Reducing Return Volatility While Preserving the Median
Summary
The document asks how to reduce the standard deviation of a negatively skewed, high-kurtosis return series while keeping its mean, median, skew, and kurtosis approximately unchanged. It explains why simply scaling observations around the mean cannot satisfy those constraints: scaling changes the distance between the mean and median, so preserving both locations while changing dispersion is not possible with a linear transform.
The responses suggest nonlinear transformations or moment matching for some univariate distributions, but caution that preserving higher moments can distort the empirical distribution and may not be meaningful. For multivariate settings, the problem is harder; entropy pooling is offered as a more general way to impose views on median and variance. Any such adjustment should be checked for an effective scenario count that is too small. The discussion gives conceptual guidance rather than a tested transformation, and challenges whether preserving both mean and median is necessary; a single robust location measure may be more appropriate.
Key ideas
- Linear rescaling cannot change dispersion while preserving both the mean and median when they differ.
- Nonlinear transforms can target location and moment constraints, but may distort the empirical distribution.
- Moment matching may be practical for some univariate distributions, while multivariate adjustments are more difficult.
- Entropy pooling can impose views on median and variance, with scenario effectiveness checked afterward.
- Consider whether one robust location statistic is sufficient instead of preserving both mean and median.
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Full text
# Transformation to reduce standard deviation without changing median
# Transformation to reduce standard deviation without changing median
Consider some negative skew and high kurtosis return time-series $X_t$. I do not know the functional form of the pdf of $X_t$ and have about 150,000 data points.
Suppose that I was to create an adjusted $X_t$ called $X_t^a$ where the mean, median, skew and kurtosis remain approximately the same as the original series, and the only main difference is that $\sigma (X_t) = 2*\sigma (X_t^a)$.
What's an appropriate transformation? I've tried:
$$ X_t^a := (X_t - E[X_t])*0.5\frac{\sigma(X_t)}{\sigma(X_t)} + E[X_t] \\ = Z*0.5\sigma(X_t) + E[X_t] \sim N(E[X_t],(0.5\sigma(X_t))^2)$$ $$Z \sim N(0,1)$$
but this severely reduces the positive median towards zero which is not what I want (I want to keep $\text{median}(X_t)$ approximately equal to $\text{median}(X_t^a)$).
## Answer by Quartz (score 5)
https://quant.stackexchange.com/a/7819
It's not possible with a simple linear transformation like the one you mentioned: since scale and thus the distance between mean and median are required to change, either the mean or the median will not be preserved. Therefore you must use nonlinear transformations, which will complicate quite a bit mantaining skew and kurtosis and imho will not be meaningful anyway (the distortions needed to mantain the statistics will do more harm than good to the empirical PDF).
Sorry if I suggest to challenge the requirements: why do you want both mean and median preserved? These statistics do not have much intrinsic value other than providing estimates of location (but not the same location though: even with full PDF knowledge they´d be different! yet one might still convert one of them to the other kind), forcing such a relationship between the two sounds like a recipe for trouble. An alternative would be to define a location statistic as some average of the two and mantain that constant in a simple linear transform. Or even better using a more robust location statistic in place of both.
## Answer by John (score 2)
https://quant.stackexchange.com/a/7834
As Quartz says it is possible to make non-linear transformations taking into account skew and kurtosis, but this is mostly is limited to univariate processes (one approach for a t distribution is to match moments). For multivariate processes, it is considerably more difficult. A more general solution is to rely on Entropy Pooling. You could take views on both the median and the variance of the process in a univariate or multivariate setting. Be careful to check the effective number of scenarios afterward in case you have inadvertently taken an extreme view (unlikely if fixing the median and reducing the standard deviation, but possible if making the standard deviation sufficiently large).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.