# Why Arithmetic Average Fails to Measure Average Percentage Return over Time

## Arithmetic average: the basic tool in statistics

Arithmetic average is a good tool for measuring central tendency of data sets which represent independent values and values taken at one point of time, for example when you’re calculating average return of a number of stocks in a given time period (see arithmetic average calculation example).

## Calculating average return over multiple time periods

On the other hand, when you have a set of data which has a time-series nature and is expressed in percentages, arithmetic average has weaknesses. The classic example is calculating average return of a stock over several periods of time. Let’s see how and why arithmetic average fails here.

## A stock investment example and arithmetic average

Let’s say you have bought a stock. Your stock performed like this in the last two years:

• Year 1: +70%
• Year 2: -50%

What is the arithmetic average of the stock’s return? The sum of +70% and -50% is +20%, divided by 2 gives us +10%. This looks like a solid return.

## Arithmetic average fails to discover the loss

How much money have you made in reality? If you have invested a million dollars, at the end of year 1 your position was worth 1.7 million. In the second year, its value has halved, and you end up at 0.85 million – less than what you’ve invested. You have lost money. The arithmetic average is misleading.

## Why is arithmetic average not working?

When you have stock performance expressed in percent, the percentage advance or decline in every particular year is measured relative to the stock’s price at the beginning of that year – which is different for each year. You are comparing incomparable numbers. 50% of 1.7 million is greater than 70% of 1 million, though it doesn’t seem so looking at the percentages only.

## Geometric average is better for averaging performance over time

When you are investing, dollars or euros matter more than percentages in the end. Whenever you have time series kind of data expressed as percentages (which is a very frequent case in finance and investing), arithmetic average is misleading. Geometric average is much more suitable here, as it automatically takes the different starting values in every single year into consideration.

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