# Use of geometric mean for average return of several indices

Can anyone give any reference for using the geometric mean to average the returns from several indices? Note, this question is not about the usual use of geometric mean to obtain the average return from a single time series. It is about averaging several indices in a single time step, so for example :-

January 2014
index 1 return = 3%
index 2 return = 5%
index 3 return = -2%


E.g.

returns = {3, 5, -2};
meanreturn = (GeometricMean[returns/100. + 1] - 1)*100


Edit

My current thought as to why the geometric mean might be used to average returns in a single time period is that it produces a lower result than the arithmetic mean, so for generally positive returns with a leptokurtic bias (shown red c/w blue normal dist.) the geometric mean would damp out the contribution of the higher returns. This seems a bit of a kluge though; any references welcome.

Averages generated from randomly generated distributed returns around a value of 1%

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why don't you just apply the geometrix mean formula directly ? – Probilitator Mar 14 '14 at 13:07
@Probilitator - I'm not sure what you mean; I have added an example though. – Chris Degnen Mar 14 '14 at 13:38
so why don't you just use that formula ? would you like to know whether such an approach is valid ? – Probilitator Mar 14 '14 at 13:48
Yes, that's the thing. I have seen this method used and I am trying to find the rationale. (I understand clearly geometric mean being used for time series.) – Chris Degnen Mar 14 '14 at 14:00
@ChrisDegnen What you're trying to do doesn't make much sense to me. You don't need a geometric average to get an average of returns cross-sectionally. Arithmetic average is fine. Weighted averages (based on market-cap or something) are also common. – John Mar 14 '14 at 14:22

I can offer an intuitive answer.

The limit when your equally weighted portfolio is continuously rebalanced will give you the geometric mean.

This is because the excess return of the better performing strategies will be allocated towards the least performing strategies, compounding high returns with low returns.

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Thanks, I'll look into it. – Chris Degnen Mar 17 '14 at 14:11
Yes, that was exactly the reason. Thanks. – Chris Degnen Mar 2 '15 at 16:14

You need the weights for each index to compute the portfolio return time series. The portfolio return would be the weighted average of the returns at each time step. Once you have a single time series of portfolio returns you can compute the the statistics.

Averaging the returns without any weights would be a purely mathematical exercise at this point. By computing geometric mean of three indices you are allocation capital to a exotic asset which needs to be structured by some BB structuring desk. Probably some banks have structured these products.

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