Geometric Mean Returns and Portfolio Rebalancing
Summary
The document asks whether the geometric mean is an appropriate way to average returns from different indices during one period. One answer connects it to an equally weighted portfolio that is continuously rebalanced: gains from stronger performers are repeatedly allocated to weaker performers, producing a geometric-mean-like result in the limiting case. Another answer stresses that an investable portfolio return requires explicit weights; with weights, calculate the weighted return for each period, then analyze that portfolio’s return series over time.
The example illustrates the calculation but does not establish that an unweighted geometric mean is a general measure of index performance. Without a specified allocation and rebalancing rule, averaging separate indices is a mathematical construction rather than a uniquely defined portfolio return. The discussion offers intuition, not a formal derivation or empirical comparison, so the method’s relevance depends on the portfolio design being modeled.
Key ideas
- A portfolio’s return for a period depends on the weights assigned to its constituent assets.
- An equally weighted portfolio that is continuously rebalanced is associated with a geometric-mean-like return in the limiting case.
- A portfolio return series can be constructed from weighted constituent returns and then analyzed over time.
- An unweighted geometric average does not by itself define an investable portfolio.
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Full text
# Use of geometric mean for average return of several indices
# 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
```
> Answer: 1.95711 %
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%
## Answer by not.so.quanty (score 3, accepted)
https://quant.stackexchange.com/a/10601
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.
## Answer by Taran (score 0)
https://quant.stackexchange.com/a/14121
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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.