Is there any relationship between the risk aversion coefficient in an individual's utility function (commonly used in portfolio optimization) and the preference for higher moments such as skewness and kurtosis? In what range, approximately, should they be? Logically, a relationship should exist, as they all somehow represent the investor's attitude towards risk.

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    $\begingroup$ Excellent question, Pasha! I'll try to do some more digging tomorrow, but here is an alternative framework that incorporates a preference for positive skewness and low kurtosis as well as low variance into a single optimization meant as an alternative to mean-variance. The author writes that "there is no clear leader amongst these alternative objective functions," so there may not be a consensus yet regarding your precise question. $\endgroup$ – Tal Fishman Aug 25 '11 at 1:56

Yes. Check out the Lower Partial Moments literature. In my view the best introduction to this is Narwrocki - A Brief History of Downside Risk Measures. Uryasev established equivalence between CVaR approach and low partial moments. If Markowitz had the tools at the time time, LPM utility functions would be the introductory optimization model as opposed to mean-variance.

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    $\begingroup$ Here is a compendium of Uryasev's papers on Conditional Value-at-Risk (CVar) from the University of Florida RMFE Lab. $\endgroup$ – Tal Fishman Aug 25 '11 at 2:22
  • $\begingroup$ Quant Guy and @sheegaon thank you for giving me clue and direction.. It really helps me. I will try to write some brief notes when I complete my research.. thank you once more $\endgroup$ – Pasha Aug 26 '11 at 14:51

There's a huge literature on this topic going back at least 30 years, and I am unfortunately not familiar enough with this literature to give you a great answer to your specific question. However, I will in this answer at least try to point you in some useful directions according to what I've found thus far.

Kurtosis, by the way, seems like it is not addressed directly as part of this literature, perhaps because it is too difficult to measure, or, because it is an even higher moment than skewness, perhaps we should attain a solid understanding of the first 3 moments before we move on to the fourth.

There are at least 3 ways I've come across to answer the question of a relationship between risk aversion and a preference for skewness.

  1. Cross-sectional relationship between equity skewness and average equity returns.
  2. Option-implied skewness and average option returns.
  3. Theory (e.g. Mean-CVaR optimization).

All of these actually try to measure the amount of expected return one gives up in order to get positive skewness, or the amount one gets in exchange for negative skewness. However, combining these estimates with popular estimates of risk aversion measured in the traditional sense could give you a precise answer.

Some references to get you started:

Morningstar recently came out with a piece entitled The Real World is Not Normal: Introducing the new frontier: an alternative to the mean-variance optimizer. It essentially summarizes their views on Mean-CVaR optimization, based on Xiong and Idzorek (2011).


Some people like to play for the "big win." Their utility function involves the excitement of the gamble, and they don't mind the negative expectation, or even the big loss, as long as there is a chance for the big win. Investments like lottery tickets and penny stocks exhibit extremely high kurtosis. So, for a person with a "gambling", "shoot the moon" - type of mindset, high kurtosis is desirable, as long as some of the extreme rare events that drive kurtosis are on the positive return side. A portfolio with extremely large positive kurtosis would be attractive to such an investor, even if the portfolio had zero skewness.

There is no difficulty with kurtosis, except that people have simply not understood it for more than a century (Pearson started it all wrong in 1905), calling it a measure of "peakedness," which is not only wrong, it also detracts from understanding its proper interpretation and use. Its proper interpretation and use is as a measure of tail extremity (e.g., the penny stock windfall or the lottery win).


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