In what follows, we will study this case too with all perimeters constant. And although there will be quite a few numbers here on this flipchart, the result will be quite trivial. And the actual value of this exercise is to see how we can generalize that to study more interesting cases. Well, the setup is like this. We assume the following parameters of the company. The market cap rate of 20%, this is the rate at which we discount. Then some parameters of the company. Earnings per share at year 1 is expected to be $15. Then pay out rate is 60%. Therefore, from this multiply we can see that dividend 1 is expected to be $9. Then clearly plowback which is in this case 1 minus this is 40%. The only thing that is missing is return on equity, on book equity in this case, which is 30%. And if this is the case these are all the numbers that for our company. That this is just an example. Now from that, we can find g, because g is plowback times ROE which is 12% in this case. And we can use Gordon's formula. We can say that P sub 0, because this is really constant growth, is $9 divided by 0.20- 0.12, which is 112.50. And just to observe that, we'll find that the P of no growth, that will be what? EPS 1 divided by 0.20, which is in this case, $15.20, which is $75. Now, from this we can conclude, I'll put it here, that PVGO, present value growth opportunities is the difference of these which is 37.50. Well, we found everything, but what I will do, I will not stop here. I will start to go deeper in how these mechanics actually yields this result. So we will follow through the investments year after year, and we will see that each investment contributes to this growth. Again, this is easier written than said, so, We will see the chart like this. So these are years, 1, 2, 3, 4, 5, 6. Well, I could stop earlier, but that makes it more understandable. So this is the result of previous investment. So this is APS 1, 15, 15, 15 forever. Now, let's analyze in detail investment at year 1. So we reinvest 40% or minus $6, and this $6 brings us 30%. This is written on equity, so this is 1.8. So here we have 1.8, 1.8 and so forth. Now, let's stop for a moment here. If there were no investment, there will be nothing on this flipchart anymore. So we will take 15, we would have divided them by 0.2 and then we would have received 75, the no growth case. So the top line in this, as you'll see triangle table is no growth case. Now see what happens as a result of investment at year 2. Now the key story is here we had 15. Now by year 2, we have grown to this combination. So now we have 16.8, so we take 30% of that and we get -6.72, this is the investment. Well, all these numbers here, they grow at the rate of 12% which is constant. So this 6.72 produces us 30% in perpetuity, which is 2.02, 2.02 and that goes forever. Now, you don't have to be real whiz to realize that here we have -7.53 and the amount is 2.26. Here it will be -8.43, and then the year 5 it will be 2.53. And then finally, here it will be -9.44. I will stop here because that now is clear. So we've created this triangle table, and see what is the bottom line? So we see that here this is the amount of cash that shareholders can pocket. This is actually the dividend, so this is 9 here, it's 1,008 here, it's 11.29 here, 12.64 and all that grows again. Here it grows at 12%. So let's see exactly the NPV of these investments. Here I put in blue to be consistent. I will put that the NPV is. So we invest -6 and then plus. We get 1.8 in perpetuity, so we divide by 0.2, and that gives us exactly 3. Now here on the same exercise, -6.72 + 2.02, divided by 0.2, and that gives you 336. Well, clearly again, here growth is at 12%, and so on and so forth. And if we then took this 3, and that is also constant growth, then we can see that PVGO will be 3 divided, Divided by 0.08, this is the difference between 20% and 12%, that will be exactly 37.5. This result we've had before. Now you can ask the question, why the hell did we study all that with these numbers? Now, see what happens? The trivial result that which we've had before, we got that because we put all these parameters of reinvestment constant. But let's take a look at this table. Let's say that at year 3 we decided to change plowback. Then in this case we would all these numbers would be different. And the corresponding PV would be different. So what we see here is that for some finite period of time, we can make whatever investments we would like. And then we can calculate the contribution of these investments to the final PVGO with the use of the corresponding horizontal line in this table. From some further point until infinity, we must make some assumptions about constant parameters, and then we'll get to something like this. So basically this approach, although it's kind of funny, because this is sort of a table that doesn't seem to be very serious. But that really allows you to go deeper into the idea of growth. You can see that lines may be different, and they can contribute differently, but they all do. And by the same token, with this table, you can easily study cases of borrowing. Let's say if in this year you borrowed, then you can invest much more. And the contribution of that would be higher. Then you will ultimately have to take into account the interest you're paying on that and the repayment of a loan at the point when it occurs. So this fun example shows to you how you can actually pinpoint growth for growth rates that are not always equal. This was for equal for the whole period of time, but like I said, for a finite time period you can play with them. This is a very powerful tool, and oftentimes this representation happens to be much clearer than the one if I used only formulas here. Again, I was giving assignments to my students for many years, and one year I realized that talking about this triangle table results in a much better rate of solutions in their assignments. So we are almost done, and in the next, the final episode of the second week, we will put everything together. We will briefly analyze what we've succeeded in so far, and where we would go from here.
