Zephyr Concepts - Skewness and Kurtosis

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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?

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

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.

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.

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

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High Kurtosis Fund Low Kurtosis Fund

Replaced With Average Monthly Return Replaced

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)

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High Kurtosis Fund,outliers replaced

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Low Kurtosis Fund,outliers replacedMarket Benchmark:S&P 500

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

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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%.

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.

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.

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

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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.

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.

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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.


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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


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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

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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.


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

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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


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-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


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


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


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%


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


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


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


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


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

Zephyr Concepts - Skewness and Kurtosis

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