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# Calculate Your Return: You May Do It Wrong

Today it’s about something very fundamental when it comes to investing money:

How to calculate the average return of your investment correctly. There is a small stumbling block here that you better jump over.

content

• Arithmetic vs. geometric mean
• The arithmetic mean is a blender when it comes to returns
• Calculate the return using the geometric mean
• Summary
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## Arithmetic vs. geometric mean

You certainly know the simple (arithmetic) average. You simply add all the values ​​in a sample and divide by the number of values. That’s pretty trivial:

Sample: 10, 15, 20, 5

Total: 50

Number of values: 4

simple average: 12.5

So far, all right, and you may be thinking, “What is he up to?” Well, let’s look at another example. This time, the annual, percentage gains of the DAX. So simply, what percentage has lost or won the German leading index in a year.

year DAX change % Change
2005 5,408.26
2006 6,596.92 1,188.66 21.98
2007 8,067.32 1,470.40 22.29
2008 4,810.20 -3,257.12 -40.37
2009 5,957.43 1,147.23 23.85
2010 6,914.19 956.76 16,06
2011 5,898.35 -1,015.84 -14.69
2012 7,612.39 1,714.04 29.06
2013 9,552.16 1,939.77 25.48
2014 9,805.55 253.39 2.65
2015 10743.01 937.46 9.56
total 95.87
Number of values 10

So, if we calculate the simple average, we get:

95.87 / 10 = 9.587 or 9.587%

Okay, on average the DAX has risen by 9.587% pa in the last 10 years. If you do that maybe for the last 20 years, then we have an average that allows us to estimate what the stock market is like.

Not correct!

Why is that wrong? Well, calculate it, what does an annual average rate of return with the above average over 10 years mean?

This corresponds to 1.09587 * 1.09587 * 1.09587 * 1.09587 * 1.09587 * 1.09587 * 1.09587 * 1.09587 * 1.09587 * 1.09587 = 1.09587 ^ 10 = 2, 5

Starting from the starting value 2015 (= final value 2014), which is at 5408.26, the DAX would now have to be 2.5 * 5408.26 = 13,520 .

At the time of this article, the DAX has never been at this value, so somewhere must be a mistake in logic.

One of the reasons lies with an old acquaintance: the compound interest. Because of this effect, it is only possible to generate considerable assets in a lifetime. It leads to a non-linear, exponential development. In other words, a euro you deposit is worth more than a euro. Sounds funny, but it is.

However, in most cases this non-linear behavior can not be correctly represented with a simple average.

### The arithmetic mean is a blender when it comes to returns

Here is an example to illustrate why the arithmetic mean is not suitable for you as an investor. Look at the following sequences:

Year 1: + 16%

Year 2: -16%

Year 3: 0%

In simple terms, nothing would have happened here, because this is 0 . What would an investor have experienced in these three years? Let’s assume it is the gains of a stock, which at the beginning (year 0 of if you will) would have been worth € 100. So the development would look like:

Year 1: € 116 (+ 16%)

Year 2: € 97.44 (-16%)

Year 3: € 97.44 (0%)

Right here lies the stumbling block at the arithmetic mean. It just does not work when it comes to returns.

Because actually you would have made loss here and not 0%.

Here’s another example with other numbers:

Year 1: 8%

Year 2: 5%

Year 3: 8%

If we calculate the simple average return, then we come to 7%. Let’s compare, for example, how would a stock with an initial value of € 100 hit the arithmetic averaged rate of return and the actual returns achieved.

year actual development arithmetic mean
1 € 108 € 107
2 € 113,40 € 114.49
3 € 121.34 € 122,50

As you can easily see, if you estimate with the simple average how the return would develop, then you systematically overestimate.

So how could you do it better?

### Calculate the return using the geometric mean

In simple terms, the geometric mean takes into account compound interest. Take the previous example: From € 100 at the beginning € 121.31 have become after three years. Overall, this equates to a return of (121.31 / 100) = 1.2134 therefore 0.2134 or even 21.34%. We are now distributing this return over three years so that, taking interest on interest rates into account, we will reach € 121.34. This is done with the following formula:

∛ (1.2134) = 1.0666 therefore 6.66%

Believe it or not, the representation of root signs in the web is quite limited and not quite correct at the top, so I’ll formulate the formula briefly in words: The third root of 1.2134.

Short check whether this is true: 1.0666 ^ 3 = 1.2134. So it’s true

Generally speaking, you simply pull the nth root out of your total return . N stands for the number of years that have passed from beginning to end. In our example, as I said, it took three years to get from € 100 to € 121.34.

Earlier we had the small table with the values ​​of the DAX over 10 years where we had found that the arithmetic mean was too optimistic or simply wrong.

The initial value of the series was 5,408.26 at the end of 2005 and the final value was 10,743.01 in 2015. The average geometric return is calculated as follows:

10 root out of 10743.01 / 5408.26 = 10 root out of 1.9864 = 1.071 ⇔ 7.1% pa on average

Short check calculation: 1,071 ^ 10 = 1,9856 fits. Attention, the values ​​are rounded to the fourth decimal place.

Here you will find the whole thing shown again in a graphic:

We note: the arithmetic mean was 9.587% pa , thus mercilessly overestimating the DAX. That was unavoidable, as we have seen, because that is the nature of the arithmetic return. The geometric mean, however, was 7.1% pa and correctly considered the compound interest effect.

On the topic average / geometric return, I have also picked out an interesting video.

Here is another general explanation of the geometric return:

Summary

For us as an investor, it is of course interesting to calculate the average return over several years. We want to know how our investments have developed over the long term, year after year. It is important to use the correct mean and that is the so-called geometric mean . Because this takes into account the non-linear development of returns.

The formula for this is: nth root of end assets / initial assets where n stands for the number of years.

With this tool and a twofold try it is easy for you to easily turn around this cliff of investing.