Fundamentals of Returns

Fundamentals of Returns

I think most investors probably want to start by looking at returns. What exactly is a return, and how do we characterize returns? Take a look at this chart.

This is a set of 12 monthly returns for two assets. There’s the blue asset, and there’s the orange asset. Now clearly, they’re both different in some way. The blue asset is a lot less volatile, and you can see that it tends to have less variation, and the orange asset is a lot more volatile. But what’s interesting is both of these assets have exactly the same monthly return of 1%, so we’re looking at 12 returns, which means 12 months, and it’s, 1% is the average return of both of these assets. But clearly, they are not behaving the same. So right away, we should be able to tell that average returns are not a good way to look at how an asset behaves. The average return can, as we’ll see, be quite misleading. And one way of trying to understand the difference between these is by looking at what would happen if you invested in these two assets. So if you look at this chart, what happens is, I’ve put $1,000 in both of these assets at the start of the year, and I look at how it’s done during the course of the year.

And you can see the blue asset sort of chugs along and does fairly well, and the orange asset is all over the place. And for a good part of the year, It looks like the orange asset might have been the smart thing to be in, but it actually ends the year lower. Now that’s interesting because remember that I told you, the average return of both of these assets is exactly the same. So the first thing that you should take away from this is, just because the average return, the average monthly return, is the same, doesn’t mean that you’re going to end up with the same amount of money. In fact, at the end of month 12, you have actually different values. So what we’re going to do is really try and understand how you characterize returns, and what is a good way of thinking about what the returns of an asset are. So let’s start at the very, very basics, which is, how do you compute the return?

The return on the asset is nothing more than the difference in the prices. That is Pt+1- Pt. That’s, let’s call it the profit or the loss. Divided by PT as a percentage, right? So let’s take an example. Let’s say you bought a stock at $10, you sold it at $12. Well, what’s the difference between the two? It’s 12- 10, that’s your profit, divided by your cost, which is $10. So 2 divided by 10, that’s 20%. Now, one thing I want to say here is that you can think of the profit as 20%, but there’s also this format that you will keep coming across which I just call 1 plus r format. And this 1 plus r format is, instead of doing Pt+1- Pt divided by Pt, you can just think of it as a ratio of Pt+1 to Pt. So, it’s Pt+1 divided by Pt and in this example of 12 being Pt+1 and 10 being Pt, it’s the ratio would be 12 divided by 10, which is 1.2. So the difference is just 1 more than this colloquial form of return, which is 20%. The reason I want you to start getting used to this Pt+1 divided by Pt or the 1 plus r type format is that when we start doing computational work, it’s really very valuable and very convenient to look at these returns in terms of 1 plus R. And very quickly in your head, you should start looking at these as very equivalent ways of describing return.

Multi-Period Return Calculation

Now let’s look at the multi-period return. So if you have two time periods, so let’s say T0 to T1 and T1 to T2. You have a return for each time period, the question is, what is the return over the combined time period?

So if you have R1 as the return between, say, T0 to T1, and if you have R2 as the return, say, between T1 and T2, if you think of it in 1 plus r format is very easy to look at the compounded return. Because what you have is, you just multiply the returns, so it’s 1+R1 times 1+R2. That gives you the total return over the, compounded return over that period, that’s also in 1 plus r format. So you just have to remember to subtract 1. So in general, let’s say you’re trying to compound it over two periods, the compounded return over two periods is (1+R1) times (1+R2)- 1.

How do you compare monthly return with, say, quarterly return or daily return? And the answer is a process called annualization, and we are going to use annualized returns for almost everything we do when we start reporting numbers, so it’s important to understand how that’s computed. The annualized return is nothing more than the return you would get if the return that you’re looking at had continued for a year. So let’s say you had a monthly return of say 1%, right? What is the return you would get over a year? Well, it’s tempting perhaps to say, well, it’s 1% per year times 12, so it’s 12 percent, but that’s not right. Why, because you have to do this 1 plus r thing, so it is 1.01 because it’s 1% per month. So 1 plus 0.01 is 1.01. Multiplied by 1.01, multiplied by 1.01, and so forth 12 times, so that’s 1.01 to the power of 12, and then subtract 1 from that, and that results in 12.68 percent, not 12%