Welcome back to Corporate Finance, last time we talked about compounding or the process of moving cash flows forward in time. Today I wanna present several useful shortcuts to compute the present value and future value of common streams of cash flows that we see often in practice, let's get started. >> Hey everybody welcome to our third lecture on the time value of money. So last time we talked about compounding, or the process of moving cash flows forward in time. [COUGH] Excuse me, to find their future value, whereas in our first lecture we moved cash flows back in time by discounting to find the present value. So what I wanna do today is I want to give you some useful shortcuts for computing the present value or the future value of some streams of cash flows that commonly arise in practice. So let's get started. The first thing I wanna talk about is an annuity. An annuity is a finite stream of cash flows of identical magnitude and equal spacing in time, and I've highlighted key elements of this definition. So here's a timeline representing an annuity. [COUGH] First key aspect or feature of an annuity is, our, cash flows of identical magnitude. These are all, all of these cash flows are the same number. Okay, so identical magnitude. Secondly, this is a finite stream of cash flow, so it ends at some point in time and that might seem like an unnecessary or obvious assumption, but you'll see that we'll deal with infinite cash flow streams in a little bit. And then the last assumption is that the spacing between the cash flows has to be equal so it's always a year. We always get the cash year after one year, two months, whatever that spacing is it has to be the same. And it turns out that this cash flow stream arises in a number of situations in practice. So insurance companies sell a product called an annuity, representing its cash flow stream. Home mortgages are an example of an annuity stream. Auto leases, certain bond payments, and amortizing loans are annuities, so it's actually fairly common in practice. Now, if we wanted to find the present value of these cash flows, we know how to do that, right? We can brute force it. We can take each cash flow and discount it back to today. All right, so imagine I had a second cash flow. The second cash flow here I could take those CF divided by 1 plus R squared, that would bring it back to today. I could take this cash flow CF over 1 plus R to the T minus 1. That would bring it, and I do that for all the cash flows, and then I could add them up here. And that would give me the present value. But that's a bit burdensome, especially when T is big. So, what I'd like to show you is a shortcut or a simple formula to compute the present value of this cash flow stream, and here it is. We take the cash flow, CF. We divide it by the discount rate and multiply it by this term in parentheses here. Now if I move the R over here I can re-express the present value of the annuity formula as just the annuity cash flow times this term here which is called an annuity factor. Okay, that will give me the present value of this cash flow stream. One thing to keep in mind though, is that in order for this formula to make sense, not only do all the features defining an annuity stream have to be true, but we're assuming that the first cash flow arrives one period from today. So for example, if my cash flow stream looked like this is an annuity stream of cash flows, but this formula is not gonna give me the present value of this cash flow stream. Actually what I would have to do is I could just add CF. Okay, because this is the present value, it's coming today. Let's do an example. How much do I have to save today to withdraw $100 at the end of the each of the next 20 years if I can earn 5% per annum? Well step one is draw a timeline. I'm trying to figure out how much I have to save today in order to pull out $100 every year over the next 20 years. Well we know how to do that by brute force. We can simply discount all of the cash flows back into today's time units and add them up. A more elegant solution that we just learned of course, is to apply our present value of annuity formula, right? The cash flow is 100, CF. The discount rate, R, is 5%, and the time of the cash flows is 20 years. Plugging all those numbers into the formula and computing, we get the present value of these cash flows is $1246.22. That's how much money I have to save today in order to do withdraw the $100 every year. Now let's turn to something called the growing annuity, which is, as the name suggests, just like an annuity, but for the fact that the cash flows are growing. So it's a finite stream of cash flows, okay, evenly spaced through time, okay. But now the cash flows aren't constant, they're growing at a constant rate g. And this type of cash flow stream pops up in a number of instances in practice. Certain income streams, for example, your work. Right, you might have imagined that your salary grows at some constant rate g, or approximately some constant rate g. Certain saving strategies. Maybe you wanna save a certain amount each year, but you want that amount to grow with your growing income stream. Incorporate finance certain project revenue and expense stream is six streams will often grow at a near constant growth rate. So it's a really useful approximation to many cash flow streams we'll come across in practice. And like our annuity stream we can represent the present value of this cash flow stream with a simple formula as follows. We take the cash flow [COUGH] as of the first period divided by the discount rate less the growth rate times this factor here. That will give us the present value of this cash flow stream right here. Okay. But remember a critical assumption is that the first cash flow arrives one period from today. Okay? Let's do an example. How much do I have to save today to withdraw $100 at the end of this year, $102.50 next year, $105.06 the year after, and so on for the next 19 years if we can earn 5% per annum? Well, let's draw a timeline, and what we see is that our first [COUGH] withdrawal of a $100 occurs one year out. Then 102.5, and on and on and on, what we can discern from this problem is that these cash flows are growing at a constant rate, g, equal to two and a half percent per annum. So this cash flow stream satisfies all of the requirements needed to use the present value of a growing annuity formula. So our first cash flow of $100, our discount rate of 5% and here's our growth rate of two and a half percent that's gonna get us a present value of $1,529.69. That's how much we would need to withdraw a $100 growing at two and a half percent every year. Now let's talk about a perpetuity. A perpetuity is just like an annuity except the cash flows go on forever, all right? We get the same amount of money, equally spaced in time forever. So where does this thing come up in practice? Well oddly enough it actually does. And something called perpetuities, or consol bonds, which exist over in the UK. And interestingly enough, the formula for this cash flow stream is very simple. It's just the cash flow divided by the discount rate. CF over R. Let's do an example. How much do you have to save today to withdraw $100 at the end of each year forever, if you can earn 5% per annum? Well the timeline looks as follows, right? $100 every year, forever. And clearly the brute force method of discounting each cash flow one at a time is never gonna work, it's just impossible. So we have to use our formula. We take the $100, divide it by the discount rate of 5%, and that gives us $2,000. We need $2,000 to be able to withdraw $100 a year forever, assuming that money can earn 5% per annum, and intuitively, what's going on is once we get out 100 years, 200 years, whatever. The present value of that money is so small it's very close to 0, which is why you don't need an infinite amount of money. And the perpetuity's cousin, a growing perpetuity, is just that, it's an infinite stream of cash flow that grows at a constant rate, g that are evenly spaced out through time. So here's a visual representation. Here's our timeline of a growing perpetuity. What's an example of a growing perpetuity in practice? Well, dividend streams are much like a growing perpetuity. They're useful approximation. Don't take it literally. Companies don't last forever. But we can treat them as such because there is no finite end date to most companies. Absent some event such as a bankruptcy or an acquisition or a takeover, something like that. So what's the formula for a growing perpetuity? Well, it's just the cash flow that we're gonna receive in the first year divided by the discount rate minus the growth rate of that cash flow. Again, we're gonna have to assume that the first cash flow arrives one year from today to use this formula, as well as [COUGH] having all the other requirements being satisfied, the cash flows being evenly spaced then end them growing at a constant rate. So, let's do a little example now. How much do you have to save today to withdraw $100 at the end of this year, 102.5 next year, 105.06the year after and so on forever, if we can earn 5% per annum? Well, let's draw our timeline. So, we're gonna get $100 here in year 1, 102.5 in year 2, that's growing at 2.5% as this 105.06 if I've written it here. So our g is 2.5%. The first cash flow comes 1 year from today there's our first cash flow. This goes on forever and the spacing is equal. So this is a growing perpetuity, to which we can apply our growing perpetuity present value formula. So we take the first cash flow, $100. Divide it by the difference between the discount rate and the growth rate, to get $4,000. In other words, if we have $4,000 today and it's earning 5% interest per annum every year thereafter, we can pull out $100 next year and have that amount grow by two and half percent every year thereafter. All right. So let's summarize. We learned a couple of useful shortcuts today we talked about an annuity and it's present value formula. We talked about a perpetuity and it's present value formula. We talked about their growing cousins, the growing annuity and the growing perpetuity in the present value formula for those guys and while that might seem somewhat esoteric and bland or boring, we also discussed some of the applications that you might see in practice, these cash flow streams arising. And where these shortcuts are really useful, more than just finding the present value or the future value, is in finding the cash flow associated with the stream. So being able to manipulate these formulas is very important. And so in the problems, you're gonna spend some time in real life contexts, or at least as close to real life as we can get, manipulating these formulas. To derive certain aspects of interest, whether it's the cash flow, or the amount of time, or the discount rate, or the growth rate. All right, so, tackle the problem said, it really brings the material to life, but be very careful, that applying these formulas takes care. Don't blindly apply them to any setting because as we discussed, certain characteristics of the cash flows have to be met in order to, or certain requirements of the formula have to be met in order to use it. So good luck with the problems.