I want to calculate the skewness and the excess kurtosis (third and fourth moment) of a (trading rule) return distribution. To calculate the skewness coefficient I'm using the quantile-based skewness measure of Hinkley (1975):

Skewness measure

Now I'm trying to find a robust quantile-based kurtosis measure (for asymmetric distributions, like the trading rule return distribution). I found one introduced by Ruppert (1987):

Kurtosis measure

However I'm not sure how to apply this (/write this as the same way of the skewness measure) on my return distribution? (What does the greek letter "eta" precisely mean in this situation?).


Hinkley, D. V. (1975). On power transformations to symmetry. Biometrika, 62(1), 101-111.

Ruppert, D. (1987). What is kurtosis? An influence function approach. The American Statistician, 41(1), 1-5.


Background information of the Adjusted Sharpe Ratio (ASR) method: The ASR (Pézier 2004): ASR

The skewness measure by Pézier (2004): Skewness

The kurtosis measure by Pézier (2004): Kurtosis

Reference: Pezier, J., Alexander, C., & Sheedy, E. (2004). Risk and risk aversion. Alexander C. & Sheedy E. The Professional Risk Managers’ Handbook, 1. Direct link: Handbook

Page 65: Adjusted Sharpe Ratio

Pages 708-711: Skewness and Kurtosis measure


1 Answer 1


The denominator is essentially just some kind of "middle" part of the distribution for the tail to be large or not so large relative to; you need something to scale the $p$-distance by. For my illustrations here I chose the middle half (interquartile range) as a reasonable default; it's the most obvious one to try.

So let's take say $p=0.01$ and $\eta=0.25$ (I don't claim that's a n ideal choice, but $\eta=0.25$ seems a reasonable default).

Then $k = R_{0.25,0.01}= \frac{q_{0.99}-q_{0.01}}{q_{0.75}-q_{0.25}}$.

This then will be larger when the tail is heavy (since the small $p$ with a heavy tail will make the numerator large) and large when the distribution is peaked (since the larger $\eta$ with a peaked distribution will make the denominator small).

Here's an illustration for the normal:

enter image description here

For that particular pair $\eta,p$ we have:

 distribution     k
   uniform       1.96
   normal        3.45
   Cauchy        31.8
   exponential   4.18

You can of course choose different values for $\eta$ and $p$.

  • $\begingroup$ Indeed (I take $p=0.01$ as well). However the struggle is in the denominator part of the formula for me. How do you define/interpret $\eta$? And what number do you assign to it (how to determine it)? I couldn't find a clear explanation. $\endgroup$
    – Wildman
    Commented Oct 28, 2015 at 9:03
  • $\begingroup$ @Sander See my edit that posted right after your comment. It is essentially just some kind of "middle" part of the distribution for the tail-range to be large or small relative to; you need something to scale the p-distance by. I chose the middle half as a reasonable default; it's the most obvious one to try. $\endgroup$
    – Glen_b
    Commented Oct 28, 2015 at 9:05
  • $\begingroup$ I understand now, thanks! Is there some "official" definition/literature for $\eta$ in this case (or is it just some scaling number)? Indeed the middle half sounds like a reasonable default. $\endgroup$
    – Wildman
    Commented Oct 28, 2015 at 9:17
  • 1
    $\begingroup$ To echo the suggestion of @kjetil b halvorsen (see comment on question), an $L$-moment measure of kurtosis is a weighted combination of all quantiles and avoids all arbitrariness of which inner and which outer pair you use. It's not going to be robust/resistant to really fat or long tails, but $L$-moments people regard this as fair enough: it's going to measure what it is supposed to. (It be much, much more robust/resistant than moment-based kurtosis.) $\endgroup$
    – Nick Cox
    Commented Oct 28, 2015 at 12:27
  • 1
    $\begingroup$ Nassim Taleb's free book Silent Risk (available from his website) has some deep meditations on risk as a function of extreme values in returns that includes developing a number of quite practical metrics for tail risk. $\endgroup$
    – user78229
    Commented Oct 28, 2015 at 13:59

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