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AN ABSTRACT OF THE RESEARCH PAPER OF
Mark Hoaglund, for the Master of Science degree in economics, presented on August 14, 2008, at Southern Illinois University at Carbondale.
TITLE: Measuring the Performance of the Hedge Fund Market
MAJOR PROFESSOR: Scott Gilbert
The objective of this study was to determine some of the characteristics of the hedge
fund market and to compare the returns of the hedge fund market to the S&P 500 by
computing various statistical measures of performance for a representative sample of
hedge funds and imparting meaning to the results. In order to achieve the objective,
Capital Asset Pricing Models, polynomial regressions, variances, correlations, and mean
averages were computed and the results were analyzed. Finally, graphs were generated
as tests for heteroscedasticity and normality in the CAPM regressions, and plausible
interpretive meaning was suggested. The collective statistical analysis concluded that the
performance of hedge funds exceeds the market impressively. Specifically, the hedge
fund market was found to be far less volatile and more profitable than the S&P 500.
Moreover, those particular funds - as distinguished from the overall hedge fund market -
with higher Sharpe Ratios were found to be both less volatile and more profitable than
the S&P 500. Thus, within the hedge fund market, investment alternatives exist which
are characterized by an overall improvement to the index fund.
i
ACKNOWLEDGEMENTS
I’d like to thank Professor Scott Gilbert for helping me throughout the process of
developing this study.
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TABLE OF CONTENTS
ABSTRACT ……………………………………………………………..i
ACKNOWLEDGEMENTS ….......................................................................................ii
LIST OF TABLES ……………………………………………………………iv
LIST OF FIGURES …......................................................................................vi
TEXT ……………………………………………………………..1
REFERENCES …………………………………………………………..123
VITA …………………………………………………………..124
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LIST OF TABLES
TABLE PAGE
Table 1…..........................................................................................................................6
Table 2…………………………………………………………………………………...11
Table 3…………………………………………………………………………………...20
Table 4…………………………………………………………………………………...23
Table 5….........................................................................................................................31
Table 6…………………………………………………………………………………...37
Table 7…………………………………………………………………………………...38
Table 8…………………………………………………………………………………...39
Table 9….........................................................................................................................92
Table 10………………………………………………………………………………….93
Table 11………………………………………………………………………………….94
Table 12………………………………………………………………………………….95
Table 13...........................................................................................................................96
Table 14………………………………………………………………………………….97
Table 15………………………………………………………………………………….98
Table 16………………………………………………………………………………….99
Table 17.........................................................................................................................100
Table 18………………………………………………………………………………...101
Table 19………………………………………………………………………………...102
Table 20………………………………………………………………………………...103
Table 21.........................................................................................................................104
Table 22………………………………………………………………………………...105
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Table 23………………………………………………………………………………...106
Table 24………………………………………………………………………………...107
Table 25.........................................................................................................................108
Table 26………………………………………………………………………………...109
Table 27………………………………………………………………………………...110
Table 28………………………………………………………………………………...111
Table 29.........................................................................................................................112
Table 30………………………………………………………………………………...113
Table 31………………………………………………………………………………...114
Table 32………………………………………………………………………………...115
Table 33.........................................................................................................................116
Table 34………………………………………………………………………………...117
Table 35………………………………………………………………………………...118
Table 36………………………………………………………………………………...119
Table 37.........................................................................................................................120
Table 38………………………………………………………………………………...121
Table 39………………………………………………………………………………...122
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LIST OF FIGURES
FIGURE PAGE
Figure 1……………………………………………………………………………………8
Figure 2……………………………………………………………………………………8
Figure 3……………………………………………………………………………………9
Figure 4……………………………………………………………………………………9
Figure 5…………………………………………………………………………………..10
Figure 6…………………………………………………………………………………..10
Figure 7…………………………………………………………………………………..14
Figure 8…………………………………………………………………………………..15
Figure 9…………………………………………………………………………………..16
Figure 10…………………………………………………………………………………16
Figure 11…………………………………………………………………………………17
Figure 12…………………………………………………………………………………18
Figure 13…………………………………………………………………………………18
Figure 14…………………………………………………………………………………19
Figure 15…………………………………………………………………………………26
Figure 16…………………………………………………………………………………26
Figure 17…………………………………………………………………………………27
Figure 18…………………………………………………………………………………28
Figure 19…………………………………………………………………………………28
Figure 20…………………………………………………………………………………29
Figure 21…………………………………………………………………………………30
Figure 22…………………………………………………………………………………30
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Figure 23…………………………………………………………………………………32
Figure 24…………………………………………………………………………………33
Figure 25…………………………………………………………………………………33
Figure 26…………………………………………………………………………………34
Figure 27…………………………………………………………………………………41
Figure 28…………………………………………………………………………………41
Figure 29…………………………………………………………………………………42
Figure 30…………………………………………………………………………………42
Figure 31…………………………………………………………………………………43
Figure 32…………………………………………………………………………………43
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Figure 35…………………………………………………………………………………45
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Figure 37…………………………………………………………………………………46
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Figure 39…………………………………………………………………………………47
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Figure 48…………………………………………………………………………………51
Figure 49…………………………………………………………………………………52
Figure 50…………………………………………………………………………………52
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Figure 73…………………………………………………………………………………65
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Figure 89…………………………………………………………………………………75
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Figure 91…………………………………………………………………………………76
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Figure 97…………………………………………………………………………………79
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Figure 98…………………………………………………………………………………79
Figure 99…………………………………………………………………………………80
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Figure 123………………………………………………………………………………..95
Figure 124………………………………………………………………………………..96
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Figure 129………………………………………………………………………………101
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Figure 135………………………………………………………………………………107
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Figure 143………………………………………………………………………………115
Figure 144………………………………………………………………………………116
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Figure 146………………………………………………………………………………118
Figure 147………………………………………………………………………………119
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Figure 148………………………………………………………………………………120
Figure 149………………………………………………………………………………121
Figure 150………………………………………………………………………………122
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Introduction
The purpose of the following study was to examine various measures of performance
of the hedge fund market, to compare the hedge fund market to the broader stock market
by way of the S&P 500 index, and to determine the implications of the hedge fund
market performance from the perspective of considering all the investigative results
collectively. The source of data used in the analysis of hedge funds was
www.hedgefund.net which is a service owned by Channel Capital Group Incorporated
that provides hedge fund news and proprietary performance data on approximately 8000
hedge funds.1 The hedge fund data were drawn by conducting a search for funds
according to the Sharpe Ratio in descending order and then selecting the performance
data from 30 hedge funds using an algorithm. By arranging the funds in terms of the
Sharpe Ratio, a sample of data more representative of the overall hedge fund market was
obtained because the data accounted better for the full spectrum of both risk and return of
the funds. Many interesting statistics began to emerge once the data was arranged in
Excel and analyzed.
The literature contained information that shared a complementary relationship with
the findings of this study, but also, that information yielded some cautionary reservations
that must be noted with respect to this study’s performance findings of hedge funds. In
an article by John Morgan on July 7, 20082, a warning was issued that the Securities and
Exchange Commission (SEC) is poised to initiate tighter regulation of the hedge fund
market depending on the political persuasions of those elected in the impending
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presidential election. If these regulatory prospects materialize, then access might be
further restricted to investors, and fewer funds may form as a result of an inability of
smaller firms to raise capital. Perhaps the lack of smaller, unstable firms might actually
improve the statistical performance results, such as those that are found in this study,
because there would be fewer firms that collapse and pull the performance data of the
hedge fund market down. However, an October 2004 publication by Burton G. Malkiel3
extensively studied many ways that hedge fund performance data artificially inflate the
true returns of the hedge fund market. For example, hedge funds that are about to close
stop reporting their performance data during the last months of their existence, and
because hedge funds, unlike mutual funds, do not have to report their performance data to
the SEC, a hedge fund only begins offering its data to a database when the fund has
established some sustainable measure of success so that the initial performance remains
unreported. Nevertheless, if a hedge fund were to be chosen judiciously, such as the
selection of one with low volatility and a proven track record, then surely the integrity of
the results will be intact since the investor would not have to be as concerned about the
hedge fund folding. An additional concern is also noted in a September 11, 20064 article
by Pascal Botteron regarding the inflated perception of hedge fund performance.
Namely, the fact that hedge funds tend to have low volatility is only true insofar as the
fund itself is solvent and viable. For example, the volatility of stocks in a company
reflects broadly disseminated reports about the welfare of the company itself, but a hedge
fund is not required to produce such information, so an imperative for wise investment is
the process of thoroughly vetting a fund. These reservations about the hedge fund market
performance must be taken into context and temper any understanding about the results.
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Models and Variables
Employed in the study of the hedge fund market were a number of statistical variables
and models which will be defined and explained next.
The Sharp Ratio, mentioned in the introduction, is defined as where E(Ri)
is the expected return of fund i, Rf is the risk free rate of return as measured by treasury
bonds, and σ is the standard deviation of the excess return as given by the entire
numerator. The Sharpe Ratio is considered a measure of the tradeoff between excess
return and risk from volatility.
The variance is a measure of the spread of the values of a random variable around the
expected value. The variance can be defined, in its most abstract sense, as
var(X) = E(X – μ)2 where X is a random variable and μ = E(X), the expected value of X.
The coefficient of correlation is a measure of the degree of association between two
variables. The coefficient always lies in the interval [-1, 1] where a high positive value
means that the two variables move closely together whereas a low negative value means
that the two variables move in opposition to each other. The coefficient of correlation is
defined differently for a population of data and a sample of data.
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2
The definition of the population coefficient of correlation in its most abstract is
ρ = where X and Y are random variables and the rho values in the
denominator are their respective population standard deviations.
The definition in its most abstract form of the sample coefficient of correlation is
r = where X and Y are random variables and the s values are their sample
standard deviations. However the definition in a form best suited for interpretation in
terms of simple regression is
r = where X and Y are random variables and
n is the number of pairs observed.
A point of clarification must be addressed in preparation for the body of the study. In
conducting the analysis of the hedge fund market, the tables for the coefficient of
correlation values were computed for a population of data in Excel because the only
Excel function available to compute correlations uses the formula for populations of data.
The only difference between the correlation formulas for populations and samples of data
is that the sample standard deviation is divided by n-1 whereas the population standard
deviation is divided by n. Consequently, the denominator of the population correlation is
smaller than the denominator of the sample correlation, so the population correlation is
larger than the sample correlation when both the sample correlation and population
correlation are applied to the same set of data. In truth, the populations of data were
3
known in this study, but these populations were often treated as samples in order to
project future trends, so whether the population correlation or sample correlation is more
desirable is a matter of interpretation. Also, the reader must know that the regressions are
based on the sample correlation formula when the discussion about the r2 = R2 values is
encountered later in the study.
The coefficient of determination, r2, is a measure of how well a regression line fits the
data. In other words, the coefficient measures the percentage of the regression that can
be explained by the regression where the remaining percentage can only be accounted for
by random error. In regression involving more than one explanatory variable, that is, in
multiple regression, the term used by convention for the coefficient of determination is
R2, and in regression involving only one explanatory variable, that is, in simple
regression, the term used is r2. However, R2 is often used interchangeably for both simple
regression and multiple regression. Since Excel used the R2 term for the simple
regressions discussed later, the reader must be aware that r2 = R2.
Average Returns
For both the S&P 500 and the individual hedge funds, each month of percentage
returns was annualized by multiplying each monthly return by 12. For the period of
January, 1995 – April, 2008, the mean average of the annualized monthly returns for the
S&P 500 index was 9.21515%. The mean average of the annualized monthly returns for
each hedge fund was obtained similarly, but care must be taken to note that many of the
hedge funds did not span the same number of months as the time period stated above that
was chosen for the S&P 500. Regardless, when these averages for the individual hedge
4
funds were themselves averaged, the result was 13.11473%, which is substantially higher
than the return for the S&P 500. Moreover, the hedge fund performance data was also
approached somewhat differently by first averaging, for any given month, across all 30
hedge funds so that, for example, in September of 1995, the average annualized monthly
performance across all the hedge funds was 37.2%. When these monthly annualized
averages were themselves averaged, the result was 16.9934%, which was even higher
than the 13.114% figure. Therefore, the average returns of the hedge fund market yielded
much higher returns than the general stock market.
Correlations
The correlations between the annualized S&P 500 monthly market returns and each of
the 30 hedge fund monthly performances were computed to determine how closely hedge
fund investments behave like the market. The correlations are shown below in
descending order of the Sharpe Ratio as explained in the introduction.
Table 1. Fund Correlations With the S&P 500__________________________________
Fund #1 -0.173464524 Fund #2 0.649193451
Fund #3 -0.044247993 Fund #4 -0.0714241
Fund #5 -0.207737278 Fund #6 0.497261555
Fund #7 0.414375015 Fund #8 0.470509021
Fund #9 0.253269275 Fund #10 0.486760127
Fund #11 0.601120061 Fund #12 -0.190548726
5
Fund #13 0.407577199 Fund #14 0.44922268
Fund #15 0.570891982 Fund #16 0.620327411
Fund #17 0.298016148 Fund #18 0.45629676
Fund #19 0.262008673 Fund #20 0.409339549
Fund #21 0.781050409 Fund #22 0.514196748
Fund #23 0.744336757 Fund #24 0.202843746
Fund #25 0.421641744 Fund #26 -0.585642472
Fund #27 0.582170503 Fund#28 0.35064364
Fund #29 0.65484412 Fund #30 -0.083269169
________________________________________________________________________
Upon inspection, the only detectable pattern in the behavior of the correlations is that, as
the fund number increases, that is, as the Sharpe Ratio decreases, the correlation between
the given fund and the market tends to grow larger. The increase in the correlation could
indicate that many of the fund selections, especially those with decreased Sharpe Ratios
that more closely resemble the volatility of the stock market, might be characterized by
investments intentionally designed to mimic the behavior of the market. In fact, the time
plots comparing the excess market returns with the excess fund returns corroborate the
suspicion that many of the selected funds were designed thusly. Consider the following
6
selected examples shown below.
Figure 1. Fund 2 Performance Comparison_____________________________________
Figure 2. Fund 6 Performance Comparison_____________________________________
7
Figure 3. Fund 10 Performance Comparison____________________________________
Figure 4. Fund 16 Performance Comparison____________________________________
8
Figure 5. Fund 23 Performance Comparison____________________________________
Figure 6. Fund 29 Performance Comparison____________________________________
9
Similarly, many of the funds appear to move negatively with the market by construct, and
the remainder, of course, appear to move neither with the market nor against the market,
and there are a significant number of these graphs seemingly unconnected to the market
movement in the 30 funds selected. The reader can observe the graphs for himself on
page 74.
Variances
Interestingly, of all the first 10 hedge funds, the average annualized monthly return
exceeded that of the market, yet, as the reader can quickly verify from the time plots, the
variances are extremely small for most of the first 10 hedge funds compared to the
variance of the market. Thus, the hedge fund investments with high Sharpe Ratios
offered both exceptionally-lower risk and higher returns than the market. Consider the
raw variance data for the first 10 hedge funds and the average hedge fund:
Table 2. Fund Variance and Average Return____________________________________
Market Variance: 2415.999957 Average Market Return: 9.21525
Ave. Fund Variance: 753.2803166 Ave. Fund Return: 16.99337751
Fund #1 Variance: 11.70683544 Average Return: 8.890983447
Fund #2 Variance: 726.1629818 Average Return: 26.14545455
Fund #3 Variance: 51.82846841 Average Return: 11.27661972
Fund #4 Variance: 453.5904889 Average Return: 14.82
Fund #5 Variance: 938.9087074 Average Return: 17.65830508
10
Fund #6 Variance: 1524.90216 Average Return: 19.965
Fund #7 Variance: 363.6100871 Average Return: 11.8096
Fund #8 Variance: 653.8928485 Average Return: 13.65818182
Fund #9 Variance: 1908.131577 Average Return: 19.17795918
Fund #10 Variance: 1129.798794 Average Return: 15.16941176
________________________________________________________________________
Notice that for the average over the entire Sharpe Ratio spectrum of funds, the variance is
only 753 compared to 2415 for the S&P 500. Such a comparatively low variance
reinforces the position that the entire hedge fund market, even when fledgling hedge
funds with low Sharpe Ratios are included in the analysis, remains far less volatile than
the stock market. Another significant characteristic of this data is that, despite the low
variability compared to the market, the average annualized monthly return for these funds
with the highest Sharpe Ratios are typically higher than those latter 20 with the lower
Sharpe Ratios. Although the fact that all but fund one of the top ten Sharpe Ratio funds
exceeded the average market return of 9.21525 could be the result of a coincidental
selection of funds, a trend seems more likely that most funds in the market with favorable
Sharpe Ratios do not merely compromise high returns with excessively low volatility.
Hence, the evidence supports the hypothesis that the hedge fund market in general forms
a powerful apparatus for generating inordinate returns.
11
Regressions
Regressions of the excess annualized monthly fund returns on the excess annualized
monthly market returns were performed for all 30 hedge funds with the intention of
examining the results for the overall volatility of the hedge fund market as measured
against the stock market and the degree to which the overall hedge fund market moves
with the stock market when, in fact, the hedge fund market actually does move with the
stock market.
The regressions revealed that the volatility of the hedge fund market and the degree to
which the hedge fund market moves with the S&P 500 depend on the perspective from
which the regression results are considered. By averaging the excess returns across each
month and then regressing those average monthly returns on the excess S&P 500 returns,
results were determined for the general hedge fund market. Consider the graph of that
regression as shown below as an overview of the data.
12
Figure 7. Regression of the Average Fund Return on the S&P 500__________________
The regression equation demonstrates that the degree of movement in the hedge fund
market is not very responsive to the S&P 500. Specifically, an increase or decrease in
S&P 500 returns of 1% corresponds to an increase or decrease, respectively, of
only .3576% in the hedge fund market. The observation must be noted that the R2 value
is .428, which means that only 42.8% of the variation in the hedge fund market is being
explained by the regression. Furthermore, examining each of the hedge fund regressions
individually yields more perspective by revealing some potential hazards, but also some
detectable trends.
13
Inspection of the regressions shows that some patterns emerge. The funds with the
highest Sharpe Ratios tend to have, in terms of absolute value, the smallest beta
coefficients because low risk implies lower volatility. The following regression graph
illustrates the effect.
Figure 8. Regression of Fund 1 on the S&P 500_________________________________
As can be seen, the beta coefficient indicates that a change of 1% in the stock market
corresponds to a change of only 0.01%. Such a small coefficient might simply reflect a
hedge fund which is volatile but which has data points that are more randomly dispersed
thereby representing a fund which is neither highly positively nor highly negatively
correlated with the S&P 500. There exist a few regressions matching that description for
which polynomial regressions were fitted to the data for somewhat better results in the
last part of this section, but for many of the regressions with extremely low beta
coefficients, the time plots confirm that the low coefficients reflect low volatility. In the
case of fund 1 shown above, the associated time plot is shown below.
14
Figure 9. Time Plot Returns Comparison Between Fund 1 and the S&P 500___________
An additional example pair of graphs is shown below for fund 3.
Figure 10. Regression of Fund 3 on the S&P 500________________________________
15
Figure 11. Time Plot Comparison Between Fund 3 and the S&P 500________________
Traversing the list of funds toward the funds with lower Sharpe Ratios leads to
regressions with beta coefficients increasing in absolute value. Ultimately, the purpose of
illustrating how the Sharpe Ratios affect the beta coefficients is to add interpretive
meaning to the average excess fund regression. For example, a citation of the .428 beta
coefficient would be remiss without attributing some of that coefficient’s meaning to the
Sharp Ratio’s effect. A few examples of the increased beta coefficients are shown below.
16
Figure 12. Regression of Fund 16 on the S&P 500_______________________________
Figure 13. Regression of Fund 23 on the S&P 500_______________________________
17
Figure 14. Regression of Fund 26 on the S&P 500_______________________________
Of greatest importance regarding the trend toward increasing beta coefficients is that,
with the exception of a single graph, the highest coefficient of any of the regressions
is .7875, and consequently, the hedge fund market is significantly less volatile than the
stock market. In order to facilitate the attainment of some sense of the extent to which
the hedge fund market trails the increases and decreases of the stock market, the top 15
correlations, in terms of absolute value, from the section entitled “Correlations” above,
have been juxtaposed below with their corresponding fund numbers and associated
regression beta coefficients.
18
Table 3. Fund Correlations and Beta Coefficients________________________________
Fund #21: Correlation: .781 Beta Coefficient: .7249
Fund #23: Correlation: .7443 Beta Coefficient: .7522
Fund #29: Correlation: .654844 Beta Coefficient: .4571
Fund #2: Correlation: .649193 Beta Coefficient: .5707
Fund #16: Correlation: .62032 Beta Coefficient: .6344
Fund #11: Correlation: .60112 Beta Coefficient: .6685
Fund #26: Correlation: -.58564 Beta Coefficient: -1.2199
Fund #27: Correlation: .58217 Beta Coefficient: .3053
Fund #15: Correlation: .57089 Beta Coefficient: .7875
Fund #22: Correlation: .514196 Beta Coefficient: .5018
Fund #6. Correlation: .49726 Beta Coefficient: .5249
Fund #10: Correlation: .48676 Beta Coefficient: .5338
Fund #8: Correlation: .4705 Beta Coefficient: .401
Fund #18: Correlation: .456296 Beta Coefficient: .4087
Fund #14: Correlation: .4492 Beta Coefficient: .7716
________________________________________________________________________
19
The underlying assumption of analyzing these values is that the funds with the highest
correlations follow the market either naturally or by design such that the results of the
analysis can be used as predictors of the degree to which the broader market of hedge
funds that mimic the S&P 500 follows the stock market. A cursory overview of the data
shows that the beta coefficients are quite high in terms of absolute value, but none of
them, except fund #26, exceeds .8 indicating that hedge funds might actually be a safer
investment than the stock market.
The reader must be cognizant of some discrepancies in the regressions and data. First,
some of the regressions are based on a limited number of performance data. This
deficiency is attributable to the fact that many of the hedge funds that were selected have
not long been in existence, so the sum of 12 data points per year is not many points to
plot over the course of three or fewer years. Second, many of the regressions have very
low R2 values. The fact that so many of the regressions have these low values is
especially disturbing because there is no immediate, non-statistical way to account for the
proportion of the regression attributable to error. There is one statistical remedy,
however, that has been employed in the regression graphs on page 57: since many of the
graphs seemed to exhibit non-linear trends, polynomial curves were fitted to the data and
generated some improvement in the R2 values. Some of the scatter plots, however, were
so widely dispersed that even polynomials of orders five and six, which most uniquely
fitted the data, did not yield much improvement in R2. Moreover, interpretation of the
polynomial regressions becomes unwieldy at the higher powers. All of the polynomial
curves are only of order two, and in most of the regressions, the coefficient of the
variable raised to the first power is greater than the beta coefficient in the linear fit, and
20
the coefficients in the polynomial regressions are either both positive or both negative, so
the reader can interpret those results as meaning that a 1% increase or decrease in the
S&P 500 corresponds to at least an increase or at least a decrease of the coefficient of the
variable raised to the first power.
Heteroscedasticity
Scatter plots were derived by first obtaining the residuals from regressing each of the
funds on the S&P 500, squaring the residuals, and then plotting those squared residuals
against the S&P 500 for the purpose of determining the presence or absence of
heteroscedasticity. The reader can see the graphs and SAS regression tables on page 91.
An inspection of the graphs shows that the presence of heteroscedasticity is very weak.
Two primary factors to explain why the variability in hedge fund performance is so
weakly related to the performance of the stock market are immediately suspects. First,
hedge fund investors are typically required to relinquish control over the money invested
for a period of six months or a year unless the investors are willing to accept a penalty for
withdrawing in the middle of that time interval, so whereas stock market investors may
invest and withdraw continually, hedge fund managers can continue their investment
strategy with impunity. Second, because access to hedge fund investing is extremely
limited, and because hedge funds are not required to report their performance to the SEC
and, by extension, the general public, hedge funds are not subject to the same nature of
stock market speculation as issuers of stock are subject to. These two aforementioned
possible reasons, however, imply only that hedge funds are facilitated, not coerced, to
lack a strong presence of heteroscedasticity with the stock market. For example, as will
be shown further into this section, if the nature of a hedge fund, perhaps by construction,
21
is to respond to the stock market, then some meaningful degree of predictive force of the
variability in a hedge fund might exist. Regardless, observation of the scatter plot for
the squared residuals associated with the average fund confirms that, although the
residuals are somewhat widely dispersed or that there are some outliers, depending on
how the graph is interpreted, there is simply no pronounced directional pattern of the
residuals other than strictly homoscedastic horizontal movement. Most of the individual
plots are constituted similarly to this average fund plot.
Before proceeding, the reader should consider the following table containing the betas
from regressing the squared residuals on the S&P 500 as discussed at the beginning of
this section. The funds, from which the residuals were originally obtained when the
funds were regressed on the S&P 500, are numbered in the left column, and the p-values
for the t-tests on the betas, obtained from regressing the squared residuals on the S&P
500, are in the rightmost column.
Table 4. Betas and p-values of Regressing the Squared Residuals on the S&P 500______
Fund #1: Beta: .02147 p-value: .3414
Fund #2: Beta: -.35204 p-value: .9110
Fund #3: Beta: -.95092 p-value: .0482
Fund #4: Beta: .29539 p-value: .9208
Fund #5: Beta: -14.92568 p-value: .0559
Fund #6: Beta: -11.25687 p-value: .3120
22
Fund #7: Beta: -.78789 p-value: .2328
Fund #8: Beta: 4.70992 p-value: .4046
Fund #9: Beta: 3.86371 p-value: .5341
Fund #10: Beta: -18.79702 p-value: .0031
Fund #11: Beta: -2.54423 p-value: .7317
Fund #12: Beta: -.31229 p-value: .7679
Fund #13: Beta: 1.80260 p-value: .9044
Fund #14: Beta: 1.99910 p-value: .9409
Fund #15: Beta: 12.07186 p-value: .5855
Fund #16: Beta: -13.25197 p-value: .1044
Fund #17: Beta: -7.80340 p-value: .6205
Fund #18: Beta: 8.00835 p-value: .2370
Fund #19: Beta: -2.39933 p-value: .1891
Fund #20: Beta: 4.56654 p-value: .5358
Fund #21: Beta: -6.43843 p-value: .0260
Fund #22: Beta: -22.24671 p-value: .0034
Fund #23: Beta: -2.16036 p-value: .3398
23
Fund #24: Beta: 3.1539 p-value: .1856
Fund #25: Beta: -.01672 p-value: .9693
Fund #26: Beta: -46.81454 p-value: .0245
Fund #27: Beta: -.02682 p-value: .9847
Fund #28: Beta: -32.74194 p-value: .1814
Fund #29: Beta: -4.2439 p-value: .2619
Fund #30: Beta: .33543 p-value: .8406
Average Fund:Beta: -3.41761 p-value: .0215
If the betas were consistently positive or consistently negative, then inference could be
made about the volatility of the hedge fund market when the stock market performed well
or poorly, but 30% of the hedge funds had positive betas which is a percentage too high
to conclude that there is a definitive trend. However, the absence of a trend in the
entirety of the hedge fund market does not imply such an absence in any individual fund,
and, in fact, fund 26 provides an excellent illustration. The graph is shown on the next
page. Fund 26 was chosen because it contains a sufficiently-large number of data points,
a large beta coefficient in terms of absolute value, and an R2 value relatively larger than
the other funds with similar numbers of data points: few data points can exaggerate the
regression results, a large beta indicates that the variability of the fund increases or
decreases with a movement in the stock market, and a higher R2 value suggests that more
24
of the changing variability in the fund is attributable to the stock market rather than
Figure 15. The Original Fund 26 Trend Line___________________________________
Figure 16. Performance Comparison For Fund 26_______________________________
25
random error. Now let the reader consider the time plot performance comparison
between fund 26 and the S&P 500. It is shown above. Whenever the S&P 500 is in a
state of decreasing, the performance of the fund tends to be extremely positive or
extremely negative. Thus, whereas the betas did not yield any trend, predicting the
variability in individual funds is possible.
Generally, the graphs tended to be homoscedastic. Some choice examples are given in
the graphs below.
Figure 17. Fund 13 Example of Homoscedasticity______________________________
26
Figure 18. Fund 15 Example of Homoscedasticity______________________________
Some of the graphs were less obviously homoscedastic for reasons that were common to
other funds with similar characteristics. Fund 24 below is the first example. There
Figure 19. Fund 24 Example of Aberrant Homoscedasticity_______________________
27
appear to be outliers present in this graph, but beware that this appearance is illusory,
because the scale on the vertical axis does not extend far compared to other graphs
suffering true outlier effects. Fund 6 below is the second example. Because there are so
Figure 20. Fund 6 Example of Insufficient Sample Size__________________________
few data points available, a solid homoscedastic or heteroscedastic trend simply cannot
be known.
Although the scatter plots do not seem to assert the presence of heteroscedasticity, the
betas for funds 3, 10, 21, 22, and 26 and for the average fund were statistically significant
at the .05 level as computed in SAS. Every one of these funds, however, becomes
statistically insignificant when an outlier is removed. Fund 21 functions as an ideal
example. The fund 21 graph is below, and the outlier can clearly be seen in the upper left
corner. When the graph is recomputed without the outlier, not only does the beta become
statistically insignificant, but also the beta, R2, and intercept are changed dramatically.
28
Figure 21. Fund 21 Squared Residuals With the Outlier vs. S&P 500_______________
Figure 22. Fund 21 Squared Residuals Without the Outlier vs. S&P 500_____________
29
The results can be seen in the graph above. The table below has been prepared to show
relevant information and the t-statistics for the data sets excluding the outliers.
Table 5. t-Statistics and Other Relevant Information for the Modified Graphs________
Fund critical t new t df(n-2) new beta new intercept new R2
#3: 1.980 > .547058 68 -.77692 48.54227 .0327
#10: 2.021 > 1.65064 31 -9.034 730.2802 .0808
#21: 1.960 > .738383 144 1.509757 741.10505 .0038
#22: 1.960 > .613519 135 -3.56885 1614.52462 .0028
#26: 1.980 > 1.43195 65 -29.0355 4401.04655 .0306
Ave. 1.960 > .123604 150 .1486 373.7932 .0001
Because the betas all become statistically insignificant when an outlier is removed from
each of the funds, the p-values generated from the data sets including the outliers are
spurious detectors of heteroscedasticity, and the trend in the data does in fact appear to be
horizontal and homoscedastic in each of these funds in the table.
Normality
In order to test whether the assumptions of the classical regression model were
satisfied, a test of normality for the residuals was conducted by regressing all 30 of the
funds and the average fund on the S&P 500 and plotting the residuals into histograms.
The graphs can be seen on page 40. With the exception of funds 1, 6, 11, 14, 18, 23, and
30
24, all of the histograms conformed reasonably well to the normal curve, and in most of
those funds which did not readily conform, there were so few residual data that
concluding that the fund residuals were either normally distributed or not normally
distributed in future trends was premature. In particular, funds 1, 2, 14, and 18 are
represented by too few residuals. A normal fit for funds 6, 11, and 24 might actually be
deemed acceptable, but the shape isn’t as pronounced as it is for the other funds. Fund 23
is the only case for which there were a sufficient number of residuals available to exclude
an obviously normal fit. Most importantly, the average fund residuals appeared to
assume a very strong normal curve shape, and because many of the arguments presented
in this study have been embodied and buttressed by the results of the average fund
regression, the weights of those arguments are more securely anchored. Some sample
graphs depicting the strong tendency toward a normal curve shape, especially for the
average fund, are shown below.
Figure 23. Fund 7 Normal Shaped Residuals___________________________________
31
Figure 24. Fund 21 Normal Shaped Residuals__________________________________
Figure 25. Fund 27 Normal Shaped Residuals__________________________________
32
Figure 26. Average Fund Normal Shaped Residuals______________________________
Conclusion
When all the aspects of hedge fund performance are assessed collectively, the hedge
fund market dramatically outperforms the stock market. The average returns discussed in
the introduction show that, purely in terms of generating profit, the hedge fund market
outstripped the S&P 500. In terms of volatility, the hedge fund market performance
again defeated the market. Specifically, the variance data showed that most of the funds,
especially for high Sharpe Ratios, were far less variable and yielded greater returns than
the S&P 500. Moreover, the regression analysis confirmed that the hedge fund market as
a whole remained less volatile than the S&P 500 and that, for those funds highly
correlated to the stock market, the performance fluctuations were more dampened than
the stock market. In addition to the performance and volatility attributes, hedge funds did
33
not exhibit any heteroscedasticity which effectively translates into more stable
expectations on returns because of the independent nature of hedge fund operations. All
of these performance qualities affirm the superiority of the hedge fund market over the
stock market.
34
Descriptive Statistics
Complete tables of the statistical measures used in the study are given here for the
reader who wishes to gain comprehensive insight into the arguments that were proposed.
35
Table 6. Performance Averages______________________________________________
S&P 500: 9.21525 Fund #1: 11.70683544
Fund #2: 26.14545455 Fund #3: 11.27661972
Fund #4: 14.82 Fund #5: 17.65830508
Fund #6: 19.965 Fund #7: 11.8096
Fund #8: 13.65818182 Fund #9: 19.17795918
Fund #10: 15.16941176 Fund #11: 15.28421053
Fund #12: 8.902702703 Fund #13: 21.87630252
Fund #14: 18.70909091 Fund #15: 19.87175258
Fund #16: 15.20047059 Fund #17: 14.57454545
Fund #18: 11.286 Fund #19: 7.771111111
Fund #20: 12.79418182 Fund #21: 12.28
Fund #22: 12.40956522 Fund #23: 9.06375
Fund #24: 7.451320755 Fund #25: 6.177391304
Fund #26: 12.76058824 Fund #27: 6.142702703
Fund #28: 8.341132075 Fund #29: 5.942608696
Fund #30: 5.215 Average Fund:16.99337751
36
________________________________________________________________________
Table 7. Variance_________________________________________________________
S&P 500: 2415.999957 Fund #1: 8.890983447
Fund #2: 726.1629818 Fund #3: 51.82846841
Fund #4: 453.5904889 Fund #5: 938.9087074
Fund #6: 1524.90216 Fund #7: 363.6100871
Fund #8: 653.8928485 Fund #9: 1908.131577
Fund #10: 1129.798794 Fund #11: 1126.69836
Fund #12: 193.7927049 Fund #13: 3808.637586
Fund #14: 3189.621818 Fund #15: 4362.841398
Fund #16: 2253.336476 Fund #17: 2108.215882
Fund #18: 993.25512 Fund #19: 250.9952252
Fund #20: 2125.89921 Fund #21: 2205.50663
Fund #22: 2575.932906 Fund #23: 985.9317532
Fund #24: 420.0005155 Fund #25: 137.1764503
Fund #26: 7179.599546 Fund #27: 254.4790703
Fund #28: 3352.015176 Fund #29: 555.4719747
37
Fund #30: 253.566817 Average Fund:753.2803166
________________________________________________________________________
Table 8. Fund Correlation with the S&P 500___________________________________
Fund #1: -0.173464524 Fund #2: 0.649193451
Fund #3: -0.044247993 Fund #4: -0.0714241
Fund #5: -0.207737278 Fund #6: 0.497261555
Fund #7: 0.414375015 Fund #8: 0.470509021
Fund #9: 0.253269275 Fund #10: 0.486760127
Fund #11: 0.601120061 Fund #12: -0.190548726
Fund #13: 0.407577199 Fund #14: 0.44922268
Fund #15: 0.570891982 Fund #16: 0.620327411
Fund #17: 0.298016148 Fund #18: 0.45629676
Fund #19: 0.262008673 Fund #20: 0.409339549
Fund #21: 0.781050409 Fund #22: 0.514196748
Fund #23: 0.744336757 Fund #24: 0.202843746
Fund #25: 0.421641744 Fund #26: -0.585642472
Fund #27: 0.582170503 Fund #28: 0.35064364
Fund #29: 0.65484412 Fund #30: -0.083269169
38
Average Fund: 0.654979938 __________________________________________
Residual Histograms
The residuals from regressing each of the funds on the S&P 500 were plotted and
placed into histograms as given here.
39
Figure 27. Fund 1 Residuals________________________________________________
Figure 28. Fund 2 Residuals________________________________________________
40
Figure 29. Fund 3 Residuals________________________________________________
Figure 30. Fund 4 Residuals________________________________________________
41
Figure 31. Fund 5 Residuals________________________________________________
Figure 32. Fund 6 Residuals________________________________________________
42
Figure 33. Fund 7 Residuals________________________________________________
Figure 34. Fund 8 Residuals________________________________________________
43
Figure 35. Fund 9 Residuals________________________________________________
Figure 36. Fund 10 Residuals_______________________________________________
44
Figure 37. Fund 11 Residuals_______________________________________________
Figure 38. Fund 12 Residuals_______________________________________________
45
Figure 39. Fund 13 Residuals_______________________________________________
Figure 40. Fund 14 Residuals_______________________________________________
46
Figure 41. Fund 15 Residuals_______________________________________________
Figure 42. Fund 16 Residuals_______________________________________________
47
Figure 43. Fund 17 Residuals_______________________________________________
Figure 44. Fund 18 Residuals_______________________________________________
48
Figure 45. Fund 19 Residuals_______________________________________________
Figure 46. Fund 20 Residuals_______________________________________________
49
Figure 47. Fund 21 Residuals_______________________________________________
Figure 48. Fund 22 Residuals_______________________________________________
50
Figure 49. Fund 23 Residuals_______________________________________________
Figure 50. Fund 24 Residuals_______________________________________________
51
Figure 51. Fund 25 Residuals_______________________________________________
Figure 52. Fund 26 Residuals_______________________________________________
52
Figure 53. Fund 27 Residuals_______________________________________________
Figure 54. Fund 28 Residuals_______________________________________________
53
Figure 55. Fund 29 Residuals_______________________________________________
Figure 56. Fund 30 Residuals_______________________________________________
54
Figure 57. Average Fund Residuals___________________________________________
55
Regressions
These graphs are the result of regressing each of the funds on the S&P 500 and then determining a regression line. In many of the graphs, polynomial regressions were also determined and plotted as curves.
56
Figure 58. Fund 1 Regression_______________________________________________
57
Figure 59. Fund 2 Regression_______________________________________________
Figure 60. Fund 3 Regression_______________________________________________
58
Figure 61. Fund 4 Regression_______________________________________________
Figure 62. Fund 5 Regression_______________________________________________
59
Figure 63. Fund 6 Regression_______________________________________________
Figure 64. Fund 7 Regression_______________________________________________
60
Figure 65. Fund 8 Regression_______________________________________________
Figure 66. Fund 9 Regression_______________________________________________
61
Figure 67. Fund 10 Regression______________________________________________
Figure 68. Fund 11 Regression______________________________________________
62
Figure 69. Fund 12 Regression______________________________________________
Figure 70. Fund 13 Regression______________________________________________
63
Figure 71. Fund 14 Regression______________________________________________
Figure 72. Fund 15 Regression______________________________________________
64
Figure 73. Fund 16 Regression______________________________________________
Figure 74. Fund 17 Regression______________________________________________
65
Figure 75. Fund 18 Regression______________________________________________
Figure 76. Fund 19 Regression______________________________________________
66
Figure 77. Fund 20 Regression______________________________________________
Figure 78. Fund 21 Regression______________________________________________
67
Figure 79. Fund 22 Regression______________________________________________
Figure 80. Fund 23 Regression______________________________________________
68
Figure 81. Fund 24 Regression______________________________________________
Figure 82. Fund 25 Regression______________________________________________
69
Figure 83. Fund 26 Regression______________________________________________
Figure 84. Fund 27 Regression______________________________________________
70
Figure 85. Fund 28 Regression______________________________________________
Figure 86. Fund 29 Regression______________________________________________
71
Figure 87. Fund 30 Regression______________________________________________
Figure 88. Average Fund Regression__________________________________________
72
Time Plots
These plots compare the performance of each of the funds and the S&P 500 over time.
73
Figure 89. Fund 1 Time Plot Comparison______________________________________
74
Figure 90. Fund 2 Time Plot Comparison______________________________________
Figure 91. Fund 3 Time Plot Comparison______________________________________
75
Figure 92. Fund 4 Time Plot Comparison______________________________________
Figure 93. Fund 5 Time Plot Comparison______________________________________
76
Figure 94. Fund 6 Time Plot Comparison______________________________________
Figure 95. Fund 7 Time Plot Comparison______________________________________
77
Figure 96. Fund 8 Time Plot Comparison______________________________________
Figure 97. Fund 9 Time Plot Comparison______________________________________
78
Figure 98. Fund 10 Time Plot Comparison_____________________________________
Figure 99. Fund 11 Time Plot Comparison_____________________________________
79
Figure 100. Fund 12 Time Plot Comparison____________________________________
Figure 101. Fund 13 Time Plot Comparison____________________________________
80
Figure 102. Fund 14 Time Plot Comparison____________________________________
Figure 103. Fund 15 Time Plot Comparison____________________________________
81
Figure 104. Fund 16 Time Plot Comparison____________________________________
Figure 105. Fund 17 Time Plot Comparison____________________________________
82
Figure 106. Fund 18 Time Plot Comparison____________________________________
Figure 107. Fund 19 Time Plot Comparison____________________________________
83
Figure 108. Fund 20 Time Plot Comparison____________________________________
Figure 109. Fund 21 Time Plot Comparison____________________________________
84
Figure 110. Fund 21 Time Plot Comparison____________________________________
Figure 111. Fund 21 Time Plot Comparison____________________________________
85
Figure 112. Fund 21 Time Plot Comparison____________________________________
Figure 113. Fund 21 Time Plot Comparison____________________________________
86
Figure 114. Fund 21 Time Plot Comparison____________________________________
Figure 115. Fund 27 Time Plot Comparison____________________________________
87
Figure 116. Fund 28 Time Plot Comparison____________________________________
Figure 117. Fund 29 Time Plot Comparison____________________________________
88
Figure 118. Fund 30 Time Plot Comparison____________________________________
Figure 119. Average Fund Time Plot Comparison_______________________________
89
Heteroscedasticity graphs and SAS output tables
In order to determine the regression graphs discussed and shown in the body of this
paper, each of the funds was regressed on the S&P 500. The SAS output tables in this
section are the result of squaring the residuals from those regressions, and then regressing
the squared residuals on the S&P 500 for the purpose of analyzing the p-values of the test
statistic for the betas. The graphs are the result of plotting the squared residuals vs. the
S&P 500.
90
Table 9. Squared Residuals on Fund 1_________________________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 79 Number of Observations Used 79
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 70.77308 70.77308 0.92 0.3417 Error 77 5954.70142 77.33378 Corrected Total 78 6025.47451
Root MSE 8.79396 R-Square 0.0117 Dependent Mean 8.48215 Adj R-Sq -0.0011 Coeff Var 103.67613
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 8.52905 0.99061 8.61 <.0001excess_market_return excess_market_return 1 0.02147 0.02244 0.96 0.3417
Figure 120. Squared Residuals against Fund 1__________________________________
91
Table 10. Squared Residuals on Fund 2________________________________________
Figure 121. Squared Residuals Against Fund 2__________________________________
92
Table 11. Squared Residuals on Fund 3________________________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 71 Number of Observations Used 71
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 113937 113937 4.05 0.0482 Error 69 1942836 28157 Corrected Total 70 2056773
Root MSE 167.80060 R-Square 0.0554 Dependent Mean 51.70812 Adj R-Sq 0.0417 Coeff Var 324.51500
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 50.63853 19.92136 2.54 0.0133excess_market_return excess_market_return 1 -0.95092 0.47272 -2.01 0.0482
Figure 122. Squared Residuals Against Fund 3__________________________________
93
Table 12. Squared Residuals on Fund 4________________________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 82 Number of Observations Used 82
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 13521 13521 0.01 0.9208 Error 80 108791378 1359892 Corrected Total 81 108804898
Root MSE 1166.14417 R-Square 0.0001 Dependent Mean 445.13084 Adj R-Sq -0.0124 Coeff Var 261.97784
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 445.98346 129.06266 3.46 0.0009excess_market_return excess_market_return 1 0.29539 2.96242 0.10 0.9208
Figure 123. Squared Residuals Against Fund 4__________________________________
Table 13. Squared Residuals on Fund 5________________________________________
94
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 59 Number of Observations Used 59
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 13202690 13202690 3.81 0.0559 Error 57 197554173 3465863 Corrected Total 58 210756862
Root MSE 1861.68275 R-Square 0.0626 Dependent Mean 883.62026 Adj R-Sq 0.0462 Coeff Var 210.68810
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 961.50702 245.63372 3.91 0.0002excess_market_return excess_market_return 1 -14.92568 7.64731 -1.95 0.0559
Figure 124. Squared Residuals Against Fund 5__________________________________
Table 14. Squared Residuals on Fund 6________________________________________
The REG Procedure Model: MODEL1
95
Dependent Variable: squaredResidual
Number of Observations Read 16 Number of Observations Used 16
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 2539867 2539867 1.10 0.3120 Error 14 32314491 2308178 Corrected Total 15 34854358
Root MSE 1519.26887 R-Square 0.0729 Dependent Mean 1074.69491 Adj R-Sq 0.0066 Coeff Var 141.36746
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 1000.51918 386.34344 2.59 0.0214excess_market_return excess_market_return 1 -11.25687 10.73116 -1.05 0.3120
Figure 125. Squared Residuals Against Fund 6__________________________________
Table 15. Squared Residuals on Fund 7________________________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
96
Number of Observations Read 150 Number of Observations Used 150
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 228957 228957 1.44 0.2328 Error 148 23604159 159488 Corrected Total 149 23833116
Root MSE 399.35893 R-Square 0.0096 Dependent Mean 291.99981 Adj R-Sq 0.0029 Coeff Var 136.76685
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 295.03138 32.70554 9.02 <.0001excess_market_return excess_market_return 1 -0.78789 0.65758 -1.20 0.2328
Figure 126. Squared Residuals Against Fund 7__________________________________
Table 16. Squared Residuals on Fund 8________________________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 55 Number of Observations Used 55
97
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 1069506 1069506 0.71 0.4046 Error 53 80292858 1514960 Corrected Total 54 81362364
Root MSE 1230.83694 R-Square 0.0131 Dependent Mean 499.47060 Adj R-Sq -0.0055 Coeff Var 246.42831
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 488.21475 166.50580 2.93 0.0050excess_market_return excess_market_return 1 4.70992 5.60560 0.84 0.4046
Figure 127. Squared Residuals Against Fund 8__________________________________
Table 17. Fund 9 Squared Residuals on the S&P 500_____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 147 Number of Observations Used 147
98
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 5573957 5573957 0.39 0.5341 Error 145 2080216229 14346319 Corrected Total 146 2085790186
Root MSE 3787.65347 R-Square 0.0027 Dependent Mean 1764.95251 Adj R-Sq -0.0042 Coeff Var 214.60371
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 1756.26783 312.71094 5.62 <.0001excess_market_return excess_market_return 1 3.86371 6.19859 0.62 0.5341
Figure 128. Fund 9 Squared Residuals Against the S&P 500_______________________
Table 18. Fund 10 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 34 Number of Observations Used 34
Analysis of Variance
99
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 10785697 10785697 10.21 0.0031 Error 32 33799777 1056243 Corrected Total 33 44585475
Root MSE 1027.73685 R-Square 0.2419 Dependent Mean 835.55238 Adj R-Sq 0.2182 Coeff Var 123.00089
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 850.42414 176.31685 4.82 <.0001excess_market_return excess_market_return 1 -18.79702 5.88230 -3.20 0.0031
Figure 129. Fund 10 Squared Residuals Against the S&P 500______________________
Table 19. Fund 11 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 38 Number of Observations Used 38
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
100
Model 1 217459 217459 0.12 0.7317 Error 36 65562859 1821191 Corrected Total 37 65780317
Root MSE 1349.51492 R-Square 0.0033 Dependent Mean 701.61839 Adj R-Sq -0.0244 Coeff Var 192.34315
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 698.60214 219.09418 3.19 0.0030excess_market_return excess_market_return 1 -2.54423 7.36285 -0.35 0.7317
Figure 130. Fund 11 Squared Residuals Against the S&P 500______________________
Table 20. Fund 12 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 74 Number of Observations Used 74
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 12574 12574 0.09 0.7679
101
Error 72 10313729 143246 Corrected Total 73 10326302
Root MSE 378.47884 R-Square 0.0012 Dependent Mean 181.82405 Adj R-Sq -0.0127 Coeff Var 208.15664
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 181.42427 44.01796 4.12 <.0001excess_market_return excess_market_return 1 -0.31229 1.05406 -0.30 0.7679
Figure 131. Fund 12 Squared Residuals Against the S&P 500______________________
Table 21. Fund 13 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 119 Number of Observations Used 119
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 999622 999622 0.01 0.9044 Error 117 8072698768 68997425 Corrected Total 118 8073698390
102
Root MSE 8306.46889 R-Square 0.0001 Dependent Mean 3136.72300 Adj R-Sq -0.0084 Coeff Var 264.81359
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 3140.00131 761.93971 4.12 <.0001excess_market_return excess_market_return 1 1.80260 14.97608 0.12 0.9044
Figure 132. Fund 13 Squared Residuals Against the S&P 500______________________
Table 22. Fund 14 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 22 Number of Observations Used 22
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 90674 90674 0.01 0.9409 Error 20 321555936 16077797 Corrected Total 21 321646610
103
Root MSE 4009.71281 R-Square 0.0003 Dependent Mean 2431.52719 Adj R-Sq -0.0497 Coeff Var 164.90512
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 2433.15191 855.14736 2.85 0.0100excess_market_return excess_market_return 1 1.99910 26.61988 0.08 0.9409
Figure 133. Fund 14 Squared Residuals Against the S&P 500______________________
Table 23. Fund 15 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 97 Number of Observations Used 97
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 32361346 32361346 0.30 0.5855 Error 95 10263473634 108036565 Corrected Total 96 10295834980
Root MSE 10394 R-Square 0.0031 Dependent Mean 2911.99771 Adj R-Sq -0.0074
104
Coeff Var 356.93929
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 2966.53641 1060.05147 2.80 0.0062excess_market_return excess_market_return 1 12.07186 22.05700 0.55 0.5855
Figure 134. Fund 15 Squared Residuals Against the S&P 500______________________
Table 24. Fund 16 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 85 Number of Observations Used 85
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 31913896 31913896 2.70 0.1044 Error 83 982792968 11840879 Corrected Total 84 1014706864
Root MSE 3441.05785 R-Square 0.0315 Dependent Mean 1369.74415 Adj R-Sq 0.0198 Coeff Var 251.21902
105
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 1330.11919 374.01473 3.56 0.0006excess_market_return excess_market_return 1 -13.25197 8.07203 -1.64 0.1044
Figure 135. Fund 16 Squared Residuals Against the S&P 500______________________
Table 25. Fund 17 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 33 Number of Observations Used 33
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 1798265 1798265 0.25 0.6205 Error 31 222881127 7189714 Corrected Total 32 224679392
Root MSE 2681.36417 R-Square 0.0080 Dependent Mean 1858.27275 Adj R-Sq -0.0240 Coeff Var 144.29336
Parameter Estimates
106
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 1857.09987 466.77148 3.98 0.0004excess_market_return excess_market_return 1 -7.80340 15.60318 -0.50 0.6205
Figure 136. Fund 17 Squared Residuals Against the S&P 500______________________
Table 26. Fund 18 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 20 Number of Observations Used 20
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 1464962 1464962 1.50 0.2370 Error 18 17623855 979103 Corrected Total 19 19088816
Root MSE 989.49635 R-Square 0.0767 Dependent Mean 746.42674 Adj R-Sq 0.0255 Coeff Var 132.56443
Parameter Estimates
Parameter Standard
107
Variable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 756.77754 221.41987 3.42 0.0031excess_market_return excess_market_return 1 8.00835 6.54703 1.22 0.2370
Figure 137. Fund 18 Squared Residuals Against the S&P 500______________________
Table 27. Fund 19 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 54 Number of Observations Used 54
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 274772 274772 1.77 0.1891 Error 52 8070837 155208 Corrected Total 53 8345609
Root MSE 393.96497 R-Square 0.0329 Dependent Mean 228.85600 Adj R-Sq 0.0143 Coeff Var 172.14535
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
108
Intercept Intercept 1 234.28782 53.76705 4.36 <.0001excess_market_return excess_market_return 1 -2.39933 1.80327 -1.33 0.1891
Figure 138. Fund 19 Squared Residuals Against the S&P 500______________________
Table 28. Fund 20 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 110 Number of Observations Used 110
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 5211721 5211721 0.39 0.5358 Error 108 1458515975 13504778 Corrected Total 109 1463727695
Root MSE 3674.88470 R-Square 0.0036 Dependent Mean 1751.62510 Adj R-Sq -0.0057 Coeff Var 209.79859
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 1765.18398 351.06564 5.03 <.0001excess_market_return excess_market_return 1 4.56654 7.35090 0.62 0.5358
109
Figure 139. Fund 20 Squared Residuals Against the S&P 500______________________
Table 29. Fund 21 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 147 Number of Observations Used 147
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 15477978 15477978 5.06 0.0260 Error 145 443496940 3058600 Corrected Total 146 458974918
Root MSE 1748.88524 R-Square 0.0337 Dependent Mean 855.09146 Adj R-Sq 0.0271 Coeff Var 204.52610
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 869.56348 144.38901 6.02 <.0001excess_market_return excess_market_return 1 -6.43843 2.86210 -2.25 0.0260
110
Figure 140. Fund 21 Squared Residuals Against the S&P 500______________________
Table 30. Fund 22 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 138 Number of Observations Used 138
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 181893582 181893582 8.87 0.0034 Error 136 2787682139 20497663 Corrected Total 137 2969575721
Root MSE 4527.43446 R-Square 0.0613 Dependent Mean 1878.57795 Adj R-Sq 0.0543 Coeff Var 241.00328
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 1930.42408 385.79360 5.00 <.0001excess_market_return excess_market_return 1 -22.24671 7.46809 -2.98 0.0034
111
Figure 141. Fund 22 Squared Residuals Against the S&P 500______________________
Table 31. Fund 23 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 32 Number of Observations Used 32
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 137752 137752 0.94 0.3398 Error 30 4391339 146378 Corrected Total 31 4529091
Root MSE 382.59373 R-Square 0.0304 Dependent Mean 424.98352 Adj R-Sq -0.0019 Coeff Var 90.02555
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 424.92681 67.63368 6.28 <.0001excess_market_return excess_market_return 1 -2.16036 2.22697 -0.97 0.3398
112
Figure 142. Fund 23 Squared Residuals Against the S&P 500______________________
Table 32. Fund 24 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 53 Number of Observations Used 53
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 472317 472317 1.80 0.1856 Error 51 13378754 262329 Corrected Total 52 13851071
Root MSE 512.18015 R-Square 0.0341 Dependent Mean 397.06678 Adj R-Sq 0.0152 Coeff Var 128.99093
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 389.16096 70.59964 5.51 <.0001excess_market_return excess_market_return 1 3.15139 2.34860 1.34 0.1856
113
Figure 143. Fund 24 Squared Residuals Against the S&P 500______________________
Table 33. Fund 25 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 92 Number of Observations Used 92
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 56.50432 56.50432 0.00 0.9693 Error 90 3414573 37940 Corrected Total 91 3414629
Root MSE 194.78115 R-Square 0.0000 Dependent Mean 110.74397 Adj R-Sq -0.0111 Coeff Var 175.88420
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 110.67599 20.38359 5.43 <.0001excess_market_return excess_market_return 1 -0.01672 0.43335 -0.04 0.9693
114
Figure 144. Fund 25 Squared Residuals Against the S&P 500______________________
Table 34. Fund 26 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 68 Number of Observations Used 68
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 243245060 243245060 5.30 0.0245 Error 66 3030197047 45912076 Corrected Total 67 3273442107
Root MSE 6775.84507 R-Square 0.0743 Dependent Mean 4642.58884 Adj R-Sq 0.0603 Coeff Var 145.94971
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 4716.28044 822.31538 5.74 <.0001excess_market_return excess_market_return 1 -46.81454 20.33866 -2.30 0.0245
115
Figure 145. Fund 26 Squared Residuals Against the S&P 500______________________
Table 35. Fund 27 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 37 Number of Observations Used 37
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 23.34205 23.34205 0.00 0.9847 Error 35 2178579 62245 Corrected Total 36 2178602
Root MSE 249.48969 R-Square 0.0000 Dependent Mean 161.82420 Adj R-Sq -0.0286 Coeff Var 154.17329
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 161.81661 41.01772 3.95 0.0004excess_market_return excess_market_return 1 -0.02682 1.38501 -0.02 0.9847
116
Figure 146. Fund 27 Squared Residuals Against the S&P 500______________________
Table 36. Fund 28 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 53 Number of Observations Used 53
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 47308462 47308462 1.84 0.1814 Error 51 1314472819 25773977 Corrected Total 52 1361781281
Root MSE 5076.80774 R-Square 0.0347 Dependent Mean 2885.10628 Adj R-Sq 0.0158 Coeff Var 175.96606
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 2922.50623 697.89935 4.19 0.0001excess_market_return excess_market_return 1 -32.74194 24.16717 -1.35 0.1814
117
Figure 147. Fund 28 Squared Residuals Against the S&P 500______________________
Table 37. Fund 29 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 23 Number of Observations Used 23
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 443061 443061 1.33 0.2619 Error 21 7000840 333373 Corrected Total 22 7443901
Root MSE 577.38489 R-Square 0.0595 Dependent Mean 303.46275 Adj R-Sq 0.0147 Coeff Var 190.26549
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 291.76784 120.81971 2.41 0.0249excess_market_return excess_market_return 1 -4.24397 3.68134 -1.15 0.2619
118
Figure 148. Fund 29 Squared Residuals Against the S&P 500______________________
Table 38. Fund 30 Squared Residuals on the S&P 500____________________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 48 Number of Observations Used 48
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 4552.07495 4552.07495 0.04 0.8406 Error 46 5119032 111283 Corrected Total 47 5123584
Root MSE 333.59151 R-Square 0.0009 Dependent Mean 244.42716 Adj R-Sq -0.0208 Coeff Var 136.47890
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 244.30955 48.15330 5.07 <.0001excess_market_return excess_market_return 1 -0.33543 1.65847 -0.20 0.8406
119
Figure 149. Fund 30 Squared Residuals Against the S&P 500______________________
Table 39. “Average” Fund Squared Residuals on the S&P 500_____________________
The REG Procedure Model: MODEL1 Dependent Variable: squaredResidual
Number of Observations Read 153 Number of Observations Used 153
Analysis of Variance
Sum of Mean Source DF Squares Square F Value Pr > F
Model 1 4413953 4413953 5.40 0.0215 Error 151 123520256 818015 Corrected Total 152 127934209
Root MSE 904.44178 R-Square 0.0345 Dependent Mean 422.30449 Adj R-Sq 0.0281 Coeff Var 214.16817
Parameter Estimates
Parameter StandardVariable Label DF Estimate Error t Value Pr > |t|
Intercept Intercept 1 432.21711 73.24420 5.90 <.0001excess_market_return excess_market_return 1 -3.41761 1.47126 -2.32 0.0215
120
Figure 150. “Average” Fund Squared Residuals Against the S&P 500_______________
References
1. http://www.hedgefund.net/marketing_index.cfm?template=aboutus.html
2. Hedge Funds Fear New SEC Regulations
Investment Management Weekly; 7/7/2008, Vol. 21 Issue 27, p6-11, 2p
John Morgan
3. Hedge Funds: Risk and Return
http://www.princeton.edu/~ceps/workingpapers/104malkiel.pdf
Burton G. Malkiel, Princeton University
4. Hedge funds: Ten years of private client investing — is it still time to invest?
Botteron, Pascal pascal.botteron@db.com
Derivatives Use, Trading & Regulation; 2007, Vol. 12 Issue 4, p301-313, 13p
121
VITA
Graduate School
Southern Illinois University
Mark V. Hoaglund Date of Birth: June 5, 1977
203 West Jackson, Desoto, Il. 62924
Southern Illinois University at Carbondale
Bachelor of Science, Mathematics, May 2005
Research Paper Title:
Measuring the Performance of the Hedge Fund Market
Major Professor: Scott Gilbert