Do hedge funds have enough capital? A value-at-risk approach $

Journal of Financial Economics 77 (2005) 219 253 www.elsevier.com/locate/econbase Do hedge funds have enough capital? A value-at-risk approach $ Anurag Gupta a,, Bing Liang b a Weatherhead School of Management, Case Western Reserve University, 10900 Euclid Avenue, Cleveland, OH 44106-7235, USA b Isenberg School of Management, University of Massachusetts, Amherst, MA 01003, USA Received 14 November 2002; received in revised form 30 May 2004; accepted 30 June 2004 Available online 5 March 2005 Abstract We examine the risk characteristics and capital adequacy of hedge funds through the Valueat-Risk approach. Using extensive data on nearly 1,500 hedge funds, we find only 3.7% live and 10.9% dead funds are undercapitalized as of March 2003. Moreover, the undercapitalized funds are relatively small and constitute a tiny fraction of 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. r 2005 Elsevier B.V. All rights reserved. JEL classification: G23; G28; G29 Keywords: Hedge funds; Value-at-Risk; Capital adequacy; Extreme value theory; Monte Carlo simulation $ We thank Stephen Brown, Sanjiv Das, Will Goetzmann, Peter Ritchken, Bill Sharpe, Ajai Singh, Jack Treynor, and especially the two referees Philippe Jorion and Andrew Lo, for comments and suggestions on earlier drafts, and seminar participants at Case Western Reserve University, London School of Economics, 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. Corresponding author. Tel.: +1 216 368 2938; fax: +1 216 368 6249. E-mail address: anurag.gupta@case.edu (A. Gupta). 0304-405X/$ - see front matter r 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jfineco.2004.06.005

220 A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 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 find five dominant investment styles in hedge funds, which when added to Sharpe s (1992) asset class factor model, can provide an integrated framework for style analysis of both buy-and-hold and dynamic trading strategies. Brown etal. (1999) examine the performance of offshore hedge funds and attribute their performance to style effects rather than managerial skills. Ackermann etal. (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 (2004) 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 of yet, relates to the risk profile of hedge funds. The debacle of Long Term Capital Management LP (LTCM) highlights 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 Value-at-Risk (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 both the VaR for each fund and 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 undercapitalized. 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 given above, we address issues related to the techniques that are appropriate for risk estimation in the hedge fund industry. We characterize the distribution of hedge fund returns, and analyze whether VaR is a better measure of risk for evaluating hedge fund capital adequacy than traditional measures such as the standard deviation of returns and leverage ratios. Since hedge 1 Lo (2001) questions the usefulness of VaR as a risk measure for hedge funds due to their dynamic risk profiles. Accordingly, we conduct several robustness tests on the effectiveness of our VaR estimation methodology.

A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 221 fund returns are strongly nonnormal, it is likely that VaR measures will differ from the usual standard deviation-based measures. We examine whether the risk characteristics of dead funds are significantly different from live funds, and whether VaR is able to capture these differences. We segment all of 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 mitigate potential survivorship and selfselection bias concerns in the hedge fund data, since the simulation tests are not contaminated by these biases. These Monte Carlo simulations capture not only the regular price movements in hedge funds, but also extreme movements under rare marketconditions. 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 1,500 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 etal. (2004) 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 2003. 2 The (3.7%) live undercapitalized funds are mostly small funds, with median net assets of $66 million, 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 1998. 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 fund risk that precede their death. This supports the use of VaR for capital estimation. We also find significant nonnormality in hedge fund returns, in terms of high kurtosis. 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.

222 A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 Traditional risk measures like normality based standard deviation or leverage ratios fail to capture the true risk in hedge fund returns. 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 of the loss distribution. 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, 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, but does not account for the differences in the volatility of assets and liabilities, nor does it account for correlations. The VaR-based capital adequacy measure is also being increasingly adopted by regulators and supervisors. The Basel Committee for Banking Supervision (BCBS) now allows commercial banks to use their own internal VaR estimates to determine their capital requirement for market risk. 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

A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 223 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 three times the 99% ten-day VaR for market risk. 3 However, the choice of individual parameters is arbitrary, and the same market risk charge number can be obtained using differentparameter combinations. 4 We use three times the 99% one-month VaR as the required equity capital for hedge funds. The time horizon used is one 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 can only use private funding, 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, however, 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 such as options, interest rate derivatives, etc. For such portfolios, estimating the VaR is a complex task, since both the non-gaussian nature of the fluctuations of the underlying assets and the nonlinear dependence of the price of the derivatives must be dealt with. Moreover, there are no data available on the position holdings of hedge funds, since this constitutes proprietary trading information. Therefore, it is not possible for us to estimate the VaR of hedge funds through a position level analysis. The best data available are that of monthly returns reported by the hedge funds, which we use to estimate the VaR 5. 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 2003. The return data go back to 3 The safety multiple of three 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 three takes care of all error introduced due to misspecification of the true distribution of returns. 4 In tests during the summer of 1998, the BCBS found this market risk charge number to be adequate for commercial banks. 5 TASS (used by Fung and Hsieh (1997) and Liang (2000)), HFR Inc. (used by Ackermann etal., 1999; Agarwal and Naik, 2004), and CISDM (used by Ackermann etal., 1999) report monthly returns. The U.S. Offshore Funds Directory (used by Brown etal., 1999) reports annual returns.

224 A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 February 1977 for some of the live funds, and July 1978 for some of the dead funds. 6 The total assets under management for live funds are about $259 billion (out of a total of about $600 billion under management across all hedge funds), making itone of the largest hedge fund databases for academic research. 7,8 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). 9 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 undercapitalization. However, we do consider a large number of dead funds, and not just those 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 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. 10 4. 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 as an extreme 6 TASS started collecting dead fund information in 1994, hence the survivorship bias in the data is likely to be greater for the years prior to 1994. 7 See Implications of the Growth of Hedge Funds Staff Report to the SEC in September 2003. 8 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. 9 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 estimate the VaR using different lengths of time. The results are very similar, therefore we report our results from the five-year window only. 10 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 notation from TASS.

A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 225 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. 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 quantile, the VaR is estimated as follows: VaR ¼ð0 R 99% ÞTNA, (1) where VaR is the 99% one month VaR, R 99% is the cutoff return at the 99% confidence level estimated using EVT, and TNA is the total net assets (equity) of a fund. This VaR is relative to a 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 to be three times this VaR number. We examine the validity of the safety multiplier of three in later sections of this paper. To evaluate capital adequacy, we compare this required capital with the actual equity capital that is backing these funds. We compute a capitalization ratio (the Cap ratio) defined as follows: Cap ¼ E actual E required. (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 undercapitalized. 12 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 TCL ¼ð0 E½RjRoR 99% ŠÞ TNA. (3) 11 We compute the VaR relative to mean returns as well, in order to check the robustness of our conclusions; the capital adequacy results are very similar. In any case, over a short horizon of a month, the expected return is likely to be small compared to the VaR, hence the adjustment for the mean return is not likely to matter in most cases. 12 Note that substituting required capital with 3 x VaR in (2), it yields Cap ¼½1=ð3 ð0 R 99% ÞÞŠ 1:

226 A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 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 can indicate how safe it is to use the standard multiplier of three recommended by the Basel Committee. Many traditional risk-based capital measures assume the return distribution to be normal, though it is often significantly nonnormal. A comparison of risk capital measures based on EVT with those based on 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 VaR ¼½ðs R 2:326ÞTNAŠ, (4) where s 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 undercapitalization 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. 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, many 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 nonnormal. 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 nonlinear than regular price fluctuations under normal circumstances. Hedge funds often 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. 14 According to Tabachnick and Fidell (1996), the standard error for skewness is roughly O(6/N), hence for skewness estimated using 60 returns, the two-standard error bounds for significance are approximately 70.63. Both 0.06 and 0.01 are within these bounds. However, some hedge fund styles such as convertible arbitrage, emerging markets, fixed income arbitrage, and event driven have negative skewness outside these bounds.

Table 1 Descriptive statistics for hedge fund returns This table presents the descriptive statistics for the monthly return distributions of hedge funds across investment styles. The data is from TASS Management Limited. There are 3,702 hedge funds, including 2,256 live funds and 1,446 dead funds as of March 2003. The reported statistics are for the return history over a five-year minimum period. There are a total of 942 live funds and 494 dead funds that meet this return history requirement. All figures reported are the medians within each style. Style Live funds Dead funds No. Mean Std. dev. Median Skew Kurt No. Mean Std. dev. Median Skew Kurt Convertible arbitrage 59 0.90 1.55 0.94 0.30 4.50 14 0.61 2.51 0.90 0.70 4.66 Dedicated short bias 10 1.05 7.91 1.01 0.14 4.00 6 0.18 6.31 0.05 0.50 3.45 Emerging markets 62 0.48 7.46 0.31 0.28 5.32 40 0.43 6.68 0.81 0.81 6.34 Marketneutral 36 0.77 2.49 0.59 0.25 4.08 10 0.30 2.22 0.35 0.19 3.58 Eventdriven 107 0.66 1.86 0.73 0.51 5.12 39 0.58 2.56 0.81 0.72 7.07 Fixed income arbitrage 32 0.64 1.83 0.97 1.47 7.61 18 0.62 2.34 0.96 3.10 15.91 Fund of funds 185 0.56 2.30 0.55 0.03 6.58 107 0.54 3.55 0.48 0.17 4.12 Global macro 40 0.63 3.64 0.34 0.45 3.92 40 0.83 5.16 0.37 0.38 4.38 Long/shortequity hedge 288 0.91 5.51 0.50 0.44 4.57 114 1.01 6.01 0.99 0.08 4.09 Managed futures 94 0.92 5.69 0.61 0.25 3.52 105 0.60 5.53 0.36 0.24 4.17 Other 29 0.78 2.55 0.88 0.04 6.10 1 0.52 3.71 0.80 5.72 40.77 Total 942 0.72 3.70 0.61 0.06 4.85 494 0.62 4.57 0.64 0.01 4.43 A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 227 ARTICLE IN PRESS

228 A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 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 2003. 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 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 undercapitalized. For the live funds, about 3.7% (35 out of 942) of the funds are undercapitalized, 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 undercapitalized than the live funds. This is somewhat consistent with the hypothesis that one of the reasons for a fund s death is undercapitalization. However, undercapitalization 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 such as 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 undercapitalization are the emerging markets funds (6 out of 62, or 9.7%), and fixed income arbitrage funds 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.

Table 2 Hedge fund VaR based on Extreme Value Theory This table presents VaR estimates for hedge funds across investment styles, for 942 live and 494 dead funds (as of March 2003). The absolute VaR is derived from VaR ¼½ð0 R 99% ÞTNAŠ; where VaR is the 99% one month VaR, R 99% is the cutoff return at 99% confidence level estimated using EVT, and TNA is the total net asset value (equity) of the fund, measured at the end of the sample period (which may not be March 2003 for dead funds). The relative VaR is the ratio of absolute VaR to fund net assets. The absolute VaR as well as fund assets are in millions of dollars. Style Live funds Dead funds No. Fund assets EVT VaR Relative VaR No. Fund assets EVT VaR Relative VaR Mean Median Mean Median Mean Median Mean Median Mean Median Mean Median Convertible arbitrage 59 317.7 94.5 18.1 3.6 5.1 3.3 14 55.0 14.7 9.1 1.3 10.4 7.2 Dedicated short bias 10 91.9 36.5 17.7 3.5 14.9 14.6 6 25.8 7.8 4.2 0.9 18.4 16.7 Emerging markets 62 85.0 29.6 19.9 5.9 21.9 20.7 40 22.5 13.2 5.4 2.5 26.1 25.1 Marketneutral 36 317.2 66.0 4.6 2.5 6.0 6.6 10 23.8 11.1 1.7 0.9 9.3 9.5 Eventdriven 107 234.1 81.0 13.2 6.1 8.3 6.7 39 149.7 30.0 28.3 4.3 17.1 10.3 Fixed income arbitrage 32 261.2 140.1 33.5 9.7 12.1 5.5 18 88.3 25.6 16.2 4.1 20.5 15.8 Fund of funds 185 158.1 52.0 10.6 3.0 8.0 7.3 107 32.7 6.5 3.5 0.8 14.0 12.1 Global macro 40 144.2 42.7 9.1 2.3 8.6 6.7 40 96.5 5.7 16.6 0.8 15.6 13.4 Long/shortequity hedge 288 190.3 45.9 19.3 5.7 13.6 12.5 114 40.1 16.7 8.2 2.6 20.0 15.8 Managed futures 94 114.8 10.1 15.4 1.3 13.3 11.7 105 20.3 2.6 1.6 0.3 18.9 16.1 Other 29 587.7 119.4 51.5 4.8 9.7 5.2 1 190.0 190.0 33.0 33.0 17.4 17.4 Total 942 198.9 53.3 16.9 4.1 11.3 9.8 494 48.1 8.5 8.0 1.06 17.9 14.6 A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 229 ARTICLE IN PRESS

230 Table 3 Undercapitalization based on VaR from EVT This table presents the Cap ratios for hedge funds using VaR from EVT, for 942 live and 494 dead funds (as of March 2003), where Cap ¼ ðe actual E required Þ=E required (the Cap ratio) represents the degree of undercapitalization, E required is the required equity that is three times the 99% one month VaR of the fund (using EVT), and E actual is the actual equity which is taken at the end of the VaR sample period. 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. Note that U-cap refers to undercapitalized funds. Style Live funds Dead funds Total funds No. U-cap % U-cap Cap ratio Total funds No. U-cap % U-cap Cap ratio Mean Median Mean Median Convertible arbitrage 59 2 3.4 14.8 7.5 14 0 0 4.7 3.6 Dedicated short bias 10 0 0 1.9 1.3 6 1 16.7 1.2 1.0 Emerging markets 62 6 9.7 0.9 0.6 40 12 30.0 0.7 0.3 Marketneutral 36 2 5.6 18.0 4.0 10 0 0 3.3 2.5 Eventdriven 107 6 5.6 5.1 3.9 39 6 15.4 2.8 2.0 Fixed income arbitrage 32 5 15.6 8.5 4.9 18 6 33.3 1.9 0.8 Fund of funds 185 1 0.5 6.7 3.6 107 4 3.7 2.5 1.7 Global macro 40 1 2.5 5.1 4.0 40 3 7.5 2.1 1.5 Long/shortequity hedge 288 8 2.8 2.7 1.7 114 11 9.6 1.5 1.1 Managed futures 94 3 3.2 2.3 1.8 105 11 10.5 1.8 1.1 Other 29 1 3.5 6.1 5.4 1 0 0 0.9 0.9 Total 942 35 3.7 5.3 2.4 494 54 10.9 2.0 1.3 A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 ARTICLE IN PRESS

A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 231 (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 undercapitalized. Similarly, in the dead funds group, emerging market funds (12 outof 40, or 30%) and fixed income arbitrage funds (6 outof 18, or 33.3%) have high levels of undercapitalization. 5.2. 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 2003. However, hedge fund risk exposures are highly dynamic, hence the Cap ratio is unlikely to be constant over time. First, hedge funds change their portfolio compositions fairly frequently. Second, 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 over 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 of 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 2003. In this manner we go back month by month to January 1995, which is as far back as the data allow us to go, in order to get a reasonably large number of funds for crosssectional analysis. In Fig. 1, we present the percentage of live funds undercapitalized, for each month, from January 1995 to March 2003. The fraction of undercapitalized live funds steadily increases from 0.49% in January 1995 (1 out of 203 live funds undercapitalized) to a maximum of 5.43% as of August 2000 (37 out of 681 live funds undercapitalized), and then reduces to 3.72% in March 2003 (35 out of 942 funds). This graph shows some important trends. The extent of undercapitalization steadily increases up to the middle of 2000, after which it declines a bit. This decline may be due to the fact that some of the undercapitalized 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 undercapitalized 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 undercapitalized over time. The second plotin Fig. 1 presents the median Cap ratios for all live funds, each month, from January 1995 to March 2003. 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 1998. 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 2003.

232 A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 6% Rolling Percentage of Undercapitalized Live Funds 5% 4% 3% 2% 1% 0% 95 Jul- 95 96 Jul- 96 97 Jul- 97 98 Jul- 98 99 Jul- 99 00 Jul- 00 01 Rolling Cap Ratios for Live Funds Jul- 01 02 Jul- 02 03 3 2 1 0 95 Jul- 95 96 Jul- 96 97 Jul- 97 98 Jul- 98 99 Jul- 99 00 Jul- 00 01 Jul- 01 02 Jul- 02 03 12% Historical Percentage of Undercapitalized Dead Funds 10% 8% 6% 4% 2% 0% 96 Jul- 96 97 Jul- 97 98 Jul- 98 99 Jul- 99 00 Jul- 00 01 Jul- 01 02 Jul- 02 03 Fig. 1. Historical Rolling Window Capitalization. We present the rolling percentage of undercapitalized live and dead funds, and the rolling median cap ratios for live funds. Each rolling window has 60 months. For example, for live funds, the first rolling window spans February 1990 to January 1995, the second spans March 1990 to February 1995, and the last one spans April 1998 to March 2003. EVT VaR is estimated from each window and cap ratios are calculated based on the EVT VaR. The rolling windows for dead funds are defined in a similar manner.

A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 233 The third plot in Fig. 1 presents the fraction of dead funds that are undercapitalized 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 undercapitalized. The percentage of undercapitalized dead funds rises sharply from 1.49% in April 1998 (1 outof 67 funds) to 11.91% in December 2000 (33 outof 277 funds). After that, it fluctuates around 11%. Again, the fraction of undercapitalized dead funds rises steeply after the Asian financial crisis of 1997 and the Russian debt crisis of 1998. 16 It is important to note that the 35 undercapitalized live funds constitute only 1.2% ($2.3 billion out of $187.4 billion) of the total net assets of the 942 live funds, indicating that a very large proportion (98.8%) of the live fund assets in our sample are not exposed to the risk of undercapitalization. Table 4 presents a statistical comparison of various characteristics of the undercapitalized funds (both live and dead) with the remaining adequately capitalized funds. Among the live funds, the 35 undercapitalized funds are significantly smaller than the remaining funds, with average net assets of $66.3 million, as compared to $201.2 million 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 such as fee structure, watermark provisions, use of derivatives, etc., the undercapitalized 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 undercapitalized and adequately capitalized funds. In fact, the average dead fund in our sample existed for about eight years, irrespective of whether it was undercapitalized or not at the time of death. These comparisons tell us that the adequately capitalized and undercapitalized funds differ in size as well as investment styles, but not in any of the other reported characteristics. 17 However, they differ significantly in their return distributions. Therefore, for making inferences about capital adequacy, we need to focus on returns rather than fund characteristics. 5.3. 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 16 An analysis of the time-series patterns of capital adequacy for individual undercapitalized funds reveals 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 undercapitalized. 17 We do 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 find the large funds, in our sample, to be significantly better capitalized than the other funds.

234 A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 Table 4 Comparative characteristics of undercapitalized funds This table compares various characteristics of undercapitalized and adequately capitalized funds. The reported statistics are for 942 live (35 are undercapitalized) and 494 (54 are undercapitalized) dead funds that meet the return history requirement of a five-year minimum period, as of March 2003. The t-test is conducted for the difference between undercapitalized and adequately capitalized funds for both live and dead categories. Variable Live funds Dead funds Adequate-cap Under-cap t-stat Adequate-cap Under-cap t-stat Mean Std. dev. Mean Std. dev. Mean Std. dev. Mean Std. dev. Equity ($m) 201.2 683.3 66.3 89.7 4.7 48.6 146.6 35.7 142.2 0.6 Cap ratio 5.5 10.8 0.2 0.2 15.8 2.3 2.9 0.2 0.2 17.9 Leverage ratio 4.5 10.6 5.3 10.5 0.3 2.7 6.9 3.8 8.9 0.8 Minimum return 11.4 9.5 39.34 16.1 8.7 13.2 9.1 37.7 16.6 10.4 Maximum return 14.1 12.6 33.0 30.3 3.1 14.5 10.9 34.1 45.0 3.1 Mean return 0.8 0.7 0.4 1.7 1.0 0.7 0.7 0.2 1.7 2.1 Median return 0.6 0.7 0.6 1.6 0.1 0.6 0.8 0.2 2.2 1.5 Std. dev. 4.5 3.4 10.2 5.5 5.2 5.0 3.2 11.1 8.3 5.2 Skewness 0.02 1.3 1.6 2.6 3.0 0.2 1.3 1.1 2.1 2.9 Kurtosis 6.4 5.3 16.3 11.1 4.5 6.1 5.7 11.8 10.7 3.8 EVT return 10.4 6.4 45.4 11.3 15.4 14.6 7.4 46.7 14.7 15.4 Age (months) 104.3 40.7 95.7 35.0 1.2 96.6 34.8 92.1 25.7 1.1 Max leverage ratio 6.4 12.6 17.3 23.3 2.3 4.2 9.9 5.1 10.9 0.5 Managementfee 1.4 0.8 1.4 0.6 0.1 1.7 1.2 1.6 1.0 0.9 Incentive fee 16.2 7.3 17.1 7.4 0.6 15.1 8.3 17.1 8.5 1.6 Leverage dummy 0.6 0.5 0.8 0.4 1.5 0.6 0.5 0.8 0.4 1.9 Watermark dummy 0.3 0.5 0.4 0.5 0.5 0.1 0.3 0.1 0.3 0.3 Lockup period (months) 2.0 4.7 3.0 5.7 0.8 0.5 2.4 0.8 2.9 0.7 Minimum investment ($m) 1.0 7.4 0.4 0.3 2.5 0.4 0.7 0.4 0.8 0.3 Advance notice period (days) 32.9 26.3 27.5 23.3 1.1 14.3 20.7 15.4 20.5 0.4 Managers personal investment 0.5 0.5 0.4 0.5 0.9 0.6 0.5 0.6 0.5 0.8 Open-end fund dummy 0.7 0.5 0.7 0.5 0.7 0.8 0.4 0.9 0.3 1.6 Open-to-public dummy 0.2 0.4 0.2 0.4 0 0.2 0.4 0.2 0.4 0.6 Derivatives trading dummy 0.2 0.4 0.3 0.5 1.1 0.3 0.4 0.3 0.4 0.3 Significantatthe 1% level. Significantatthe 5% level. Significantatthe 10% level. capitalization by running a cross-sectional regression of Cap ratios on various fund characteristics and styles. For robustness, we choose several models that include different subsets 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 (unlike incentive fee), and is directly deducted from the total assets in order to obtain net assets; hence, it is negatively correlated with the Cap ratio. Younger funds may not be well established,

A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 235 Table 5 Cross-sectional regression of cap ratios on fund characteristics This table presents the regression results of cap ratios for live funds (as of March 2003) on various fund characteristics. The regression equation is LogðCap i Þ¼a 0i þ a 1i logðsizeþþa 2i logðageþþa 3i ðmfeeþþa 4i ðifeeþ þ a 5i ðleverageþþa 6i ðwatermarkþþa 7i ðlockupþþ X10 b ji ðdummy j Þ, where log(size) is the natural logarithm of fund equity as of March 2003, log(age) is the natural logarithm of fund age from the first month of reported returns to March 2003, and dummy j ( j ¼ 1 to 10) represent ten style dummy variables. Robust p-values are reported in parentheses. Variable Model 1 Model 2 Model 3 Model 4 Model 5 Intercept 0.378 0.058 0.528 0.692 0.589 (0.379) (0.898) (0.219) (0.097) (0.172) Log(size) 0.150 0.153 0.117 0.114 0.119 (0.000) (0.000) (0.000) (0.000) (0.000) Log(age) 0.184 0.233 0.222 0.250 0.240 (0.018) (0.004) (0.002) (0.001) (0.001) Managementfee 0.046 0.049 0.088 0.092 (0.153) (0.135) (0.008) (0.006) Incentive fee 0.005 0.003 0.004 (0.194) (0.373) (0.272) Leverage ratio 0.003 0.003 0.003 0.003 (0.189) (0.157) (0.141) (0.175) Watermark dummy 0.088 0.055 0.072 (0.159) (0.317) (0.195) Lockup (months) 0.006 0.002 0.002 (0.326) (0.735) (0.697) Convertible arb 0.612 0.612 0.608 (0.000) (0.000) (0.000) Shortbias 0.392 0.360 0.363 (0.163) (0.205) (0.199) Emerging markets 0.926 0.961 0.933 (0.000) (0.000) (0.000) Marketneutral 0.491 0.470 0.481 (0.006) (0.009) (0.008) Eventdriven 0.031 0.038 0.033 (0.839) (0.802) (0.828) Fixed income arb 0.037 0.022 0.028 (0.840) (0.908) (0.881) Fund of funds 0.204 0.132 0.203 (0.173) (0.365) (0.176) Global macro 0.131 0.083 0.122 (0.455) (0.636) (0.491) Long/short 0.479 0.463 0.474 (0.001) (0.001) (0.001) Managed futures 0.164 0.285 0.167 (0.315) (0.070) (0.309) Adj. R-square (%) 10.6 11.2 31.4 30.9 31.1 Significantatthe 1% level. Significantatthe 5% level. Significantatthe 10% level. j¼1

236 A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219 253 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 is one, since many hedge funds do notuse borrowed funds. 18 Therefore, it is unlikely that leverage ratios will convey any relevant information about hedge funds risk profiles, 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 ¼ 0.06) for live funds and 0.09 (p-value ¼ 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 undercapitalized based on the VaR estimated by assuming normality. If a simple standard deviationbased 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 undercapitalization 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. 18 As reported by TASS, only 67% of live funds (1,510 outof 2,256) and 72% of dead funds (1,037 outof 1,446 funds) are levered, as of March 2003. The restof the funds have no directborrowing or indirect leverage through short selling or derivatives trading. 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.