Do Hedge Funds Have Enough Capital? A Value-at-Risk Approach *

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1 Do Hedge Funds Have Enough Capital? A Value-at-Risk Approach * Anurag Gupta Bing Liang April 2004 *We thank Stephen Brown, Sanjiv Das, Will Goetzmann, David Hseih, Kasturi Rangan, Peter Ritchken, Bill Sharpe, Ajai Singh, Jack Treynor, and two anonymous referees for comments and suggestions on earlier drafts, and the seminar participants at Case Western Reserve University, University of Massachusetts at Amherst, Virginia Tech., the 2003 European Finance Association Meetings in Glasgow, the 2003 Western Finance Association Meetings in Los Cabos, the 2003 Q- Group fall seminar in Scottsdale, the 2001 FMA European Meetings in Paris, and the 2001 FMA meetings in Toronto. Bing Liang acknowledges a summer research grant from the Weatherhead School of Management, Case Western Reserve University. We also thank TASS Management Limited for providing the data. We remain responsible for all errors. Department of Banking and Finance, Weatherhead School of Management, Case Western Reserve University, Cleveland, OH Phone: (216) , Fax: (216) , anurag.gupta@case.edu. Department of Finance and Operations Management, Isenberg School of Management, University of Massachusetts, Amherst, MA Phone: (413) , Fax: (413) , bliang@som.umass.edu.

2 Do Hedge Funds Have Enough Capital? A Value-at-Risk Approach Abstract We examine the risk characteristics and capital adequacy of hedge funds through the Value-at-Risk approach. Using extensive data on nearly fifteen hundred hedge funds, we find that only 3.7% live and 10.9% dead funds are under-capitalized as of March Moreover, the undercapitalized funds are relatively small and constitute a tiny fraction of the total fund assets in our sample. Cross-sectionally, the variability in fund capitalization is related to size, investment style, age, and management fee. Hedge fund risk and capitalization also display significant time variation. Traditional risk measures like standard deviation or leverage ratios fail to detect these trends. JEL Classification: G23; G28; G29 Keywords: Hedge funds; Value-at-Risk; Capital adequacy; Extreme value theory; Monte Carlo simulation.

3 1. Introduction The hedge fund industry is one of the fastest growing sectors in finance, due to limited regulatory oversight, flexible investment strategies, and performance based fee structures. The rapid growth in this area has captured the attention of both academics and practitioners. This has led to several studies that analyze hedge fund performance, examine survivorship bias issues, and investigate the reasons for differences in fund performance across styles. These studies include Fung and Hsieh (1997), who argue that the highly dynamic hedge fund investment strategies can provide an integrated framework for style analysis. Brown, Goetzmann, and Ibbotson (1999) examine the performance of offshore hedge funds and attribute their performance to style effects rather than managerial skills. Ackermann, McEnally, and Ravenscraft (1999) conclude that hedge funds outperform mutual funds. Liang (1999) finds that hedge fund investment strategies are different from those of mutual funds. Agarwal and Naik (2003) propose a general asset class factor model comprising of option-based and buy-and-hold strategies to benchmark hedge fund performance. All of the above studies analyze hedge fund performance relative to certain benchmarks. An important question, unanswered as yet, is about the risk profile of hedge funds. The debacle of Long-Term Capital Management LP (LTCM) highlighted the need for more academic studies in hedge fund risk exposure and capital adequacy. In this paper, by doing extensive research on a large hedge fund database, we address the following primary questions: How risky are hedge funds, in general? To what extent are they adequately capitalized? What are the time-series patterns in the levels of capitalization in the hedge fund industry? How is fund capitalization related to the various fund characteristics? Like Jorion (2000), we propose a VaR approach, since VaR not only measures the maximum amount of assets a fund can lose over a certain time period with a specified probability, but can also be used to measure the equity capital needed to cover those losses. 1 We analyze the VaR for each fund, its distribution across all funds, and compute a VaR based estimate of required equity capital for each fund. This required equity is then compared to the actual fund equity to determine how many hedge funds are under-capitalized. We also study fund risk and capitalization on a dynamic basis in order to examine their time series variation and analyze the determinants of fund capitalization. Extensive robustness checks are conducted to ensure that our results and inferences are reliable. In addition to the primary questions, we address issues related to the techniques that are appropriate for risk estimation in the hedge fund industry. We characterize the distribution of 1 Lo (2001) questions the usefulness of VaR as a risk measure for hedge funds since their risk profiles change over time. We conduct several robustness tests on the effectiveness of our VaR estimation methodology, to ensure that we arrive at robust results. 1

4 hedge fund returns, and analyze whether VaR is a better measure of risk for evaluating hedge fund capital adequacy than traditional measures like the standard deviation of returns and leverage ratios, especially if there is significant non-normality in the return distribution. We examine whether the risk characteristics of dead funds are significantly different from live funds, and if VaR is able to capture these differences. We segment all our results by investment styles, in order to understand if there are significant risk and capital adequacy differences in hedge funds across styles. Finally, we conduct carefully designed Monte-Carlo simulation tests in order to address potential survivorship and self-selection bias concerns in the hedge fund data. These Monte Carlo simulations capture not only the regular price movements in hedge funds but also extreme movements under rare market conditions. To the best of our knowledge, our paper is the first one to address capital adequacy and risk estimation issues in the entire hedge fund industry. Although Jorion s study is the first one to apply the VaR methodology to hedge funds, he examines only a single fund, while we examine nearly fifteen hundred hedge funds. Fung and Hsieh (2000) examine hedge fund performance and risk in some major market events/crisis. However, they adopt a traditional mean-variance approach, which is not effective in capturing hedge fund risk. In contrast, we use the VaR approach to study hedge fund risk. Lhabitant (2001) reports factor model based VaR figures for some hedge funds. However, he does not use the return information directly in estimating the VaR, and does not examine any capital adequacy issues. Getmansky, Lo, and Makarov (2003) examine the illiquidity exposure of hedge funds. They focus on liquidity risk while we study the market risk and capital adequacy issues in the hedge fund industry. We find that, in our sample, a majority of hedge funds (96.3% of the live and 89.1% of the dead funds) are adequately capitalized as of March The (3.7%) live under-capitalized funds are mostly small funds, with median net assets of $66m, which together constitute only 1.2% of the total net assets in our sample. Cross-sectionally, the variability in fund capitalization is related to size, investment style, age, and management fee. In particular, the convertible arbitrage and market neutral funds are better capitalized than the emerging markets, long/short equity, and managed futures funds. On a dynamic basis, we document a significant drop in fund capitalization after the Russian debt crisis in Robustness tests based on Monte Carlo simulation and backtesting evidence support our findings, hence the application of VaR (using Extreme Value Theory) for inferring capital adequacy seems valid. For dead funds, we find that the estimated VaR increases by an average of 74% over the two years immediately preceding the fund s death, while no such trend is observed for live funds, indicating that our estimated VaR is effective in capturing some elements of hedge 2 Our results are subject to the caveat that some of the largest hedge funds (like LTCM) are not in our sample, since they do not report data to any vendor. Therefore, our inferences cannot be readily generalized to the entire hedge fund industry, and are applicable only to the funds in our sample. 2

5 fund risk that lead to their death. This supports the use of VaR for capital estimation. We also find significant non-normality in hedge fund returns, in terms of high kurtosis. Traditional risk measures like normality based standard deviation or leverage ratios fail to capture the true risk in hedge fund returns. Since the debacle of LTCM, there has been an extensive debate amongst the regulatory agencies and market participants regarding hedge fund risk and regulation. 3 While our study, due to data limitations, cannot provide conclusive evidence for or against increased hedge fund regulation, we do present an extensive, systematic study of the risk characteristics and capital adequacy of hedge funds for which data is available, shedding some light on these regulatory issues as well. This paper is organized as follows. Section 2 explains the concept of VaR and its application in determining capital requirements. Section 3 describes the data. Section 4 explains the research methodology. The empirical results for capital adequacy and robustness tests are presented in section 5. Section 6 provides the Monte Carlo simulation results. Section 7 concludes. 2. Value-at-Risk and capital adequacy VaR is a measure of the worst loss that can happen over a target horizon with a given confidence level. If c is the selected confidence level, VaR corresponds to the 1-c lower tail. It is calculated in dollar amounts and is designed to cover most, but not all, of the losses that a risky business might face. Therefore, it has the intuitive interpretation of the amount of economic or equity capital that must be held to support that level of risky business activity. In fact, the definition of VaR is completely compatible with the role of equity as perceived by financial institutions - while reserves or provisions are held to cover expected losses incurred in the normal course of business, equity capital is held to provide a capital cushion against any potential unexpected losses. Since all unexpected losses cannot be covered with 100% certainty, the level of this capital cushion must be determined within prudent solvency guidelines. This definition of risk capital encompasses a broader concept of risk than the traditional leverage ratios, which only depend on the liabilities side of the balance sheet. Firms with any level of leverage may have significant risk of not being able to continue with their business, if they hold very risky assets. This is especially true for financial firms (including hedge funds), which hold traded assets that must be marked-to-market periodically. The potential for losses on these assets, 3 Market participants argue that, barring some exceptions, hedge funds in general operate within prudent solvency norms. Subjecting them to excessive regulation would stifle their ability to implement optimal investment strategies, thereby compromising returns for investors as well as hindering the flow of capital across markets and asset classes. On the other hand, regulators use LTCM as an example of how bad things could get if hedge funds are allowed to continue to have a free rein. They argue that since hedge funds face hardly any regulatory constraints, they could indulge in excessive risk taking the way LTCM did. This could have disastrous consequences for the stability of financial markets worldwide, and create enormous systemic risk. Therefore, they are pressing for increased regulation of hedge funds. 3

6 in relation to the equity capital, is the most important determinant for capital adequacy of such firms, not their leverage ratio. Of course, leverage magnifies the impact of such losses. The VaR-based capital adequacy measure is also being increasingly adopted by regulators and supervisors. The Basel Committee for Banking Supervision (BCBS) now allows some financial institutions to use their own internal VaR estimates to determine their capital requirement. The Derivatives Policy Group (DPG) formed by the Securities and Exchange Commission (SEC) in 1994 also makes similar recommendations to broker-dealers that conduct an OTC derivatives business. Therefore, the use of a VaR measure to study capital adequacy is extremely relevant and is in line with the norms and guidelines in place for various financial institutions. There are three main decision variables in estimating VaR - the confidence level, a target horizon, and an estimation model. If the objective of estimating VaR is to estimate risk capital requirements, the confidence level should be chosen to be high enough so that there is very little probability of failure. The target horizon is related to the liquidity of the positions in the portfolio. It should reflect the amount of time necessary to take corrective action, if something goes wrong and high losses occur, and should correspond to the time necessary to raise additional funds to cover losses. The VaR model would, of course, determine the accuracy of the VaR estimate. There is considerable uncertainty in choosing these variables, and the choice is often arbitrary. For commercial banks, the BCBS (1996) stipulates a capital requirement of 3-times the 99% tenday VaR for market risk. 4 However, the choice of individual parameters is arbitrary, and the same market risk charge number can be obtained using different parameter combinations. 5 We use 3 times the 99% 1-month VaR as the required equity capital for hedge funds. The time horizon used is 1-month instead of ten days because hedge funds are quite different from commercial banks. As pointed out by Jorion (2000), commercial banks are closely supervised by regulators, hence they can react to potential difficulties much sooner. Hedge funds are far less regulated and are not allowed to raise funds from the public, hence they would have a harder time raising additional capital when needed. Therefore, for hedge funds, the target horizon should at least be longer than that for banks. It must be recognized, though, that the choice of target horizon is still arbitrary, as in the case of the BCBS guidelines. 3. Data Hedge funds often have complex portfolios including nonlinear assets like options, interest rate derivatives, etc. For such portfolios, estimating the VaR is a difficult task, since both the non- 4 The safety multiple of 3 is to provide extra capital cushion for keeping the probability of bankruptcy reasonably low, and to take care of estimation biases and model misspecification in VaR estimation. In fact, as Stahl (1997) shows using Chebyshev s inequality, a maximum correction factor of 3 takes care of all error introduced due to misspecification of the true distribution of returns. 5 In tests during summer 1998, the BCBS found this market risk charge number to be adequate. 4

7 Gaussian nature of the fluctuations of the underlying assets and the non-linear dependence of the price of the derivatives must be dealt with. Moreover, there is no data available on the position holdings of hedge funds, since it constitutes proprietary trading information. Therefore, it is not possible for us to estimate the VaR of hedge funds by doing a position level analysis. The best data available is monthly returns reported by the hedge funds, which we use to estimate the VaR as accurately as possible. 6 We use the hedge fund dataset from TASS Management Limited (hereafter, TASS), which contains monthly return data on 3,702 hedge funds, including 2,256 survived and 1,446 dissolved funds, as of March The return data goes back till February 1977 for some of the live funds, and till July 1978 for some of the dead funds. The total assets under management for live funds are about $259 billion, making it one of the largest hedge fund databases for academic research. 7 We use a five-year return history as the minimum time period required to estimate the VaR, leaving us with 1,436 funds (942 live and 494 dead funds). 8 This also ensures that, at least for the live funds, the returns we use to estimate the VaR overlap with some of the most turbulent times in financial markets, starting with the Asian currency crisis of 1997, the Russian debt crisis and LTCM debacle of 1998, and the stock market crash from 2000 onwards. The minimum return history requirement may introduce survivorship bias in our VaR estimation, by throwing away younger (and potentially riskier) funds. Therefore, we may underestimate the true degree of under-capitalization. However, we do consider a large number of dead funds, not just funds that have survived. These 494 dead funds help us identify the differences, if any, in the risk profiles of live versus dead funds. This significantly mitigates the survivorship bias in our study. Our data is categorized by 11 fund styles, as defined by TASS. These styles are convertible arbitrage, dedicated short bias, emerging markets, market neutral, event driven, fixed income arbitrage, fund of funds, global macro, long/short equity hedge, managed futures, and others. This classification allows us to study hedge fund risk and capital adequacy by investment styles. For leverage information, TASS reports two numbers - the average leverage ratio and the maximum leverage ratio. The average leverage provides a measure of the historical leverage ratio 6 TASS (used by Fung and Hsieh (1997) and Liang (2000)), HFR Inc. (used by Ackermann, McEnally, and Ravenscraft (1999), and Agarwal and Naik (2003)), and CISDM (used by Ackermann, McEnally, and Ravenscraft (1999)) report monthly returns. The U.S. Offshore Funds Directory (used by Brown, Goetzmann, and Ibbotson (1999)) reports annual returns. 7 Liang (2000) indicates that the TASS data has some advantages over the other databases since it contains more dissolved funds and is more accurate in describing fund characteristics. 8 The five-year period for live funds ends March 2003, while it ends at the last month for dead funds, whenever they die. For robustness, we also estimated the VaR using different lengths of time. The results were very similar, therefore we report our results from the five-year window only. 5

8 on average, while the maximum leverage indicates the largest capacity up to which a fund can be levered. We use the average leverage of funds for leverage analysis throughout this paper Research methodology We estimate VaR in order to determine the capital requirement for hedge funds. Equity capital, by definition, is the capital reserve required to bear unexpected losses. Most of the unexpected losses arise due to extreme events in financial markets. Therefore, the estimation of capital requirements can be considered to be an extreme value problem. While estimating VaR, we focus on the behavior of the return distribution in the left tail. Extreme Value Theory (EVT) provides a firm theoretical foundation to model and estimate tail related risk, and hence VaR. 10 In Appendix A.1, we explain EVT and how it can be applied to estimate the 99-percentile return in the left tail of the return distribution. Using this return, the VaR is estimated as follows: where VaR R 99% TNA ( R ) TNA VaR 0 (1) = 99% = 99% 1-month VaR, = Cut off return at 99% confidence level estimated using EVT, = The total net assets (equity) of a fund. This VaR is relative to zero return, which specifies the absolute dollar loss, instead of the VaR from the mean return, which is the dollar loss relative to the expected return over the target horizon. We use VaR relative to zero since there may be significant biases and errors in the estimated mean returns for hedge funds, which would introduce another source of error in our VaR estimate. In addition, for the purposes of determining equity capital, it is critical to measure the absolute dollar loss that the fund might incur over the target horizon, rather than the shortfall from expected returns. 11 The capital requirement is then taken as 3 times this VaR number. We examine the validity of the safety multiplier of 3 in later sections of this paper. To evaluate capital adequacy, we compare this required capital with the actual equity capital backing these funds. We compute a capitalization ratio (the Cap ratio) defined as follows, 12 9 TASS uses two different notations for leverage: 1:1 means no leverage (asset to equity ratio of 1), while 100% means a fund has borrowed 100% of equity, resulting in an asset to equity ratio of 2. Throughout this paper we define leverage as the ratio of total assets to total equity, consistent with the notations from TASS. 10 da Silva and Mendes (2003) find that the extreme value method of estimating VaR is a more conservative approach to determining capital requirements than traditional methods. 11 We computed the VaR relative to mean returns as well, in order to check the robustness of our conclusions, and the capital adequacy results were very similar Note that substituting required capital with 3 x VaR in (2), it becomes Cap = 1. 3 (0 R ) 99% 6

9 Eactual Erequired Cap = (2) E required A Cap ratio less than zero implies that the actual equity is not sufficient to cover the risk of the portfolio as per the VaR approach, hence the fund is under-capitalized. In addition to VaR, we also estimate the tail conditional loss (TCL, or expected shortfall). TCL measures the potential size of the expected loss if it exceeds the VaR. The minimum capital required should be sufficient to cover the losses, if an extreme loss occurs. A 99% VaR only tells us the minimum loss that can be expected 1% of the time - it does not tell us anything about how large the loss might be, if it occurs. TCL provides an estimate of how large this loss might be, on average, hence it can be useful in determining capital adequacy. In Appendix A.2, we provide details on estimating the TCL using EVT. The TCL is defined as: ( E[ R R < R ]) TNA TCL 0 (3) = 99% The ratio of TCL to VaR can provide a more objective basis of determining the appropriate capital multiplier that should be used in conjunction with VaR, and indicate how safe it is to use the standard multiplier of 3 recommended by the Basel Committee. Many traditional risk based capital measures assume the return distribution to be normal, though it is often significantly non-normal. A comparison of risk capital measures based on EVT with those based on the normality would highlight the error introduced by assuming normality. Therefore, we re-estimate the 99% VaR of each fund assuming the return distribution to be normal, as follows: 13 [( 2. ) ] VaR = σ R 326 TNA (4) where σ R is the standard deviation of fund returns. The Cap ratio is computed in a manner similar to that in the EVT approach. The differences in the levels of under-capitalization using the EVT VaR and the standard deviation based VaR can be attributed solely to the departures from normality in the actual return distribution of hedge funds. 13 Note that this VaR is relative to mean returns, not relative to zero returns. This can only bias our results in favor of the standard deviation based VaR, since it will be higher than the VaR from zero returns. 7

10 5. Results and robustness tests 5.1. The capital adequacy of hedge funds Table 1 presents the descriptive statistics of hedge fund returns by investment styles. All figures are the medians across funds in the same style. Several inferences can be drawn from this table. First, live funds outperform dead funds in most styles. The median live fund earns an average monthly return of 0.72%, compared with 0.62% for a median dead fund. These results are consistent with Liang (2000), who documents that one of the reasons that funds die is poor performance. Dead funds also exhibit higher volatility of returns than live funds. Second, hedge fund returns do not exhibit a high level of skewness, except for some styles within dead funds. 14 However, all investment styles show high kurtosis above three, which indicates that hedge fund returns have fat tails and more extreme return values, thus making their return distributions significantly non-normal. For example, fixed income arbitrage funds have a median kurtosis of 7.61 (15.91) for live (dead) funds, while emerging markets funds have a median kurtosis of 5.32 (6.34) for live (dead) funds. This is consistent with hedge fund risk being more event-driven and non-linear than regular price fluctuations under normal circumstances. Hedge funds often implement opportunistic trading strategies and bet on major markets events worldwide. Their returns are heavily affected by these events, hence extreme positive (as in the famous attack on the Sterling by George Soros funds in 1992) and negative (as in the downturn for LTCM in 1998) returns may be realized. Therefore, using just the second moment to measure hedge fund risk is inappropriate, and we turn to VaR based on EVT for evaluating hedge fund risk in this paper. Table 1 also shows that there are significant differences in hedge fund return distributions across investment styles, hence a study by fund styles is more insightful than just an aggregated study that groups all hedge funds together. 15 Table 2 presents statistics for fund sizes and absolute VaR numbers across styles for both live and dead funds, in order to understand the magnitude of the dollar values in question. In the live funds group, the average fund size ranges from only $85.0 million for the emerging markets style to $317.7 million for the convertible arbitrage style, as of March Because of the differences in fund size across investment styles, the average estimated VaR ranges from only $4.6 million for the market neutral funds to $33.5 million for the fixed income arbitrage style. It is not surprising to find that dead funds are generally smaller than live funds. Dead funds lose capital because of poor performance, or they are unable to reach a critical mass, so they die. Because fund assets differ, a VaR relative to fund assets is more appropriate than the absolute VaR for comparison 14 According to Tabachnick and Fidell (1996), the standard error for skewness is roughly (6/N), hence for skewness estimated using 60 returns, the two standard errors bound for significance is approximately ±0.63. Both 0.06 and 0.01 are within this bound. 15 An analysis of 1,368 younger funds (705 live and 663 dead), with return history between two and five years, reveals that their first four moments are similar to those for funds older than five years. Hence these younger funds, which are excluded in this study, do not appear to be riskier than the older funds. 8

11 purposes. When analyzing the VaR as a percentage of fund size, we find that generally, dead funds have higher relative VaRs than live funds, which reflects the higher risk implicit in dead funds. For example, the median (mean) relative VaR is 9.8% (11.3%) for the live funds, compared with a higher 14.6% (17.9%) relative VaR for the dead funds. Across styles, the dedicated short bias, emerging markets, fixed income arbitrage, long/short equity hedge, and managed futures styles are particularly riskier than the other styles for both live and dead funds. The main results for capital adequacy are presented in Table 3. We find that very few hedge funds (both live and dead) in our sample are under-capitalized. For the live funds, about 3.7% (35 out of 942) of the funds are under-capitalized, while the corresponding fraction is 10.9% (54 out of 494) for the dead funds. The median (mean) Cap ratio is 2.4 (5.3) for live funds, compared with 1.3 (2.0) for dead funds. On average, dead funds are more under-capitalized than the live funds. This is somewhat consistent with the hypothesis that one of the reasons for a fund s death is undercapitalization. However, under-capitalization does not appear to be the primary reason for fund death, since nearly 90% of the dead funds had adequate equity capital right until the fund exit date. Therefore, other reasons like poor performance, mergers and acquisitions, voluntary withdrawals, etc., may contribute more to the demise of a hedge fund than capital inadequacy. For the live funds, the only styles with significant levels of under-capitalization are the emerging markets funds (6 out of 62, or 9.7%), and fixed income arbitrage funds (5 out of 32, or 15.6%). Most of the other styles, even riskier fund styles with very high kurtosis in fund returns (like fund of funds) have an extremely small number of funds that are under-capitalized. Similarly, in the dead funds group, emerging market funds (12 out of 40, or 30%) and fixed income arbitrage funds (6 out of 18, or 33.3%) have high levels of under-capitalization Time series variation in capital adequacy The capital adequacy results in the previous section present a snapshot of the hedge fund industry as of March However, hedge fund risk exposures are highly dynamic, hence the Cap ratio is unlikely to be constant over time. Firstly, hedge funds change their portfolio compositions fairly frequently. Secondly, market conditions change over time, so even for a static portfolio, its risk profile is likely to change. Therefore, in addition to computing the static Cap ratios, we go back in time and estimate the Cap ratios for all available funds for 60-month rolling windows for fund returns. For example, for February 2003, we analyze all the live and dead funds as of February 2003 with at least 60 months return data, and estimate the Cap ratio for each one of them. Hence there may be some fund that is categorized as a live fund as of February 2003, but as a dead fund in March In this manner we go back month by month, till January 1995, which is as far back as the data allows. 9

12 In Fig. 1, we present the percentage of live funds under-capitalized, for each month, from January 1995 till March The fraction of under-capitalized live funds steadily increases from 0.49% in Jan 1995 (1 out of 203 live funds under-capitalized) to a maximum of 5.43% as of August 2000 (37 out of 681 live funds under-capitalized), and then reduces to 3.72% in March 2003 (35 out of 942 funds). This graph shows some important trends. The extent of under-capitalization steadily increases till the middle of 2000, after which it declines a little bit. This decline may be due to the fact that some of the under-capitalized funds that were alive in 2000 may now be dead, hence they will not show up in the live database. There is a steep increase in the fraction of live funds under-capitalized during the third quarter of 1997, just after the Asian financial crisis, and this uptrend continues through the Russian debt crisis of 1998 and beyond. In aggregate, it appears that there is a clear increase in the fraction of live funds that are under-capitalized over time. The second plot in Fig. 1 presents the median Cap ratios for all live funds, each month, from January 1995 to March This figure is consistent with the previous figure the median fund appears to be less capitalized now, than it was during the years prior to the Russian debt crisis and the LTCM debacle in Fall The steep decline in the median Cap ratio from 2.76 in July 1998 to 1.99 in August 1998 is due to the big market movements during Fall 1998, which are included in all the moving return windows for subsequent months. These time series trends in the level of capitalization of live funds reveal much more information about the dynamic risk levels in the hedge fund industry than just a static analysis as of March The third plot in Fig. 3 presents the fraction of dead funds that are under-capitalized at any point in time in the past. Therefore, as of a particular month, we analyze the dead fund database and select funds that had at least five years of return history before death. Of these funds, we examine how many are under-capitalized. The percentage of under-capitalized dead funds rises sharply from 1.49% in April 1998 (1 out of 67 funds) to 11.91% in December 2000 (33 out of 277 funds). After that, it fluctuates around 11%. Again, the fraction of under-capitalized dead funds rises steeply after the Asian financial crisis of 1997 and the Russian debt crisis of It appears that inadequate capital may be the reason for the demise of some of the funds in recent years, and capital adequacy may be a concern for more funds now, than it has been before. 16 It is important to note that the 35 under-capitalized live funds constitute only 1.2% ($2.3b out of $187.4b) of the total net assets of the 942 funds, indicating that a very large proportion (98.8%) of the live fund assets in our sample are not exposed to the risk of under-capitalization. Table 4 presents a statistical comparison of various characteristics of the under-capitalized funds (both live and dead) with the remaining adequately capitalized funds. Amongst the live funds, the 35 under-capitalized funds are significantly smaller than the remaining funds, with average net 16 An analysis of the time series patterns of capital adequacy for individual under-capitalized funds revealed that funds go above and below the threshold of adequate capital fairly often a fund that is undercapitalized at one point in time may not always remain under-capitalized. 10

13 assets of $66.3m, as compared to $201.2m for the remaining 907 funds. Their mean Cap ratio is -0.2, indicating that on average, they have only 80% of the required equity. They exhibit significantly higher volatility, negative skewness, and kurtosis of returns. However, they are not significantly younger than the adequately capitalized funds. In addition, on other attributes like fee structure, watermark provisions, use of derivatives, etc., the under-capitalized live funds are not significantly different from the adequately capitalized funds. For dead funds as well, there is no significant difference in fund attributes, including age, between under-capitalized and adequately capitalized funds. In fact, the average dead fund in our sample existed for about 8 years, irrespective of whether it was under-capitalized or not at the time of death. These comparisons tell us that, in terms of reported characteristics (except for size), there is no significant difference between adequately capitalized and under-capitalized funds. 17 They largely differ only in their return distributions. Therefore, for making inferences about capital adequacy, we need to focus on returns rather than fund characteristics The determinants and traditional measures of capital adequacy In Table 4, we present a univariate comparison of adequately capitalized funds with undercapitalized funds. However, there may be more than one factor that affects a fund s capitalization. Therefore, we conduct a multivariate analysis on capitalization by running a crosssectional regression of Cap ratios on various fund characteristics and styles. For robustness, we choose several models that include different sub-sets of fund variables. These regression results in Table 5 show that fund size is positively related to the Cap ratio, while age and management fee are negatively related to the Cap ratio. Most large funds have enough equity capital to support their activities. Therefore, in general, they are adequately capitalized. Management fee is asset based rather than performance based so it can be viewed as a fixed cost to the assets, hence it is negatively correlated with the Cap ratio. Younger funds may not be well established, so they may be more cautious in their investment strategies in order to build up a good reputation in the early stages of their profession. These factors may result in a negative correlation between age and the Cap ratio. Across styles, the convertible arbitrage and market neutral funds are better capitalized than the emerging markets, long/short equity hedge, and managed futures funds. These results are consistent with those from the univariate analyses in Table 4. Can the inferences from VaR based capital measures be arrived at by just observing the leverage ratios of these hedge funds? If leverage can capture risk the way VaR does, then there is no need for these calculations. However, that is not the case. The median leverage ratio for all fund styles 17 We did a similar statistical comparison between 35 large live funds (net assets of $1 billion or more) and the remaining 907 live funds, since the failure of large funds can have a potentially large impact on financial markets. We found the large funds, in our sample, to be significantly better capitalized than the other funds. 11

14 is one, since many hedge funds do not use borrowed funds. 18 Therefore, it is unlikely that leverage ratios will convey any relevant information about hedge fund risk profile, and their true risk of failure. The correlation coefficient between VaR based Cap ratios and the average leverage ratios is found to be 0.06 (p-value=0.06) for live funds and 0.09 (p-value=0.04) for dead funds. Although these correlations are statistically significant the magnitudes are economically trivial. In addition, in the analysis in Table 5, after controlling for other factors, the leverage variable is insignificant. Therefore, there is not much information in the leverage ratios of hedge funds that can be related to their risk profiles and hence capital adequacy. This further reinforces the need to have capital estimation procedures based on VaR, instead of leverage. Table 6 reports the number and the percentage of funds that are under-capitalized based on the VaR estimated by assuming normality. If a simple standard deviation based risk measure can capture risk adequately, then there is no need for more complex EVT based measures. 19 Comparing the results of Table 6 with those of Table 3, we find that using the VaR based on normality leads to an underestimation of capital requirements, especially for dead funds. Standard deviation based VaR is able to detect under-capitalization in 2.4% of the live funds, and in only 3.0% of the dead funds, while the corresponding numbers are 3.7% and 10.9% respectively using the EVT VaR. This is not surprising, because assuming normality ignores the fat tails of the hedge fund return distribution, which in turn underestimates the risk of extremely low return realizations. This is especially true for dead funds, which have higher kurtosis and more negative skewness of returns. EVT captures the probability of occurrence of extreme negative returns better, hence it provides a more accurate measure of VaR. Therefore, reliance on traditional standard deviation based risk measures alone could lead to funds keeping a lower capital cushion than that dictated by their true risk profiles, thereby leading to a higher probability of failure. 5.4 Robustness tests Tail Conditional Loss (TCL) and the 99.94% VaR To examine the validity of using 3 as the safety multiplier for capital requirement, we estimate the tail conditional loss (TCL) for each fund. The ratio of TCL to VaR is compared with the safety multiplier of 3 in order to test whether it is adequate or not. Table 7 presents the TCL/VaR ratios by hedge fund styles. The median ratios for live and dead funds are 1.35 and 1.37, respectively. In fact, for none of the funds was the TCL/VaR ratio greater than 3. Hence for all the funds in our sample, even if the VaR is breached, the expected loss is likely to be less than 3 times the VaR. Therefore, the safety multiplier of 3 appears to be appropriate for examining capital adequacy. 18 As reported by TASS, only 67% of live funds (1,510 out of 2,256) and 72% of dead funds (1,037 out of 1,446 funds) are levered. 19 Note that one could use the standard deviation of returns along with a fat tailed distribution (such as a Student t-distribution) to alleviate at least some of the problems due to high kurtosis. 12

15 In addition to TCL, we also report the ratio of the 99.94% VaR to the 99% VaR for the funds using EVT. 20 This provides us with a further indication of whether the Basel ratio of 3 is safe. Table 7 presents the 99.94%/99% VaR ratios. The median ratios for live and dead funds are 1.87 and 2.06, respectively. Again, none of the mean or median ratios exceeds 3, for any fund style. In particular, for 94.1% of the live funds and 92.7% of the dead funds, this ratio is less than 3. For robustness, we recomputed the Cap ratios using the maximum of the 99.94% VaR and 3 times the 99% VaR, as the required equity capital. We found that 37 live funds (instead of 35 earlier) and 57 dead funds (instead of 54) were under-capitalized, which is very similar to the percentage of funds under-capitalized using 3 times the 99% VaR as the required capital VaR confidence intervals We construct (95%) confidence intervals on our (99%) VaR estimates, in order to estimate the tail quantile estimation errors. These confidence intervals provide important information about the reliability of the point estimates of VaR. Since we use EVT to estimate the VaR, we use profile likelihood methods to estimate the confidence intervals on these VaRs, as explained in Appendix A.3. Given the relative uncertainty about quantiles in the tail of the distribution, the upper bound of the confidence interval for the VaR provides a reasonable upper limit for how high the VaR could be in reality. A ratio of the upper bound of the confidence interval to the VaR itself can then be compared with the BCBS multiplier of 3 to examine how often the standard multiplier of 3 fails to capture the true risk of extremely low returns. Table 8 presents the upper and lower confidence intervals on VaR, as a percentage of the estimated VaR. The median live fund has an upper (lower) confidence bound that is 2.02 (0.52) times the VaR. The bounds are similar for dead funds. The upper bounds are much farther away from the VaRs than the lower bounds. This is consistent with the intuition that there is much more uncertainty as we go deeper into the tails of the return distribution. However, on average, these upper bounds are about twice of the VaR estimate, hence using the Basel multiplier of 3 appears to be reasonably safe for estimating capital requirements. In fact, for only 9.2% of the live funds (87 out of 942) and 12.6% of the dead funds (62 out of 494) was the upper confidence bound greater than 3 times the VaR. When we re-estimate the Cap ratios by defining the required equity to be the maximum of 3 times the VaR and the upper bound of the VaR confidence interval, we find 4.0% of live funds (38 out of 942) and 11.9% of dead funds (59 out of 494) to be undercapitalized, which is only marginally higher than the under-capitalization using 3 times the VaR. Hence our primary capital adequacy results are not sensitive to the sample size issue. 20 The default probability implied by maintaining a capital reserve equal to the 99.94% worst loss is consistent with the expected default rates for investment grade institutions. See Jorion (2000) for details. 13

16 The effectiveness of VaR as a risk measure for hedge funds Standard deviation is ruled out as an appropriate measure for hedge fund risk, due to the nonnormality in hedge fund returns. Leverage ratio is also ruled out as the correct measure since it does not proxy for asset risk in any meaningful way. However, even VaR can be used as an appropriate risk measure for hedge funds only if it can effectively detect the changing risk patterns over time. In general, the risk for dead funds increases toward the fund death date, either because of bad performance that leads to loss of capital and investor withdrawals, or because of the excessive risk shifting incentives that occur when a fund has been performing poorly in the past. If VaR is an appropriate risk measure for hedge funds, then it should capture these risk characteristics, and we should be able to detect a significant increasing trend for VaR over time, for dead funds, as the fund death date approaches. On the other hand, we should not expect such a systematic trend in the VaRs for live funds. Therefore, we analyze the dead funds with a return history of at least seven years. We compute their VaRs over three rolling time periods of 60 months each - the five years immediately preceding the fund death (window 3), the five years ending one year before the fund death date (window 2), and the five years ending two years before the fund death date (window 1). For comparison, we estimate the same three rolling VaRs for live funds as well, the only difference being that the time horizon for live funds is the same for all funds, ending March Note that there may be some trends in estimated fund VaRs just due to the different magnitudes of nonfund specific risk over different periods. In Fig. 2, we present the results for the 591 live and 272 dead funds that have at least a sevenyear return history. The box plots present the ratios for window 2 VaR to window 1 VaR, and the ratios for window 3 VaR to window 2 VaR, for live and dead funds, respectively. The mean (median) for dead funds is 1.27 (1.17) for the ratios of window 2 VaR to window 1 VaR, and 1.37 (1.25) for the ratios of window 3 VaR to window 2 VaR. These ratios show that the VaR of hedge funds, estimated using 60 monthly returns, increases significantly in the two years prior to a fund s death. In fact, over those two years, just prior to a fund s death, the average increase in the VaR of the fund is 74%. VaR appears to be effective in capturing the components of risk of hedge funds that lead to its death. In contrast, the corresponding mean (median) ratios for live funds are much lower at 1.09 (1.04) for the ratios of window 2 VaR to window 1 VaR, and 1.11 (1.02) for the ratios of window 3 VaR to window 2 VaR. For the two-year period, for live funds, the average increase in VaR is only 21%. However, there are many funds for which the recent VaRs are lower than before as well. The slightly increasing trend of VaR for live funds has been observed due to the increasing volatility in financial markets in recent years. Hence, adjusting for the impact of these recent 14

17 events, there should be no significant trend in the VaRs of live funds over time, while the dead funds indicate a clear, sharp increase in VaR as the fund death date approaches. 6. Monte Carlo simulation tests In the previous section, even though we include a large number of dead funds in our analysis, some degree of survivorship and sample selection biases are inevitable. There may be some funds that have extremely high volatility, skewness and kurtosis of returns, but they may not be in our sample either due to insufficient return history, or because they are not in the TASS database at all. Therefore, while it is useful to conduct robustness checks on the funds in our sample, it is important to fundamentally examine our methodology in a context that is free of these biases and can cover a wide range of market swings and rare events with large magnitude and volatility. We accomplish this using Monte Carlo simulation experiments, where we simulate returns for a wide range of hedge fund profiles, and then examine the accuracy of VaR estimates using the same methodology that is used for all hedge funds. We analyze six different fund profiles a median live and a median dead fund, representing median funds across our entire sample; a median fixed income live and a median fixed income dead fund, representing median funds from the style with the maximum skewness and kurtosis; and an extreme live and an extreme dead fund, representing the most extreme skewness and kurtosis amongst live and dead funds (both of these extreme funds happen to be fixed income funds). Table 9 presents the first four moments of returns for these representative funds. The intuition behind picking these fund profiles is to test our methodology on funds with moderate skewness and kurtosis in returns as well as on those with the most extreme skewness and kurtosis The return generating process We simulate two different processes a Geometric Brownian Motion (GBM) for the asset value process (implying normally distributed returns), and the GBM process augmented by a jump process (jump diffusion model, JD), to generate skewness and kurtosis in returns (Z): K Z = x + yi, i= 0 2 where x ~ N( α, σ ), K ~ Poisson( λ), 2 N( µ H, σ H ) w. p. yi ~ 2 N( µ L, σ L ) w. p. p p H L = 1 p In a standard JD model, the jump frequency has to be unrealistically high to match the skewness and kurtosis observed in the data. In our process, the number of jumps (K) is controlled by λ; however, each jump can be one of two types one jump (L) occurs with greater frequency, but has a smaller magnitude and volatility. The other jump (H) occurs very rarely, but with a much higher magnitude and volatility. This specification provides enough free parameters to match the moments of the process to actual fund moments, and still provide for large, infrequent jumps. H (5) 15

18 The mean, variance, skewness, and kurtosis of this process, estimated using the methodology outlined in Das and Sundaram (1999), are as follows: E [ Z] = α + λ[ p µ + p µ ] Var Kurt H H L L [ Z] = σ + λ[ ph ( µ H + σ H ) + pl ( µ L + σ L )] λ [ ] [ ph ( µ H + 3µ Hσ H ) + pl ( µ L + 3µ Lσ L )] = { σ + λ[ ph ( µ H + σ H ) + pl ( µ L + σ L )]} λ [ ] [ ph ( µ H + 6µ Hσ H + 3σ H ) + pl ( µ L + 6µ Lσ L + 3σ L )] Z = { σ + λ[ p ( µ + σ ) + p ( µ + σ )]} Skew Z H This JD process has eight free parameters. However, we only have four fund moments to match the process moments for estimating the parameters. 21 Therefore, we must specify four parameters, and then estimate the remaining four parameters to match the moments. We fix λ, p H, and the coefficients of variation (c.v.) of the two normal distributions from which jumps occur (σ H /µ H, and σ L /µ L ), at realistic levels, and then vary them across a wide range to ensure that our results are robust. We fix λp H it to be once every 10 years, and λp L at once every year. This gives us values of λ=11/120, and p H =1/11. The alternate λ values we consider are: H H L L L (6) λ=11/360 extreme jump once every 30 years, moderate jump once every 3 years, λ=11/240 extreme jump once every 20 years, moderate jump once every 2 years, λ=11/60 extreme jump once every 5 years, moderate jump once every 6 months, λ=11/36 extreme jump once every 3 years, moderate jump once every 3.6 months. Similarly, we fix the coefficient of variation of both the jumps at 0.3, but run the simulation experiments for c.v. values of 0.1, 0.2, 0.4, and 0.5 as well, for all λ values. In summary, for each of the six hedge fund profiles, we run simulations for five different jump intensities, and five different c.v. for the jumps. Therefore, in all, we conduct simulation tests for 150 different sets of parameters for the JD process. The estimated JD process parameter values for the six base cases (λ=11/120, p H =1/11, σ H /µ H =0.3, and σ L /µ L =0.3) are presented in Table 9. These parameter values clearly indicate the type of process that is generating the returns for that particular hedge fund type. For example, for the extreme dead fund, the reported parameters indicate that in order to generate the observed skewness (-6.29) and kurtosis (56.20), the GBM process must be augmented by a jump with a mean size of -47.0% occurring on average once every 10 years, plus another jump with a mean size of -13.1% occurring on average once every year. 21 In theory, one could match higher moments (fifth and beyond) as well, but they do not make any intuitive sense, and it reduces the whole exercise to just mathematical fitting, hence we avoid it. 16

19 6.2. Simulation results Once the parameters of the two processes are estimated, we simulate sixty hedge fund returns for each process, and estimate the 99 percentile return by applying EVT on these sixty returns. The primary objective is to examine whether, based on sixty return observations, the EVT methodology is able to statistically distinguish the JD process from the GBM process. 22 We repeat this experiment for the 150 sets of parameter values for the JD process, to ensure that our results are not sensitive to any particular specification of the process. The results, based on 30,000 simulation runs, are reported in Table 10 (one for each of the six hedge fund types). 23 The estimate of the 99 percentile return and its standard error are reported in the table. Results show that the EVT methodology is able to clearly distinguish between the JD process and the GBM process, even when only sixty returns are available. For example, for the median live fund, the 99 percentile EVT return for the JD process is -8.03% (with a standard error of 0.01%), while the 99 percentile EVT return for the corresponding GBM process is -6.43% (with a standard error of 0.01%), which is significantly lower than the GBM return at any level of significance. Table 10 also presents the several results for robustness checks on our EVT VaR approach. The scaled confidence intervals show that for the median live fund, the simulation based 95% confidence interval on the 99% VaR is [0.62, 1.59]. For the extreme dead fund, which has the maximum skewness and kurtosis across all our funds, this confidence interval is [0.33, 1.89]. These confidence intervals are similar to those obtained for individual funds using profile likelihood methods in the previous section. In addition, all the TCL/VaR ratios are well below two, implying that the use of three as the safety multiplier is appropriate. The column VaR 180 represents the ratio of 99% VaR using 60 return observations to the 99% VaR using 180 return observations. The objective of computing this ratio is to examine the bias, if any, in our VaR estimates due to reliance on a return history of five years. One may argue that a five-year window may be too short for estimating the true VaR, and could ignore the impact of catastrophic events that may be observed over time periods longer than five years. 24 Although there is slight underestimation using a five-year return history than a fifteen-year history, the difference is only marginal. The column EmpVaR represents the ratio of EVT VaR to the empirical VaR, both from 180 return observations. For the JD process, half of the six ratios are below one and the other half are above one. The errors are not systematic, 22 In comparing the JD process with the GBM process, we use the same stream of random numbers for the two processes in each simulation run, so that the differences between the two are more tightly attributable to the differences in the return processes, rather than to simulation error. 23 Increasing the number of simulation runs to 100,000 gave almost identical results. 24 However, the most recent five-year period used in our study (April 1998 to march 2003) covers three major financial market events - the Asian currency crisis of 1997, the Russian debt crisis of 1998, and the equity market crash from March 2000 onwards. Therefore, the hedge fund returns exhibited during these five years include return observations during fairly volatile times, with some very extreme observations. 17

20 which implies that our EVT approach is appropriate and the estimated VaRs are very close to the true VaRs from the empirical distributions Backtesting the VaR estimation approach Berkowitz and O Brien (2002) question the statistical accuracy of the internal VaR models used by six large U.S. commercial banks. 25 Based on statistical backtests, they examine whether the bank VaR models are able to predict the 99 th percentile trading P&L. How risk measures relate to actual performance over time is critical in evaluating the validity of the risk estimation model. Therefore, in this section, we examine the out-of-sample accuracy of our VaR estimates, and explore how reliable they are in forecasting future extreme returns. We do backtesting in a formal statistical framework that consists of comparing the history of VaR estimates with actual realized returns in the next period, to determine whether the frequency of exceedances is in line with the predicted confidence level of the VaR. For example, a 99% VaR should be exceeded 1% of the time, so we should expect an extreme return lower than the 99% VaR in 1 out of 100 months, on average. If the actual number of VaR violations is significantly different from the predicted number of violations, then the VaR estimation approach is not valid. Statistically, the VaR model verification using failure rates is a sequence of Bernoulli trials, where the outcome of each trial is binary - either the actual loss exceeds the VaR level, or it does not. Let p be the predicted failure rate of the model (p=1-c, where c is the confidence level for the VaR). The number of failures x (out of total trials T) follows a binomial probability distribution: f ( x) T p x x T x = (1 ) p (7) This binomial distribution is used to test the null hypothesis that the frequency of failures is equal to the predicted failure rate of the model. The 95% confidence regions for a powerful test of this hypothesis can be estimated using the tail points of the log-likelihood ratio: T N N [( 1 p) p ] + 2ln ( N ) T N {[ ] ( N ) } N LR = 2 ln 1 (8) T T where N is the actual number of exceedances in T trials. 26 This likelihood ratio is asymptotically distributed Chi-square with one degree of freedom under the null hypothesis that p is the true 25 It is not surprising that Berkowitz and O Brien (2002) find that a simple GARCH model does better than the internal models. This may be explained by the parameter restrictions imposed by the Basel rules, which force the internal VaR models to be slow moving. Jorion (2002) explains this in detail. 26 See Kupiec (1995) and Jorion (2001) for details. Also, please note that the 95% confidence region denotes the coverage of the confidence interval for the predicted number of violations, and is unrelated to the VaR confidence level of 99%. Therefore, on average, 5% of the funds should fail this backtest. 18