Measuring Hedge Fund Performance

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

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LIST OF FIGURES

FIGURE PAGE

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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

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
xiii

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.

xiv

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.

E ( Ri ) − Rf
The Sharp Ratio, mentioned in the introduction, is defined as where E(Ri) is
σ

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.

xv

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 known in this study,

3

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 funds were themselves

4

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_____________________________________

7

Figure 2. Fund 6 Performance Comparison_____________________________________

Figure 3. Fund 10 Performance Comparison____________________________________

8



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

10

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










________________________________________________________________________

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

11

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.

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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.

13

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.

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

14

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.

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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________________________________

16

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.

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Figure 12. Regression of Fund 16 on the S&P 500_______________________________


18


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.

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Table 3. Fund Correlations and Beta Coefficients________________________________

Fund #21: Correlation: .781 Beta Coefficient: .7249






Fund #26: Correlation: -.58564 Beta Coefficient: -1.2199




Fund #6. Correlation: .49726 Beta Coefficient: .5249





________________________________________________________________________

20

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 the coefficients in the

21

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, is to

22

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

23

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

24

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 of the changing variability in

25

the fund is attributable to the stock market rather than

Figure 15. The Original Fund 26 Trend Line___________________________________

26

Figure 16. Performance Comparison For Fund 26_______________________________

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.

27

Figure 17. Fund 13 Example of Homoscedasticity______________________________

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

28

Figure 19. Fund 24 Example of Aberrant Homoscedasticity_______________________

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

29

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.

Figure 21. Fund 21 Squared Residuals With the Outlier vs. S&P 500_______________

30

Figure 22. Fund 21 Squared Residuals Without the Outlier vs. S&P 500_____________

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________

new intercept new R2
Fund critical t new t df(n-2) new beta

#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

31

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 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.

32

Figure 23. Fund 7 Normal Shaped Residuals___________________________________

Figure 24. Fund 21 Normal Shaped Residuals__________________________________

33

Figure 25. Fund 27 Normal Shaped Residuals__________________________________

Figure 26. Average Fund Normal Shaped Residuals______________________________

34

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 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.

35

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.

36

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

37

________________________________________________________________________

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

38

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

39

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.

40

Figure 27. Fund 1 Residuals________________________________________________


41



42



43



44


Figure 36. Fund 10 Residuals_______________________________________________

45



46



47



48



49



50



51



52



53



54



55

Figure 57. Average Fund Residuals___________________________________________

56

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.

57

Figure 58. Fund 1 Regression_______________________________________________

58


59



61



63



65


Figure 67. Fund 10 Regression______________________________________________

67



69



71



73



75



77



79



81



83



84



85

Figure 88. Average Fund Regression__________________________________________

86

Time Plots
These plots compare the performance of each of the funds and the S&P 500 over time.

88

Figure 89. Fund 1 Time Plot Comparison______________________________________


90



91



92



93


Figure 98. Fund 10 Time Plot Comparison_____________________________________

94

Figure 99. Fund 11 Time Plot Comparison_____________________________________

Figure 100. Fund 12 Time Plot Comparison____________________________________

95



96



97



98



100



102



103



104



105



106

Figure 119. Average Fund Time Plot Comparison_______________________________

107

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.

108

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 Standard
Variable Label DF Estimate Error t Value Pr > |t|

Intercept Intercept 1 8.52905 0.99061 8.61 <.0001
excess_market_return excess_market_return 1 0.02147 0.02244 0.96 0.3417

Figure 120. Squared Residuals against Fund 1__________________________________

109

Table 10. Squared Residuals on Fund 2________________________________________

Figure 121. Squared Residuals Against Fund 2__________________________________

110

The REG Procedure
Model: MODEL1



Sum of Mean

Model 1 113937 113937 4.05 0.0482
Error 69 1942836 28157
Corrected Total 70 2056773

Dependent Mean 51.70812 Adj R-Sq 0.0417
Coeff Var 324.51500

Parameter Estimates

Parameter Standard

Intercept Intercept 1 50.63853 19.92136 2.54 0.0133
excess_market_return excess_market_return 1 -0.95092 0.47272 -2.01 0.0482


111

The REG Procedure
Model: MODEL1



Sum of Mean

Model 1 13521 13521 0.01 0.9208
Error 80 108791378 1359892

Coeff Var 261.97784

Parameter Estimates

Parameter Standard




112

The REG Procedure
Model: MODEL1



Sum of Mean

Model 1 13202690 13202690 3.81 0.0559
Error 57 197554173 3465863

Coeff Var 210.68810

Parameter Estimates

Parameter Standard



The REG Procedure
Model: MODEL1

113




Sum of Mean

Model 1 2539867 2539867 1.10 0.3120
Error 14 32314491 2308178

Coeff Var 141.36746

Parameter Estimates

Parameter Standard



The REG Procedure
Model: MODEL1

114



Sum of Mean

Model 1 228957 228957 1.44 0.2328
Error 148 23604159 159488

Coeff Var 136.76685

Parameter Estimates

Parameter Standard



The REG Procedure
Model: MODEL1


115


Sum of Mean

Model 1 1069506 1069506 0.71 0.4046
Error 53 80292858 1514960

Coeff Var 246.42831

Parameter Estimates

Parameter Standard



Table 17. Fund 9 Squared Residuals on the S&P 500_____________________________
The REG Procedure
Model: MODEL1



116

Sum of Mean

Model 1 5573957 5573957 0.39 0.5341
Error 145 2080216229 14346319

Coeff Var 214.60371

Parameter Estimates

Parameter Standard


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



Sum of Mean

117

Model 1 10785697 10785697 10.21 0.0031
Error 32 33799777 1056243

Coeff Var 123.00089

Parameter Estimates

Parameter Standard


Figure 129. Fund 10 Squared Residuals Against the S&P 500______________________

The REG Procedure
Model: MODEL1



Sum of Mean

Model 1 217459 217459 0.12 0.7317

118

Error 36 65562859 1821191

Coeff Var 192.34315

Parameter Estimates

Parameter Standard



The REG Procedure
Model: MODEL1



Sum of Mean

Model 1 12574 12574 0.09 0.7679
Error 72 10313729 143246

119

Coeff Var 208.15664

Parameter Estimates

Parameter Standard



The REG Procedure
Model: MODEL1



Sum of Mean

Model 1 999622 999622 0.01 0.9044
Error 117 8072698768 68997425

120

Coeff Var 264.81359

Parameter Estimates

Parameter Standard



The REG Procedure
Model: MODEL1



Sum of Mean

Model 1 90674 90674 0.01 0.9409
Error 20 321555936 16077797


121

Coeff Var 164.90512

Parameter Estimates

Parameter Standard



The REG Procedure
Model: MODEL1



Sum of Mean

Model 1 32361346 32361346 0.30 0.5855
Error 95 10263473634 108036565

Root MSE 10394 R-Square 0.0031
Coeff Var 356.93929

122

Parameter Estimates

Parameter Standard



The REG Procedure
Model: MODEL1



Sum of Mean

Model 1 31913896 31913896 2.70 0.1044
Error 83 982792968 11840879

Coeff Var 251.21902

Parameter Estimates

123

Parameter Standard



The REG Procedure
Model: MODEL1



Sum of Mean

Model 1 1798265 1798265 0.25 0.6205
Error 31 222881127 7189714

Coeff Var 144.29336

Parameter Estimates

Parameter Standard

124




The REG Procedure
Model: MODEL1



Sum of Mean

Model 1 1464962 1464962 1.50 0.2370
Error 18 17623855 979103

Coeff Var 132.56443

Parameter Estimates

Parameter Standard

Measuring Hedge Fund Performance

Measuring Hedge Fund Performance

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