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

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

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

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

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

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

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

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

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

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

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

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

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

Â this project, from this project are going to be.

Â And now, here comes the big difference.

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

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

Â Now we equate this to zero and

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

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

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

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

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

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

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

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

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

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

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

Â the internal rate of return of evaluating a project.

Â So now we're solving for

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

Â That's not an easy thing to solve.

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

Â You need a scientific calculator,

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

Â It may get very complicated.

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

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

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

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

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

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

Â that is what we have to deal with now.

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

Â the internal rate of return.

Â As we said before, and

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

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

Â You need some sort of software with this.

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

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

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

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

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

Â There may be more than one, and

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

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

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

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

Â It may have more than one.

Â It may have no solution at all.

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

Â now keep in mind that this is more difficult to

Â solve mathematically speaking than simply calculating a net present value.

Â The rule is actually fairly simple,

Â because the intuition once again is fairly simple.

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

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

Â can expect from this particular project.

Â Exactly as we discussed from bonds before.

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

Â our pocket to buy the bond at the market price.

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

Â then we backed out.

Â We calculated beginning from the market price and

Â the expected cash flows, our mean annual return.

Â Well, this is identical, absolutely identical to that.

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

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

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

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

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

Â is going to be expressed in annual terms.

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

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

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

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

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

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

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

Â for now we still thinking that,

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

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

Â this project because this is basically like burning money up.

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

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

Â Companies don't want to do that, either.

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

Â That is exactly what this rule tells you.

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

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

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

Â Otherwise don't.

Â So the intuition is getting simple.

Â The devil again is, is in the details.

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

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

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

Â the more you would want to invest in it.

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

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

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

Â But for now, let me say that in principle it

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

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

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

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

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

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

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

Â the project, the more I want to go for

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

Â 7:33

If it ever happens, and, and it might happen actually more than once,

Â that you have a conflict between what the NPV rule recommends and

Â what the IRR rule recommends, you always want to fall back on the NPV rule.

Â And the reason you want to fall back on the NPV rule is,

Â is that mathematical reason that we actually mentioned in passing.

Â We didn't get into the details, but

Â that mathematical reason that we mentioned before.

Â When you're calculating it at present value, that is not a tricky calculation.

Â It may be a messy calculation.

Â It may be cumbersome calculation, but it's simply calculation of a present value,

Â there's no mathematical complexity involved there.

Â When you're trying to solve the expression for the IRR, you may run into trouble.

Â You may run into situations where you have no solutions,

Â where you have many solutions.

Â And therefore it gets a little bit tricky, and

Â again we're going to discuss a couple of examples very soon but

Â as far as what matters real, or right now is concerned do remember that.

Â If you calculate a project's NPD, if you calculate a project, a, a project's IRR,

Â and it happens to be the case, because it doesn't have to be the case,

Â but if it happens to be the case that the NPD tells you go for this project, and

Â the IRR tells you don't go for this project, or the other way around,

Â always fall back, always rely on the recommendation of the NPV.

Â So that's important that you keep in mind.

Â In any case of conflict, you want to fall back, you want to follow the NPV rule.

Â 9:05

Why are we saying all this?

Â Let me start from the end of what I want to cover in the next few minutes.

Â I don't want to tell you that you shouldn't use the IRR.

Â I don't want to tell you that the IRR is very problematic and

Â very cumbersome and therefore you should actually forget about it.

Â What I want to tell you is you should be careful when you use the IRR.

Â And in particularly, I want to tell you that you should know the structure of

Â the cash flows that you're dealing with.

Â And this will be more clear in a minute.

Â But let me start with an example that actually illustrates why you have to

Â be careful with the IRR.

Â Let's take a look at at these numbers.

Â Very simple project that you see.

Â Only three cash flow.

Â We have to put down 100 million today.

Â We expect to get 260 million a year from now.

Â And 165 million a year, two years from now.

Â And we're dealing with a company whose discount rate is 12%.

Â So if we have those cash flows and

Â a discount rate of 12%, well, we can do two things.

Â We can calculate the net present value, but because we're talking about

Â shortcomings, or problems with the IRR let, let's focus on the IRR.

Â And, here's a picture for you to take a look at.

Â Remember the definition of the IRR is the solution of that expression,

Â the IRR is the solution that gives you an NPV equal to zero.

Â That basically means that if I'm plotting different discount rates on the horizontal

Â axis, as the picture shows you, and I look at the NPV on the vertical axis,

Â for each discount rate I can calculate what the NPV would be.

Â And for some of those discount rates the NPV will be equal to zero.

Â Well in those cases, that's exactly the NPV that tells me that that is the IRR,

Â because remember that by definition, the IRR is the rate,

Â the discount rate that makes the net present value equal to zero.

Â Now you can see what the problem is with the picture.

Â That there's two times in which the line, the blue line crosses the horizontal axis.

Â And that means that there are two solutions for that equation.

Â And that means that there's two instances in which the net present value is

Â equal to zero.

Â Not just one, but two.

Â Now let me put down,

Â if in case you can't see the picture very clearly, what those IRR are, are.

Â One is an IRR of 10%, which is the one on the left, and

Â the other is an IRR of 50%, which is the one on the right.

Â And here comes the problem.

Â We're dealing with a company whose discount rate is 12%.

Â So what do we do?

Â We do not invest in this project, because the IRR is 10% and

Â therefore lower than 12%, or we do invest in this project

Â because the IRR is 50% and the discount rate is 12%.

Â That's the problem, and we cannot tell one or the other.

Â We cannot tell that one IRR is better, more accurate, or superior to the other.

Â There's no mathematical or intuitive way of doing that.

Â So here you see what the problem is.

Â We may be facing a situation in which the expression that we

Â need to solve doesn't have one solution but more than one.

Â In our extremely simple case, that solution is actually there's two

Â solutions, and they happen to be one below and one above our discount rate.

Â So what do we do in situations like that?

Â Exactly what we said before.

Â We need to fall back on the NPV.

Â And if you calculate the NPV of those cash flows and

Â you discount them at the rate of 12%, then you're going to get an NPV of 0.6 million,

Â or, or $600,000, and that number is positive, and

Â that tells you that you should ho, go ahead with this project.

Â So problem number one of the internal rate of return, we can have more than one.

Â In this very, very simple case, we actually have two, but

Â you can actually have many more,

Â depending on how complex, how long is the expression for the, for the NPV.

Â Just as an aside, and you know,

Â this, this doesn't really go to the heart of what we're discussing here, but

Â if you're wondering, what makes you know, what is the problem here?

Â Why do we have two solutions?

Â Well, you know, if you have a clean case, in which you

Â have a whole bunch of negative signs and then a whole bunch of positive signs.

Â In other words, you have changes in the signs of

Â the cash flows only once from negative to positive and from positive to negative,

Â then that guarantees that you're going to have only one IRR.

Â But of course you know, suppose that you're putting down some money to

Â start a project, you start receiving money out of the project.

Â But eventually three, four, five years down the road you need

Â to actually put down some money to add fresh capital to the project,

Â to maybe invest in buildings that have depreciated or

Â the machine that has depreciated and now you have another negative cash flow.

Â