Geometric standard deviation

[globalarbtrader]

@kut2k2 and others; see if you can help with this

http://quant.stackexchange.com/questions/25371/geometric-means-standard-deviation-and-sharpe-ratios

GAT

 

[kut2k2]

https://en.m.wikipedia.org/wiki/Pythagorean_means

Don't know where GM ~ AM - V/2 comes from but it doesn't make sense dimensionally.

HTH

 

[Simples]

@kut2k2 and others; see if you can help with this

http://quant.stackexchange.com/questions/25371/geometric-means-standard-deviation-and-sharpe-ratios
GAT

Convert x to log-space using log(x+1)=y:
log(2+1) = ~0,47712125471966243729502790325512
log(3+1) = ~0,60205999132796239042747778944899

Convert y from log-space using (10^(y))-1=x:
(10^(0,47712125471966243729502790325512))-1 = 2
(10^(0,60205999132796239042747778944899))-1 = 3

Is there anything preventing you from converting to any log-scale, do all linear operations you would otherwise naively do and then convert back?
Less headache involved and should be independent of chosen log-value.

Doing multiplications on price instead, ie. for geometric calculations, can more easily lead to overflows and errors.

 

[globalarbtrader]

Convert x to log-space using log(x+1)=y:
log(2+1) = ~0,47712125471966243729502790325512
log(3+1) = ~0,60205999132796239042747778944899

Convert y from log-space using (10^(y))-1=x:
(10^(0,47712125471966243729502790325512))-1 = 2
(10^(0,60205999132796239042747778944899))-1 = 3

Is there anything preventing you from converting to any log-scale, do all linear operations you would otherwise naively do and then convert back?
Less headache involved and should be independent of chosen log-value.

Doing multiplications on price instead, ie. for geometric calculations, can more easily lead to overflows and errors.

The issue is that you may not have access to the individual returns you're trying to calculate statistics on. For example I might say that returns are gaussian with some mean and standard deviation (or with some skew, kurtosis...); and I then want to know the approximate geometric mean and standard deviation without for example generating a bunch of random gaussian returns and then measuring the geometric moments directly.

GAT

 

[kut2k2]

Addendum:

I personally prefer

GM = exp(mean(ln(1+rt))) - 1

GS = exp(std(ln(1+rt))) - 1

 

[globalarbtrader]

https://en.m.wikipedia.org/wiki/Pythagorean_means

Don't know where GM ~ AM - V/2 comes from but it doesn't make sense dimensionally.

HTH

It's from the approximation ln(1+x) ~ x - 0.5 * x^2

GAT

 

[panzerman]

Since prices are assumed to move via Geometric Brownian Motion, the geometric mean is what you should use. Although the difference in accuracy between arithmetic, geometric, and harmonic means is small and probably will not make a difference to your trading methodology or returns.

 

[globalarbtrader]

Since prices are assumed to move via Geometric Brownian Motion, the geometric mean is what you should use. Although the difference in accuracy between arithmetic, geometric, and harmonic means is small and probably will not make a difference to your trading methodology or returns.

Yes

I don't think you've actually read my question since you haven't answered it, but I agree with you

GAT

 

[nonlinear5]

Although the difference in accuracy between arithmetic, geometric, and harmonic means is small and probably will not make a difference to your trading methodology or returns.

The difference between the geometric and arithmetic means becomes large (and one might say it becomes profound) when the returns are leveraged. If you make 70% gain in one period, followed by the 70% loss in the next period, the arithmetic mean is 0%, while the geometric mean is -28.6%. This divergence between arithmetic and geometric means becomes critical when optimizing for leverage. Specifically, arithmetic averages give you a straight line with respect to leverage (i.e. there is no preference for any particular level of leverage), while geometric averages give you a nice concave unimodal curve, with a well defined maximum.

 

[Sergio77]

Geometric mean is not an unbiased estimator of future returns. It just computes final values accurately.