In this session, we're going to be looking at yields. So let's go straight in or what is a yield? Imagine I've got the present value and the future value of some cash flows. This means I know how much I'm going to invest today, the present value, and I know how much I'm going to get back in the future, the future value.

I also know how long we're going to have to wait between the present value and the future value. And this is N or the number of years. What I don't know is the percentage return that I'm going to earn between the two, and I want to try and calculate that percentage return. There's a formula that we can use here, and it's a rearrangement of the present value formula. Let's have a read through it. R, which is our rate of return or yield equals the future value divided by the present value, all in brackets to the power 1 over N, and that 1 over N also needs to be in little brackets, otherwise it'll all go wrong. Then it's very easy to forget the minus 1 at the end, and that minus 1 needs to be outside of all of those brackets. Otherwise it will be inside the powering and everything will go wrong. So just be careful if you are using a calculator or Excel bracket work is very important here and accuracy with your brackets.

Now what do all these things mean? Remember, future value and present value N being the number of years an I and R, both ways of referring to the yield or interest. These are the terms that you'll need to be able to calculate the yield.

Let's look through a question together. If 100 is invested now and 200 will be returned in 10 years, what's the yearly return or yield? Let's try and put those figures into our diagram. I know that my present value is a hundred and I know my future value is 200. I also know that N is going to be 10. When we put those figures into the equation, then it will read 200, which is the future value divided by 100, the present value, those need to be in brackets.

Then everything in the brackets will be to the power of 1 over 10 being 1 over N and that 1 over 10 needs some little brackets of its own.

And then right at the end we minus 1 and that minus 1 needs to be outside all the brackets. And if you plug that into your calculator or Excel properly, 7.2% will come out. And the meaning of that is that the yield or IRR as we'll see later, 7.2% on an annual basis. Let's move on to the IRR, otherwise known as the internal rate of return. When we were talking about yield earlier, we were talking about two figures, so the present value and the future value separated by time. And then we were finding the implied growth between them, which we called the yield that could use a rearranged version of the present value formula that we saw in another module.

When you've got two cash flows, that's exactly what you can do and you can keep it quite simple. But when you've got multiple cash flows, which is very common in a project or in evaluation and lots of other work that financial professionals might do, it can be very difficult to calculate your return or your yields using the method that we used earlier. If we look at a quick example, I've invested 124.3 here and then I receive 50, 50 and 50 back over the ensuing three years. And the question is, what's my rate of return? This is very difficult, but there is a way to get through it. What we can do is try and calculate the present value of these three 50s iteratively. And that means to go round many times and effectively trial and error. So if we try and calculate the present value first using a discount rate of 8%, we'll get a present value of the three 50s of 128.9. That's gone a little bit too far because it doesn't match the 124.3, which was the input or the present value.

The IRR will be reached when the present value of the inflows, the three 50s matches the present value of the outflow.

And because the outflow is at times 0, the present value, the outflow is just 124.3. Now, if I move on and do the next iteration and I try a discount rate of 9% at a discount rate of 9%, we get a present value of the three 50s of 126.6, which is close but still not close enough. So if I move the discount rate on to 10%, excellent, the present value is 124.3.

What this means is I've invested 124.3, we've got a return of 50, 50 and 50 on time, one, two, and three. But if you subject these to a discount factor of 10% and the present value of those three 50s gets you back to the same starting point 124.3, this means the rate of return has been 10%. The yield of this project could be said to be 10%.

Now we'll see that Excel has a way to speed up that iteration, But kind of under the hood. What Excel is doing is that iterative process we've just explored.

In these three workouts we're going to be looking at the yield or IRR, and we're going to be looking at it in two main ways. Firstly, we'll look at a sort of single cash flow growing and that's what workouts one and two take a look at and they explore some different language and ideas. And then workout three is a bit more developed and will get us exploring an Excel shortcut. And what it does is it looks at more complicated projects, especially cash stream C, where we have what might be called lumpy cash flows, which are going up and down and aren't just a single figure.

If we start with workout one, you can see that what we've got is a present value so that's the value today of 55 and we've got a future value of 100. And so you can see the money has just short of doubled, but you can't just say, well, it's something like a 100% return. And that's because the return needs to be on an annual basis and this has taken 10 years.

So what we need to do is we need to work to the discount rate. And effectively what we're doing is reversing the discount rate equation or compounding equation depending on how you think about it. What we do is we say, what is the relationship between the ending point and the beginning point? And you can see that in there is that kind of a 100% or thereabouts growth that I talked about earlier.

But we then need to reflect the fact that that happened not just in one leap that happened in 10 leaps. And so you end up with a 1 over N relationship there and you just need to be really careful with your brackets. Now if I press enter there, you'll see that we've got a very high return and this is a very common error that people make, myself included. It's very easy just to forget that minus 1 or to perhaps have the minus 1 by accident within one of those brackets. So just be careful with that minus 1. And you can see that what we've got there is a nice 6.2% and that reflects the fact that it took 10 years to get from that one figure to the other. Now, workout two really just does the same thing but explores language a bit. So rather than calling it present value, we're now calling it the investment. And rather than calling it future value, we're calling it say the payout or the final amount. And then rather than calling it an investment horizon, we're calling it an investment duration. Okay? Now these are really the same ideas. You've got present value, future value an N, and although it says here, annual return could be called a discount rate, so we're looking for R or I. And so we're doing the same thing. We're saying careful with our brackets, where do we end up? What's the relationship with the starting point? How long did it take us to get there? And then remembering that minus 1. So here, although the relationship is colossal on the face of it, because it takes so long to get from the one to the other, the annual return isn't quite as spectacular as you might imagine it to be.

If we move down to out three, you can see that this one's more complex. We have three projects, A, B, and C. Okay? And these are three distinct projects with different cash flows. And then these projects also have more interesting and complex timing. So you can see that each of the projects has what you might call a purchase price or investment of 100. And then all the projects have some sort of payoff, be it in year four or in the case of C kind of spread out. And then one of the projects has a bit of a top up investment in year three, or maybe that's some sort of decommissioning or cleanup. Okay? And so project C much more interesting in terms of cash flows and it would be very, very difficult to use this above method to work out the yields of these three projects. So we might worry and say, oh no, how are we gonna do this? But it's okay Excel actually has a built-in iterative tool. It's got a formula equals IRR, which we'll use now. So if we start using that formula, we would say equals IRR, we would open the bracket and you can see it's suggesting things at the top. It might be quite small for you, but you can see values and we're gonna point it at the values within the project from time 0 onwards. Okay? It needs 0 there rather than gaps. It's not very good at looking at gaps. So just be careful with gaps, make sure there's 0s. And then what it's gonna do is interpret each one of those columns as a year.

Now there is, you can see another term called guess, but we are not gonna fill that in. We don't need it. We can just hit it, enter. And you can see that Excel has really nicely come up with an IRR or rate of return here of 18.92%. And what I can do is I can take that whole block and I can copy it down because if we look, it will now pick up the cash streams from B and the cash streams from C quite nicely. And if we take a look at it, you can see that they're wildly variable.

Probably the deciding factor for cash stream C is that massive payoff, which is very early in the project of 500. And that means it's got a huge return.