Let's now discuss the important feature of projects, namely their time difference, length, or time interaction. Now we will discuss the following. Let's say that a company has some cars that carry couriers, and they deliver, let's say they deliver pizzas. Now the company would like to make a decision to buy some of these cars, and it is making the choice between the two options. And let's say that the black car serves only two years, and then it has associated costs of purchase and then costs of annual maintenance for gasoline, for the labor of the driver, and so on and so forth. And the other car has a useful life of three years and also the associated costs with that. Now, at first glance, it's difficult to compare them because what happens if we choose the black car, what happens in year three? Well you can say, in this simplistic case, what you can do is, you can take a six year long project that will contain two cars of blue color and three cars of the black color. Then, you'll be able to compare two projects of the same length. But clearly, what we would like to do is to be able to compare projects of lengths that are not like three to two but of whatever length. And how can we do that? Well, let's suppose for simplicity, for a moment, that there is no discounting, that the opportunity cost of money is here zero. Then, if we have all costs associated with these projects, you could take them together and then divide by the number of years that would give you the sort of an average annual cost. But we know that, in the real world, there is time value of money, and there is discounting, so we have to somehow find an equivalent to that with the use of present values of these costs. Now, I specifically gave you an example of these cars that carry couriers, that there is no difference in the revenues here. So these cars, they provide the same service and the only thing that we care about is the cost. Now, how can we approach this? Well let's analyze the first black project in some great detail. Again for simplicity we will say that to buy this car costs C0 and, in year one we incur costs C1, and in the year two costs C2. And again, all these costs are after taxes. Now what's the PV of costs? Well, first of all, you know all these costs that have the same sign. So we ignore the positive things, so we will say this is C0 plus C1 over one plus 'r' plus C2 over one plus 'r' squared. As always when we study some specific feature, we ignore the differences in others. So here, we assume that 'r' is constant, it is same for these two years. Otherwise it would have been 'r' one and 'r' two, which is also not that much of a problem, but that unfortunately in our next step will pose a problem. So these are the PV of costs. Now, let's analyze this project in the following equivalent way. What if instead of buying this car, we would lease it, and would pay the same amount of money each year, that would include, all costs? It would include insurance, it would include gasoline, that would include the labor of the driver. So in this case, we would say that the equivalent project will be like this. In year one, we incur that big, I would put EAC. That stands for equivalent annual cost and the same amount will be here and nothing here. And then we would say that, the PV of these equivalent costs is equal to, this is an annuity. This will be EAC over 'r', times one minus one over one plus 'r' the squared. We know that from our first week. So, if we assume that these PVs are the same, then we can say that this is an equivalent project to that. See here, we know C0, C1, C2 and 'r', so we can calculate this. We calculate all this stuff and then we get the EAC. So now, I would say that for each project, we calculate equivalent annual cost, that is sort of the average cost of implementing this project where we are taking the account discounting and present value. Well, let's flip over and see what we do for the second one. Well, for the blue project. We have exactly the same story, but here PV, will be equal to, I will put caps, C0 cap plus C1 cap over one plus 'r' plus C2 cap over one plus 'r' squared plus C3 cap over one plus 'r' to the third power. Again, that would be equivalent to EAC two divided by 'r' and here one minus one over one plus 'r' to the third power. So basically, we can identify this EAC two and now what we'll do with projects? We will just compare the EAC and then, the only thing that we will need here is that sometimes, these projects may have some residual value. So, but it's easy to do so because we could always add as a final bullet payment that will be negative if it has value because all these are costs, so they are all negative. But if there is any kind of addition, then we can easily employ residual value. So, we have now identified a very simple way to compare projects of different length, provided that they are the projects of the same risk because here the important thing is that this 'r', that you use to compare these things. It's important to use the same R here in this equation because you can always say, well, if for any reason, we use one 'r' for blue projects, and the other 'r' for black projects, then the result may still be the same equivalent annual cost, but most often, it's better to compare projects of equivalent risks because this way, you're positive not to make any aggravating assumption or any misinterpretation, because clearly for example, remember I said that for any reason we would have had different 'r' here, the question would arise, what 'r' we have to employ here? Well, we would have to come up with a certain equivalent to the, let's say yield to maturity. And that would be much more cumbersome, and not only and not primarily in terms of calculations, but rather in terms of ideology. So, we will say that, if we're comparing projects of equivalent risk and the only difference is the length, and for these projects it's a good assumption to think that 'r' stay constant over the life of this project then, this approach with the calculation and comparison of EAC produces clear and easy to use results. In the next episode, I will show to you how the use of EAC can lead to some corporate decisions. We will compare two different heating systems, that will be very much realistic project to analyze.