The suitability of Beta as a measure of market-related risks for alternative investment funds presented to the Graduate School of Business of the University of Stellenbosch in partial fulfilment of the requirements for the degree of Master of Business Administration by FLORIAN BÖHLANDT Subject: Statistics Lecturer: Prof. Eon Smit Date: 17 April 006 1
The suitability of Beta as a measure of market-related risks for alternative investment funds Report for Statistics STUDENT NUMBER : 14959747 SURNAME: BÖHLANDT INITIALS : FMB TELEPHONE NUMBER: 07 70 9058 SUBJECT: Statistics (INCLUDING THIS PAGE) 9 NUMBER OF PAGES LECTURER: Prof. Eon Smit COURSE: MBA FULL-TIME 006 DUE DATE : 18/04/006 CERTIFICATION I certify the content of the assignment to be my own and original work and that all sources have been accurately reported and acknowledge, and that this document has not previously been submitted in its entirety or in part at any educational establishment. SIGNATURES: Florian Martin Böhlandt VIR KANTOORGEBRUIK / FOR OFFICE USE DATUM ONTVANG: DATE RECEIVED :
Declaration I herewith declare this work to be my own, that I have acknowledged all the sources I have consulted in this assignment itself and not only in the bibliography, that all wording unaccompanied by a reference is my own, and that no part of this assignment can be found on the internet, any published source, or in any other document that has been submitted to any university in partial of full satisfaction of the requirements for a subject or course or degree. I acknowledge that if any part of this declaration is found to be false I shall receive no marks for this assignment, and I shall not be allowed to complete this module, and that charges can be laid against me for plagiarism before the Central Disciplinary Committee of the University. Florian Böhlandt Student-No. 14959747 3
The suitability of Beta as a measure of market-related risks for alternative investment funds Test statistics...5 Testing for normality...6 Testing for autocorrelation...7 Conclusion...8 Sources...9 4
Test statistics Many portfolio managers claim that the risks associated with funds can be assessed by estimating the linear relationship between certain funds and the overall market portfolio. When deciding on how to diversify portfolios, managers tend to rely upon Beta-coefficients and the Capital Asset Pricing Model (Sharpe) to compare the dependency of one investment on market movements with the dependency of another investment. It is questionable, however, whether Beta can be used to determine the systematic risks of managed alternative funds. Two core reasons have been identified why Beta is not fit to be used as the sole determinant of systematic risk: 1. The distributions of returns on certain alternative funds can be negatively skewed (higher probability for fat tails ). The overall chances for extreme or total losses are higher due to the high proportion of leverage and derivatives employed. Consequently, the performance of some alternative funds are not normally distributed. Returns of certain asset classes can be correlated over time. Fund managers tend to smooth the performance of funds to create a more stable performance. Managerial compensation is usually linked to the performance of the fund (the outperformance of a market index). High-water marks for compensation can be an incentive to even the performance of funds, thus resulting in effects of autocorrelation Normality and independence of the error variables are two conditions for the reliability of Beta as an estimate for systematic risks. Two tests will be conducted to approximate whether the requirements for Beta as a suitable measure of systematic risk are being met: 1. Jarque-Bera test to estimate the significance of deviations of skewness and kurtosis (as measures of normality) from normal distribution. Durbin-Watson test to estimate the significance of the effects of first-order autocorrelation Two funds have been chosen to be represented in this study. The Hedge Funds Net Asset Value Index Investable Fund (HEDGNAV) consists of the majority of European Hedge Funds; the Hedge Global Macro Index Investable Fund (HEDGGLMA) incorporates a large number of alternative investment funds operating worldwide 1. The two funds employ alternative trading strategies, leverage and derivatives to hedge their positions. It must be noted that the funds in this study are fund of funds. This approach allows incorporating a large number of funds into the study. However, it is reasonable to say that if non-normality and 1 Indices provided by Hedge Fund Research as Investable Index Funds 5
autocorrelation were observed with fund of funds, a study of single funds would be likely to produce similar results. Fund of funds are expected to mitigate the effects of non-normal distributions or serial correlation occurring within single funds. Using investable fund of funds to draw inferences about alternative single funds can lead to survivorship bias (closed or insolvent funds are usually not included into investable funds). Incorporating the effects of survivorship bias is beyond the scope of this study. However, it is reasonable to infer that returns of non-survivors showed greater effects of autocorrelation and were in most cases negatively skewed. Hence, the conclusions drawn from this study disregarding the effects of survivorship bias are still valid. Testing for normality To determine the goodness of fit of Beta as a determinant of the systematic risks of alternative investment funds, two null-hypotheses and two alternative-hypotheses are being postulated: H 0;1 = The returns of alternative investment funds are normally distributed H A;1 = The returns of alternative investment funds are not normally distributed And H 0; = The returns of alternative investment funds show no sign of first-order autocorrelation H A; = The returns of alternative investment funds show sufficient evidence of serial correlation If the slope coefficient Beta was reliable indicator of the systematic risk associated with alternative investment funds, the tests should not produce any results that reject the nullhypotheses. In order to determine whether returns of alternative investment funds are indeed normally distributed, a Jarque-Bera test will be conducted. The Jarque-Bera test determines the probability of a given number of returns being normally distributed with a skewness of S and an excess-kurtosis of K: Fung, William; Hsieh, David A.: Is mean-variance analysis applicable to hedge funds? ; pp. 54-57 6
( K ) S BJ = N + 3 6 4 With N = number of observations; S = skewness; K = Excess-kurtosis After calculating the test statistics, the results are compared to the critical chi-squared value for two degrees of freedom at a 99% confidence level in order to determine whether alternative investment funds are normally distributed. The results can be seen below: χ.010 ; Skewness Kurtosis Jarque-Bera HEDGGLMA Index -0,04 1,5805 8,0887 9,103 HEDGNAV Index 0,0914 1,4395 9,7713 9,103 According to the test the Null-hypothesis is rejected for the HEDGNAV Index Investable Fund. The fund shows evidence that deviations of skewness and kurtosis are too great to infer normality. Thus, we cannot assume normality when using the slope of the regression equation as a determinant for the systematic risk of the fund. For the HEDGGLMA Index, no evidence can be found to infer that the Null-hypothesis is not true. Testing for autocorrelation A simple regression analysis is conducted to calculate the systematic risk (Beta) for both funds. We expect the beta to be low; the market-related risk of alternative investments should be close to zero. In other words, alternative investments are expected not to be correlated with the performance of the overall market. The market portfolio will be represented by the Standard&Poor s 500 index. The results can be seen below: For HEDGNAV Index Investable Fund: ANOVA Coefficients Standard Error Intercept 0,0077 0,004 SPX Index 0,786 0,0518 For HEDGGLMA Index Investable Fund: ANOVA Coefficients Standard Error Intercept 0,0114 0,0038 SPX Index 0,1959 0,0831 7
As expected, alternative investment funds respond only weakly to changes in the overall market. However, effects of correlation can distort the reliability of Beta. Hence, we need to test for the effects of autocorrelation. We conduct a Durbin-Watson test to assess serial correlation 3 : d = n ( ) ei ei 1 i= n ei i= 1 With n = number of observations; e = consecutive residuals After calculating the test statistic, the results are compared with the critical values of the Durban-Watson statistic. We have one degree of freedom (one independent variable in the regression model) and are testing at a 95% confidence level. The results are depicted in the table below: Durbin-Watson d L d U HEDGGLMA Index 1,8739 1,65 1,69 HEDGNAV Index 1,7731 1,65 1,69 Both test statistics lie within the rejection region. Thus, we reject the Null-hypothesis and conclude that there is enough evidence to infer that first-order correlation does exist. Consequently, Beta may not be the best representative for systematic risks associated with alternative investment funds. Conclusion After conducting the two tests, there is sufficient evidence to infer that Beta is not a reliable measure of the systematic risks associated with alternative investment funds. Thus, investors may arrive at wrong conclusions when basing there choices of portfolio diversification upon Beta. The literature offers a range of measures to include the effects of non-normality and autocorrelation into the calculation and to adjust Betas accordingly 4. However, these methods have not been proven to be effective yet and go beyond the scope of this report. Investors should be aware of the effects of autocorrelation and non-normal distribution when assessing the risks of alternative investments funds. The benefits of weakly market-correlated investments can be offset by these effects. 3 Keller, G.; Warrack, B.: Statistics for Management and Economics ; pp. 681-687 4 Scholes, M.; Williams, J.: Estimating Betas from Nonsynchronous Data 8
Sources Fung, William; Hsieh, David A.: Is mean-variance analysis applicable to hedge funds? ; Economic Letters; vol. 6; 1999 Keller, G.; Warrack, B.: Statistics for Management and Economics ; Sixth Edition; Brooks/Cole (003) Scholes, M.; William, J.: Estimating Beta from Nonsynchronous Data ; Journal of Financial Economics; vol. 7; 1977 Index Data provided by Hedge Fund Research (HEDGNAV Index; HEDGGLMA Index) and Reuters (SPX Index) 9