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Simple Application II: Present Value of an Annuity

Simple Application II: Present Value of an Annuity
We are still with annuities, and I did some examples with you with future values. But yes, there's some structure provided by annuities, but when I saw them first, I said, what is this? Well, how will it work? But that's, it's very useful, even the assumption that you're putting away fixed amount every year or paying fixed amount is very realistic because people want to invest in ways that are understandable, simple, not changing every period. And then when you think about, a PMT, it derives we'll see in a second, from loans. And you take a loan, and then pay cash flows in the future, and typically they are fixed in quantity. So that's where the word PMT comes from.
So let's get started. Now I'm going to shift a little bit, and first do the concept of present value of annuities. So please stare at the screen and what you'll see is the same structure of an annuity. Well, three things are happening. You need to know R and we'll introduce it. You need to know N which in this case is conceptual cases three, and you need to know cash flow C right? So what I'm going to do is go a little bit quickly, because you've seen this before, so the cash flow at time zero is zero. Years left to discount. Now, it's the opposite, right? So, you're discounting. You're not carrying forward. You don't need to discount zero.
Any number of years because it's today. So remember, this is a time line you can write it out like 1 2 3, C, C, C. And now where you looking for the value here not at future value. So you're looking at PV, not FV. Okay? So how much do you have to discount the first? No cash flow, no discounting. This one year.
This two years. This three years. Let's do it. Because we know how to do one period. This will be what? Zero? This will be C, not multiplied by 1+r but divided by 1+r. I'm putting it in parenthesis because if you want to do this kind of mathematical operation and say in excel, you need stuff that belongs together in divisions and all in parenthesis. Most of you probably know that. So this will be divided by (1 + r) square. So if you didn't put (1 + r) in parenthesis, it will think r is being squared only, but that's not the case. This divided by (1 + r) cubed.
You see how simple this is, in spite of my bad writing, you have a very clear idea of what's going on. What I'm going to do is take a little time, write it out as a formula and explain one more time before we do examples. So the PV, of an annuity, annuity, is what? C over 1 plus r, sometimes I write cap r, small case r, don't worry about it. C over one plus r squared, plus C over one plus r, cubed.
Right? This is what the formula is. Again when you notice, you can take C outside and what do you get? 1 + r + 1 over 1 + r squared + 1 over 1 + r cubed. My writing is getting better as we go along. [INAUDIBLE] I'm very impressed with myself. So this is called Present Value Annuity Factor. Symbols used may be different, even by me, but you know it's an annuity. And it's a function of, what, r and n.
Right? What's cool about it is it has three things in it because how many cash flows are there? Three, but they are very simple, they're adjective. You have first cash flow discounted, second cash flow discounted, third cash flow discounted. To stare at this, this numbers or at the back of textbooks. Still, but you should be using Excel.
So if I asked you what is the value of getting one buck a hundred times in the future with an interest rate of say two percent, you should be able to do this very quickly. Not in your head because then you're being insane. I can say that to you because we are talking to each other, however, Excel is wonderful. It does it very quickly.
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