How present value works

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 $1.06^{10} = 1.790847$, 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.

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

$F$ is a lump due after $n$ periods, $PMT$ a level end-of-period payment, and $i$ the rate per period. At a zero rate the present value is just the undiscounted sum, because nothing is being given up by waiting.

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 $12t$ 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. The calculator above opens on the $10,000 lump because that is the first worked example.

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, $i = 0.05/12$, the closed form $PMT \times (1 - (1+i)^{-n}) / i$ 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 $1+i$. 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 20-year wait does not shrink 240 payments of $500 all the way down to a 20-year lump of the same size, because most of the payments arrive much sooner than year 20. That is why the stream is worth $75,762.66 rather than the present value of one distant lump equal to the undiscounted sum of the payments. The stream and the lump are different series.

Face $1,000 in 5 years, plus $30 at the end of each half-year, discounted at 5 percent compounded semiannually, is a lump plus a stream. The period rate is $0.05/2 = 0.025$, and $n = 10$. The face is worth $781.20 today. The ten coupons are worth $262.56. Add them and the package is $1,043.76.

It prices above face because the coupon rate, 6 percent a year, sits above the 5 percent yield. That is the same fact the bond price calculator reports with the coupons derived from a coupon rate rather than typed in as a payment. Duration, how far that present value moves when the yield moves, is the bond duration calculator.

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 0 percent case is not a worked example on this page; it is the identity the formula collapses to when $i = 0$.

That sensitivity is duration's starting point: how far the present value moves when the rate moves. 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. On a coupon bond the coupons pull some of the weight forward.

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.

Match the rate to the cash flows. Amounts written in today's prices need a real rate; amounts that already include price rises need a rate that still contains inflation. Mixing them is the Fisher-identity error the real return calculator exists to catch.

Discounting a monthly stream at an annual rate, or a yearly lump at a monthly rate, without converting the rate to the period, prices a different series. $500 a month for 20 years at 5 percent is not $500 \times (1 - 1.05^{-20}) / 0.05$. That formula would be 20 annual payments of $500, and it is not the stream the second worked example prices.

The $i$ and the $n$ 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.

The APR against APY calculator is the compounding conversion if you are holding a nominal rate and an effective yield and need them on the same footing before you discount.

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.

This is educational material, not financial advice. The figures throughout are teaching cash flows: a $10,000 lump, a $500 monthly stream, and a small bond of $1,000 face plus $30 coupons. They are there so every published number can be re-derived.