NPV & IRR

Discount a project's dated cash flows to one number today — and learn when the rate-based shortcut lies.

The idea

The net present value of an investment is the value of its dated cash flows, measured in today's dollars. We cannot add flows at different dates directly, because a dollar later is worth less than a dollar now; so we discount each flow to today, then add:

$\mathrm{NPV} = \sum_{t} \frac{C_{t}}{(1+r)^{t}}$

Here $C_{t}$ is the cash flow at year $t$, with the initial outlay $C_{0}$ entered as a negative number, and $r$ is the discount rate — the return the same money could earn elsewhere. A positive NPV means the flows are worth more today than they cost, so the decision rule is to take every project with $\mathrm{NPV} > 0$.

The internal rate of return of the same flows is the discount rate at which the NPV equals zero — the project's break-even rate. Take a project when its IRR exceeds the return available elsewhere. For a single project with one outflow followed by inflows, the two rules agree; between two competing projects they can rank differently, and then the NPV, measured in dollars, decides.

Ways to work on it

• Walkthrough. Compute net present values, interpret their sign, and solve for the internal rate of return.
• Practice. NPVs of short flow streams, exact IRRs, and accept-or-reject calls.
• Hardest. Rank two projects when NPV and IRR disagree, and find the rate where the ranking flips.

Not sure where to start? Take the ten-question placement test.