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.