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Now for one of the most perplexing concepts, which I thought couldn't possibly be useful, but is more useful than anything I apply. [LAUGH] You can be very naive. And this is the concept of perpetuity. And I have a very complex accent so I say perpetuity, other people say perpetuity so differently, it's a tongue twister for me. But anyway, what is a perpetuity? Think about perpetual, it's forever. So I said, how can anything be forever? So it turns out a perpetuity is this. It is multiple payment that goes on forever. So, let me just draw the timeline.
You have 1, 2, and some period n. C, C, C. If you notice it can be with or without growth. Growth in what? We'll soon do more complex problems. In an annuity, it stops at n, and the Cs were fixed, yeah? But the Cs could grow, and I'll explain that in a second. But perpetuity doesn't stop at ten it keeps going on C, C, C, C, C, C forever.
And turns out it looks very complex, right? So what is the present value of perpetuity? It's C / (1 + r), C / (1+r) square, how many times? A gazillion times. This, by the way, this curly little monster, is my way of writing infinity. So if you look at this, this looks like, mind-boggling, even Excel couldn't do it If you have the patience and the time, and you're nerdy enough. This is actually very easy to solve, and it's because it's geometric progression, right? So you can do it and answer will turn out to be C / r. So if a perpetuity is fixed forever, actually the formula is very simple. At C / r. And why is this useful?
Turns out that there's something called a consol in England, which pays one pound of whatever forever, and you can buy it.
But that's not the only use. It's a very powerful basis of almost all valuation of things called stocks.
So think of this. An annuity is applicable to something that is a fixed life. And that's why we did things like loans. But a perpetuity is applicable to things like stocks where what a stock is, and we'll see this later in the class, people thought, why teach stocks to an introductory class? Because it's so beautiful. [LAUGH] And it's so applicable to thinking that, you can, I'm using stocks as a concept that'll pull you in anyways. So, what is a stock? When you buy a stock in a company, there's no contract between you and them.
You buy a share, and you give some money, and hope to get in return money, not just once, if you hold it for a long time, but forever. And that payment is called a dividend. So the c becomes a d, that's all. But the neat thing about stocks is that it's not a contract. But it wants to live forever. And therefore, all valuation of stocks and long term companies, like companies are also long term, fit together very well. I'll get into that, but I don't want to get too excited right now, but the thing I wanted to tell you is that if a cashflow lasts approximately forever, and a CCC, the formula is pretty straightforward.
So very quickly, if I get $10 every year for the rest of eternity and the interest rate is 10%, what's the present value? You can do it in your head. 10/0.1, don't divide by 10, 0.1. It's 100. You see how simple it is? So a lot of really, really thoughtful people who know all the details of finance, use such shortcut formulas in a context which is applicable, right. Not in a very short term loan, but in the long term context.
Examples of perpetuities. So the first example of perpetuities is a stock.
You can think of company.
You can think of any long term stuff, because it's very powerful. So, for example, if I gave you 10 bucks forever at, and the r was 10%, you would have.
Because it's 10 over 0.1. The beauty of it is yes, companies want to last forever, stocks want to last forever. But suppose you have a very long term horizon, and suppose m was 13.
You can approximate a perpetuity, so do it for yourself. Please. So do = PV. And by the way, I haven't done it. And just take time. 0.10. And instead of infinity, what you're trying to do is P. PV over payment, right, how many times? 30 times.
And what is the PMT? 10 bucks. If you press Return, this you'll find to be a very, very close approximation of this.
And the reason is very simple. First the longer your horizon, 30 is much longer than 5, and the higher your interest, the closer the applicability, or perpetuity formula. I was looking for it, C / R.
And the reason is very simple. Because of the power of discounting, anything happening beyond year 30, years 31 is what? Its impact to today's PV of the whole perpetuity is very low. So the present value of the 31st year's cash flow is very, very low. Most of the value is coming from the earlier cash flows. And that's true, especially if 0.1 is a pretty high interest rate, if that's the case. I want to do couple more things, and then take a break, right?
Because we are coming towards the ends of what I would call simple applications, and I want to show you the power of perpetuity simply because it seems like the simplest application, and it's applicable to very complex things, like long term stuff, stocks and bonds. So there's one more thing about perpetuities, and that is suppose the first number was C but the second one was C(1+g) and the third, let's write.
And the third was C(1+g) squared. 3, and this went down forever. There is a name for some animal like this, and it's called growth stock. So if you buy a company that's growing very fast or growing for a while, as opposed to a company that's not growing and is paying our dividend fixed amount cash flow every year. This kind of company is like a really aggressive growth technology company. The formula of the perpetuity turns out to be C1, because now you have to write C1, right, because C1 is not equal to C2. So what is C2? C2 is growing at growth rate g. And suppose this is 2%, that will give you a sense of what I'm talking about, right?
So, if say inflation is 2%, you would hope your cash flows grow at 2%, but suppose inflation is close to 0, which is the environment in many places today, you still want your cash flows to grow if you're generally growing. So it will be C1 / r-g. Many people say, but this formula doesn't make any sense if g is equal to r or greater than r. Point is yes, it doesn't make any sense, so don't use it. So, use what? The long formula. And use the short form when it's applicable, and we'll see later that's exactly what we'll do. But this, again, is so powerful.
I actually believe if you understand this, this, and this, you know all of finance, because that's the beauty. These capture the power of compounding, the power of valuation, especially as long term complex things. And the notion is, the more complex the thing, the less valuable it is. And in the future is to try to be very precise about C1 to C3, because you're going to be wrong anyways, right? So you're making a decision today that's impacting you forever. Investing in a company, starting a company, whatever. The more uncertain you are, actually, the more shortcuts like this really help you think through the problem. But you have to know discounted cash flows the details to make sense of it.
So, before I end this segment, I wanted to highlight one thing. I will put all the formulas, and I'm clicking through so if you stare at the screen for a little bit, there are two pages with the formulas here, I don't want you to worry about them. Because they are a lot and we talked about most of them, some others we'll talk about as we do more complex problems. I'm going to post for you as notes. I'm doing that because it is very tedious to keep writing these formulas. I also am doing that because I think you will probably not use the formulas by hand, you use Excel. But Excel needs to know the formulas. Right?
So they are built into Excel for the most part, and whenever the formulas are not built in, I will work with Excel to make you understand what's going on.
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