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I still have problems to exactly understand the difference between geometric and arithmetic mean. I know that e.g. for returns, the arithmetic mean can be wrong (e.g. if I start with 100 $ and if my stock then goes up +10%, and then from 110 it goes down -10%, the mean of the return would be 0, however there is a loss).

So I know the problem lies in the fact, that the arithmetic mean "rewards" volatility. But why is this so. Can you explain in simple words, why the arithmetic mean is not appropriate for the example in brackets and why the geometric mean is the better measure?

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    $\begingroup$ I trust it's obvious, but the ideas of arithmetic and geometric mean transcend statistics, and certainly finance, so no general explanation can be phrased in terms of returns or volatility. So, I suggest, you should rephrase this. Either you are focusing on financial applications, and you want to know when and whether and why one measure is preferable; or you want an entirely general discussion (as your title implies). $\endgroup$ – Nick Cox Dec 10 '14 at 16:16
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e.g. if I start with 100 $ and if my stock then goes up +10%, and then from 110 it goes down -10%, the mean of the return would be 0,

This is not necessarily about arithmetic or geometric mean. This is also about simple or continuous return. Consider this:

$100(1+0.1)(1-0.1)=100(1-0.01)=99$

$100 e^{0.1}e^{-0.1}=100$

In the first case I assumed that your 10% return is a simple return. In the second case I assumed it's continuous return. Both are used a lot in finance in different situations.

Going back to the arithmetic and geometric returns, the rule of thumb is that if you're using the return for a single period forecast, then you use arithmetic return. If you're using it for multi period return, i.e. with compounding, use geometric return. This is not an absolute rule like NEwton's law of mechanics, of course

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At heart, geometric means are what you want to work with because what you get back from investment is multiplicative - if you invest $1$ for two periods, getting $(1+r_1)$ and $(1+r_2)$ you end up with the product of the two single period amounts, $(1+r_1)(1+r_2)$ (since $(1+r_1)$ is available to invest after 1 period).

The arithmetic mean would be what you would use if instead you ended up with $(1+r_1+r_2)$ (which would happen if you only reinvested $1$ after the first period, losing the compounding).

So the appropriate averages are based on products not sums.

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    $\begingroup$ It's a bit more complicated in finance. Let's say I'm calculating my overnight value-at-risk. In this case I'm interested in the distribution of overnight returns. I may attempt to estimate based on historical values. In this case the arithmetic average is more appropriate. Also, one must be careful with which definition of a return was used in the first place, simple or continuous. $\endgroup$ – Aksakal Dec 10 '14 at 16:40
  • $\begingroup$ @Aksakal Thanks, while I'm aware of at least some of these subtleties, I was attempting to convey a straightforward motivation of the central reasoning for how the geometric mean arises in this situation. It can be difficult to know the extent to which one details the "... but" part of these intuitive explanations (such as bias under some situations of what may seem the intuitive choice), if at all. It's useful to mention the existence of such issues. $\endgroup$ – Glen_b Dec 10 '14 at 21:34
  • $\begingroup$ I actually agree with your motivation. Geometric mean captures the concept of compounding in finance, i.e. accumulation of consecutive returns. Hence, it's used a lot. $\endgroup$ – Aksakal Dec 10 '14 at 21:37

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