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Simple Application I: Future Value of an Annuity

Simple Application I: Future Value of an Annuity
We just finished the power of discounting. We had done power of compounding, power of discounting, and we found how impactful the passage of time is, right? So if we got something like $87 million.
And I remind you the negative, positive sign of Excel depends on whether you're getting money or losing money. So 87 million at 5% and about less than 1 million, 900,000 or so, if the interest rate is 15%, obviously inverse of the future value Right? Now let us go into what I would call as more realistic applications. We will go in a stepwise manner. I will not include everything. That will confuse you in a problem. But we'll go stepwise. But the interesting thing is even though I'm going stepwise, the power of each concept I'm telling you is so remarkable that you can apply even the simplest of applications in many contexts.
Let's start off with the one thing which we assumed until now Is now going to happen. So we drew a timeline, and remember in the timeline we did go from period 1, we went up to n. But the assumption we made till now was either something was happening here at time 0 or something was happening here In other words we had one time payments, either today or in the future. What i'm going to introduce now is the notion of multiple payments. Because most problems in life are most issues, or most concepts, or most applications involve multiple payments. Right, so we'll start off as I said in a very structured way with something called an annuity.
An annuity is some payment that you get we call it cash flow c which is fixed in continuity typically. And lasts for end period, so here what's going on is instead of getting. Say are paying $100 once, you are doing it multiple times and you will see applications of it as I do them. But first since you know present value and future value now let's just think about it conceptually. So I will go back and forth between application and concepts, once you know some concept I'm gonna go to the concept next time and then application and so on So multiple payments, let's talk about the concept a little bit.
We'll call the payments C, but now this is related to Excel. In Excel, they don't call it cash flows right, they call it PMT, and it is a cash flow, and it's called payment because it's like a payment every period. And you'll see where all of this is coming from. It's coming from basically the structure of the loan. Okay, so if you stare at the graph on top, what I want to do is what I said. Conceptually lay out what is going on. The cash flow in a typical annuity and here convention is very important doesn't start at time zero, and the assumption is the first cash flow is occurring at the end of the first period.
Now, the period could be one year long, six months long, whatever, depends on the problem. So what we are going to do is we are going to have a simple annuity which lasts
n is equal to three which is important because you want to know how many periods. And in each period you're getting a pmt of C. C will be a number in our problems obviously right? And I would like, again, to start off with a concept of future value because future value is quite interesting. So, in order to figure out future value, now you have to conceptually think of it this way. Well, how many years left from period 0 to the future?
Years left, 3. How many years left from the period one to the future? Now the future, remember, is ending at time 3. And period one, whenever we say a period, we mean end of the period. This causes a lot of Anxiety among people. That's why drawing a timeline is important. This is 1 and this is 0. What this gives you a sense is another way of writing the timeline. And I'm going to be a little bit painfully slow here because we are assuming we have beginners not used to doing this. So the C, C, C. This is how it's working, right? So how many periods from 0 to this? Three. How many periods from 0 to 1 to 3? Two.
2 to 3? One. At 3? Zero. So that's the structure presumed about an annuity. Of course, you can change that structure depending on the nature of the problem, but you can go very far with this kind of structure even though it's pretty [INAUDIBLE]. So let's see what the future value will look like and we'll do it slowly and then we'll take a break and come back to applications.
This future value will be zero. And the reason is, at time zero, nothing is happening in a standard annuity. What is the future value of the cash flow you're getting at time one. Remember time one means what? At this time. At the end of the first year. It's be C (1+r) how many, squared. Remember, and the key is this number because you're carrying this C forward two periods. How much are they carrying this C? One period, C(1+r). And how much are you carrying this last C? No amount, just C itself. So when you're doing future value for annuity you're essentially carrying forward cash flows now.
But the last one isn't period three and therefore, it doesn't need to be carried forward much. So this gives you a sense of the concept and the formula, future value of an annuity,
becomes C(1+r)square + C(1+r) + C. Again, this is the 1 at the end of the first period, future value of the 1 at the end of the second period, and finally, future value of the third 1 is in period three. Notice that you can take C.
You can rewrite this and this is the formula for an annuity. And in the textbooks, what you'll find is At the back, this number's calculate. And it's called Future Value Annuity Factor. And it's a function of two things, r and n, number of of periods.
So not surprisingly, again, r matters because it's time value of money, but time matters too and it's three years. The key difference. Between just simple future value is that you get payment once today and you carry it forward, or a cash flow once. Here it's happening three times.
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