NPV calculator and net present value formula

Enter what the project costs today, the cash it returns each later year, how many years it runs and the rate you discount at. The result is the net present value: everything the project pays in the future, valued in today's money, minus what it costs now.

The table under the result shows the working year by year. Each year's cash is divided by $(1+r)^t$, so the same $3,000 counts for less every year further out. Watch the bottom rows as you raise the rate. Distant money is where the value drains away, and it is the part people miss when they judge a project on total cash received.

$$NPV = \sum_{t=0}^{n} \frac{CF_t}{(1+r)^t}$$

Read it one year at a time. Cash arriving $t$ years out is divided by $(1+r)^t$, the discount factor for that year. At 8 percent the factor for year 1 is $1/1.08 = 0.9259$, for year 2 it is $1/1.08^2 = 0.8573$, and by year 5 it is down to $0.6806$. Multiply each year's cash by its factor, add the results, subtract the cost.

The cost sits at $t = 0$. Its divisor is $(1+r)^0 = 1$, so it is already in today's money and enters at full size with a minus sign in front of it.

When every year pays the same amount there is a shortcut, the annuity factor $\frac{1 - (1+r)^{-n}}{r}$. At 8 percent over five years that factor is 3.992710, and $3,000 times 3.992710 is the same $11,978.13 the year-by-year sum produces. Round the factor to 3.9927 first and the product comes out three cents light, which is why the digits you carry are part of the answer. The shortcut saves typing; it does not change the arithmetic.

The rate is not a quote you look up. It is the return the same money could earn in its next best use at similar risk, which is why two people can study one project, do the arithmetic correctly, and still disagree.

Raising the rate shrinks NPV on any project that spends first and collects later, and it hits the far years hardest. The five-year project worth $1,978.13 at 8 percent is worth $814.33 at 12 percent, less than half as much, without a cent of the cash flows changing. The exception is a project carrying a large cost in a late year, a mine to fill in or a plant to strip out: the higher rate discounts that cost too, so it can lift the NPV rather than cut it.

Push the rate high enough and NPV crosses zero. That crossing point is the internal rate of return, which the IRR calculator solves for. The two measures are one equation asked in opposite directions: NPV fixes the rate and returns a value, IRR fixes the value at zero and returns a rate.

Discounting is compounding run backwards. The same growth factor that turns money into a bigger number in the compound interest calculator is the divisor here.

A positive NPV means three things at once: the project returns the outlay, it pays the rate you demanded on every year the money is tied up, and there is a surplus left over. The number is that surplus, measured in today's money.

A negative NPV does not mean the project loses cash. It means the cash comes back too slowly, or too small, to clear the return you asked for. Zero is the honest boundary, where the project earns your rate exactly and nothing more.

Two projects only compare this way if they are measured in the same money over the same span. Cash flows written in today's prices need a discount rate with inflation stripped out; cash flows that already include price rises need one that includes it. The real return calculator converts between the two.