Well, let's use our understanding of EEC to the analysis of the real comparison of two projects. Well, we are comparing two heating systems. One will be gas, the other will be electric and that will produce the following table. So this is the electric system. Here will be cost, this is cost of installation, the initial C sub 0. This will be useful life. That will be in years. Costs will be in, Thousands of dollars. And then here will be annual cost, Before taxes. Now for the electric system, we will have 1.2 million for installation. 5 is the useful life, and then $50,000 a year before tax. This is the amount that they have to spend to support it. Now, we assume that after five years it have to be completely replaced. Well, that maybe too tough an assumption but this is all done for illustrative purposes. Now, there is also gas heating system that costs more than 2 million, but that serves for 8 years and requires only 20,000 pre-tax to support its operations. Now, in order to proceed we have to add some more information. First of all, the tax rate T, we will take it to be 30%. Then the discount rate r, we'll take as 10%, because this is heating, so these are infrastructure. So the rate of return required by investor is quite moderate. And then we also, to be kind of more advanced, we will add that the inflation rate is expected to be 2%. And with respect to both systems are obviously depreciated, and we will take the linear depreciation, With no salvage value. So that basically means that if you take the useful life, then you take the overall cost and divide it by the number of years, and then each year you have the same depreciation charge. So this is the set of information we need, and then, we will now set up the cash flow schedule. So for whatever system, it looks like this. Here, this is C sub 0. This is 1.2 million for the case of electric, and 2 million for the case of gas, and these years until its end. Now, the good news in this example is that if we took the linear depreciation with no salvage value, then the depreciation charge for all years is the same. Now, see what happens here. Here, we have C times 1- t, this is the annual expense, this is the tax rate, and this is real. Because we have inflation, and these costs are likely to grow with inflation. Here, however, we have depreciation tax shield, D times T, and this is nominal. So that is the same for all years. So I will just, I'll just put like this, and then I'll put like that. So we can easily put the overall formula for the PV of costs. Again, that will be equivalent for the other system. So P of course is what? This is C sub 0, plus depreciation tax shield. Now, here the important thing is that if we used this sign, then the DT comes with a negative sign. So it will be not plus, but it will be -DT, and here will be the estimation side of k from 1 to T, which T is here. And then this is 1/1 + r to the kth power. And then to that you have to add + C, 1- T, also the summation sign but here it's a little bit worse because now these are real. So they have to be blown up with inflation or we can use a real rate, so here would be 1 + i/1 + r to the kth power. So this is the PV of cost and, Well this is a cumbersome formula but there's nothing very bad about that, and then we have to replace that. By the equivalent annual costs so then we forget about all this, and let's say put at each point this equivalent annual cost. That actually will go here, and we will say that this is equal to equivalent annual cost divided by r cap, which is real divided by 1- 1 divided by 1 + r cap to the Tth power. And what is r cap? r cap is equal to 1 + r divided by 1 + i- 1. So r cap is real in this case, and we said that r was 10%, i is 2%, so here we will have r cap is equal to approximately 7.84% So we are sort of prepared and now we will have to just rewrite this formula. And in order to make it less cumbersome we will introduce the annuity factor. Well, not introduce, we will use it. We will say that A (rN) which is the annuity factor, is equal to 1 over r, 1- 1 over 1 + r to the Nth power. So this is the present value of an annuity of just $1 for N periods, yet discounted at the rate r. Then, we will write the following equation. We will put all costs here, C sub 0- DT. Here comes the annuity factor of r, N. Then +, C /1- T annuity factor of r cap N, because this is real. And that should be equal to EAC times annuity factor again of r cap N. So if you divided by the annuity factor then you'll get the final formula that equivalent annual cost is equal to c sub 0 divided by a of r cap N, -, DT. And here comes the ratio of two annuity factors, one of nominal and one of the real rate. And then + C/1- T. So if we know these rs, and again here, I would not put all these, for example a of 10 5 is 3.791, but I will put only the answer, that equivalent annual cost 1 will be equal to 266,184. And the equivalent annual cost 2 will be equal to 290,700. So I will put that is the electric system, and that is the gas system. So our answer is very simple. We can see that on the basis of accrued annual cost, we vote for the electric system that is less costly. Although, it works for only five years and in five years it must be replaced with another one. Indeed, this is an example, but this is widely used by the people because you can always argue that we wish we knew what happens in five years. Maybe sometimes we would prefer to be able to lock in the heating system for a longer period of time. But here we are basing our choice only and specifically on the idea of equivalent annual cost, which is not very far from what people do in reality. So you've seen that what was important in this example. The important thing was that, number one, we used equivalent annual cost as the overall approach. Number two, we took into account depreciation and depreciation tax shield. Number three, we took into account inflation, so we were careful in our treatment of nominal cash flows and real cash flows, that is why you see these factors of r and r Cap. So in this analysis, we combined our knowledge of what must be done with cash flows and therefore discount rates to come up with the correct use of an NPV criteria. So although that's a simple thing, but this is like a micro case that is very much like the real evaluation case for investment projects. And the only problem is that oftentimes, people have to really work hard to get with these C sub 0. Well, C sub 0is kind of an easy thing, but with Cs, then with these Ds, and then, with proper rates. So we are now arriving at the final episode of this week, which we'll do a short wrap up. And we'll come closer to the next week in which we will study r's in greater detail. And now in the final episode of this week, we will put things together, what we've studied so far this week.