And I'm going to set that equal to 0.

And now I'm simply going to say, what rate sets that equal to 0?

And all we have to do is solve for the IRR.

And the spreadsheet's going to help us do that.

It's hard to do that with pen and paper.

But it's easy, actually, to do it in Excel or in a spreadsheet model

or on a financial calculator.

They usually have IRR functions built in.

And so now we're going to solve this polynomial equation for the IRR.

And that's going to tell us, if I had smacked

down the cash flows, at what r would it drive all the value out of the project.

We could think about this relationship graphically.

So if I think about a project that's got positive NPV if I don't

do any discounting-- so what I'm going to do

is I'm going to graph the value of the net present value, how big the NPV is,

against how hard I'm smashing it down, the discount rate.

As I smash that cash flow down, that NPV down,

with higher and higher discount rates, that NPV is going to come down

until at some point, it crosses 0.

The point at which that crosses 0, the discount rate that

sets the NPV equal to 0, is the IRR.

Now look at what we've done.

We've bisected the project into discount rates higher, where

the NPV is negative, and discount rates where the NPV is positive.

If my discount rate for this project really ought to be here--

let's say this is 5%-- and my IRR is 10%,

what is the IRR rule really told me?

What it's told me is look, any time the discount rate is below the IRR,

it's a positive NPV project.

So in a sense, IRR is just like NPV, except that I've

smushed it into a percentage place where the discounting just turns it negative.

And so that gives us a nice, because now I can say hey,

the return on this project, the IRR, is 10%.

And that might be easier than saying the net present value is 1,613,672,

which might be a hard number to figure out, because it's a big nominal number.

And I don't really know what it means.

But I tell you the return is 10%, you're like,

10% relative to a 5% cost on capital.

That's a good project.

That's what IRR does for us.

We could think that through in terms of a simple example.

Let's say I were spending $9,364 in order

to generate cash of $10,000 in year one, and $1,000 in year two.

If I didn't do any discounting at all and just added up the cash flows,

that would be 1,636.

36 If I discounted the cash flows at 10%,

I would discount the $10,000 back one period at 10%,

discount the $1,000 back two periods at 10%,

give me a net present value of 553.

If I instead of discounting at 10%, discounted at 20% instead of 10%,

that would give me a net present value of minus 336.

So what must have happened in the middle?

Somewhere that net present value crossed from positive to negative.

If we solve it, that actually happens at exactly 16%.

That's the trade off where discounting it hard enough, just

to draw all the value out of it.

It's actually a really easy thing to do in Excel.

If I take the example that I just showed you and look at those cash flows,

take the present value of those cash flows, again discounting the $10,000,

discounting the $1,000, computing the net present value,

summing them all up to get 553, all I have to do in Excel is says equals IRR.

Go and grab that set of cash flows right there, which

is what the cell is doing right there, and built into Excel

is an easy IRR function.

And so it spits it right out.

It solves that polynomial equation, finds the root, sets it to 0,

and spits out the 16% for me.

So it can be conceptually difficult to think about solving

big equations that have lots of roots.

But Excel does it for us, solving that, setting that NPV equal to 0.

Finding that 16% is relatively easy.

To wrap up, internal rate of return is very similar to NPV.

But it scales that NPV into a percent.

It's a more intuitive measure, because it gives us

a sense of what kind of rate of return the project is yielding.

And it accounts for the timing, the opportunity cost,

and the risk of the project in a very similar way to what NPV does.

So IRR is one of our good capital budgeting tools.

We should always compute it alongside of our NPV.