Research Paper Mark Hoaglund

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This is a copy, saved in Word 1997-2003 format, of my research paper that was submitted to and accepted by the graduate school for the Master of Science in Economics. It pertains to the hedge fund market using proprietary data from hedgefund.net.

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Research Paper Mark Hoaglund

  1. 1. 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
  2. 2. ACKNOWLEDGEMENTS I’d like to thank Professor Scott Gilbert for helping me throughout the process of developing this study. ii
  3. 3. TABLE OF CONTENTS ABSTRACT ……………………………………………………………..i ACKNOWLEDGEMENTS ….......................................................................................ii LIST OF TABLES ……………………………………………………………iv LIST OF FIGURES …......................................................................................vi TEXT ……………………………………………………………..1 REFERENCES …………………………………………………………..123 VITA …………………………………………………………..124 iii
  4. 4. 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 iv
  5. 5. 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 v
  6. 6. 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 vi
  7. 7. 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 Figure 33…………………………………………………………………………………44 Figure 34…………………………………………………………………………………44 Figure 35…………………………………………………………………………………45 Figure 36…………………………………………………………………………………45 Figure 37…………………………………………………………………………………46 Figure 38…………………………………………………………………………………46 Figure 39…………………………………………………………………………………47 Figure 40…………………………………………………………………………………47 Figure 41…………………………………………………………………………………48 Figure 42…………………………………………………………………………………48 Figure 43…………………………………………………………………………………49 Figure 44…………………………………………………………………………………49 Figure 45…………………………………………………………………………………50 Figure 46…………………………………………………………………………………50 Figure 47…………………………………………………………………………………51 vii
  8. 8. Figure 48…………………………………………………………………………………51 Figure 49…………………………………………………………………………………52 Figure 50…………………………………………………………………………………52 Figure 51…………………………………………………………………………………53 Figure 52…………………………………………………………………………………53 Figure 53…………………………………………………………………………………54 Figure 54…………………………………………………………………………………54 Figure 55…………………………………………………………………………………55 Figure 56…………………………………………………………………………………55 Figure 57…………………………………………………………………………………56 Figure 58…………………………………………………………………………………58 Figure 59…………………………………………………………………………………58 Figure 60…………………………………………………………………………………59 Figure 61…………………………………………………………………………………59 Figure 62…………………………………………………………………………………60 Figure 63…………………………………………………………………………………60 Figure 64…………………………………………………………………………………61 Figure 65…………………………………………………………………………………61 Figure 66…………………………………………………………………………………62 Figure 67…………………………………………………………………………………62 Figure 68…………………………………………………………………………………63 Figure 69…………………………………………………………………………………63 Figure 70…………………………………………………………………………………64 Figure 71…………………………………………………………………………………64 Figure 72…………………………………………………………………………………65 viii
  9. 9. Figure 73…………………………………………………………………………………65 Figure 74…………………………………………………………………………………66 Figure 75…………………………………………………………………………………66 Figure 76…………………………………………………………………………………67 Figure 77…………………………………………………………………………………67 Figure 78…………………………………………………………………………………68 Figure 79…………………………………………………………………………………68 Figure 80…………………………………………………………………………………69 Figure 81…………………………………………………………………………………69 Figure 82…………………………………………………………………………………70 Figure 83…………………………………………………………………………………70 Figure 84…………………………………………………………………………………71 Figure 85…………………………………………………………………………………71 Figure 86…………………………………………………………………………………72 Figure 87…………………………………………………………………………………72 Figure 88…………………………………………………………………………………73 Figure 89…………………………………………………………………………………75 Figure 90…………………………………………………………………………………75 Figure 91…………………………………………………………………………………76 Figure 92…………………………………………………………………………………76 Figure 93…………………………………………………………………………………77 Figure 94…………………………………………………………………………………77 Figure 95…………………………………………………………………………………78 Figure 96…………………………………………………………………………………78 Figure 97…………………………………………………………………………………79 ix
  10. 10. Figure 98…………………………………………………………………………………79 Figure 99…………………………………………………………………………………80 Figure 100………………………………………………..………………………………80 Figure 101………………………………………………………………………………..81 Figure 102………………………………………………………………………………..81 Figure 103………………………………………………………………………………..82 Figure 104………………………………………………………………………………..82 Figure 105………………………………………………………………………………..83 Figure 106………………………………………………………………………………..83 Figure 107………………………………………………………………………………..84 Figure 108………………………………………………………………………………..84 Figure 109………………………………………………………………………………..85 Figure 110………………………………………………………………………………..85 Figure 111………………………………………………………………………………..86 Figure 112………………………………………………………………………………..86 Figure 113………………………………………………………………………………..87 Figure 114………………………………………………………………………………..87 Figure 115………………………………………………………………………………..88 Figure 116………………………………………………………………………………..88 Figure 117………………………………………………………………………………..89 Figure 118………………………………………………………………………………..89 Figure 119………………………………………………………………………………..90 Figure 120………………………………………………………………………………..92 Figure 121………………………………………………………………………………..93 Figure 122………………………………………………………………………………..94 x
  11. 11. Figure 123………………………………………………………………………………..95 Figure 124………………………………………………………………………………..96 Figure 125………………………………………………………………………………..97 Figure 126………………………………………………………………………………..98 Figure 127………………………………………………………………………………..99 Figure 128………………………………………………………………………………100 Figure 129………………………………………………………………………………101 Figure 130………………………………………………………………………………102 Figure 131………………………………………………………………………………103 Figure 132………………………………………………………………………………104 Figure 133………………………………………………………………………………105 Figure 134………………………………………………………………………………106 Figure 135………………………………………………………………………………107 Figure 136………………………………………………………………………………108 Figure 137………………………………………………………………………………109 Figure 138………………………………………………………………………………110 Figure 139………………………………………………………………………………111 Figure 140………………………………………………………………………………112 Figure 141………………………………………………………………………………113 Figure 142………………………………………………………………………………114 Figure 143………………………………………………………………………………115 Figure 144………………………………………………………………………………116 Figure 145………………………………………………………………………………117 Figure 146………………………………………………………………………………118 Figure 147………………………………………………………………………………119 xi
  12. 12. Figure 148………………………………………………………………………………120 Figure 149………………………………………………………………………………121 Figure 150………………………………………………………………………………122 xii
  13. 13. 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
  14. 14. 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
  15. 15. 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
  16. 16. 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,
  17. 17. 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
  18. 18. 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
  19. 19. 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
  20. 20. 6 selected examples shown below. Figure 1. Fund 2 Performance Comparison_____________________________________
  21. 21. 7 Figure 2. Fund 6 Performance Comparison_____________________________________ Figure 3. Fund 10 Performance Comparison____________________________________
  22. 22. 8 Figure 4. Fund 16 Performance Comparison____________________________________ Figure 5. Fund 23 Performance Comparison____________________________________
  23. 23. 9 Figure 6. Fund 29 Performance Comparison____________________________________ 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
  24. 24. 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 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 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
  25. 25. 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.
  26. 26. 12 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.
  27. 27. 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
  28. 28. 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.
  29. 29. 15 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________________________________
  30. 30. 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.
  31. 31. 17 Figure 12. Regression of Fund 16 on the S&P 500_______________________________ Figure 13. Regression of Fund 23 on the S&P 500_______________________________
  32. 32. 18 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.
  33. 33. 19 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 ________________________________________________________________________
  34. 34. 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
  35. 35. 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
  36. 36. 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
  37. 37. 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
  38. 38. 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
  39. 39. 25 the fund is attributable to the stock market rather than Figure 15. The Original Fund 26 Trend Line___________________________________
  40. 40. 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.
  41. 41. 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
  42. 42. 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
  43. 43. 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_______________
  44. 44. 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
  45. 45. 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.
  46. 46. 32 Figure 23. Fund 7 Normal Shaped Residuals___________________________________ Figure 24. Fund 21 Normal Shaped Residuals__________________________________
  47. 47. 33 Figure 25. Fund 27 Normal Shaped Residuals__________________________________ Figure 26. Average Fund Normal Shaped Residuals______________________________
  48. 48. 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.
  49. 49. 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.
  50. 50. 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
  51. 51. 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
  52. 52. 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
  53. 53. 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.
  54. 54. 40 Figure 27. Fund 1 Residuals________________________________________________ Figure 28. Fund 2 Residuals________________________________________________
  55. 55. 41 Figure 29. Fund 3 Residuals________________________________________________ Figure 30. Fund 4 Residuals________________________________________________
  56. 56. 42 Figure 31. Fund 5 Residuals________________________________________________ Figure 32. Fund 6 Residuals________________________________________________
  57. 57. 43 Figure 33. Fund 7 Residuals________________________________________________ Figure 34. Fund 8 Residuals________________________________________________
  58. 58. 44 Figure 35. Fund 9 Residuals________________________________________________ Figure 36. Fund 10 Residuals_______________________________________________
  59. 59. 45 Figure 37. Fund 11 Residuals_______________________________________________ Figure 38. Fund 12 Residuals_______________________________________________
  60. 60. 46 Figure 39. Fund 13 Residuals_______________________________________________ Figure 40. Fund 14 Residuals_______________________________________________
  61. 61. 47 Figure 41. Fund 15 Residuals_______________________________________________ Figure 42. Fund 16 Residuals_______________________________________________
  62. 62. 48 Figure 43. Fund 17 Residuals_______________________________________________ Figure 44. Fund 18 Residuals_______________________________________________
  63. 63. 49 Figure 45. Fund 19 Residuals_______________________________________________ Figure 46. Fund 20 Residuals_______________________________________________
  64. 64. 50 Figure 47. Fund 21 Residuals_______________________________________________ Figure 48. Fund 22 Residuals_______________________________________________
  65. 65. 51 Figure 49. Fund 23 Residuals_______________________________________________ Figure 50. Fund 24 Residuals_______________________________________________
  66. 66. 52 Figure 51. Fund 25 Residuals_______________________________________________ Figure 52. Fund 26 Residuals_______________________________________________
  67. 67. 53 Figure 53. Fund 27 Residuals_______________________________________________ Figure 54. Fund 28 Residuals_______________________________________________
  68. 68. 54 Figure 55. Fund 29 Residuals_______________________________________________ Figure 56. Fund 30 Residuals_______________________________________________
  69. 69. 55 Figure 57. Average Fund Residuals___________________________________________
  70. 70. 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.
  71. 71. 57 Figure 58. Fund 1 Regression_______________________________________________
  72. 72. 58 Figure 59. Fund 2 Regression_______________________________________________
  73. 73. 59 Figure 60. Fund 3 Regression_______________________________________________ Figure 61. Fund 4 Regression_______________________________________________
  74. 74. 60
  75. 75. 61 Figure 62. Fund 5 Regression_______________________________________________ Figure 63. Fund 6 Regression_______________________________________________
  76. 76. 62
  77. 77. 63 Figure 64. Fund 7 Regression_______________________________________________ Figure 65. Fund 8 Regression_______________________________________________
  78. 78. 64
  79. 79. 65 Figure 66. Fund 9 Regression_______________________________________________ Figure 67. Fund 10 Regression______________________________________________
  80. 80. 66
  81. 81. 67 Figure 68. Fund 11 Regression______________________________________________ Figure 69. Fund 12 Regression______________________________________________
  82. 82. 68
  83. 83. 69 Figure 70. Fund 13 Regression______________________________________________ Figure 71. Fund 14 Regression______________________________________________
  84. 84. 70
  85. 85. 71 Figure 72. Fund 15 Regression______________________________________________ Figure 73. Fund 16 Regression______________________________________________
  86. 86. 72
  87. 87. 73 Figure 74. Fund 17 Regression______________________________________________ Figure 75. Fund 18 Regression______________________________________________
  88. 88. 74
  89. 89. 75 Figure 76. Fund 19 Regression______________________________________________ Figure 77. Fund 20 Regression______________________________________________
  90. 90. 76
  91. 91. 77 Figure 78. Fund 21 Regression______________________________________________ Figure 79. Fund 22 Regression______________________________________________
  92. 92. 78
  93. 93. 79 Figure 80. Fund 23 Regression______________________________________________ Figure 81. Fund 24 Regression______________________________________________
  94. 94. 80
  95. 95. 81 Figure 82. Fund 25 Regression______________________________________________ Figure 83. Fund 26 Regression______________________________________________
  96. 96. 82
  97. 97. 83 Figure 84. Fund 27 Regression______________________________________________ Figure 85. Fund 28 Regression______________________________________________
  98. 98. 84 Figure 86. Fund 29 Regression______________________________________________ Figure 87. Fund 30 Regression______________________________________________
  99. 99. 85 Figure 88. Average Fund Regression__________________________________________
  100. 100. 86 Time Plots These plots compare the performance of each of the funds and the S&P 500 over time.
  101. 101. 87
  102. 102. 88 Figure 89. Fund 1 Time Plot Comparison______________________________________ Figure 90. Fund 2 Time Plot Comparison______________________________________
  103. 103. 89
  104. 104. 90 Figure 91. Fund 3 Time Plot Comparison______________________________________ Figure 92. Fund 4 Time Plot Comparison______________________________________
  105. 105. 91 Figure 93. Fund 5 Time Plot Comparison______________________________________ Figure 94. Fund 6 Time Plot Comparison______________________________________
  106. 106. 92 Figure 95. Fund 7 Time Plot Comparison______________________________________ Figure 96. Fund 8 Time Plot Comparison______________________________________
  107. 107. 93 Figure 97. Fund 9 Time Plot Comparison______________________________________ Figure 98. Fund 10 Time Plot Comparison_____________________________________
  108. 108. 94 Figure 99. Fund 11 Time Plot Comparison_____________________________________ Figure 100. Fund 12 Time Plot Comparison____________________________________
  109. 109. 95 Figure 101. Fund 13 Time Plot Comparison____________________________________ Figure 102. Fund 14 Time Plot Comparison____________________________________
  110. 110. 96 Figure 103. Fund 15 Time Plot Comparison____________________________________ Figure 104. Fund 16 Time Plot Comparison____________________________________
  111. 111. 97 Figure 105. Fund 17 Time Plot Comparison____________________________________ Figure 106. Fund 18 Time Plot Comparison____________________________________
  112. 112. 98 Figure 107. Fund 19 Time Plot Comparison____________________________________ Figure 108. Fund 20 Time Plot Comparison____________________________________
  113. 113. 99
  114. 114. 100 Figure 109. Fund 21 Time Plot Comparison____________________________________ Figure 110. Fund 21 Time Plot Comparison____________________________________
  115. 115. 101
  116. 116. 102 Figure 111. Fund 21 Time Plot Comparison____________________________________ Figure 112. Fund 21 Time Plot Comparison____________________________________
  117. 117. 103 Figure 113. Fund 21 Time Plot Comparison____________________________________ Figure 114. Fund 21 Time Plot Comparison____________________________________
  118. 118. 104 Figure 115. Fund 27 Time Plot Comparison____________________________________ Figure 116. Fund 28 Time Plot Comparison____________________________________
  119. 119. 105 Figure 117. Fund 29 Time Plot Comparison____________________________________ Figure 118. Fund 30 Time Plot Comparison____________________________________
  120. 120. 106 Figure 119. Average Fund Time Plot Comparison_______________________________
  121. 121. 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.
  122. 122. 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__________________________________
  123. 123. 109 Table 10. Squared Residuals on Fund 2________________________________________ Figure 121. Squared Residuals Against Fund 2__________________________________
  124. 124. 110 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 Standard Variable Label DF Estimate Error t Value Pr > |t| 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 Figure 122. Squared Residuals Against Fund 3__________________________________
  125. 125. 111 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 Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 445.98346 129.06266 3.46 0.0009 excess_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________________________________________
  126. 126. 112 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 Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 961.50702 245.63372 3.91 0.0002 excess_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
  127. 127. 113 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 Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 1000.51918 386.34344 2.59 0.0214 excess_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
  128. 128. 114 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 Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 295.03138 32.70554 9.02 <.0001 excess_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
  129. 129. 115 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 Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 488.21475 166.50580 2.93 0.0050 excess_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 Analysis of Variance
  130. 130. 116 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 Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 1756.26783 312.71094 5.62 <.0001 excess_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 Sum of Mean Source DF Squares Square F Value Pr > F
  131. 131. 117 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 Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 850.42414 176.31685 4.82 <.0001 excess_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 Model 1 217459 217459 0.12 0.7317
  132. 132. 118 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 Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 698.60214 219.09418 3.19 0.0030 excess_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 Error 72 10313729 143246 Corrected Total 73 10326302
  133. 133. 119 Root MSE 378.47884 R-Square 0.0012 Dependent Mean 181.82405 Adj R-Sq -0.0127 Coeff Var 208.15664 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 181.42427 44.01796 4.12 <.0001 excess_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
  134. 134. 120 Root MSE 8306.46889 R-Square 0.0001 Dependent Mean 3136.72300 Adj R-Sq -0.0084 Coeff Var 264.81359 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 3140.00131 761.93971 4.12 <.0001 excess_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 Root MSE 4009.71281 R-Square 0.0003 Dependent Mean 2431.52719 Adj R-Sq -0.0497
  135. 135. 121 Coeff Var 164.90512 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 2433.15191 855.14736 2.85 0.0100 excess_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 Coeff Var 356.93929
  136. 136. 122 Parameter Estimates Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 2966.53641 1060.05147 2.80 0.0062 excess_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 Parameter Estimates
  137. 137. 123 Parameter Standard Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 1330.11919 374.01473 3.56 0.0006 excess_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 Parameter Standard
  138. 138. 124 Variable Label DF Estimate Error t Value Pr > |t| Intercept Intercept 1 1857.09987 466.77148 3.98 0.0004 excess_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 Variable Label DF Estimate Error t Value Pr > |t|

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