Now let's talk about the annuities. Well this is a very widespread shortcut. So the annuities looks like this. So, looks very much like perpetuity in the beginning, but it has- so we have CCCC but not forever. But up to the point T. Now, the formula to find the PV of perpetuity can- over the new item, I'm sorry, can be very easily derived, if we use the following trick. If we prolong this timeline for infinity, and then, we'll have CCC up there. And then, to make it equal to this, we have from to point T plus one, we have to add another perpetuity. Starting here, and it will be minus C, minus C, minus C, and so on. So, that will be clearly zero all the way through. But the good news is, that we would know the PV of the top perpetuity. We would know the PV of the negative. This perpetuity and the only thing is to discount the PV of this law perpetuity back T times. So, the formula would look like this, that the PV of an annuity is equal C over r, and here comes one minus one, one plus r to power T. Well, you can see that if C goes to infinity that disappears and we are back to perpetuity, which is great as that which is consistent. Now, the first question is, why are we studying this? Well, one clear answer is that there- do we exist certain important financial instruments? The cash flows of which are very close to this. Well, the most well-known application of annuity is a fixed-rate mortgage. So, it's structured that well the home buyer takes this loan from the bank, then you repays this loan with equal installments over a long period of time. We will study of the value of a fixed-rate mortgage in the next episode, we will use of some merical example. But it's worthwhile using these things. And now we can see how the annuity formula is applied. Then it's not only a mortgage because some life insurance contracts also make these payments and then, this approach is very helpful in finding the use of installment payments. And there is another thing, which doesn't seem to be a very much on the surface but that is extremely important. Let's say that, you are comparing some investment options and let's say, that their revenue stream is the same, and you are making usually is based on the lower costs. And oftentimes, if this projects have different length, and if some of these costs are different at various points, you might be interested in finding inequivalent. So that your project would look like that, let's say couple of years and every year, you incur the same costs. So that it has a special name that's called the equivalent annual cost, and we study that in greater detail in a third week. But for now, the important thing is that in order to come up what these equivalent annual costs are, people also use these the lesion approach for an annuity. And I would say that in annuity is probably the most widely used shortcut. Although it's kind of simplistic. And again let me remind you, that holds only if r is equal to is constant because for example, we can also say that coupon payments of a bond, they also look like this. Well, here is another bill of payment of the- with you with your face value at the very end. But as you will see, later on in this course in the second week actually, then even before the bonds that do not involve a lot of risk, this assumption is poor. And although, we will identify a very special parameter of a bond namely yield to maturity that will be sort of at replacement of this assumption was that this is very vaguely. Said now, but whether we will arrive at this point, we will see what I mean. But still, it's a widely used thing. Now, it doesn't seem to be very clear how this formula works, until we study that in some detail and on some example. So, as an exercise for both you and me, will- what we will do in the next episode, we will apply this valuation approach to find the payments that are associated with a fixed-rate mortgage. Not only will we do that, but we will see that the fact that the mortgage is really a long term loan of that sort of forces, the borrower, to pay a lot of money in the form of end of the interest payments. And we will see that this is extremely dependent on these r's and I will have two options of a lower r and of a higher r. And you will see how the contribution of this interest plays a significant role when r's grow. So, let's now take a pause and in the next episode, we will come back to a fixed-rate mortgage in some greater detail.