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LineOfBestFit
Topic Author
Posts: 60
Joined: August 3rd, 2012, 1:09 am

### Skewness & Kurtosis of Correlation?

I'm wondering if there is any standard calculation for the degree of skewness and kurtosis for the joint distribution in a correlation analysis? I seem to only come upon highly specialized approaches in my searches; I'm only looking to see if there is a plain general formula for this.

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Skewness & Kurtosis of Correlation?

For a given distribution skewness and kurtosis are different characteristics. For a sample distribution the situations looks even more uncertain. These are characteristics of third and forth moments. Correlations is the second order characteristics, ie these are independent. You probably could not find the papers which analyse their relationship for general distribution. For a say normal distribution you can calculate these characteristics and for a general case we can make a conclusion how this general distribution is close to the chosen one in terms of correlation, skewness and kurtosis.

pcaspers
Posts: 698
Joined: June 6th, 2005, 9:49 am

### Skewness & Kurtosis of Correlation?

There seem to be the notions of coskewness and cokurtosis (see wikipedia), I never worked with them though. Maybe that is in the direction what you are looking for ?

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Skewness & Kurtosis of Correlation?

QuoteOriginally posted by: outrunon my site is a blogpost about the sample correlation distribution, ..it's not clear what type of distribution (and its higher moments) you're looking for.I probably lost something in probability but as far as remember term correlation relates to two random variables. When I read "a set of correlated random number" on your site I misunderstand whether you mean set of random numbers. Nevertheless correlation relates to the second moment of the random variable if the two random variables are the same. Notions skewness and kurtosis are third and fourth moments notions. And it seems to me that second , third, and fourth moments for a general distributions do not dependable to each other.

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Skewness & Kurtosis of Correlation?

Thanks, outrun. It is clear what do you mean. But it is not quite clear how does one can specifies correlation for a given bivariate sample.On the other hand in original message it was talk about correlation which is characteristics of dependence of at least two random variables while skewness and kurtosis is characteristics of a distribution of a single random variable.

list1
Posts: 1696
Joined: July 22nd, 2015, 2:12 pm

### Skewness & Kurtosis of Correlation?

X and Y are two random variables. For me S ( X , X , Y ) is like correlation coefficient between Y and $X^2$ and do not have any relationship to skew or kurtosis for X or Y
Last edited by list1 on September 22nd, 2015, 10:00 pm, edited 1 time in total.

LineOfBestFit
Topic Author
Posts: 60
Joined: August 3rd, 2012, 1:09 am

### Skewness & Kurtosis of Correlation?

pcaspers and outrun -- thank you very much. This is what I'm looking for. Shame on me, as I have come across coskew and cokurtosis before, but these didn't come to mind right away for some reason. But these are the exact metrics that I need.In a nutshell, I am trying to produce matrices which show the coskew and cokurtosis of pairs of assets and describe the degree of diversification benefit that any individual asset may truly bring to a portfolio. Since the joint distribution between two assets, described by correlation, is assumed to be distributed normally, the use of correlation in a standard mean-variance optimization could lead to suboptimal allocations if significant nonzero coskewness and cokurtosis existed across a wide range of pairings for the given list of constituent assets involved.Again, great thanks for putting me in the right direction.

purbani
Posts: 89
Joined: July 14th, 2002, 3:00 am

### Skewness & Kurtosis of Correlation?

I think this may be what you are looking for. It is also possible to generate an analogue to the Pearson Product Moment Correlation matrix for the four moment case using those co-skewness and co-kurtosis tensor matrices however it does not scale very well obviously. See Four Moment Decomposition and Four Moment Decomposition Spreadsheet as well as Incremental and Marginal VaR for the Four Moment case for an example of use. Note however that in order to allow the portfolio Four Moment or Modified Volatility to be calculated from the Modified correlation and covariance matrices as per the normal we have had to embed both the current portfolio weights and confidence level which means it can not easily be used for optimisation purposes but is exact for risk decomposition purposes. Taking the square root of the diagonal of the Modified Covariance Matrix will recover the individual component Modified Volatilities in the same way as doing so for the 2 moment Pearson Product Moment Covariance matrix will recover the 'normal' or 2 moment vols. Other examples notably the ones in R use a simplification for computational speedup and do not recover the exact result.