NPV Explained: Formula, Discount Rate, and Worked Examples

A dollar next year is worth less than a dollar today, because a dollar today can be invested. Net present value, NPV, applies that idea to a whole project: it discounts every future cash inflow back to its present value at a chosen rate, adds them up, and subtracts the initial investment. If the result is positive, the project earns more than the discount rate. If it is negative, it earns less. If it is zero, it earns exactly the discount rate. This guide explains the discount rate, works two examples by hand, and shows why NPV beats simpler rules such as payback period.

The formula

C₀ is the initial investment, made today, so it is not discounted. C₁ through C_n are the cash flows in each future period, and r is the discount rate per period. Each future cash flow is divided by one plus the rate raised to the number of periods away, which shrinks distant cash flows more than near ones. The calculator on this site runs the same sum for any cash flow list and rate.

Present value is the same idea in miniature. At a 10% discount rate, $400 in one year is worth $400 divided by 1.10, which is $363.64 today. In two years it is $400 divided by 1.21, which is $330.58. In three years it is $400 divided by 1.331, which is $300.53. The further away the money, the less it is worth today, and NPV is just this calculation applied to every cash flow and added up.

Worked example 1: three equal inflows, two verdicts

A project costs $1,000 today and returns $400 at the end of each of the next three years. At a 10% discount rate, the present values are $363.64, $330.58, and $300.53, totalling $994.75. NPV is $994.75 − $1,000 = −$5.25, slightly negative: at 10%, the project just fails.

At an 8% discount rate, the present values are $370.37, $342.94, and $317.53, totalling $1,030.84. NPV is $1,030.84 − $1,000 = +$30.84, positive: at 8%, the same project passes. Nothing about the project changed, only the rate, which is why the discount rate must be defended. It usually represents the cost of capital, the return the money could earn elsewhere at similar risk, or a hurdle rate set by the investor.

Worked example 2: uneven flows and the payback trap

A project costs $5,000 today and returns $1,000 in year one, $2,000 in year two, and $4,000 in year three. The payback period, the time to recover the investment in undiscounted cash, is between two and three years, since $1,000 plus $2,000 is $3,000 and the third year adds $4,000. Payback says accept. At 10%, the present values are $909.09, $1,652.89, and $3,005.26, totalling $5,567.24, so NPV is +$567.24, and NPV agrees here. But raise the rate to 20%: the present values become $833.33, $1,388.89, and $2,314.81, totalling $4,537.03, so NPV is −$462.97, and NPV rejects a project payback accepts. Payback ignores both the time value of money and every cash flow after payback; NPV ignores neither.

Choosing the discount rate honestly

The discount rate should reflect the risk of the cash flows and the opportunity cost of the money. A government-backed project can use a low rate; a speculative venture needs a high one. Using a rate that is too low flatters every project into acceptance; using one that is too high rejects good projects. Sensitivity analysis is the honest response: compute NPV at several rates and see how far the rate can move before the verdict flips. The rate at which NPV is exactly zero is the internal rate of return, IRR, and reporting NPV alongside IRR gives a fuller picture than either alone.

Also mind the cash flows themselves. NPV is only as good as the forecasts fed into it, and forecasts are where projects are usually oversold: revenues assumed too high, costs assumed too low, and terminal values assumed too generous. A positive NPV built on heroic forecasts is not a good project; it is a good story.

Mistakes: nominal versus real, and double counting

First, mixing nominal cash flows, which include inflation, with a real discount rate, which excludes it, or vice versa. Both sides must be nominal or both real. Second, double counting financing: if cash flows are already after interest, the discount rate should not also include the cost of debt, or the project is charged twice. Third, ignoring working capital and terminal value: a project that ties up inventory or leaves equipment with resale value has cash flows at the end that belong in the sum. Fourth, comparing NPVs of projects with different lives without adjusting: a five-year project and a ten-year project are not directly comparable on NPV alone.

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