Why a dollar later is worth less than a dollar now
Present value is the amount you would accept today instead of a specified payment in the future. It is smaller than the future amount for one structural reason: money in hand can be put to work, so a dollar received in five years must be discounted by whatever five years of earning power is worth to you.
That earning power is the discount rate, and choosing it is the entire judgement in a present value calculation. Everything else is arithmetic. If you would otherwise pay down a 7% loan, 7% is your rate. If you would buy a Treasury yielding 4.3%, that is your rate. For a company appraising a project, the rate is the return investors require on capital of that risk, which the WACC calculator estimates.
The decisions this settles are common and consequential. A defined-benefit pension offers $2,400 a month for life or $410,000 as a lump sum — which is larger? A lottery advertises a $100 million jackpot paid over 30 years alongside a much smaller cash option. A landlord offers a rent-free month now against a higher rent later. In every case the offers are only comparable once both sides are converted to the same date, and today is the natural date to pick.
The formula, and why it is the future value formula inverted
Compounding multiplies by (1 + i) once per period. Discounting divides by it once per period. So the present value of a single future amount is FV ÷ (1 + i)^n, which is more usually written FV(1 + i)^−n. The quantity (1 + i)−n is the discount factor, tabulated as PVIF, and it is always between zero and one for a positive rate.
For a stream of level payments, discount each one on its own date and add them. Payment 1 arrives after one period and is divided by (1 + i); payment n is divided by (1 + i)n. Summing that geometric series gives the closed form PMT · [(1 − (1 + i)^−n) / i], the annuity discount factor PVIFA. The cash flow table on this page performs the same sum term by term, so you can watch the later payments contribute steadily less.
Two properties of that annuity factor are worth internalising. First, it is always less than n for a positive rate, and equal to n when the rate is zero — it is effectively the number of payments you are getting credit for. Second, it converges. As n grows without bound, (1 + i)−n approaches zero and the factor approaches 1 ÷ i. That is why a perpetuity is worth PMT ÷ i, and it explains why extending a 20-year annuity to 30 years adds far less value than the extra ten payments suggest.
The timing switch multiplies the payment term by (1 + i). In an annuity due the first payment arrives immediately and is not discounted at all, which is how pensions, leases and insurance premiums usually work. Getting this wrong understates a pension's value by one period of interest.
Worked example: $1,000 a year for ten years plus $50,000 at the end, discounted at 5%
These are the calculator's defaults, with payments at the end of each year.
- Rate as a decimal. i = 5 ÷ 100 = 0.05.
- Compound factor. 1.0510 = 1.6288946.
- Discount factor. 1 ÷ 1.6288946 = 0.6139133. A dollar in year 10 is worth 61.4 cents now.
- PV of the lump sum. $50,000 × 0.6139133 = $30,695.66.
- Annuity discount factor. (1 − 0.6139133) ÷ 0.05 = 0.3860867 ÷ 0.05 = 7.7217349. Ten payments earn credit for 7.72 of them.
- PV of the payments. $1,000 × 7.7217349 = $7,721.73.
- Present value. $30,695.66 + $7,721.73 = $38,417.39.
- Nominal cash. $50,000 + 10 × $1,000 = $60,000, so the value given up to waiting is $60,000 − $38,417.39 = $21,582.61, which is 36% of the headline number.
Check one row of the table by hand to confirm the mechanism: the payment in year 3 contributes $1,000 ÷ 1.053 = $1,000 ÷ 1.157625 = $863.84, while the identical payment in year 10 contributes only $613.91.
How to read the result, and how to choose the rate
Read the present value against whatever you are being offered instead. If a buyer offers $40,000 today for the stream in the worked example and your genuine alternative use of money earns 5%, take the cash — it beats the $38,417 the stream is worth to you. If your alternative earns only 3%, redo it: 50,000 × 0.744094 + 1,000 × 8.530203 = $45,734.90, and the offer is now poor. The rate is not a detail; it is the decision.
Choosing the rate. Use the return on the best alternative available to you at comparable risk. For a household, that is usually the rate on debt you could repay instead, because repaying an 8% loan is a guaranteed 8% return. For a safe, contractually certain stream such as a government pension, a government bond yield of matching maturity is the defensible choice. Do not use a high equity-style rate to discount a certain payment; that mixes a risk adjustment into a time adjustment and understates the value.
The value-given-up figure. This is the gap between the headline nominal amount and what the offer is really worth, and it is the number that lottery and structured-settlement advertising depends on you not computing. In the worked example it is 36% of the total. That fraction rises steeply with both the rate and the horizon.
Sensitivity. Always re-run at a rate one or two points either side. Present value is far more sensitive to the rate than most people expect, particularly for long streams, because the discount factor is exponential in the horizon. If a decision flips between 4% and 6%, you do not have a decision — you have a coin toss dressed as arithmetic, and you should look for other grounds to choose.
Present value annuity factors, PVIFA(i, n)
| Rate per period | n = 5 | n = 10 | n = 20 | n = 30 |
|---|---|---|---|---|
| 2% | 4.713460 | 8.982585 | 16.351433 | 22.396456 |
| 4% | 4.451822 | 8.110896 | 13.590326 | 17.292033 |
| 6% | 4.212364 | 7.360087 | 11.469921 | 13.764831 |
| 8% | 3.992710 | 6.710081 | 9.818147 | 11.257783 |
| 10% | 3.790787 | 6.144567 | 8.513564 | 9.426914 |
Each cell is (1 − (1 + i)^−n) ÷ i. Compare the n = 20 and n = 30 columns: at 10% the extra ten payments add under one full payment of value, because the factor is converging on 1 ÷ i = 10.
Mistakes that produce the wrong present value
- Mixing the rate and the period. Entering an annual rate alongside a monthly period count is the most common error and understates the present value severely. Divide the annual rate by the number of periods per year first, and use the effective annual rate output to check.
- Using ordinary timing for a pension. Most pensions and leases pay at the start of the period. Getting the timing wrong changes the payment component by a factor of (1 + i) — it leaves the lump-sum component untouched, so the error in the headline figure is smaller than (1 + i) whenever a lump sum is also present. On a pure annuity it is the full (1 + i): trivial at 0.5% a month, worth 8% of the value at 8% a year.
- Discounting a certain payment at a risky rate. The discount rate should reflect the risk of the cash flow, not your general appetite for return. A guaranteed government payment discounted at 10% will always look worthless, and that conclusion is an artefact of the rate.
- Forgetting inflation and then double-counting it. Either discount nominal cash flows at a nominal rate or real cash flows at a real rate. Discounting nominal payments at a real rate overstates value; the reverse understates it.
- Ignoring tax. A pension payment and a lump sum are frequently taxed differently, and the comparison must be made after tax. Run the calculation on after-tax amounts on both sides.
- Treating a life-contingent stream as a fixed annuity. A pension paid for life is not an n-period annuity; its value depends on mortality as well as on the discount rate. This calculator values a fixed number of payments, which is a reasonable approximation only if you fix n at a realistic life expectancy and understand what you have assumed.
The perpetuity shortcut
When payments continue indefinitely, the annuity factor collapses to 1 ÷ i and the present value is simply PMT ÷ i. At 5%, a perpetual $1,000 a year is worth $20,000 today. This is worth remembering as an upper bound: no finite annuity at that rate can ever be worth more, so if someone values a 25-year stream of $1,000 at 5% above $20,000, the arithmetic is wrong. The exact 25-year figure is $14,093.94, which is 70% of the perpetuity value — the remaining 30% sits in the payments from year 26 onwards.
Where present value sits among the other tools
Present value is the foundation of essentially all valuation. A bond's price is the present value of its coupons and principal. A share's intrinsic value under a dividend discount model is the present value of its dividends. A project's worth is the present value of its cash flows less the outlay, which is what the net present value calculator computes — NPV is this calculation applied to an irregular cash flow stream with the initial investment subtracted.
The related question “what rate makes the present value equal the price” is answered by the internal rate of return calculator. Applied to a bond, that rate is its yield to maturity; applied to a project, it is the return the project earns on the capital tied up in it. The question “how long until the discounted inflows repay the outlay” is the discounted payback period calculator.
Running the same relation forwards rather than backwards gives the future value calculator, and expressing an observed change as an annual rate gives the CAGR calculator. All four are rearrangements of one equation with five variables; which one you reach for depends only on which variable you do not know.
Two limits of this page are worth stating. It assumes a single flat discount rate for every period, whereas real yield curves slope, so a strictly correct valuation discounts each date at its own spot rate. And it assumes the payments are certain. Where they are not — a startup's projections, a royalty on an unproven product — the right treatment is to probability-weight the cash flows and discount at a rate reflecting the remaining risk, rather than to inflate the discount rate until the answer feels conservative.