal cash flows, or the reverse, is the Fisher-identity error the real return calculator exists to catch.

Keep reading

This page is educational material, not financial advice. The figures come from the formula shown and assume the inputs you enter hold for the whole term. Your own rate, fees, taxes and timing will differ, so treat the output as arithmetic to check a decision against, not as a recommendation.