An average is given to a series of numbers for us to attempt to make order from chaos. There are several ways to calculate an average, though the term is most commonly associated now with the arithmetic mean. This mean is calculated as the sum of numbers divided by the number of numbers in a series. Other forms of average are the mode and median. When it comes to investments averages become more sophisticated, taking into account both the pattern of the numbers, and how adding (and withdrawing) from the investment at certain times will impact the return.

This post will explore averages, from the arithmetic mean, to the geometric mean (time weighted return) and the money weighted return.

## Calculating a simple return

An investment of $10,000 is worth $14,000 after two years, what is the average return?

- $14,000-10,000=$4,000/2= $2,000
- $2,000/$10,000 = 20%

So you could say that the average return is 20%, the ‘average’ you are using here is arithmetic mean.

However, we need averages for variable annual returns, and the arithmetic mean is not the best choice for forecasting:

## Calculating average return from variable annual returns

Investing $10,000 at the start of year 1

- Year 0 = $10,000
- Year 1 = $12,500
- Year 2 = $14,000

We use Year zero to indicate the value when the investment starts, after 12 months have passed we reach our year 1 return.

- Year 1 growth to $12500 = 25% return (12500-10000)/10000=25%
- Year 2 growth to $14,000 = (14000-12500)/12500=12%

### The Arithmetic Mean

The arithmetic mean of these two number is 18.5%. If you invested $10,000 at a fixed 18.5% your investment would be worth:

- Year 0 = $10,000
- Year 1 = $11,850
- Year 2 = $14,042

The additional $42 doesn’t seem like much, but over a period of time the ‘creep’ compounds and pushes estimates off target. To address this we use the geometric mean.

### The Geometric Mean

To calculate the geometric mean we then apply the following calculation to the returns: [Year 1 return*Year 2 return] √2. Note that the divisor is the number of periods, so if we were looking at **5** years it would be (Y1*Y2*Y3*Y4*Y5)√**5**. Back to our example:

(1.25)*(1.12)√2 = 18.32%

Using this number, we get the following progression

- Year 0 = $10,000
- Year 1 = $11,832
- Year 2 = $13999.62

As you can see, this is a very close estimate to the actual $14,000. The geometric mean is also called the time weighted return, and it is the number used to describe the performance of the asset itself over time. However, this too has limitations, which is why we also have the money weighted return.

## The Money Weighted Return

The money weighted return studies cashflows to and from the investment. IE it looks at the investment in relation to how you invest in it. If we were to look at that same time weighted example, while it ‘averaged’ an 18.32% return from the geometric mean, and 18.5% from the arithmetic mean, it still earned 25% in year 1 and 12% in year 2.

With that in mind, if you had put money into the investment in different ways, you would have seen a different result. Examples:

So, what you can see here is that you can average a return with some accuracy by using the geometric mean, which gives an average for the investment, but that same investment can offer a different return to different people depending on how and when they invest.

## The role of Standard Deviation

Averages without standard deviation can be misleading. If we know the standard deviation, we are able to make a risk assessment more accurately. If we look at arithmetic mean, we can see this in action:

So while both Investment 1 and Investment 2 share a common ‘average’ there is a lot more to consider about the two when you consider the dispersion around that average. Investment 1 here is reflective of a fixed income product, such as a bond, whereas Investment 2 is a good reflection of a highly volatile equity, likely a single stock. The standard deviation introduces risk when we consider the money weighted return, because if we buy in after Year 1 in the above example, the average return would be 1 ( or 0.73% geometrically). This concept of risk and reward from average returns is the essence of modern portfolio theory.