# Arithmetic vs. Geometric Rate of Return. Calculations and Explanation

When you're calculating the rate of return for stock or for an index like the S&P 500 you have to be very careful to understand what type of return it is you're calculating. Are you calculating an arithmetic return or a geometric compounded annual rate of return? That matters and I'm gonna explain why in this article.

## Arithmetic Rate of Return

So let's take the arithmetic rate of return first, so let's go ahead and calculate that if we would just take 25% return for year 1, we have the rate of return 25% we add that and then we subtract out 40 for year 2, as we have a loss of 40% and then we add 30 for year 3. These are our rates of return for some stock. Let's say and then we divide that by 3 which is the number of years, and that's going to give us a rate of return of 5% if we have used the arithmetic method.

## Geometric or Compounded Rate of Return

Now if we were to use the geometric or compounded rate of return we might get a very different rate of return. So let me go ahead and I'll just calculate that, I have already written an article on Geometric or Compounded Rate of Return and if you don't quite follow it I suggest you read the article that I wrote on that. So we'll take the (1.25 ✕ 0.6  1.3) and the reason this, 0.6 might throw you off is basically when we lose 40% of our investment that's the same way as saying we're keeping 60% of it or 0.6. So we're gonna take all that but then we're going to raise that to power, we're gonna raise that to a power which is 1 divided by the number of periods and there are three years here, so we're gonna raise it to the 1/3 power. Then from that whole thing, we are gonna subtract 1. Now that is going to give us -.008. I'm just gonna round that number, it is basically -0.01 which is the same as saying -1%.

## What is the Difference Between Arithmetic and Geometric Return?

Now if you've noticed immediately in one situation when we do the arithmetic rate of return we actually have a positive 5% but when we use the geometric or compounded rate of return it's actually negative. It's not just a different rate of return it's actually a negative number because we've lost money and you might be wondering how could this be. Well, basically the arithmetic rate of return does not consider volatility. It doesn't take into consideration the fact, it's just averaging these three rates of return but you have to remember for example after year 2 when you've had that negative 40% that's affecting the balance. When you're using the arithmetic return it's not really considering any of the volatility and how the balance that earning, whatever the rate is it's not considering how that balance changes over time. It's just taking basically the average rate of return so that's really what's driving the difference.

## When should the Arithmetic Rate of Return be Used?

So you might wonder then why would we ever use the arithmetic rate of return? Well, we want to use the arithmetic rate of return when we're looking at future performance. When we're trying to say "What do we think is going to be the expected rate of return for this particular stock or this particular portfolio of assets?" When you're looking you know forward-looking information you're trying to compute an expected return, then it's actually not going to be that problematic to use the arithmetic rate of return.

Now when you're looking backward, when you're looking at historical information so let's say you're talking about a mutual fund and you want to look at that mutual fund's performance over time then you want to be using the compounded annual rate of return. You might even see something in your mutual fund prospectus it says something like your compounded annual growth rate (CAGR) etc and that's this geometric rate of return. You can call it geometric or call it compounded but that's the idea is, this is understanding that because of volatility the balance that is earning a rate of return is going to change over time, it changes from year 1 to year 2 from year 2 to year 3. The balance is changing and so you're having a bigger or smaller balance the next year that's earning that rate of return. So if you really want to evaluate a mutual fund or some stock or you know that's SP 500 and you want to look at historical information then you really need to be looking at this geometric compounded rate of return rather than the arithmetical.