Let's now study some special cases. Well the first case is well-known and is called the perpetuity. Well this is the following cash flow pattern. So this is point zero, this is point one, two, three, and so on and that goes infinitely. And the cash flow stream is like this, at point one you get C, at point two you get the same C, here we have the same C and then you keep receiving these C's forever. Then, will also assume for simplicity that will this cash flow stream, we can associate the rate of return R that is constant overtime, this is a strong assumption, and then also R is positive. Now, that seems to be really an example from of that is specifically made up. But there are some important insights to get even from that, well first of all, the question goes, how much money will you have received over the life of this cash flow stream? And the answer is, you will have an infinite amount of cash because you'll keep receiving C, let's say every month or every year. Well, but the thing is that because of these future cash flows, they contribute progressively less to the PV of this perpetuity. I claim that the PV of this perpetuity is finite and that will show you how to get to this point. Well, we know the general formula that PV is equal to C over one plus R, plus C over one plus R squared, plus C over one plus R this is a K point plus and that goes forever. Well, I will rewrite that and I'll take C over one plus R out and then in braces, I'll get one plus one, over one plus R, plus one over one plus R, that's strictly speaking will be K minus first power and then like this. Well, the good news is that this long serious in braces this is the geometric progression. And for R greater than zero this whole thing becomes one over one minus one over one plus R. All of this is one plus R over R and multiplying we get a great result that I'll put up here. We will see that for perpetuity PV is equals to C over R, unbelievable. Now, you can say, "Well, you'd played with numbers to use some algebra. So what?" But what I will say, I will give you two examples of the uses of perpetuities. One example is that although these instruments are quite exotic but there still exist some government bonds issued by the British government, as early as in the second part of the 18th century, they are called Consols. So, these are the bonds we have infinite maturity. So the owner of this consol every six months gets a fixed amount of cash. So these consols are perpetuities and they are traded in the market. And if you saw their prices, you would see that their market prices are very close to this PV. So people use this formula to value these consols. So we can see that, even the most simplistic example already serves us as a valuation procedure. Well, but let me give you a, I wouldn't say more realistic because consols are quite realistic, but they're much more important use of perpetuities. Let's say that you evaluate a real company and I'm jumping a little bit beyond what we know right now but we will go back to that very quickly in this course. But the general approach is then you have some forecast horizon for which you forecast these cash flows and then you use the NP formula in direct form. But then these companies they have the infinite life. So you have to make some assumptions from point T to infinity, what happens to the expected cash flows of this company? And the most conservative approach says that this is going to be a perpetuity. So starting from point T the company does not grow. So, this part that sometimes is called tail, oftentimes contributes to the value of the company and this is the main contribution because for example, if you made some significant investments over this forecast horizon then some of cash flows might well be negative but that'll feed your base and then will secure the stable cash flows forever. So, when investment banks or consultants or people who work with these companies, when they do valuation for whatever reason, let's say for when preparing an IPO or when working on a strategic merger transaction or any other thing, they do valuation with valuing these tales until these tales oftentimes, they use the same approach, so this is not just a made up examples. Another thing that I would like to put here in blue, let's say that the other thing is growing perpetuity. We have exactly the same but here we have C but then starting from point two we have C times one plus G. Here we have C times one plus G squared. So we can say that our cash flow is C at point K is equal to C times one plus G to K minus first power. Well, G is taken to be less than R. It can be easily understood that although, there are sometimes explosive growth, that companies grow very fast, but that cannot go forever. So for the long term perspective, the rate of return must always be higher than the rate of growth. And then, for this growing perpetuity we can get the very simple formula, we can say that the PV of that will be able to C over R minus G or clearly if G is zero we are back to this formula. Again, there's growth in perpetuity formula is widely used in corporate valuation because sometimes people would like to play not overly conservative scenario. Well, you can always say, the most conservative scenario is to say that company will make nothing. Well, these assumptions are made only if something really is likely to happen. So the company will stop losing cash flow but that is not going to be a very interesting company. Normal of this no growth case is actually a base or sort of the conservative scenario. But then people can say, "Well, what if the company's cash flows they grow at a slow but at a constant rate? Let's say, three percent forever while R is greater" Then, you'll get back to this formula and you will evaluate this tail with the use of the same very simple shortcut. So I'm wrapping up this episode because in the next I will go further and we'll talk about annuities and their use. But for now, we can see that although we did make some fundamental simplifying assumptions. First of all this one, then here also I have to put that G is constant otherwise these formulas they don't work and then we said that they're all positive and there is this inequality. So there have been quite a few assumptions but the results they are worth while making these assumptions because these results are being used and not without success in the vast area of corporate valuation.
