So what's a perpetuity? A perpetuity is a repeated payment that's made to infinity the same amount every time. And this might sound unrealistic, but certain financial instruments do behave like this. Even your bank account behaves like this theoretically. And dividend streams can be seen to behave like this if you assume that they'll go on into the distant future. So a perpetuity looks at a stream of cash flows that's predicted to go on forever and then tries to discount them back to today. And this is quite helpful for things like working out how much you could borrow if the interest was seen as perpetual. And there's a formula to help us do that. The formula to work out the present value of a perpetuity is to take the payment and divide it by R or the interest rate. And remember that the interest rate could be denoted by R or I. We're gonna use these interchangeably. Now this is a stable perpetuity and there's a second very common perpetuity.

In the second perpetuity, we are not going to have the same repeated payment every year. We're going to have a repeated payment that grows by a stable amount, maybe 2% every year.

This is helpful because of things like inflation or perhaps a growth in a company which leads to its dividends growing. And so you'll find that growing perpetuities are very common in the work that we do on valuations. Now, the growth changes the equation just a bit. I still want to find the present value of the growing perpetuity. And so I'll use the basic formula we explored before, but in this case, the formula will have a denominator. So the bottom of R minus G or I minus G if you want to use I instead.

So what is an annuity? An annuity is a regular payment made every year just like a perpetuity, but for a finite period. Now remember, the perpetuity was paid infinitely. Our annuity is only going to be made for a finite period, maybe 20 years. And good examples of this are finite bonds or pensions where this terminology is very common. A good example might be a 20 year bond. I'm going to borrow some money today and I'll pay the same interest amount every single year for 20 years. The question is how much should I borrow today? And that's calculated as the present value of the interest payments, which are the repeated payments. So we need to calculate the present value. Let's read through the formula together. The present value of the annuity is 1 minus 1 plus the interest rate to the power of the number of years, all divided by the interest rate times by the regular payments. And of course, be very careful with your brackets here.

There's a second type of annuity that's also very common. Instead of having a stable payment made every year, we're going to have one that grows by a stable amount. So it might grow by 1% or 2% each year. So for example, take a 10 year bond, but the interest payments increase by inflation.

This changes the formula slightly. Let's have a read through that second formula. The present value of a growing annuity starts off with something slightly familiar. It starts off with the present value of the perpetuity payments divided by the interest rate minus growth. So it starts off with something that looks like a growing perpetuity. We then multiply this by 1 minus 1 plus growth over 1 plus interest, all to the power N. And of course, again, be very careful with your brackets.

Now the second equation, you can think of it as a growing perpetuity that has the end chopped off. That's kind of how it works. In these workouts, we're going to examine how to do perpetuities and annuities, and we're going to look at non-growing, so stable and then growing versions of each of those. You can see there's four workouts which correspond to the four different types. So you have a stable perpetuity, a growing perpetuity, a stable annuity, and a growing annuity. And you can see that I have snipped in sort of copied and pasted the equations from the earlier videos. And that's just to make our discussion a bit easier. So if we start with the first one, they're asking us for the present value of the below cash flows. You can see that there's an annual cash flow. So this is gonna repeat itself forever, and we should judge that cash cashflow using a 10% cost of capital, which is R. Okay, so we have the payment and we have R, and we can plug those numbers in. And so then we can say equals, we can take the payment or cashflow and we can divide it by R. And there it is. We have the present value of that perpetuity. And that makes sense because if you had a thousand in the bank account today and that bank account had a yield of 10%, then that bank account would then pay out a hundred every year forever. The second workout is very similar to the first and has virtually the same numbers, but notice that we now have a growth rate. Okay? They've given us the first year cashflow and notice they've called it first year cashflow. That's important to the equation. We've then got that cashflow growing and we've got G, and then we've got the same R. So again, let's plug those numbers into the equation and we can say equals payment or cashflow divided by now, we're just careful with our brackets. And then we put R minus G and that's it. And you can see that with the growth factored in the present value of this set of cash flows is higher. And that makes sense because the cash flows will be growing over time. So they're more valuable in today's terms.

The third workout asks us for a stable annuity. And you can see it's very common to have annuities associated with pensions because they're a common area in our personal lives where we would encounter a stable cashflow, say 50,000 per year if we're lucky, and it would have a time. So, unfortunately all good things must end and this pension is predicted to end in 20 years time from its beginning. And we're gonna judge that on a 4% discount rate. And we're being asked what would be the amount that would be needed in the pension pot to achieve that performance? Okay, so if we start building our equation, then we've got careful with your brackets 1 minus, careful with your brackets. I know I keep repeating that, but it's very easy to make errors with brackets here. 1 plus and then R.

Okay, now you can see in the diagram we've got a little N there. Now that's to the power, okay? And it wants minus N, so it's gonna be minus N. So careful with that. We've then got a close bracket divided by, and then we've got divided by R at the bottom. And then we've got times the payment.

And you can see that to achieve 50,000 a year for 20 years at 4%, you would need a huge pension pot of 679,000 pounds or dollars. And that is the present value of that annuity.

The final workout takes a look at annuities and the numbers are quite different now. And notice we've got a cashflow in year one. And again, it's careful to show it's in year one. That's what the payment wants or that PMT wants. We've then got a limited number of years, so this is an annuity as opposed to a perpetuity. We've got a discount rates that would be R within the equation to the right, but we now have a growth rate, which is G. And you can see that the equation for this one is very complex. So you know, if you ever did need to do this, you'd probably end up googling it or looking it up in your notes. And so if we carefully start, we would say the first thing is the payment and divided by open bracket, R minus G bracket. And then you can see that we've got square brackets. Now let's not use square brackets in Excel. Let's just use more round brackets. It's just the different types of brackets. Brackets to try and create clarity here. And then we've got 1 minus, we've got another set of brackets, and then interestingly, another set of brackets. So that represents the squiggly bracket and then the round bracket in the diagram to the right. And we've got 1 plus G divided by 1 plus R.

And then we just need to be careful, we need to close another set of brackets. And then finally we'd put that all to the power N. Okay, then we hit enter.

And you can see that to achieve these cash flows, which will be 90, but then growing by 2% every year, we would need to have an investment of 762.5 today.