What is IRR (Internal Rate of Return)? How to Calculate Internal Rate of Return?

In this article, I'm going to be talking about IRR otherwise known as the Internal Rate of Return. Now the internal rate of return is a way of evaluating a project and coming up with a decision to accept or reject that project. 

So this is very similar in a way to NPV (Net Present Value). Now when we think about NPV we think, if the NPV is greater than zero (0) then we should accept the project.

That's our decision rule, so IRR is is kind of related in that sense and that ultimately it's going
to lead to a decision rule, but first in order to understand what IRR is let's kind of work a problem here with NPV and then show how we do it with IRR.

So let's see an example here:
We have a project and this project is going to have this just going to be a one-year timeline and at the beginning of year or at your zero right now we're going to have a cash outflow we're going to invest $100 and then at the end of year one we're going to receive a cash flow of $130. 

So this is a very simple project here paying out $100 and then getting $130 back at the end. Now in order to calculate this with our NPV the other way, holding off  IRR for a second and thinking about NPV, we're going to need to know the discount rate. Little (r) we'll call that and so let's just say that the discount will let's say it's 8%.

Right now I'm just briefly going to go through how we would do this under NPV and refer to the NPV article if you haven't seen this yet. The NPV is going to be negative 100 that cash outflow plus 130, but we're going to take that $130 over (1 + r).

1 plus (+) that discount rate and we need to do that because of the time value of money. So ultimately that NPV is going to be 130 over 1.08 with this point .08 that's just that 8% discount rate we're converting it to a decimal and adding 1 to it.

Ultimately this is going to give us a net present value to this project of $20.37 so now if we think about our decision rule, well is this greater than zero? (20.37 > 0) Well yes, it is so, then according to the NPV you would accept this project and go ahead with it, but let's think about IRR now. H
ow IRR different from NPV? Well, ultimately what we're going to do with IRR? So we are going to set the NPV equal to zero (0). 
So that we're going to have that negative one hundred (-100) which is the cash outflow and then we're going to have 130 over, and this time I'm going to have one plus (1+) but I'm going to put big (R) instead of little (r).

We're still going to have little (r) over here being our 8% discount rate. You say okay well what is this big (R) then? What is the difference between big (R) and little (r)? Well, what we're going to do is, we're going to try and solve this and figure what is the big (R) that would make this equation make the NPV is equal to zero.
And then once we've determined what (R) is, we're going to take it and compare it to that 8% and that's going to be our decision rule.

So, let's go ahead, and let's just solve this. So ultimately we're setting the NPV equal to zero and then we're solving for the (R) that would make this equation true. So ultimately you can do some algebra here but you'll end up with an (R) that is equal to 0.30. Because we're going to convert that to a percent just so that we can easily compare these two. So we've got the big (R) is 30% and then the little (r) is 8%.

So what does that tell us? well, we know that 30% is greater than 8%  if the big (R), as our decision rule if big (R) is greater than little (r) then we should accept that project.

Okay, that's our decision rule. Now let's think about what this means here conceptually. What we're trying to do is we're saying the NPV is kind of our break-even point. So when the NPV is zero (0) what does that mean? That means that wealth is neither being added nor destroyed at the firm because you're just earning this (r =8%) opportunity cost or cost of care. However, we want to think about you're just getting that 8%. So this NPV when it's zero that's our breakeven point and we're saying what return would we have to have. What big (R) would there need to be in order to get the NPV to equal zero, to get to that break-even point and we see that we have a return thereof 30% would make the NPV equal to zero. 

So we say okay well let's compare that 30% to what our opportunity cost of the capital is or our discount rate. However, we want to think about it and let's see if it's larger and then we look and we say clearly "yes". So ultimately we think of IRR as the discount rate that makes the NPV equal to zero that makes NPV equal to zero to get us to that break-even point we just ultimately say instead of calculating the NPV directly (which we did under the NPV decision rule) we're going to set NPV equal to zero and then solve for the big (R) that would make this equation true, where we find that big R that makes the NPV equal to zero. compare it to our discount rate and if it's greater than we accept the project.

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