# Arithmetic and Geometric Mean Returns

There is more than one way of both assessing mean returns and assessing risk and we are going to look at two definitions on each side. Let’s go back to our data set.

 Year USA Spain Egypt World 2004 10.7% 29.6% 126.2% 15.8% 2005 5.7% 4.9% 161.6% 11.4% 2006 15.3% 50.2% 17.1% 21.5% 2007 6.0% 24.7% 58.4% 12.2% 2008 -37.1% -40.1% -52.4% -41.8% 2009 27.1% 45.1% 39.7% 35.4% 2010 15.4% -21.1% 12.4% 13.2% 2011 2.0% -11.2% -46.9% -6.9% 2012 16.1% 4.7% 47.1% 16.8% 2013 32.6% 32.3% 8.2% 23.4% AM 9.4% 11.9% 37.2% 10.1% GM 7.6% 7.9% 21.4% 7.7%

The definition of the arithmetic mean return

To compute the arithmetic mean return you basically add up all the observations and divide that sum of observations by the number of observations. So you do that for the US, you add up those ten annual returns and you divide by ten or you do the same thing for Spain or for Egypt or for the world market, those are the numbers that you are going to get. 9.5 for the US, 11.9 for Spain, 37.2 for Egypt, and 10.1 for the world market.

What is the meaning of these numbers?

Well, the meaning of these numbers, and here is one first important idea. For you to keep in mind, is that this arithmetic mean return is something that we do not use a whole lot in finance. Most of the time when we talked about, the mean return of an asset we mean something else. But this arithmetic mean return more often than not, is actually not what we mean. I’m not saying that this is a useless number. I’m saying that when we describe the performance of an asset, this is not the relevant number that we tend to use.

What does that 9.4 percent mean?

Well 9.4%, first of all, it has the same interpretation as any average. There have been high returns, low returns, positive returns, negative returns when you average all those you get 9.4. So basically, you know, that is a number that maybe its usefulness is to compare it to the number for Spain, to compare it to the number of Egypt, to compare to the number of the world market, or any other asset that you may want to compare it to and see which one is higher, which one is lower. Of course it will be an incomplete comparison, because we need to bring risk into the equation, but one possibility, you know, in the same way, we could actually be measuring the age, of the, all the people taking these smoke, and compare it to the age of the people taking some other mock. And to make a distinction, you know, where are the younger people or where are the older people, or where are the taller people or where are the skinnier people. We could make all those comparisons, and, you know the usefulness of that comparison will depend on how interesting is the question. But that you actually post, but that arithmetic mean return is nothing but the average of everything that happened. High returns, low return, positive returns, negative returns, average all that, and you get, in the case of the US for example 9.4. percent. Now 9.4% some people were also referred to it as the return in the typical period. And again this goes back to the same idea. There have been returns that have been high or low, positive or negative in the typical average period you get a 9.4. Percent of return and that is as far as this measure, of mean return goes there’s not much more to say. So if we go back to the two definitions, just to actually have them there. It is again we look back, we take the average of the returns and that is simply what it is, and number two. Given that the returns have been high, low, positive, or negative in the typical period they’ve been at that particular number which we refer to, as the arithmetic mean return.

Definition of Geometric mean return