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Page 1 of 23
Skewness and Kurtosis by Marc Odo, CFA, CAIA, CFP
Modern Portfolio Theory taught investors to focus upon risk just
as much as return when investing. The tradeoff between return and
risk dominated financial thinking for many years. However, two
recent developments over the last decade or so have led investors
to believe that the return versus risk trade-off, while useful, was
also incomplete. First, the dot-com crash and the credit crisis
left investors wondering just how often 100 Year Storms actually
occur. Standard measures of risk didnt seem to prepare investors
for the extreme nature of the two bear markets in the decade of the
2000s. Second, the rapid growth in hedge funds and other forms of
alternative investments resulted in a profusion of products with
return patterns that didnt always fit well in to standard
definitions of return and risk. Something was missing. While the
financial world might have been operating with a less-than-complete
toolkit, those with a mathematical or statistical background knew
exactly what was missing. Traditional statistical analysis uses
four, not two, metrics to quantify and describe the distribution
characteristics of a stream of data. Those four metrics, or moments
of the distribution, are:
1. Return 2. Volatility 3. Skewness 4. Kurtosis
Although well-established in statistical theory, skewness and
kurtosis are often ignored or misunderstood in performance
analysis. This is not surprising as skewness and kurtosis are
difficult to understand and mathematically sophisticated. This
paper seeks to answer the following questions in an
easy-to-understand manner:
1. What are skewness and kurtosis? 2. What do skewness and
kurtosis tell the investor? 3. What are typical numbers for
skewness and kurtosis? 4. How does the investor use skewness and
kurtosis in the context of a search or due diligence? 5. Is there a
single metric that sums up all four moments of the distribution
(Omega)? 6. How are distributions of hedge funds different from
traditional investments?
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Skewness & Kurtosis Zephyr Associates, Inc. I. BACKGROUND
Most people are familiar with the idea of the normal or bell-shaped
distribution. In this construct most of the observations in a data
series are clustered around the mean, but some observations fall
away from the central tendency of the distribution. The further
from the mean, the fewer the occurrences. If the individual points
in a data set fall into a normal or Gaussian distribution, they are
fairly predictable, and the shape of that bell curve is
well-defined.
Figure 1. Source: Wikipedia.
Under these conditions, definitive statements can be made about
the distribution of the data points.
1. Deviations from the mean are predictable. In a normal
distribution, 68.26% of all observations fall within +/- one
standard deviation from the mean, 95.44% of all observations occur
within +/- two standard deviations of the mean, and 99.73% of all
observations fall within +/- three standard deviations of the mean
value.
2. The distribution itself is symmetrical. The count and
placement of observations are equal both above and below the mean
value. In other words, the left side of the bell is the mirror
image of the right side of the bell.
3. Tail events are rare. Extreme deviations from the mean, while
not impossible, happen with a predictable (in)frequency.
There is considerable debate within the financial world as to
just how closely capital market returns fit this idealized normal
model. Skewness and kurtosis, the focus of this paper, are measures
of the last two points above - the symmetry of the distribution and
tail events. II. DEFINITIONS Skewness Defined There are two ways to
think about skewness. One way of thinking about skewness is that it
compares the length of the two tails of the distribution curve.
Another way of thinking of skewness is that it measures
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Skewness & Kurtosis Zephyr Associates, Inc. whether or not
the distribution of returns is symmetrical around the mean. The two
are related, because if the distribution is impacted more by
negative outliers than positive outliers (or vice versa) the
distribution will no longer be symmetrical. Therefore, skewness
tells us how outlier events impact the shape of the
distribution.
Figure 2. Source: Wikipedia.
Granted, it is entirely possible that extremely large losses in
a return series are balanced out by extremely large gains of equal
size and occurrence, in which case the distribution will remain
symmetrical with outlier events on either side. However, in the
real world of investing this isnt very likely. The extreme negative
tail events tend to be crashes and market meltdowns, whereas gains
are more modest and slower to accumulate. In these cases, where the
tails fall to the far left of the distribution, the distribution is
described as being negatively skewed. A distribution dominated by
outliers to the right of the distribution is called positively
skewed. While the image in Figure 2 above is useful, in order to be
analytical a number is needed. Skewness is quantified via the
following formula:
, , 1 2
Where: n = period = return in period n = standard deviation A
skewness value of 0 informs us that the distribution is perfectly
symmetrical. Negative or positive values indicate negative or
positive skew, respectively. In section IV we will take a look at
different asset classes and different time frames in order to
understand some typical values for skewness. It stands to reason
that an investor would prefer a positive skew, avoiding the losses
associated with negative tails.
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Skewness & Kurtosis Zephyr Associates, Inc. Kurtosis Defined
Kurtosis is often described as the fatness of the tails of a
distribution. In other words, kurtosis tells us if the risk of the
distribution is dominated by outlier events - those extreme events
distant from the average return. In recent times, people have taken
to calling outliers black swans, the idea being that in nature a
black swan is a rare and unusual occurrence1.
Figure 3. Source: Onlinestatbook.com
A distribution that has fat tails is known as leptokurtic. In
Figure 3 above, the upper image is leptokurtic, with a high peak in
the center and the risk coming in the tails. A distribution without
many observations in the tails is known as platykurtic, as seen in
the lower of the two examples. Here most of the observations fall
in a moderate band and there arent predominant tails. A perfectly
normal bell-shaped distribution is called mesokurtic. The formula
for calculating kurtosis is:
, , 1 1 2 3
3 1
1 3
Where: n = period = return in period n = standard deviation A
normal, bell-shaped mesokurtic distribution has a neutral value of
0.02. A fat-tailed, leptokurtic distribution has a positive value,
whereas a platykurtic distribution without much in the tails has a
value less than zero. A key idea in understanding kurtosis is that
kurtosis tells you where the standard deviation is coming from, not
what the overall level of standard deviation is. If a manager has a
standard deviation of, say, 18% over the last decade, was that
standard deviation generated by the observations frequently
1 The Black Swan: The Impact of the Highly Improbable; Nassim
Nicholas Taleb. 2 It is worth mentioning there are differing
conventions on how kurtosis is scaled. The actual calculation for
kurtosis is represented by the first half of the equation. However,
this produces a neutral value of 3.0 as the baseline value. Because
3.0 is a bit unusual to use as a starting point, the second half of
the equation is often added to scale the baseline value to 0.0.
Zephyrs StyleADVISOR uses this convention where 0.0 represents the
baseline, mesokurtic distribution.
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Skewness & Kurtosis Zephyr Associates, Inc. bouncing back
and forth within a moderate-sized range (i.e. a platykurtic,
no-tail situation, like the lower image in Figure 3)? Or
alternatively did the vast majority of the data points fall within
a tight, narrow band, and the 18% standard deviation was generated
by only a few very extreme observations (like the upper image in
Figure 3)? The following example illustrates this point. In this
situation I started with two real sets of data. Both have similar
annualized returns and standard deviations over 10 years. However,
one has a very low kurtosis of 0.63, meaning most of the
observations occurred within a moderate band. The other has a high
kurtosis of 14.97 with most of the observations tighter around the
center and a few observations driving the standard deviation
happening in the fat tails. High Kurtosis Fund Low Kurtosis Fund
Annualized 10-yr Return 0.52% 0.29% Standard Deviation 12.78%
12.55% Kurtosis 14.97 0.63 Average Monthly Return 0.116% 0.904%
Best Month/Worst Month Return +6.01%/-24.19% +8.06%/-12.40%
Table 1.
Figure 4. At this point what I did was replace the best and
worst individual monthly returns in the dataset with the arithmetic
average return of each time series. We would expect to see little
change in standard deviation for the manager with the low kurtosis,
since the outliers in a low kurtosis situation have only a marginal
impact on the overall distribution. Alternatively, replacing the
best and worst observations with the
Zephyr StyleADVISOR Zephyr StyleADVISOR: Zephyr Associates,
Inc.Histogram of ReturnsJanuary 2001 - December 2010
Per
cent
age
of M
onth
s (%
)
0
5
10
15
20
25
30
35
Returns Range (%)< -26 -26 to -24 -24 to -22 -22 to -20 -20
to -18 -18 to -16 -16 to -14 -14 to -12 -12 to -10 -10 to -8 -8 to
-6 -6 to -4 -4 to -2 -2 to 0 0 to 2 2 to 4 4 to 6 6 to 8 8 to 10
> 10
High Kurtosis Fund Low Kurtosis Fund
Replaced With Average Monthly Return Replaced
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Skewness & Kurtosis Zephyr Associates, Inc. average return
in a high kurtosis series has an extreme impact. The standard
deviation plunges from 12.78% to 10.00%, because those few extreme
observations were responsible for driving the overall standard
deviation. High Kurtosis Fund (mod) Low Kurtosis Fund (mod)
Modified Annualized 10-yr Return 2.77% 0.86% Modified Standard
Deviation 10.00% 11.63% Modified Kurtosis 0.70 -0.01
Table 2.
Figure 5. Eliminate a few extreme observations out of a
high-kurtosis data series and the whole story changes. III. THE KEY
TO UNDERSTANDING SKEWNESS AND KURTOSIS At this point the analyst
might feel that the answer is simple. It would seem that the
investor would prefer:
1. Positive skewness, with the shape of the distribution
favoring the positive tails 2. Negative kurtosis, with less risk
being driven by the tails
However, this interpretation is overly simplistic and
potentially misleading. Why? Because skewness and kurtosis are not
stand-alone statistics. In isolation they are meaningless. Skewness
helps us understand returns, but we must first know what the
returns are. Kurtosis describes the nature of the standard
deviation, but we must know what the standard deviation is.
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.Risk / ReturnJanuary 2001 - December 2010 (Single
Computation)
Ret
urn
-15%
-10%
-5%
0%
5%
10%
15%
Standard Deviation
0% 2% 4% 6% 8% 10% 12% 14% 16%
High Kurtosis Fund
High Kurtosis Fund,outliers replaced
Low Kurtosis Fund
Low Kurtosis Fund,outliers replacedMarket Benchmark:S&P
500
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Skewness & Kurtosis Zephyr Associates, Inc. Some examples
might help illustrate these points. We start with skewness. In
Table 3 below we see two managers. One has a positive skew, one has
a negative skew. If the investor were only looking for managers
with a positive skew, Positive Skew Manager would be a viable
candidate and Negative Skew Manager would be screened out. Skewness
Mean Return (annualized) Negative Skew Manager -0.85 9.62% Positive
Skew Manager +0.95 2.51%
Table 3. However, the overall return of Negative Skew Manager is
much higher. Positive Skew Manager might be positively skewed
having a couple of very outstanding up months, but the cost of such
is that the overall return is much lower. Negative Skew Manager
lacked one or two home run months, but was able to consistently
post modest gains in the majority of its months. The histogram
labeled Figure 6 illustrates this contrast:
Figure 6.
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.Histogram of ReturnsJanuary 2001 - December 2010
Perc
enta
ge o
f Mon
ths
(%)
0
5
10
15
20
25
30
35
Returns Range (%)
< -30 -30 to -25 -25 to -20 -20 to -15 -15 to -10 -10 to -5
-5 to 0 0 to 5 5 to 10 10 to 15 15 to 20 20 to 25 25 to 30 30 to 35
35 to 40 40 to 45 45 to 50 50 to 55 55 to 60 > 60
Negative Skew, but Good Returns Positive Skew, but Low
Returns
55% of observations are in 0%-5%or 5%-10% buckets.
Only about 35% of observations are in 0%-5% or5%-10%
buckets.
One single month saw a return of +59.3%.
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Skewness & Kurtosis Zephyr Associates, Inc. It is entirely
possible that one would run return and skewness calculations for a
large number of managers and see the following combinations as
displayed in Figure 7 to the right. At first glance this is
potentially a bit overwhelming and the analyst might not know where
to focus their efforts. Keep in mind the investor would probably
prefer high returns to low returns, and positively skewed
distributions rather than negatively skewed distributions.
Therefore, the sweet spot on this grid is the northwest quadrant.
Obviously then the southeast quadrant is to be avoided, with its
low returns and negative skewness. The analysis becomes more
interesting and nuanced in the northeast and southwest quadrants,
where one must balance one favorable characteristic versus one
unfavorable characteristic, and determine the proper trade-off.
Figure 7.
High Return, Positively Skewed:
Optimal Result
High Return, Negatively
Skewed
Low Return, Positively Skewed
Low Return, Negatively
Skewed
This construct should be familiar to most analysts, as the same
2x2 grid is often used to compare the first two moments of the
distribution, return and volatility. In the classic layout seen in
Figure 8 the northwest quadrant represents the best of both worlds.
The northeast quadrant is the aggressive quadrant, as the tradeoff
of the extra return is increased risk. The southwest quadrant is
the conservative quadrant, where the lower return is offset by the
benefit of lower risks. And once again, the place to avoid would be
the southeast quadrant which fails to deliver on both fronts.
Figure 8.
High Return, Low Volatility: Optimal Result
High Return, High Volatility
Low Return, Low Volatility
Low Return, High Volatility
The same trade-off concept applies to kurtosis. Generally
speaking the low or even negative kurtosis seen by Manager A is
desirable to a fat-tailed, positive kurtosis situation. However,
the overall level of risk of Manager A is significantly higher than
Manager B, regardless of whether or not the risk is to be found in
the tails of the distribution or clustered around the mean. It
seems unlikely that an investor would prefer a doubling of the
overall risk to acquire low or negative kurtosis. Kurtosis Standard
Deviation Manager A -1.00 16.29% Manager B 5.36 8.53%
Table 4.
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Skewness & Kurtosis Zephyr Associates, Inc. Looking at a
broad set of funds or managers, it is entirely possible to see four
different types of volatility-kurtosis combinations, as graphed
here. Again this was set up so that the ideal spot is the northwest
quadrant, with low overall volatility, coupled with an absence of
tail risk (i.e. a low kurtosis). What one would hope to avoid is
the combination seen in the southeast quadrant, where the overall
absolute risk or volatility is high, and moreover that risk is
driven by tail events. A manager in the northeast quadrant has the
advantage of having a low overall volatility, but the problem is
when that risk occurs it happens during those extreme periods. An
investor in northeast quadrant might be lulled in to complacency
and think risk is lower than it actually is, then be surprised
during those rare occasions when the risk jumps. Finally, a manager
in the southwest quadrant would tend to have high overall
volatility, but the investor would at least know via the low
kurtosis that the volatility is somewhat predictable and to be
expected.
Figure 9.
Low Volatility, Low Kurtosis: Optimal Result
Low Volatility, High Kurtosis
High Volatility, Low Kurtosis
High Volatility, High Kurtosis
So do the above examples undermine the importance of skewness
and kurtosis? After seeing examples where a positively skewed
dataset can have a low return and a negative kurtosis manager has
high overall volatility, does that mean we should go back to the
beginning and only focus on return and risk? No, that is not the
lesson here at all. The intent of the above examples is to
illustrate the point that skewness and kurtosis are useful, but
only useful in understanding the nature of the returns and risks.
Let us now look at real world data to see what kind of insight we
can gather when looking at all four moments of the distribution
together. IV. ASSET CLASSES We will start off looking at the four
moments of the distribution for the broad asset classes. I used the
following indices for the nine asset classes listed.
Common Period 1/88-12/10 Asset Class Index Large Cap Stocks (US)
S&P 500 Small Cap Stocks (US) Russell 2000 Int'l Developed MSCI
EAFE Emerging Markets MSCI Emerging Mkts Invst Grade Bonds (US)
Barclays U.S. Aggregate High Yield Bonds (US) Barclays U.S. Corp
High Yield REITs FTSE Nareit All Reits Commodities S&P GSCI
Hedge Funds HFN Fund of Funds Aggr
Table 5.
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Skewness & Kurtosis Zephyr Associates, Inc. The common
period for the nine asset classes starts on January 1st, 1988, as
that was the inception date of the MSCI Emerging Markets index. The
histograms of returns in Figure 10 below show 23 years of data,
binned in to 1% increments3.
Figure 10.
Common Period 1/88-12/10 Asset Class Return Standard Dev
Skewness Kurtosis Large Cap Stocks (US) 9.78% 14.93% -0.62 1.15
Small Cap Stocks (US) 10.10% 19.20% -0.59 1.08 Int'l Developed
5.80% 17.62% -0.41 1.06 Emerging Markets 14.06% 24.16% -0.69 1.76
Invst Grade Bonds (US) 7.34% 3.95% -0.19 0.44 High Yield Bonds (US)
8.78% 9.14% -0.96 8.64 REITs 9.47% 17.98% -0.92 8.75 Commodities
6.88% 21.16% -0.18 2.25 Hedge Funds 9.97% 5.43% -0.12 5.07
Table 6. Skewness Observations What we see here is not
surprising. Every asset class is negatively skewed, as outliers
tend to occur to the far left of the distributions during market
meltdowns. Investment grade bonds, commodities, and hedge
3 Detailed, zoomed-in versions of these graphs appear in
Appendix 1.
Zephyr StyleADVISOR Zephyr StyleADVISOR: Zephyr Associates,
Inc.Large Cap: S&P 500
Per
cent
age
of M
onth
s (%
)
Small Cap: Russell 2000
Per
cent
age
of M
onth
s (%
)
International: MSCI EAFE
Perc
enta
ge o
f Mon
ths
(%)
Emerging: MSCI Emerging
Per
cent
age
of M
onth
s (%
)
Bonds: Barclays Agg
Per
cent
age
of M
onth
s (%
)
0
5
10
15
20
25
30
35
< -4 -4 to -3 -3 to -2 -2 to -1 -1 to 0 0 to 1 1 to 2 2 to 3
> 3
HY Bonds: Barclays HY
Perc
enta
ge o
f Mon
ths
(%)
Real Estate: NAREIT
Per
cent
age
of M
onth
s (%
)
Commodities: GSCI
Per
cent
age
of M
onth
s (%
)
Hedge Funds: HFN FoF
Perc
enta
ge o
f Mon
ths
(%)
0
5
10
15
20
25
30
35
< -7 -7 to -6 -6 to -5 -5 to -4 -4 to -3 -3 to -2 -2 to -1 -1
to 0 0 to 1 1 to 2 2 to 3 3 to 4 4 to 5 5 to 6 6 to 7 7 to 8 >
8
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Skewness & Kurtosis Zephyr Associates, Inc. funds tend to
have distributions closest to being symmetrical (i.e. a skewness of
0.0), but all nine asset classes are negatively skewed to some
extent. This is to be expected when we think about what we are
trying to analyze here - the individual outlier observations. The
below table looks at the individual monthly data points of each
asset class and summarizes the average return, the median return,
and then the top three and worst three months over the time frame
January 1988 to December 20104.
Common Period 1/88-12/10
Asset Class Average Median Top Three Worst Three
Large Cap Stocks (US) 0.87% 1.29% 11.44% (12/91)
9.78% (3/00) 9.75% (5/90)
-16.80% (10/08) -14.46% (8/98) -10.87% (9/02)
Small Cap Stocks (US) 0.96% 1.73% 16.51% (2/00) 15.46% (4/09)
12.46% (9/10)
-20.80% (10/08) -19.42% (8/98) -15.10% (7/02)
Int'l Developed 0.60% 0.89% 15.61% (10/90) 12.96% (4/09) 12.58%
(7/89)
-20.17% (10/08) -14.42% (9/08) -13.91% (9/90)
Emerging Markets 1.35% 1.52% 18.98% (4/89) 17.15% (5/09) 16.66%
(4/09)
-28.91% (8/98) -27.35% (10/08) -17.49% (9/08)
Invst Grade Bonds (US) 0.60% 0.66% 3.87% (5/95) 3.73% (12/08)
3.52% (1/88)
-3.36% (7/03) -2.60% (4/04) -2.47% (3/94)
High Yield Bonds (US) 0.74% 0.95% 12.11% (4/09) 10.94% (2/91)
7.68% (12/08)
-15.91% (10/08) -9.31% (11/08) -7.98% (9/08)
REITs 0.90% 1.17% 27.98% (4/09)
15.58% (12/08) 12.22% (8/09)
-30.23% (10/08) -21.51% (11/08) -19.46% (2/09)
Commodities 0.74% 0.72% 22.94% (9/90) 19.67% (5/09) 16.89%
(3/99)
-28.20% (10/08) -14.84% (11/08) -14.41% (3/03)
Hedge Funds 0.81% 0.82% 8.49% (6/88) 5.77% (5/89) 5.42%
(12/99)
-6.31% (10/08) -6.01% (9/08) -4.86% (8/98)
Table 7. The worst of the worst months are more extreme than the
best of the best months, particularly in the equity asset classes.
This is what we mean by negative skewness. Kurtosis Observations
High yield bonds provide a great illustration of kurtosis. A
zoomed-in image is provided in Figure 11 below to illustrate just
what we mean when we say kurtosis describes where the standard
deviation is coming from, not what the overall level of standard
deviation happens to be.
4 It is a shame that the MSCI Emerging Markets Index has an
inception date of January 1988. By using the common period Jan 88
Dec 10 the impact of the October 1987 crash is excluded. The fact
that the S&P 500 lost almost 20% in October 1987 and another
12% in November 1987 would certainly impact the skewness and
kurtosis numbers. The impact of October 1987 is seen in the
following section when we look at decade-by-decade analysis.
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Skewness & Kurtosis Zephyr Associates, Inc.
Figure 11. In Figure 11 above, we see the vast majority of the
observations are in the middle of the distribution. The two middle
bars representing individual monthly returns of 0%-1% and 1%-2%
account for over 50% of the observations. Most of the time returns
are bouncing back and forth within this tight little range.
However, the standard deviation that does occur (9.14%) is driven
primarily by the outlier events. This results in a very high
kurtosis number of 8.64, meaning that the risk that does exist
exists in the tails. Contrast this with the large cap stocks of the
S&P 500 in Figure 12. This distribution is pretty close to
normal.
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.
High Yield Bonds: Barclays U.S. Corp High YieldPe
rcen
tage
of M
onth
s (%
)
0
5
10
15
20
25
30<
-16
-16
to -1
5
-15
to -1
4
-14
to -1
3
-13
to -1
2
-12
to -1
1
-11
to -1
0
-10
to -9
-9 to
-8
-8 to
-7
-7 to
-6
-6 to
-5
-5 to
-4
-4 to
-3
-3 to
-2
-2 to
-1
-1 to
0
0 to
1
1 to
2
2 to
3
3 to
4
4 to
5
5 to
6
6 to
7
7 to
8
8 to
9
9 to
10
10 to
11
11 to
12
12 to
13
> 13
Over 50% of observations in 0%-1%or 1%-2% buckets.
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Skewness & Kurtosis Zephyr Associates, Inc.
Figure 12. Yes, over the last 23 years there have been some
outlier events in the S&P 500, and unfortunately those tended
to be on the negative side (thus giving us a negative skew). Yes,
there has been risk in the form of standard deviation, which at
14.93% is noticeably higher than the 9.14% standard deviation of
high yield bonds. However, looking at the above histogram it does
not appear that risk is concentrated in the tails; it looks like
the observations are fairly bell-shaped. Therefore the kurtosis of
the S&P 500 is much lower at 1.15, meaning that standard
deviation is more spread out across the distribution. Finally, let
us review these two asset classes side-by-side and sum up what we
see in all four moments of the distribution in Figure 13 and Table
8.
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.
Large Cap: S&P 500Pe
rcen
tage
of M
onth
s (%
)
0
2
4
6
8
10
12
14
16<
-17
-17
to -1
6
-16
to -1
5
-15
to -1
4
-14
to -1
3
-13
to -1
2
-12
to -1
1
-11
to -1
0
-10
to -9
-9 to
-8
-8 to
-7
-7 to
-6
-6 to
-5
-5 to
-4
-4 to
-3
-3 to
-2
-2 to
-1
-1 to
0
0 to
1
1 to
2
2 to
3
3 to
4
4 to
5
5 to
6
6 to
7
7 to
8
8 to
9
9 to
10
10 to
11
11 to
12
> 12
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Skewness & Kurtosis Zephyr Associates, Inc.
Figure 13.
Common Period 1/88-12/10 Asset Class Return Standard Dev
Skewness Kurtosis Large Cap Stocks (US) 9.78% 14.93% -0.62 1.15
High Yield Bonds (US) 8.78% 9.14% -0.96 8.64
Table 8. The annualized return is similar between large cap
stocks and high yield bonds. The overall volatility, as measured by
standard deviation, is lower for high yield bonds and is
illustrated by the fact that red bars are more densely packed in a
smaller band. Both distributions are negatively skewed, as the
left-side, negative tail stretches further than the right-side,
positive tail. Finally, the kurtosis is higher for high yield
bonds, as the standard deviation of high yield bonds is driven by
the tails. One final note on the broad asset classes. In addition
to looking at the nine asset classes over the common time period of
January 1988 to December 2010, I also broke out the metrics by
decade. This data can be seen in Appendix #2. Interestingly, no
clear trends appear. The most obvious statement that can be made
revolves around the returns table. The returns for virtually all
asset classes were spectacular in the 1980s and 1990s; the returns
of the 2000s were dismal. These are well-known facts. But looking
to the second, third, and fourth moments of the distribution, it is
difficult to make any broad, sweeping statements about any of them.
Standard deviations remain remarkably stable throughout the
decades, with few exceptions. No clear trends can be seen in
skewness numbers. Kurtosis readings also dont show any clear trends
through the decades. If anything, the kurtosis numbers for the
broad equity
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.
S&P 500 vs. High Yield BondsPe
rcen
tage
of M
onth
s (%
)
0
5
10
15
20
25
30<
-17
-17
to -1
6
-16
to -1
5
-15
to -1
4
-14
to -1
3
-13
to -1
2
-12
to -1
1
-11
to -1
0
-10
to -9
-9 to
-8
-8 to
-7
-7 to
-6
-6 to
-5
-5 to
-4
-4 to
-3
-3 to
-2
-2 to
-1
-1 to
0
0 to
1
1 to 2
2 to
3
3 to
4
4 to 5
5 to
6
6 to
7
7 to
8
8 to
9
9 to 1
0
10 to
11
11 to
12
12 to
13 > 13
-
Skewness & Kurtosis Zephyr Associates, Inc. classes were
much higher in the 1980s, because the October 1987 crash had such
an extreme impact on the decade. Of the four moments of the
distribution, only return shows clear differences between the
decades. V. PEER GROUPS Next we turn our attention to peer groups.
What are the typical ranges of skewness and kurtosis for the
various asset classes? What are the implications when searching for
superior managers within each asset class?
Skewness 1/88-12/10
Large Cap
Small Cap
Intl Emg Mkts
Invst Bond
HY Bond
REIT HF
Funds in Univ 227 40 22 0 59 34 3 4 5th -0.11 -0.19 -0.51 N/A
0.06 -0.82 N/A N/A 25th -0.45 -0.38 -0.54 N/A -0.26 -1.07 N/A N/A
50th -0.58 -0.51 -0.62 N/A -0.42 -1.27 N/A N/A 75th -0.69 -0.63
-0.69 N/A -0.63 -1.46 N/A N/A 95th -0.90 -0.80 -0.77 N/A -1.29
-1.65 N/A N/A Index -0.62 -0.59 -0.41 -0.69 -0.19 -0.96 -0.92
-0.12
Table 9. Skewness Observations What stands out in these numbers
is the fact that just about the entire universe tends to be
negatively skewed across the major asset classes. Even the 5th
percentile managers still have a distribution skewed slightly by
the outliers to the left-of-center. This relationship is also seen
in the decade-by-decade results, seen in Appendix #3a. Again
applying our practical knowledge of the markets to these numbers,
this is not surprising. When those outlier events tend to occur,
more often than not they tend to be negative events. Skewness
Take-Aways As stated in Section III, this is not the end of the
world. As long as the absolute returns are at an acceptable level,
a slightly negative skew is probably acceptable. For practical
purposes, what does this mean if we are to do a search on
skewness?
1. First of all, establishing a filter to eliminate all funds
with a negative skew will in all likelihood eliminate the vast
majority of managers. One must accept that at least some negative
skewness is inevitable in the capital markets.
2. Second, the reference point used if one is to incorporate
skewness in a search or analysis will probably have to be relative.
One will likely need to set up filters so that the skewness is less
negative than the benchmark or universe median. Having a
hard-target skewness number as a cutoff would likely be
useless.
3. Another alternative could be that the analyst uses skewness
as a red-flag test. Rather than saying that skewness should be less
negative than the index, another possibility would be to filter out
only those managers with the worst skewness numbers- those whose
distributions were really impacted by negative events.
-
Skewness & Kurtosis Zephyr Associates, Inc.
4. Finally, if one does opt to use skewness as part of a search,
it should likely be a secondary metric compared to the more
traditional return and risk metrics. The analyst might use it in a
tie-breaker role if managers are evenly matched with the other
metrics.
Table 10 below summarizes the range of kurtosis results. The
order is reversed so that the highest-kurtosis funds are towards
the bottom. The decade-by-decade results are again in Appendix
#3b.
Kurtosis 1/88-12/10
Large Cap
Small Cap
Intl Emg Mkts
Invst Bond
HY Bond
REIT HF
Funds in Univ 227 40 22 0 59 34 3 4 5th 0.68 0.81 1.09 N/A 0.46
5.59 N/A N/A 25th 1.20 1.29 1.45 N/A 1.13 7.08 N/A N/A 50th 1.52
1.83 1.68 N/A 2.02 8.42 N/A N/A 75th 1.97 2.63 2.00 N/A 3.66 10.03
N/A N/A 95th 3.64 3.81 2.95 N/A 7.99 14.10 N/A N/A Index 1.15 1.08
1.06 1.76 0.44 8.64 8.75 5.07
Table 10. Kurtosis Observations What might surprise people at
first glance is that the kurtosis numbers are higher for the fixed
income asset classes rather than the equity asset classes. But then
applying what weve discussed previously, the results make sense.
Again, kurtosis describes where the standard deviation is coming
from, not the overall level of standard deviation. For the equity
asset classes, standard deviation is higher overall, but kurtosis
is low as the standard deviation is fairly evenly spread across the
distribution. Fixed income returns, on the other hand, can be
described better by the saying, when it rains, it pours. The vast
majority of time conditions are rather staid and predictable with
the monthly return of fixed income likely being generated by the
interest portion of the total return. However, there are the
occasional interest rate shocks or flight to quality panics when
credit spreads diverge, and the principal value of the fixed income
investments is impacted. Most of the volatility of fixed income
occurs in those rare environments. Kurtosis Take-Aways How might an
analyst then incorporate kurtosis in a search process? The
recommendations on screening for kurtosis are rather similar to
those for skewness.
1. Negative kurtosis, while desirable, is extremely rare.
Filtering on managers to seek out only those with negative kurtosis
will likely eliminate the entire field.
2. Like skewness, the reference point used when analyzing
kurtosis will likely have to be a relative point, such as lower
than the index or lower than the median of the universe. A
hard-target kurtosis number is meaningless.
3. Finally, and most importantly, kurtosis must be used in
conjunction with standard deviation. All kurtosis does is provide
detail about the nature of the standard deviation. If the absolute
level of standard deviation is not known, kurtosis is worthless.
Again, the idea of using kurtosis as a tie-breaker role if managers
have similar standard deviations makes sense.
-
Skewness & Kurtosis Zephyr Associates, Inc. VI. SAMPLE
SEARCH So what if we were to apply these ideas to an actual search?
Over the ten year period from January 2001 to December 2010, the
large cap S&P 500 index has the following results:
Common Period 1/01-12/10 Large Cap US Stocks Return Standard Dev
Skewness Kurtosis S&P 500 Index 1.41% 16.38% -0.64 0.81
Table 11. What if we were to apply the following filters to all
large cap stock funds5 with ten year track record?
Returns must be greater than the S&P 500s return Standard
deviation must be less than the S&P 500s standard deviation
Skewness must be greater than the S&P 500s skewness Kurtosis
must be less than the S&P 500s kurtosis
In this particular search one can see just how stringent these
tests are. Of the starting field of 812 funds, only 17 remain at
the end of these four tests. Keep in mind we havent even applied
most of the traditional filters one would use (e.g. alpha,
information ratio, expense ratio, pain index, etc). Simply focusing
on the four moments of the distribution eliminates almost 98% of
the field.
Figure 13. At this point the analyst might understandably find
all of this information overwhelming, with four different metrics
and different relative breakpoints. The analyst might be tempted to
look for a simpler metric that captures all four moments of the
distribution in to one summary number. Fortunately there is a
metric that does an admirable job of capturing return, standard
deviation, skewness, and kurtosis in one number. That metric is
known as Omega. 5 All mutual funds in Morningstar US Mutual Fund
database with a 10-year track record through 12/31/10 classified as
Large Value, Large Blend, or Large Growth, filtered on distinct
portfolio.
-
Skewness & Kurtosis Zephyr Associates, Inc. Omega is a
metric developed by Con Keating and William Shadwick in 2002. This
measure, accompanied with by the S-shaped cumulative distribution
graph is a great way of summarizing all four moments of the
distribution and both are available in StyleADVISOR. Figure 14
below shows a cumulative distribution graph, sorting the monthly
returns of the S&P 500 from January 1988 to December 2010 from
worst to first. The standard deviation can be seen in the width of
the S-curve. Comparing the length of the negative tail versus the
length of the positive tail illustrates the skewness of the
distribution. Finally, comparing the bulk of the observations in
the observations towards the middle against the few observations in
the tails is a way to see the kurtosis. If one were to divide the
area in green above the minimum acceptable return (MAR) by the area
in red below the MAR, one gets the metric known as Omega.
Figure 14. More thorough discussions on the top-down metric
Omega can be found on Zephyrs website via the links below: Omega by
Marc Odo, CFA, CAIA, CFP Omega Explained by Thomas Becker, Ph.D.
The intent of this paper is to increase the understanding of the
moments of the distribution from the bottom-up. Hopefully these
papers will be quite complimentary to each other. VII. HEDGE FUND
FOCUS Up until now the emphasis of this study has been on the
overall capital markets, with hedge funds being just one of many
asset classes examined. However, it has long been said that hedge
funds are unique
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.Cumulative Distribution of ReturnsJanuary 1988 - December
2010
Perc
enta
ge o
f Ret
urns
Les
s Th
an o
r Equ
al to
x
0%
20%
40%
60%
80%
100%
Monthly Return (MAR = 0.00%)
-15% -10% -5% 0% 5% 10%
S&P 500
-
Skewness & Kurtosis Zephyr Associates, Inc. because their
return distributions are not normally distributed. Complicating
matters, the term hedge fund is wide-reaching and encapsulates many
radically different investing strategies. In this last section of
the paper we break out the hedge fund space into specific
categories to look at their distribution characteristics. In Table
12a we look at the four moments of the distribution over the last
five years for various categories in the Lipper TASS database6. For
comparative purposes, the same information was run for various
long-only asset classes over the same five-year period in Table
12b. Return Observations We start with the first moment of the
distribution, simple annualized returns.
Return Conv Arb
Emg Mkt
Eq Mkt Ntrl
Event Driven
Fixed Arb
FoF Global Macro
Lng/Sht Eq Hedge
Mgd Fut
Multi-Strat
Funds in Univ 41 201 84 149 68 1565 97 749 221 267
5th 17.23 29.06 16.07 17.54 18.36 19.69 27.08 18.37 21.30
24.65
25th 11.39 16.87 10.40 9.09 11.05 7.04 12.27 11.36 14.62
19.35
50th 6.78 9.53 6.19 5.96 4.64 4.10 9.14 7.51 10.08 8.43
75th 3.63 4.48 3.26 2.28 1.84 1.75 4.06 3.60 4.89 3.83
95th 0.17 -2.87 -2.06 -1.99 -3.91 -2.18 0.13 -1.73 0.25
-1.78
Table 12a. Return
Large Cap Small Cap Internl Emerging Interm Bond High Yield Real
Estate
Funds in Univ 1070 448 278 69 273 118 61 5th 6.01 9.00 6.88
15.02 7.67 9.45 5.62 25th 3.52 6.10 4.30 13.06 6.35 8.04 3.52 50th
2.25 4.36 2.94 11.06 5.64 7.32 2.76 75th 1.17 2.45 1.74 10.00 4.91
6.42 1.16 95th -0.92 -0.51 -0.35 5.98 2.54 4.22 -3.68
Table 12b. The first thing that jumps out when looking at the
universes of the hedge fund categories against the traditional
categories is the difference in dispersions of returns. The gap
between the top 5th and bottom 95th percentiles is fairly tight in
the traditional asset classes, typically somewhere between 7% and
10% separating the best from worst equity funds and the gap being
about 5% for fixed income funds. Its also worth noting that the
sizes of the universes are much larger for the traditional asset
classes. Contrast this with all 10 categories in the Lipper TASS
database. The gap separating best-from worst is double-digits
across the boards, and in many cases in excess of 20%. This is
likely due to a couple of factors. First , more dispersion is
expected due to the fact that hedge funds can invest in pretty much
any manner they
6 Lipper TASS was used due to the fact that their categories are
relatively broad and deep. Some of the other hedge fund databases
have narrowly defined categories without many constituents,
limiting the usefulness of establishing percentile ranges. Also,
the analysis was run for the five year period ending December 31st,
2010. Due to the massive growth of the hedge fund industry over the
last decade, the sample sizes were too small for meaningful
analysis if run for anything longer than five years. Finally,
indices were not analyzed as hedge fund indices are usually simply
the average of the universe constituents within each category.
-
Skewness & Kurtosis Zephyr Associates, Inc. see fit. Second,
the leverage sometimes employed by hedge funds can have quite an
impact upon returns, for better or worse. This wide dispersion of
results from best-to-worst is also seen in the other three moments
of the distribution. Looking at the centers of the distributions,
one sees that median returns for the various hedge fund strategies
are by and large better than the median returns for traditional
asset classes over the five years through 12/31/10. One should keep
in mind the various biases in the hedge fund databases like
survivorship bias, selection bias, backfill bias, and illiquidity
bias, but overall most hedge fund returns have been better than
traditional returns recently. Standard Deviation Observations Next
we look at the volatility of the funds and the splay of standard
deviations.
Standard Deviation
Conv Arb
Emg Mkt
Eq Mkt Ntrl
Event Driven
Fixed Arb
FoF Global Macro
Lng/Sht Eq Hedge
Mgd Fut
Multi-Strat
Funds in Univ 41 201 84 149 68 1565 97 749 221 267
5th 8.05 6.54 4.55 3.52 2.52 5.51 6.66 6.29 6.62 6.59
25th 10.39 12.24 7.70 6.32 9.08 7.80 10.63 11.20 12.81 11.83
50th 12.04 20.88 11.90 10.45 13.17 14.45 13.93 15.66 16.21
15.68
75th 14.46 31.66 15.56 15.65 17.52 16.18 19.61 20.72 24.61
18.81
95th 26.90 42.07 19.05 26.74 27.11 20.50 27.18 30.81 40.50
28.17
Table 13a. Standard Deviation Large Cap Small Cap Internl
Emerging
Interm Bond High Yield Real Estate
Funds in Univ 1070 448 278 69 273 118 61 5th 15.35 19.18 19.50
26.09 3.18 9.59 26.45 25th 17.55 21.46 21.49 27.19 3.70 10.92 30.84
50th 18.38 22.58 22.25 28.62 4.30 12.18 31.93 75th 19.61 23.86
23.44 29.78 5.10 13.19 32.93 95th 22.76 26.89 25.51 31.63 7.76
15.14 35.45
Table 13b. The same general comments about returns can also be
made of the standard deviations of the hedge fund universes. Again,
the gap between best and worst is much wider in the alternative
space. The best hedge funds have very low risks while the worst
funds have stomach-churning standard deviations. As with the return
dispersions, the idiosyncratic nature of hedge fund strategies and
the possibility of leverage are likely responsible. Similar to the
observations for returns, the median hedge funds tend to look
better than the median long-only managers. Another interesting
observation is that even in hedge fund categories that are thought
to be well-diversified there is a tremendous amount of variation
within those categories. The 5th and 95th percentile standard
deviations for Fund-of-Funds and Multistrategies are 5.51%-20.50%
and 6.59%-28.17%, respectively. Also, the volatility for the Equity
Market-Neutral and Long/Short Equity Hedge categories
-
Skewness & Kurtosis Zephyr Associates, Inc. are higher than
one might expect, especially as one looks at the lower breakpoints
of the universe. It would be foolish to think of these strategies
as riskless. Skewness Observations We now turn our attention to the
main focus on this paper, the third and fourth moments of the
distribution.
Skewness Conv Arb
Emg Mkt
Eq Mkt Ntrl
Event Driven
Fixed Arb
FoF Global Macro
Lng/Sht Eq Hedge
Mgd Fut
Multi-Strat
Funds in Univ 41 201 84 149 68 1565 97 749 221 267
5th 0.10 0.97 0.76 0.51 0.86 0.19 1.38 0.83 1.04 0.73
25th -1.21 0.08 -0.04 -0.25 -0.03 -0.45 0.38 0.00 0.44 -0.34
50th -2.16 -0.45 -0.52 -0.97 -0.69 -0.86 0.02 -0.38 0.05
-0.88
75th -2.44 -1.05 -0.98 -1.38 -1.11 -1.21 -0.58 -0.71 -0.28
-1.18
95th -3.56 -2.63 -2.03 -3.15 -2.38 -2.16 -1.21 -1.47 -1.11
-2.11
Table 14a. Skewness
Large Cap Small Cap Internl Emerging Interm Bond High Yield Real
Esate
Funds in Univ 1070 448 278 69 273 118 61 5th -0.44 -0.24 -0.47
-0.52 0.71 -0.84 -0.31 25th -0.69 -0.51 -0.63 -0.67 0.04 -1.45
-0.45 50th -0.82 -0.62 -0.71 -0.75 -0.55 -1.64 -0.60 75th -0.90
-0.75 -0.82 -0.87 -1.07 -1.90 -0.68 95th -1.11 -0.97 -0.99 -1.07
-1.89 -2.21 -1.00
Table 14b. We see some trends continue, but we also recognize
new characteristics when observing the dispersion of skewness
values. Like before, the gaps between the best and worst tend to be
rather wide with hedge funds, much wider than one sees in the
traditional space. As in our previous analysis of skewness for
traditional investments, there appears to be a prevalence of
managers that have a negative skew, where the extremities of the
negative tails outweigh the impact of the positive tails. Like
before, if an analyst were to screen out all managers with a
negative skew, this filter would likely eliminate the majority of
the hedge funds under analysis. That being said, there is a much
higher preponderance of positively skewed distributions when
looking at hedge fund strategies. At the 5th percentile, all ten of
the hedge fund categories are in positive territory, and two of the
categories (Global Macro and Managed Futures) are positive at the
median. In our previous analysis of skewness of the long-only side,
we surmised that the negative skewness seen almost universally
across all traditional asset classes was due to the fact that the
extreme negative months tend to occur when markets are melting down
and positive returns are more modest. Hedge funds, with their
non-benchmark strategies and ability to make unsystematic bets, are
overall less prone to negative skewness. However, that doesnt mean
hedge funds are guaranteed not to have extremely bad months and
negative skewness. Again, the worst hedge fund managers in the 75th
and especially 95th percentiles have skewness
-
Skewness & Kurtosis Zephyr Associates, Inc. numbers much
worse than the worst traditional long-only managers. As the old
saying goes, Live by the sword, die by the sword. Focusing on the
category with the best skewness, Global Macro strategies purport to
be able to avoid major losses and capture the best opportunities by
being able to go anywhere and invest in anything. The fact that
that there are more positively skewed Global Macro hedge funds than
negatively skewed funds suggests that over the last five years a
significant number of Global Macro funds have done a decent job of
delivering on that idea. Moreover, the upper reaches of the Global
Macro peer group has a very high skewness (the 5th percentile is
1.38), which indicate that outlier, home run months drive the
positive tails of the Global Macro strategies. Kurtosis
Observations Finally we look at kurtosis. While we can say that
generally speaking it would be desirable to have a negative
kurtosis number (meaning the standard deviation is not driven by
the extreme tails) it is very rare to see any product, hedge or
traditional, display that trait.
Kurtosis Conv Arb
Emg Mkt
Eq Mkt Ntrl
Event Driven
Fixed Arb
FoF Global Macro
Lng/Sht Eq Hedge
Mgd Fut
Multi-Strat
Funds in Univ 41 201 84 149 68 1565 97 749 221 267
5th 3.80 0.07 -0.13 0.14 0.51 0.03 -0.03 -0.23 -0.64 0.21
25th 4.74 1.11 0.74 1.23 1.43 0.92 0.49 0.56 -0.32 1.31
50th 8.34 2.44 1.64 2.58 2.00 2.00 1.11 1.24 0.50 2.40
75th 11.53 5.44 4.23 5.66 5.60 3.28 1.79 2.49 1.92 3.52
95th 18.79 12.98 10.48 13.50 20.99 7.83 5.98 7.06 6.11 10.54
Table 15a. Kurtosis
Large Cap Small Cap Internl Emerging Interm Bond High Yield Real
Esate
Funds in Univ 1070 448 278 69 273 118 61 5th 0.38 0.23 0.61 1.20
0.37 4.61 2.86 25th 0.93 0.78 1.09 1.52 1.52 6.83 3.05 50th 1.24
1.12 1.43 1.75 2.87 7.69 3.38 75th 1.69 1.68 1.87 2.32 5.15 8.97
3.73 95th 2.58 2.80 2.74 3.12 8.92 11.74 4.55
Table 15b. Once again, the dispersion within categories between
best and worst is vast and the worst-of-the-worst hedge funds have
very high kurtosis numbers. Moreover, we see that the typical
ranges between categories are quite different. The diversified
strategies (Fund-of-funds and Multistrategy) and long-short
strategies (Equity Market Neutral and Long/Short Equity Hedge) tend
to have reasonable kurtosis numbers, while the arbitrage strategies
run higher. The hedge fund categories with the highest kurtosis
readings are the convertible arbitrage and fixed income arbitrage
strategies. The kurtosis numbers are especially pronounced at the
lower reaches of the
-
Skewness & Kurtosis Zephyr Associates, Inc. universe, as at
the 95th percentile we see kurtosis numbers of 18.79 and 20.99,
respectively. Applying what we know about arbitrage strategies
gives color to these numbers. Arbitrage strategies seek to exploit
small pricing anomalies between similar or identical investments.
These pricing differences are small and fleeting, so therefore a
much greater degree of leverage tends to be used in order to make
these opportunities worth exploiting. On a day-to-day basis in
normal environments exploiting these arbitrage opportunities can be
thought of riskless, but occasionally a big, macro event will come
along and wallop these strategies. Examples of macro events would
include a drying up of liquidity, a ban on short sales, or a flight
to quality (all of which occurred in 2008, for example). Some
people use the analogy, Picking up nickels in front of steamroller,
to describe this kind of strategy. Those managers with kurtosis
numbers around 20.0 failed to avoid the steamroller. Overall Hedge
Fund Observations Summing up what weve seen in this section, there
are a few key important takeaways. While overall return, standard
deviation, skewness, and kurtosis readings look favorable for hedge
funds, one must be cautious. There is a significant distance
between the best and worst hedge funds across all four moments of
the distribution, and those hedge funds with the worst numbers
perform worse than traditional investments by a large margin. One
should be careful when looking at hedge fund indices. Most hedge
fund indices are an average of all the hedge funds within a
particular category, and if weve learned anything from this paper
its that averages dont tell the whole story. Outliers can have a
big impact. In addition, the survivorship bias, backfill bias, and
other biases impact hedge fund indices. While hedge funds can bring
positive benefits to a portfolio, they are not a magic bullet that
will solve all of an investors concerns. Some of the fundamental
lessons of modern portfolio theory, i.e. the usefulness of
diversification and looking at how the total portfolio behaves,
remain just as important even if hedge funds are rolled into the
equation. SUMMARY Due to its very nature, tail risk is difficult to
understand. By definition, tail events do not happen very
frequently so we dont have a lot of experience with them. The
central tendency of the distribution should be the primary focus of
an analysts efforts, as more often than not most observations fall
close to mean. However, when tail events do occur, either on the
positive or negative side, they tend to trigger the emotional
responses to the market- greed or fear, respectively. Skewness and
kurtosis are systematic, well-defined ways of applying an
analytical framework to tail risk. If well-understood, skewness and
kurtosis will hopefully minimize the impact of emotions during rare
periods of extreme conditions.
-
Skewness & Kurtosis Appendix Page 1 of 7
APPENDIX #1
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.
Large Cap: S&P 500
Perc
enta
ge o
f Mon
ths
(%)
0
2
4
6
8
10
12
14
16
< -1
7
-17
to -1
6
-16
to -1
5
-15
to -1
4
-14
to -1
3
-13
to -1
2
-12
to -1
1
-11
to -1
0
-10
to -9
-9 to
-8
-8 to
-7
-7 to
-6
-6 to
-5
-5 to
-4
-4 to
-3
-3 to
-2
-2 to
-1
-1 to
0
0 to
1
1 to
2
2 to
3
3 to
4
4 to
5
5 to
6
6 to
7
7 to
8
8 to
9
9 to
10
10 to
11
11 to
12
> 12
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.
Small Cap: Russell 2000
Perc
enta
ge o
f Mon
ths
(%)
0
2
4
6
8
10
< -2
1
-21
to -2
0
-20
to -1
9
-19
to -1
8
-18
to -1
7
-17
to -1
6
-16
to -1
5
-15
to -1
4
-14
to -1
3
-13
to -1
2
-12
to -1
1
-11
to -1
0
-10
to -9
-9 to
-8
-8 to
-7
-7 to
-6
-6 to
-5
-5 to
-4
-4 to
-3
-3 to
-2
-2 to
-1
-1 to
0
0 to
1
1 to
2
2 to
3
3 to
4
4 to
5
5 to
6
6 to
7
7 to
8
8 to
9
9 to
10
10 to
11
11 to
12
12 to
13
13 to
14
14 to
15
15 to
16
> 16
-
Skewness & Kurtosis Appendix Page 2 of 7
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.
International: MSCI EAFEPe
rcen
tage
of M
onth
s (%
)
0
2
4
6
8
10
12
< -2
1
-21
to -2
0
-20
to -1
9
-19
to -1
8
-18
to -1
7
-17
to -1
6
-16
to -1
5
-15
to -1
4
-14
to -1
3
-13
to -1
2
-12
to -1
1
-11
to -1
0
-10
to -9
-9 to
-8
-8 to
-7
-7 to
-6
-6 to
-5
-5 to
-4
-4 to
-3
-3 to
-2
-2 to
-1
-1 to
0
0 to
1
1 to
2
2 to
3
3 to
4
4 to
5
5 to
6
6 to
7
7 to
8
8 to
9
9 to
10
10 to
11
11 to
12
12 to
13
13 to
14
14 to
15
> 15
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.
Emerging: MSCI Emerging
Perc
enta
ge o
f Mon
ths
(%)
0
2
4
6
8
10
< -2
9-2
9 to
-28
-28
to -2
7-2
7 to
-26
-26
to -2
5-2
5 to
-24
-24
to -2
3-2
3 to
-22
-22
to -2
1-2
1 to
-20
-20
to -1
9-1
9 to
-18
-18
to -1
7-1
7 to
-16
-16
to -1
5-1
5 to
-14
-14
to -1
3-1
3 to
-12
-12
to -1
1-1
1 to
-10
-10
to -9
-9 to
-8-8
to -7
-7 to
-6-6
to -5
-5 to
-4-4
to -3
-3 to
-2-2
to -1
-1 to
00
to 1
1 to
22
to 3
3 to
44
to 5
5 to
66
to 7
7 to
88
to 9
9 to
10
10 to
11
11 to
12
12 to
13
13 to
14
14 to
15
15 to
16
16 to
17
17 to
18
> 18
-
Skewness & Kurtosis Appendix Page 3 of 7
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.
Bonds: Barclays AggPe
rcen
tage
of M
onth
s (%
)
0
5
10
15
20
25
30
35
< -4
-4 to
-3
-3 to
-2
-2 to
-1
-1 to
0
0 to
1
1 to
2
2 to
3 > 3
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.
HY Bonds: Barclays HY
Perc
enta
ge o
f Mon
ths
(%)
0
5
10
15
20
25
30
< -1
6
-16
to -1
5
-15
to -1
4
-14
to -1
3
-13
to -1
2
-12
to -1
1
-11
to -1
0
-10
to -9
-9 to
-8
-8 to
-7
-7 to
-6
-6 to
-5
-5 to
-4
-4 to
-3
-3 to
-2
-2 to
-1
-1 to
0
0 to
1
1 to
2
2 to
3
3 to
4
4 to
5
5 to
6
6 to
7
7 to
8
8 to
9
9 to
10
10 to
11
11 to
12
12 to
13
> 13
-
Skewness & Kurtosis Appendix Page 4 of 7
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.
Real Estate: NAREITPe
rcen
tage
of M
onth
s (%
)
0
2
4
6
8
10
12
< -3
1-3
1 to
-30
-30
to -2
9-2
9 to
-28
-28
to -2
7-2
7 to
-26
-26
to -2
5-2
5 to
-24
-24
to -2
3-2
3 to
-22
-22
to -2
1-2
1 to
-20
-20
to -1
9-1
9 to
-18
-18
to -1
7-1
7 to
-16
-16
to -1
5-1
5 to
-14
-14
to -1
3-1
3 to
-12
-12
to -1
1-1
1 to
-10
-10
to -9
-9 to
-8-8
to -7
-7 to
-6-6
to -5
-5 to
-4-4
to -3
-3 to
-2-2
to -1
-1 to
00
to 1
1 to
22
to 3
3 to
44
to 5
5 to
66
to 7
7 to
88
to 9
9 to
10
10 to
11
11 to
12
12 to
13
13 to
14
14 to
15
15 to
16
16 to
17
17 to
18
18 to
19
19 to
20
20 to
21
21 to
22
22 to
23
23 to
24
24 to
25
25 to
26
26 to
27
> 27
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.
Commodities: GSCI
Perc
enta
ge o
f Mon
ths
(%)
0
2
4
6
8
10
< -2
9-2
9 to
-28
-28
to -2
7-2
7 to
-26
-26
to -2
5-2
5 to
-24
-24
to -2
3-2
3 to
-22
-22
to -2
1-2
1 to
-20
-20
to -1
9-1
9 to
-18
-18
to -1
7-1
7 to
-16
-16
to -1
5-1
5 to
-14
-14
to -1
3-1
3 to
-12
-12
to -1
1-1
1 to
-10
-10
to -9
-9 to
-8-8
to -7
-7 to
-6-6
to -5
-5 to
-4-4
to -3
-3 to
-2-2
to -1
-1 to
00
to 1
1 to
22
to 3
3 to
44
to 5
5 to
66
to 7
7 to
88
to 9
9 to
10
10 to
11
11 to
12
12 to
13
13 to
14
14 to
15
15 to
16
16 to
17
17 to
18
18 to
19
19 to
20
20 to
21
21 to
22
> 22
-
Skewness & Kurtosis Appendix Page 5 of 7
APPENDIX #2
Return Asset Class
1980s 1990s 2000s Common Period
1/88-12/10 Large Cap Stocks (US) 17.55% 18.21% -0.95% 9.78%
Small Cap Stocks (US) 14.52% 13.40% 3.51% 10.10% Int'l Developed
22.77% 7.33% 1.58% 5.80% Emerging Markets N/A 11.04% 10.11% 14.06%
Invst Grade Bonds (US) 12.43% 7.69% 6.33% 7.34% High Yield Bonds
(US) N/A 10.72% 6.72% 8.78% REITs 12.51% 8.10% 10.18% 9.47%
Commodities 10.67% 3.89% 5.05% 6.88% Hedge Funds 21.82% 14.04%
4.85% 9.97%
Standard Deviation Asset Class
1980s 1990s 2000s Common Period
1/88-12/10 Large Cap Stocks (US) 16.39% 13.43% 16.13% 14.93%
Small Cap Stocks (US) 20.48% 17.27% 21.55% 19.20% Int'l Developed
17.51% 17.15% 17.86% 17.62%
Zephyr StyleADVISOR Zephy r Sty leADVISOR: Zephy r Associates,
Inc.
Hedge Funds: HFN FoFPe
rcen
tage
of M
onth
s (%
)
0
5
10
15
20
25
30
35
< -7
-7 to
-6
-6 to
-5
-5 to
-4
-4 to
-3
-3 to
-2
-2 to
-1
-1 to
0
0 to
1
1 to
2
2 to
3
3 to
4
4 to
5
5 to
6
6 to
7
7 to
8 > 8
-
Skewness & Kurtosis Appendix Page 6 of 7
Emerging Markets N/A 23.85% 24.89% 24.16% Invst Grade Bonds (US)
8.45% 3.91% 3.83% 3.95% High Yield Bonds (US) N/A 7.22% 11.46%
9.14% REITs 12.85% 12.09% 23.52% 17.98% Commodities 13.76% 17.58%
25.31% 21.16% Hedge Funds 11.95% 4.63% 5.26% 5.43%
Skewness Asset Class
1980s 1990s 2000s Common Period
1/88-12/10 Large Cap Stocks (US) -0.80 -0.63 -0.57 -0.62 Small
Cap Stocks (US) -1.43 -0.85 -0.41 -0.59 Int'l Developed -0.31 -0.16
-0.77 -0.41 Emerging Markets N/A -0.90 -0.66 -0.69 Invst Grade
Bonds (US) 0.62 -0.15 -0.45 -0.19 High Yield Bonds (US) N/A -0.19
-0.99 -0.96 REITs -0.65 0.12 -1.06 -0.92 Commodities -0.47 1.03
-0.48 -0.18 Hedge Funds 0.12 -0.30 -1.27 -0.12
Kurtosis Asset Class
1980s 1990s 2000s Common Period
1/88-12/10 Large Cap Stocks (US) 4.12 1.77 0.91 1.15 Small Cap
Stocks (US) 6.76 1.91 0.64 1.08 Int'l Developed 0.43 0.69 1.80 1.06
Emerging Markets N/A 2.61 1.32 1.76 Invst Grade Bonds (US) 2.56
0.10 1.32 0.44 High Yield Bonds (US) N/A 7.08 6.32 8.64 REITs 3.40
0.57 6.26 8.75 Commodities 0.80 3.78 1.30 2.25 Hedge Funds 2.62
3.32 5.21 5.07
APPENDIX #3a
Skewness Large Cap Small Cap International Emg Mkts 1990s 2000s
1990s 2000s 1990s 2000s 1990s 2000s Funds in Univ 255 744 47 298 33
227 1 52 5th 0.02 -0.22 0.09 0.38 -0.08 -0.33 NA -0.50 25th -0.30
-0.50 -0.41 -0.17 -0.38 -0.64 NA -0.61 50th -0.48 -0.61 -0.63 -0.43
-0.45 -0.74 NA -0.68
-
Skewness & Kurtosis Appendix Page 7 of 7
75th -0.65 -0.72 -0.78 -0.63 -0.66 -0.83 NA -0.75 95th -0.95
-0.90 -0.98 -0.91 -0.90 -1.02 NA -0.87 Index -0.63 -0.57 -0.85
-0.41 -0.16 -0.77 -0.90 -0.66
Skewness Invst Bond HY Bond REITs Hedge Funds 1990s 2000s 1990s
2000s 1990s 2000s 1990s 2000s Funds in Univ 69 214 36 87 6 42 14
233 5th 0.22 0.01 -0.07 -0.85 NA -0.70 NA 0.60 25th -0.01 -0.36
-0.51 -1.11 NA -0.79 NA -0.23 50th -0.15 -0.66 -0.75 -1.35 NA -0.95
NA -0.86 75th -0.25 -0.98 -1.01 -1.52 NA -1.07 NA -1.52 95th -0.45
-1.71 -1.45 -1.93 NA -1.30 NA -3.17 Index -0.15 -0.45 -0.19 -0.99
0.12 -1.06 -0.30 -1.27
APPENDIX #3b
Kurtosis Large Cap Small Cap International Emg Mkts 1990s 2000s
1990s 2000s 1990s 2000s 1990s 2000s Funds in Univ 255 744 47 298 33
227 1 52 5th 0.53 0.40 0.56 0.37 0.75 1.02 NA 0.94 25th 1.20 0.87
1.49 0.97 1.20 1.54 NA 1.18 50th 1.71 1.23 1.92 1.50 1.55 1.90 NA
1.53 75th 2.22 1.81 2.56 2.34 2.32 2.46 NA 1.85 95th 4.51 3.11 3.48
4.04 3.89 3.56 NA 2.78 Index 1.77 0.91 1.91 0.64 0.69 1.80 2.61
1.32
Kurtosis Invst Bond HY Bond REITs Hedge Funds 1990s 2000s 1990s
2000s 1990s 2000s 1990s 2000s Funds in Univ 69 214 36 87 6 42 14
233 5th -0.04 0.59 1.66 4.74 0.64 5.59 NA 0.73 25th 0.20 1.36 3.23
6.04 0.70 6.20 NA 2.15 50th 0.36 2.22 4.24 7.64 0.93 6.69 NA 3.83
75th 0.72 3.97 5.92 9.37 1.37 7.07 NA 7.66 95th 2.25 9.33 7.68
12.98 1.80 8.88 NA 16.25 Index 0.10 1.32 7.08 6.32 0.57 6.26 3.32
5.21