Combining time series - arithmetic, geometric or logarithmic mean

[Pehr]

My question regards combining several time series into just one series. I have four commodity time series (monthly nominal returns of ags, precious, industrials and energy) and want to combine these (equally weighted) into just one commodity time series. What is the best way of doing this? Either you can just take an average (arithmetic mean) of all four series, or you can take the geometric or the logarithmic mean. What is to prefer? Advantages/disadvatages with the three ways of computing the mean?

Now the series are combined by arithmetic mean and the problem that arises is that when I'm doing portfolio optimization (by markowitz mean-variance optimization) I sometimes can get portfolios above the frontier (over optimal portfolios). This I think is due to that for the calculation of the efficient frontier geometric mean is used. I guess I need to have some consistensy regarding this. What is the "correct" way is of adding these kind of time series together and why is that alternative to prefer over the others?

Another question I have is very related to this. If I want to calculate an annualized return (combining 12 monthly returns), what mean should I then use? I guess geometric mean would be correct for doing this. As of now I calculate (in excel) a one year return as (GEOMEAN(A1:A1000)^12) - 1, where A1:A1000 would be the entire series.

Thank you in advance!

 

[Alexander Krosky]

Why would you want to average them to do a mean-variance optimization? The entire point of it is to get you the weights of the different assets in your optimal portfolio. Something isn't right here.

For the mean-variance optimizer you want to be using simple returns of the individual assets. For visualizing the performance of the assets, you want to be looking at the cumsum of the returns (equity line), not the price series.

 

[Tsotne]

I have played with MPT thousand times up till now and never touched arithmetic average. What exactly it is - one doomed measure of expectation for equally weighted variables. Use Geometric mean instead wherever needed. I'll upload the .xls file containing Markowitz portfolio later. (sine I see you are using excel, possibly solver for optimization)

 

[Pehr]

Why would you want to average them to do a mean-variance optimization? The entire point of it is to get you the weights of the different assets in your optimal portfolio. Something isn't right here.

For the mean-variance optimizer you want to be using simple returns of the individual assets. For visualizing the performance of the assets, you want to be looking at the cumsum of the returns (equity line), not the price series.

Well, because I want my portfolio to consist of Equities, Commodities, T-bills, Bonds and Credit. I don't want any subindecies in the presentation of my portfolio.

I have the monthly returns for the asset classes in geometric form. I then want to compute the annualized return, and there is a few ways of doing this. Either I take the geometric mean of this series, i.e. ((1+GEOMEAN)^12)-1 in Excel. Or I can take the logarithms of the returns and then take the arithmetic mean (AVERAGE function in Excel) and multiply with 12. The latter is how it is done now in the optimization program. What is the correct way of doing this?

Also, if I want to calculate the standard deviation, how should I do? Now it is calculated by taking the STDEV function in Excel (which is arithmetic standard deviation) of the logarithm of the returns and multiply by SQRT(12). Another way is to compute the geometric standard deviation of the return series. This is done by the commands (10^STDEV(LOG("time series")-1)*SQRT(12). These two methods will give me roughly the same results but still there will be a difference. What is to prefer for computing standard deviation?

I have played with MPT thousand times up till now and never touched arithmetic average. What exactly it is - one doomed measure of expectation for equally weighted variables. Use Geometric mean instead wherever needed. I'll upload the .xls file containing Markowitz portfolio later. (sine I see you are using excel, possibly solver for optimization)

Please do upload the file, I would appreciate that.
Also, I would like you to look at the questions I wrote to Alex Krause.

 

[Tsotne]

I'll be at home in 5 hours and I'll upload immediately.

 

[Alexander Krosky]

In that case I would recommend using volume-weighted arithmetic average of the returns if you have the volume data. Anything else is going to be ignoring volume and that's very bad.

 

[Pehr]

In that case I would recommend using volume-weighted arithmetic average of the returns if you have the volume data. Anything else is going to be ignoring volume and that's very bad.

What is volume-weighted aritmetic average? Volume data?
Can you elaborate a bit more around this? How would I in my case go about to implement this?
Why is it bad to ignore volume?

 

[Alexander Krosky]

In your particular case, you're trying to construct an index which represents the asset class. You need to normalize by volume, market cap, or some similar metric. Whichever securities you're looking at within each asset class have totally different volumes. If you were constructing an equities portfolio of say Apple, Google, and Lebanon Citizens National Bank, does it seem right to assign all of them equal weighting?

On a fundamental level, you can't assign equal weighting because it's unrealistic for actual trading. You can jam a lot of shares into Apple and Google, but not Lebanon Citizens National Bank. Therefore its inflated weighting in your portfolio is entirely unrealistic.

What is the granularity of your data? You just get the volume data which corresponds to it. The return for each security is multiplied by its volume and divided by the sum of all of the securities multiplied by their volumes to normalize to one.

 

[Pehr]

In your particular case, you're trying to construct an index which represents the asset class. You need to normalize by volume, market cap, or some similar metric. Whichever securities you're looking at within each asset class have totally different volumes. If you were constructing an equities portfolio of say Apple, Google, and Lebanon Citizens National Bank, does it seem right to assign all of them equal weighting?

On a fundamental level, you can't assign equal weighting because it's unrealistic for actual trading. You can jam a lot of shares into Apple and Google, but not Lebanon Citizens National Bank. Therefore its inflated weighting in your portfolio is entirely unrealistic.

What is the granularity of your data? You just get the volume data which corresponds to it. The return for each security is multiplied by its volume and divided by the sum of all of the securities multiplied by their volumes to normalize to one.

Okay, thank you for your input. I will have that in mind while constructing the index.

Do you have any input regarding my questions for the methods to calculate mean for the annualized return and standard deviation of the returns?

 

[Tsotne]

I'll recall the exact mean calculation formula I had done somewhere in macro...

 

[Pehr]

I'll recall the exact mean calculation formula I had done somewhere in macro...

I can't seem to find it. Exactly where in the sheets/code can I find the calculation formula for the mean?

 

[Tsotne]

I can't seem to find it. Exactly where in the sheets/code can I find the calculation formula for the mean?

Ive been using Geometric mean there. I shall find the sheet I have done them in. Before, use the simple geomean, that's the correct way...

http://www.effisols.com/basics/MVO.htm

http://www.aaii.com/computerized-investing/article/mean-variance-optimization.mobile

Both in here geomean are used and explained, it should be useful.