Net Present Value of a Uniform Annual Cash Flow

NPV asks whether a project beats the cost of the money it uses. When the annual cash flow is the same every year, the whole stream collapses into one factor and the arithmetic fits on a napkin: $\mathit{NPV} = A(1-(1+i)^{-n})/i - C_0$. Thirty thousand dollars spent to save eight thousand a year for five years, money at 10%, gives a present worth of $30,326 and an NPV of $326. Positive, so it clears the hurdle, but only just.

Compare that verdict with simple payback, which reports 30000/8000 = 3.75 years and looks comfortable. Payback ignores the cost of capital entirely, and here that cost eats essentially the whole apparent gain. When two proposals are close, payback and NPV routinely disagree, and NPV is the one that corresponds to money.

Notice the variable this page cannot solve for. Setting NPV to zero and solving for $i$ is the internal rate of return, and there is no closed form: it is the root of a degree-$n$ polynomial, which every spreadsheet finds by iteration. This catalog does not ship iterative brains, so rather than fake an inverse, solve for the project life or the annual cash flow instead, or step the rate by hand until NPV crosses zero. IRR has a second problem worth knowing anyway. A cash flow that changes sign more than once can have several rates that all make NPV zero, and none of them means what people assume.