Hey! Who are you saying has a ‘fat tail’?

Aside from being a wonderful source of sophomoric humour about bizarre medical conditions, “kurtosis” is also a useful way of mathematically describing what many people say will be the downfall of hedge funds – their performance during “extreme events”.  But unfortunately, kurtosis (a.k.a. “fat tails”) is entering the mainstream as a poorly understood measure.

Kurtosis refers to the extent to which a return distribution has “outliers”.  A fund with a low kurtosis would be more pointed than the familiar bell-shaped normal distribution.  With more of its returns grouped around the middle, one might be excused for thinking that such a fund is “safer” or more “predictable” than a fund with a high kurtosis (i.e. which tend to experience more severe returns on both the upside and downside).

A recent Morningstar advertisement is exemplary of this common assumption.  The half-page ad is titled “The Skinny on Fat Tails” and was published recently in a popular trade magazine.  It contains statements about kurtosis that are technically accurate, but misleading such as:

“Fat tails indicate that the probability of extreme losses or gains is higher than ‘normal’.”

“A hedge fund with a high positive excess kurtosis has a higher probability of extreme returns than a fund with no positive excess kurtosis and a normal distribution.”

“Hedge funds with a negative skewness and excess positive kurtosis experience more frequent and extreme negative returns than a normal distribution.”

In reality, however, little can be concluded by the simple observation of high or low kurtosis.  In fact a highly kurtotic (“fat tailed”) distribution might actually experience less drawdowns of a given size than a fund with a low kurtosis (i.e. whose return distribution has “thin tails”).  What Morningstar does not point out is that a “fat tailed” distribution only experiences more extreme losses or gains than a normal distribution…with the same volatility.

The truth is that hedge funds with excess kurtosis can often have lower overall volatility than something with normally distributed returns (like, say, an individual stock long-only mutual fund).  The calculation of kurtosis essentially involves dividing a raw number by the fund’s standard deviation.  So if the standard deviation (the denominator) is low, then the final kurtosis number is going to be high.  Similarly, a highly volatile fund might easily have a lower kurtosis than a low volatility fund.  In other words, kurtosis only refers to the shape of the return distribution, not the actual “probability of extreme returns” as Morningstar suggests.

(This statistical phenomenon is similar to the point recently made by Tobias Adrian at the New York Fed about the limitations of correlation.  Correlation, he shows, is another variable that describes the world – not in absolute terms –  but compared to funds’ own volatilities.)

The winners of the inaugural AIMA Canada Research Award in 2005, Peter Klein and Todd Brulhart originally made this argument and even suggested a modification to the traditional method of calculating kurtosis in order to prevent such confusion.  In an easier-to-read version of this paper, the two are quite clear:

“…despite what is written in most textbooks, higher kurtosis cannot always be interpreted as indicating greater risk of extreme events.”

Author and UBS hedge fund guru Alexander Ineichen expanded on this point further in his book Asymmetric Returns – going as far as suggesting that “volatility is more important than excess kurtosis”. Said Ineichen:

“(Kurtosis) is often used as an indication that there is “higher” probability of a far-from-equilibrium event.  However, it does not address the issue of how severe the undesirable event might be; nor does it address the consequences for the investor.”

In fact, in a recently published summary of Ineichen’s remarks to the “World Hedge Fund Forum” in March (hosted here with special permission from UBS), UBS revisits this point using the following chart and accompanying explanation:

“…The arrow in the graph points to two occurrences where a multi-strategy fund lost between 65% and 70% of its value in one month. One of these two occurrences was Amaranth’s September 2006 return of -69.8%. (The other occurrence as well as the one observation in the -80% to -85% return bucket was from an obscure fund that we believe managed less than one million in assets and closed in November 1996.) Given that the graph is based on 28,420 monthly multi-strategy returns, we can back out a probability of 0.007% for the probability of a monthly loss in the -65% to -70% return bucket.

“This is, by coincidence, nearly identical to one of the 500 constituents in the S&P 500 experiencing such a loss in a given month…In the case of multi-strategy funds, there were four occurrences (of losses) exceeding 50%. This means the probability is in the neighborhood of 0.0141%…we can expect a monthly loss of 50% or more every 592 years.”

“How do these probabilities compare to the stock market? The probability of losing 50% or more with a S&P 500 constituent is 0.0600%. This means, if we assume we only hold one S&P 500 constituent for one month, we can expect to lose 50% or more once every 139 years. (Note that the probability of losing 50% or more over three, six or twelve months is an entirely different story.)”

The report does concede that, although they have higher risk of downside, stocks are usually purchased in highly diversified baskets (although perhaps too diversified they hint).  But still, UBS concludes with this food for thought:

“Former Harvard president Derek Bok was once quoted saying: ‘If you think education is expensive, try ignorance.’

“By examining the graph above, could one not rephrase the quote to: ‘If you think hedge funds are risky, try stocks.'”

Morningstar should be lauded for offering new statistical measures for a new investing paradigm.  But when they suggest that fat tails necessarily equal more risk, they should add an important qualifier: ceteris paribus.

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  1. Jon
    May 11, 2007 at 8:52 am

    I think this is a good post, and investors should consider as many moments of the return distribution and they can reasonably develop expectations for! However, your statement that

    “…kurtosis only refers to the shape of the return distribution, not the actual “probability of extreme returns” as Morningstar suggests.”

    is simply false. I understand you ‘get to the point’ of ceteris paribus conditions later in the post, but that statement is simply misleading.


  2. Tristram Lett
    May 11, 2007 at 10:10 am

    Two expansions of what you posted that are relevant: If you break a normal distribution synmmetrically up into thirds, kurtosis is when less observations are in the 1/3 to 2/3 portion than the normal and more in the first 1/3 and last 1/3. Therefore high kurtosis may say nothing about fat tails.

    Secondly kurtosis is notoriously unstable-one observation among 60 can alter the number significantly. To stabilize the measure you need to calculate the L-moment which bounds the set between 0 and 1 in quantile space as opposed to the unbounded set in probability space.

  3. Jon
    May 13, 2007 at 7:03 pm

    Mr. Lett:

    Your comments are technically true, in that I can increase the excess kurtosis of a distribution without affecting the shape of the tails – I believe maybe that my statement in the comments section was too strong, and I should have been more careful with my wording. On that note, is there research that you could direct me to that suggests that any observed kurtosis in ‘hedge fund’ returns is due to returns clustering around the mean without any changes in the shape of tails of the distribution (when compared to, say, a normal dist.)? I think that empirically, with financial market returns, positive excess kurtosis does imply fat tails.


  4. Tris Lett
    May 14, 2007 at 4:40 pm

    I will get back to you on that point. Sorry it took so long to respond, but I could not see a response to the article, including my own.

  5. Jon
    May 15, 2007 at 8:40 am

    No problem – I’ll check in later.


  6. Tristram Lett
    May 15, 2007 at 2:19 pm

    As it turns out the point we are discussing has been of interest to some academics, but to date no one has come up with a proof that what I said is true, though clearly it is not a new thought. So I graciously concede the point to you…

  7. Jon
    May 16, 2007 at 8:57 am

    I’ve enjoyed the discussion – it’s been quite educational for me.

    Technical matters aside, I agree with your sentiment that kurtosis is not just a hedge fund issue. It is unlikely that any financial asset has a return distribution that is truly normal, and so using kurtosis as a means of warning about hedge fund investing is sort of a weak argument.



  8. JD
    June 13, 2007 at 6:56 pm

    Did I miss something? The article suggests that a -50% return on stocks or multi strat funds is a 1 in 139 or 1 in 592 year event. -100% losses happen all the time. LTCM and Amaranth occured a bit closer than 592 years apart.

    Stop using normal distributions to consider the finer details of probabilities.

  9. Matt
    June 14, 2007 at 6:33 pm

    I think normal distributions have some value as they can help describe most general asset returns, but the fat tail needs to be considered and the notion that large unexpected drops can occur needs to be remembered.

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