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Okay, now let's park the NPV for a minute now.

We'll get back to the NPV when we work out an example a few minutes from now.

Let's go to the other rule or the other tool.

And the other tool, as we said, is the IRR, or internal rate of return.

The expression for the IRR is the one you're seeing in the screen, and,

and that expression from the, for the IRR, as you see, on the left hand side has

exactly what we had before on the right hand side on the N, of the NPV expression.

In other words, what you have on the left hand side of the IRR expression is

basically an NPV and what you had on the right hand side is zero.

So basically what we're doing here is we need to input the cash flows of

the project in exactly the same way as we've done them before.

That is, I need to known what my initial investment is going to be.

I need to make a forecast of what the cash flows that I expect for

this project, from this project are going to be.

And now, here comes the big difference.

Now we don't discount those cash flows at the discount rate,

at the cost of capital, whatever might be the discount rate.

Now we equate this to zero and

what we solve for are those IRRs that you're seeing on the screen.

Now as we said before, if you remember, when we calculated in session, three,

the return that we would get from a bond by buying it at the market price and

holding it until maturity, that was more or less a number we calculated.

We didn't actually give it the name internal rate of return when we

discussed bonds, but one thing, an interesting thing to relate here

that's something that we're doing now with something that we've done before is that

the yield to maturity of a bond is exactly the internal rate of return of the bond.

That is, given the price and given the expected cash flows of a bond,

there's a mean annual return that you're going to get by solving an expression that

is identical to the expression that you have in front of you for

the internal rate of return of evaluating a project.

So now we're solving for

those IRR's, and the same thing that we said before for bonds applies here.

That's not an easy thing to solve.

Certainly this is not something that you can do by hand.

You need a scientific calculator,

you need Excel, you need some tool to help you solve that equation.

It may get very complicated.

If you have only a couple of periods, then it may not be all that complicated, but

whenever you have three, four, five periods, not only it

gets complicated to actually solve for the IRR, but you may encounter problems,

some of which we're going to discuss a couple of minutes from now.

So, point number one that is the expression of the internal rate of return.

Mathematically it's more difficult to solve than an impressing value, but

that is what we have to deal with now.

So a few things to keep in mind about this expression for

the internal rate of return.

As we said before, and

this is important that you keep in mind, do not minimize this.

This is not easy to solve, so do not try to do this by hand.

You need some sort of software with this.

Excel would do but you need some sort of software to, to do this.

And this second thing why I want to emphasize that it's mathematically complex

is because, you know, we typically think of what is the IRR of the project, and

the key word there is the, that is what is the IRR of the project.

Well, it doesn't have to be the IRR.

There may be more than one, and

again, that is one thing that we actually will discuss in a minute.

Of course, it's going to be problematic if we get more than one but

it is possible with the equation that you have in front of you.

Nothing guarantees that that equation is going to have only one solution.

It may have more than one.

It may have no solution at all.

So we'll get back to these issues in just a second, but for, for

now keep in mind that this is more difficult to

solve mathematically speaking than simply calculating a net present value.

The rule is actually fairly simple,

because the intuition once again is fairly simple.

Once we solve the for, for that IRR, once we solve for

that internal rate of return, that is some sort of the mean annual return that you

can expect from this particular project.

Exactly as we discussed from bonds before.

Remember, for in the case of bonds we took some money out of

our pocket to buy the bond at the market price.

We held the bond until maturity and expected to receive those cash flows, and

then we backed out.

We calculated beginning from the market price and

the expected cash flows, our mean annual return.

Well, this is identical, absolutely identical to that.

So basically, when we're solving the expression, for the IRR,

what we're saying is we're comparing the initial investment that I

have to make in this project with the cash flows I expect to get out of this project,

and what we're basically calculating is the return that I get from this project.

Now this return, if those cash flows are annual cash flows,

is going to be expressed in annual terms.

So it'll be a mean annual return that you expect from investing in this project.

Now, whether or not you're going to go ahead with this project or

not, goes back to first principles that we mentioned before.

That is, if you're calculating here the return of this project, you don't want

to invest in anything that doesn't give you at least the cost of raising funds to

invest in the project, and that is exactly what we call the cost of capital before.

So the rule is very simple, if the IRR is higher than the discount rate and

for now we still thinking that,

that discount rate is the cost of capital then you invest in this project.

If the IRR is lower than the discount rate, you do not want to invest in

this project because this is basically like burning money up.

It's like borrowing money at 5% and then investing money at 3%.

Well, that's something you don't want to do.

Companies don't want to do that, either.

So if you borrow at 5%, you want to invest at something that gives you more than 5%.

That is exactly what this rule tells you.

All right, so at the end of the day, it's a very intuitive rule.

It tells you if the return of the project is higher than the cost of

raising funds to invest in the project, go for it.

Otherwise don't.

So the intuition is getting simple.

The devil again is, is in the details.

So in terms of competing projects is when it gets a little tricky.

And it gets a little tricky, because it would seem to make sense to think that

the higher the IRR, that is, the higher the mean annual return of the project,

the more you would want to invest in it.

And sometimes that is true, and sometimes that is not true.

And if you're confused as to why that may not be true,

we're going to see an example in a minute why that may be the case.

But for now, let me say that in principle it

seems to be the case that the higher the return on the project the better.

In other words, if we're comparing project A and project B, and the IRR of project

A is higher than the IRR of project B, then we should go for project A.

And in many cases, that is true, but there are cases in which that is not true, and

we'll get to a specific example about that a couple minutes from from now.

That's why for now let me just say that, the rule has some loopholes.

So again in principle it looks like it makes sense that the higher the return on

the project, the more I want to go for

it, but we're going to see a couple of counterexamples, to that.