do hedge funds have enough capital? a value-at-risk approach · e-mail address:...

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Journal of Financial Economics 77 (2005) 219–253

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

Anurag Guptaa,�, Bing Liangb

aWeatherhead School of Management, Case Western Reserve University, 10900 Euclid Avenue,

Cleveland, OH 44106-7235, USAbIsenberg 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 Value-

at-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

- see front matter r 2005 Elsevier B.V. All rights reserved.

.jfineco.2004.06.005

k Stephen Brown, Sanjiv Das, Will Goetzmann, Peter Ritchken, Bill Sharpe, Ajai Singh, Jack

especially the two referees Philippe Jorion and Andrew Lo, for comments and suggestions on

s, and seminar participants at Case Western Reserve University, London School of

University of Massachusetts at Amherst, Virginia Tech., the 2003 European Finance

eetings in Glasgow, the 2003 Western Finance Association Meetings in Los Cabos, the 2003

l seminar in Scottsdale, the 2001 FMA European Meetings in Paris, and the 2001 FMA

oronto. Bing Liang acknowledges a summer research grant from the Weatherhead School of

, Case Western Reserve University. We also thank TASS Management Limited for providing

remain responsible for all errors.

nding author. Tel.: +1216 368 2938; fax: +1 216 368 6249.

dress: [email protected] (A. Gupta).

www.elsevier.com/locate/econbase

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A. Gupta, B. Liang / Journal of Financial Economics 77 (2005) 219–253220

1. Introduction

The hedge fund industry is one of the fastest growing sectors in finance, due tolimited regulatory oversight, flexible investment strategies, and performance-basedfee structures. The rapid growth in this area has captured the attention of bothacademics and practitioners. This has led to several studies that analyze hedge fundperformance, examine survivorship bias issues, and investigate the reasons fordifferences in fund performance across styles. These studies include Fung and Hsieh(1997), who find five dominant investment styles in hedge funds, which when addedto Sharpe’s (1992) asset class factor model, can provide an integrated framework forstyle analysis of both buy-and-hold and dynamic trading strategies. Brown et al.(1999) examine the performance of offshore hedge funds and attribute theirperformance to style effects rather than managerial skills. Ackermann et al. (1999)conclude that hedge funds outperform mutual funds. Liang (1999) finds that hedgefund investment strategies are different from those of mutual funds. Agarwal andNaik (2004) propose a general asset class factor model comprising of option-basedand buy-and-hold strategies to benchmark hedge fund performance.

All of the above studies analyze hedge fund performance relative to certainbenchmarks. An important question, unanswered as of yet, relates to the risk profileof hedge funds. The debacle of Long Term Capital Management LP (LTCM)highlights the need for more academic studies in hedge fund risk exposure andcapital adequacy. In this paper, by doing extensive research on a large hedge funddatabase, we address the following primary questions: How risky are hedge funds, ingeneral? To what extent are they adequately capitalized? What are the time-seriespatterns in the levels of capitalization in the hedge fund industry? How is fundcapitalization related to the various fund characteristics?

Like Jorion (2000), we propose a Value-at-Risk (VaR) approach, since VaR notonly measures the maximum amount of assets a fund can lose over a certain timeperiod with a specified probability, but can also be used to measure the equity capitalneeded to cover those losses.1 We analyze both the VaR for each fund and itsdistribution across all funds, and compute a VaR-based estimate of required equitycapital for each fund. This required equity is then compared to the actual fund equityto determine how many hedge funds are undercapitalized. We also study fund riskand capitalization on a dynamic basis in order to examine their time-series variationand analyze the determinants of fund capitalization. Extensive robustness checks areconducted to ensure that our results and inferences are reliable.

In addition to the primary questions given above, we address issues related to thetechniques that are appropriate for risk estimation in the hedge fund industry. Wecharacterize the distribution of hedge fund returns, and analyze whether VaR is abetter measure of risk for evaluating hedge fund capital adequacy than traditionalmeasures such as the standard deviation of returns and leverage ratios. Since hedge

1Lo (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.

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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 fromthe usual standard deviation-based measures. We examine whether the riskcharacteristics of dead funds are significantly different from live funds, and whetherVaR is able to capture these differences. We segment all of our results by investmentstyles in order to understand if there are significant risk and capital adequacydifferences in hedge funds across styles. Finally, we conduct carefully designedMonte Carlo simulation tests in order to mitigate potential survivorship and self-selection bias concerns in the hedge fund data, since the simulation tests are notcontaminated by these biases. These Monte Carlo simulations capture not only theregular price movements in hedge funds, but also extreme movements under raremarket conditions.

To the best of our knowledge, our paper is the first one to address capitaladequacy and risk estimation issues in the entire hedge fund industry. AlthoughJorion’s study is the first one to apply the VaR methodology to hedge funds, heexamines only a single fund, while we examine nearly 1,500 hedge funds. Fung andHsieh (2000) examine hedge fund performance and risk in some major marketevents/crisis. However, they adopt a traditional mean–variance approach, which isnot effective in capturing hedge fund risk. In contrast, we use the VaR approach tostudy hedge fund risk. Lhabitant (2001) reports factor model-based VaR figures forsome hedge funds. However, he does not use the return information directly inestimating the VaR, and does not examine any capital adequacy issues. Getmanskyet al. (2004) examine the illiquidity exposure of hedge funds. They focus on liquidityrisk while we study the market risk and capital adequacy issues in the hedge fundindustry.

We find that, in our sample, a majority of hedge funds (96.3% of the live and89.1% 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 $66million, 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, investmentstyle, age, and management fee. In particular, the convertible arbitrage and marketneutral funds are better capitalized than the emerging markets, long/short equity,and managed futures funds. On a dynamic basis, we document a significant drop infund capitalization after the Russian debt crisis in 1998.

Robustness tests based on Monte Carlo simulation and backtesting evidencesupport our findings, hence the application of VaR (using Extreme Value Theory)for inferring capital adequacy seems valid. For dead funds, we find that theestimated VaR increases by an average of 74% over the two years immediatelypreceding the fund’s death, while no such trend is observed for live funds, indicatingthat our estimated VaR is effective in capturing some elements of hedge fund riskthat precede their death. This supports the use of VaR for capital estimation. Wealso find significant nonnormality in hedge fund returns, in terms of high kurtosis.

2Our 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.

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Traditional risk measures like normality based standard deviation or leverage ratiosfail to capture the true risk in hedge fund returns.

This paper is organized as follows. Section 2 explains the concept of VaR andits application in determining capital requirements. Section 3 describes the data.Section 4 explains the research methodology. The empirical results for capitaladequacy and robustness tests are presented in Section 5. Section 6 provides theMonte 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 agiven confidence level. If c is the selected confidence level, VaR corresponds tothe 1�c lower tail of the loss distribution. It is calculated in dollar amounts and isdesigned 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 equitycapital that must be held to support that level of risky business activity. In fact, thedefinition of VaR is completely compatible with the role of equity as perceived byfinancial institutions—while reserves or provisions are held to cover expected lossesincurred in the normal course of business, equity capital is held to provide a capitalcushion against any potential unexpected losses. Since all unexpected losses cannotbe covered with 100% certainty, the level of this capital cushion must be determinedwithin prudent solvency guidelines.

This definition of risk capital encompasses a broader concept of risk than thetraditional leverage ratios, which only depend on the liabilities side of the balancesheet. Firms with any level of leverage may have significant risk of not being able tocontinue with their business if they hold very risky assets. This is especially true forfinancial firms (including hedge funds), which hold traded assets that must bemarked-to-market periodically. The potential for losses on these assets, in relation tothe equity capital, is the most important determinant for capital adequacy of suchfirms, not their leverage ratio. Of course, leverage magnifies the impact of suchlosses, but does not account for the differences in the volatility of assets andliabilities, nor does it account for correlations.

The VaR-based capital adequacy measure is also being increasingly adopted byregulators and supervisors. The Basel Committee for Banking Supervision (BCBS)now allows commercial banks to use their own internal VaR estimates to determinetheir capital requirement for market risk. The Derivatives Policy Group (DPG)formed by the Securities and Exchange Commission (SEC) in 1994 also makessimilar recommendations to broker-dealers that conduct an OTC derivativesbusiness. Therefore, the use of a VaR measure to study capital adequacy isextremely relevant and is in line with the norms and guidelines in place for variousfinancial institutions.

There are three main decision variables in estimating VaR—the confidence level, atarget horizon, and an estimation model. If the objective of estimating VaR is toestimate risk capital requirements, the confidence level should be chosen to be high

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enough so that there is very little probability of failure. The target horizon is relatedto the liquidity of the positions in the portfolio. It should reflect the amount of timenecessary 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. Thereis considerable uncertainty in choosing these variables, and the choice is oftenarbitrary.

For commercial banks, the BCBS (1996) stipulates a capital requirement of threetimes the 99% ten-day VaR for market risk.3 However, the choice of individualparameters is arbitrary, and the same market risk charge number can be obtainedusing different parameter combinations.4 We use three times the 99% one-month

VaR as the required equity capital for hedge funds. The time horizon used is onemonth instead of ten days because hedge funds are quite different from commercialbanks. As pointed out by Jorion (2000), commercial banks are closely supervised byregulators, hence they can react to potential difficulties much sooner. Hedge fundsare far less regulated and can only use private funding, hence they would have aharder time raising additional capital when needed. Therefore, for hedge funds, thetarget 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 BCBSguidelines.

3. Data

Hedge funds often have complex portfolios including nonlinear assets such asoptions, interest rate derivatives, etc. For such portfolios, estimating the VaR is acomplex task, since both the non-Gaussian nature of the fluctuations of theunderlying assets and the nonlinear dependence of the price of the derivatives mustbe dealt with. Moreover, there are no data available on the position holdings ofhedge funds, since this constitutes proprietary trading information. Therefore, it isnot possible for us to estimate the VaR of hedge funds through a position levelanalysis. The best data available are that of monthly returns reported by the hedgefunds, which we use to estimate the VaR5.

We use the hedge fund dataset from TASS Management Limited (hereafter,TASS), which contains monthly return data on 3,702 hedge funds, including 2,256survived and 1,446 dissolved funds, as of March 2003. The return data go back to

3The 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.4In tests during the summer of 1998, the BCBS found this market risk charge number to be adequate for

commercial banks.5TASS (used by Fung and Hsieh (1997) and Liang (2000)), HFR Inc. (used by Ackermann et al., 1999;

Agarwal and Naik, 2004), and CISDM (used by Ackermann et al., 1999) report monthly returns. The U.S.

Offshore Funds Directory (used by Brown et al., 1999) reports annual returns.

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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 atotal of about $600 billion under management across all hedge funds), making it oneof the largest hedge fund databases for academic research.7,8

We use a five-year return history as the minimum time period required to estimatethe VaR, leaving us with 1,436 funds (942 live and 494 dead funds).9 This alsoensures that, at least for the live funds, the returns we use to estimate the VaRoverlap with some of the most turbulent times in financial markets, starting with theAsian 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 ourVaR estimation by throwing away younger (and potentially riskier) funds. Therefore,we may underestimate the true degree of undercapitalization. However, we doconsider 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 oflive 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 areconvertible arbitrage, dedicated short bias, emerging markets, market neutral, eventdriven, fixed income arbitrage, fund of funds, global macro, long/short equity hedge,managed futures, and others. This classification allows us to study hedge fund riskand capital adequacy by investment styles.

For leverage information, TASS reports two numbers—the average leverage ratioand the maximum leverage ratio. The average leverage provides a measure of thehistorical leverage ratio on average, while the maximum leverage indicates the largestcapacity up to which a fund can be levered. We use the average leverage of funds forleverage 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

6TASS started collecting dead fund information in 1994, hence the survivorship bias in the data is likely

to be greater for the years prior to 1994.7See ‘‘Implications of the Growth of Hedge Funds’’ Staff Report to the SEC in September 2003.8Liang (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.9The 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.10TASS 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.

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value problem. While estimating VaR, we focus on the behavior of the returndistribution in the left tail. Extreme Value Theory (EVT) provides a firm theoreticalfoundation 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 the99-percentile return in the left tail of the return distribution. Using this quantile, theVaR is estimated as follows:

VaR ¼ ð0 � R99%Þ � TNA, (1)

where VaR is the 99% one month VaR, R99% is the cutoff return at the 99%confidence level estimated using EVT, and TNA is the total net assets (equity) of afund.

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 theexpected return over the target horizon. We use VaR relative to zero since there maybe significant biases and errors in the estimated mean returns for hedge funds, whichwould introduce another source of error in our VaR estimate. In addition, for thepurposes of determining equity capital, it is critical to measure the absolute dollarloss that the fund might incur over the target horizon, rather than the shortfall fromexpected returns.11 The capital requirement is then taken to be three times this VaRnumber. We examine the validity of the safety multiplier of three in later sections ofthis paper.

To evaluate capital adequacy, we compare this required capital with the actualequity capital that is backing these funds. We compute a capitalization ratio (the Cap

ratio) defined as follows:

Cap ¼Eactual � Erequired

Erequired. (2)

A Cap ratio less than zero implies that the actual equity is not sufficient to cover therisk 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 expectedshortfall). 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 extremeloss occurs. A 99% VaR only tells us the minimum loss that can be expected 1% ofthe 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 beuseful in determining capital adequacy.

In Appendix A.2, we provide details on estimating the TCL using EVT. The TCLis defined as

TCL ¼ ð0 � E½RjRoR99%�Þ � TNA. (3)

11We 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.12Note that substituting required capital with 3 x VaR in (2), it yields Cap ¼ ½1=ð3 � ð0 � R99%ÞÞ� � 1:

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The ratio of TCL to VaR can provide a more objective basis of determining theappropriate capital multiplier that should be used in conjunction with VaR, and canindicate how safe it is to use the standard multiplier of three recommended by theBasel Committee.

Many traditional risk-based capital measures assume the return distribution to benormal, though it is often significantly nonnormal. A comparison of risk capitalmeasures based on EVT with those based on normality would highlight the errorintroduced by assuming normality. Therefore, we re-estimate the 99% VaR of eachfund assuming the return distribution to be normal, as follows:13

VaR ¼ ½ðsR � 2:326Þ � TNA�, (4)

where sR is the standard deviation of fund returns. The Cap ratio is computed in amanner similar to that in the EVT approach. The differences in the levels ofundercapitalization using the EVT VaR and the standard deviation-based VaR canbe attributed solely to the departures from normality in the actual return distributionof 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 investmentstyles. All figures are the medians across funds in the same style. Several inferencescan 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 with0.62% for a median dead fund. These results are consistent with Liang (2000), whodocuments that one of the reasons that funds die is poor performance. Dead fundsalso exhibit higher volatility of returns than live funds. Second, many hedge fundreturns do not exhibit a high level of skewness, except for some styles within deadfunds.14 However, all investment styles show high kurtosis above three, whichindicates that hedge fund returns have fat tails and more extreme return values, thusmaking their return distributions significantly nonnormal. For example, fixedincome 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 nonlinearthan regular price fluctuations under normal circumstances. Hedge funds often

13Note 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.14According 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.

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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

Market neutral 36 0.77 2.49 0.59 0.25 4.08 10 0.30 2.22 0.35 �0.19 3.58

Event driven 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/short equity 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

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implement opportunistic trading strategies and bet on major markets eventsworldwide. 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) andnegative (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 weturn to VaR based on EVT for evaluating hedge fund risk in this paper. Table 1 alsoshows that there are significant differences in hedge fund return distributions acrossinvestment styles, hence a study by fund styles is more insightful than just anaggregated study that groups all hedge funds together.15

Table 2 presents statistics for fund sizes and absolute VaR numbers across stylesfor both live and dead funds, in order to understand the magnitude of the dollarvalues 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 convertiblearbitrage style, as of March 2003. Because of the differences in fund size acrossinvestment styles, the average estimated VaR ranges from only $4.6 million for themarket neutral funds to $33.5 million for the fixed income arbitrage style. It is notsurprising to find that dead funds are generally smaller than live funds. Dead fundslose 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 moreappropriate than the absolute VaR for comparison purposes. When analyzing theVaR as a percentage of fund size, we find that generally, dead funds have higherrelative 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. Acrossstyles, the dedicated short bias, emerging markets, fixed income arbitrage, long/shortequity hedge, and managed futures styles are particularly riskier than the other stylesfor both live and dead funds.

The main results for capital adequacy are presented in Table 3. We find that veryfew hedge funds (both live and dead) in our sample are undercapitalized. For the livefunds, about 3.7% (35 out of 942) of the funds are undercapitalized, while thecorresponding 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 issomewhat consistent with the hypothesis that one of the reasons for a fund’s death isundercapitalization. However, undercapitalization does not appear to be the primaryreason for fund death, since nearly 90% of the dead funds had adequate equitycapital right until the fund exit date. Therefore, other reasons such as poorperformance, mergers and acquisitions, voluntary withdrawals, etc., may contributemore to the demise of a hedge fund than capital inadequacy.

For the live funds, the only styles with significant levels of undercapitalization arethe emerging markets funds (6 out of 62, or 9.7%), and fixed income arbitrage funds

15An 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.

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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 ¼ ½ð0 � R99%Þ � TNA�; where VaR is the 99% one month VaR, R99% 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.


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

Market neutral 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

Event driven 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/short equity 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

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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

ðEactual � ErequiredÞ=Erequired (the Cap ratio) represents the degree of undercapitalization, Erequired is the required equity that is three times the 99% one month

VaR of the fund (using EVT), and Eactual 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.


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

Market neutral 36 2 5.6 18.0 4.0 10 0 0 3.3 2.5

Event driven 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/short equity 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

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(5 out of 32, or 15.6%). Most of the other styles, even riskier fund styles with veryhigh kurtosis in fund returns (like fund of funds), have an extremely small number offunds that are undercapitalized. Similarly, in the dead funds group, emerging marketfunds (12 out of 40, or 30%) and fixed income arbitrage funds (6 out of 18, or33.3%) 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 thehedge fund industry as of March 2003. However, hedge fund risk exposuresare highly dynamic, hence the Cap ratio is unlikely to be constant over time. First,hedge funds change their portfolio compositions fairly frequently. Second, marketconditions change over time, so even for a static portfolio, its risk profile islikely to change. Therefore, in addition to computing the static Cap ratios, we goback in time and estimate the Cap ratios for all available funds over 60-month rollingwindows for fund returns. For example, for February 2003, we analyze all the liveand dead funds as of February 2003 with at least 60 months of return data, andestimate the Cap ratio for each one of them. Hence, there may be some fund that iscategorized as a live fund as of February 2003 but as a dead fund in March 2003. Inthis manner we go back month by month to January 1995, which is as far back as thedata allow us to go, in order to get a reasonably large number of funds for cross-sectional 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 fundssteadily increases from 0.49% in January 1995 (1 out of 203 live fundsundercapitalized) to a maximum of 5.43% as of August 2000 (37 out of 681 livefunds undercapitalized), and then reduces to 3.72% in March 2003 (35 out of 942funds). This graph shows some important trends. The extent of undercapitalizationsteadily increases up to the middle of 2000, after which it declines a bit. This declinemay be due to the fact that some of the undercapitalized funds that were alive in2000 may now be dead, hence they will not show up in the live database. There is asteep increase in the fraction of live funds undercapitalized during the third quarterof 1997, just after the Asian financial crisis, and this uptrend continues through theRussian debt crisis of 1998 and beyond. In aggregate, it appears that there is a clearincrease in the fraction of live funds that are undercapitalized over time.

The second plot in Fig. 1 presents the median Cap ratios for all live funds,each month, from January 1995 to March 2003. This figure is consistent with theprevious figure—the median fund appears to be less capitalized now than itwas during the years prior to the Russian debt crisis and the LTCM debacle inFall 1998. The steep decline in the median Cap ratio from 2.76 in July 1998 to 1.99 inAugust 1998 is due to the big market movements during Fall 1998, which areincluded in all the moving return windows for subsequent months. These time-seriestrends in the level of capitalization of live funds reveal much more information aboutthe dynamic risk levels in the hedge fund industry than just a static analysis as ofMarch 2003.

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Rolling Percentage of Undercapitalized Live Funds

0%

1%

2%

3%

4%

5%

6%

Jan-95

Jul-95

Jan-96

Jul-96

Jan-97

Jul-97

Jan-98

Jul-98

Jan-99

Jul-99

Jan-00

Jul-00

Jan-01

Jul-01

Jan-02

Jul-02

Jan-03

Historical Percentage of Undercapitalized Dead Funds

0%

2%

4%

6%

8%

10%

12%

Jan-96

Jul-96

Jan-97

Jul-97

Jan-98

Jul-98

Jan-99

Jul-99

Jan-00

Jul-00

Jan-01

Jul-01

Jan-02

Jul-02

Jan-03

Rolling Cap Ratios for Live Funds

0

1

2

3

Jan-95

Jul-95

Jan-96

Jul-96

Jan-97

Jul-97

Jan-98

Jul-98

Jan-99

Jul-99

Jan-00

Jul-00

Jan-01

Jul-01

Jan-02

Jul-02

Jan-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.


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The third plot in Fig. 1 presents the fraction of dead funds that areundercapitalized at any point in time in the past. Therefore, as of a particularmonth, we analyze the dead fund database and select funds that had at least fiveyears of return history before death. Of these funds, we examine how many areundercapitalized. The percentage of undercapitalized dead funds rises sharply from1.49% in April 1998 (1 out of 67 funds) to 11.91% in December 2000 (33 out of 277funds). After that, it fluctuates around 11%. Again, the fraction of undercapitalizeddead funds rises steeply after the Asian financial crisis of 1997 and the Russian debtcrisis 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 sampleare not exposed to the risk of undercapitalization. Table 4 presents a statisticalcomparison of various characteristics of the undercapitalized funds (both live anddead) with the remaining adequately capitalized funds. Among the live funds, the 35undercapitalized funds are significantly smaller than the remaining funds, withaverage net assets of $66.3 million, as compared to $201.2 million for the remaining907 funds. Their mean Cap ratio is �0.2, indicating that on average, they have only80% of the required equity. They exhibit significantly higher volatility, negativeskewness, and kurtosis of returns. However, they are not significantly younger thanthe adequately capitalized funds. In addition, on other attributes such as feestructure, watermark provisions, use of derivatives, etc., the undercapitalized livefunds are not significantly different from the adequately capitalized funds. For deadfunds as well, there is no significant difference in fund attributes, including age,between undercapitalized and adequately capitalized funds. In fact, the average deadfund in our sample existed for about eight years, irrespective of whether it wasundercapitalized or not at the time of death.

These comparisons tell us that the adequately capitalized and undercapitalizedfunds differ in size as well as investment styles, but not in any of the other reportedcharacteristics.17 However, they differ significantly in their return distributions.Therefore, for making inferences about capital adequacy, we need to focus onreturns 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 fundswith undercapitalized funds. However, there may be more than one factor thataffects a fund’s capitalization. Therefore, we conduct a multivariate analysis on

16An 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.17We 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.

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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

Management fee 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

��Significant at the 1% level.��Significant at the 5% level.�Significant at the 10% level.


capitalization by running a cross-sectional regression of Cap ratios on various fundcharacteristics and styles. For robustness, we choose several models that includedifferent subsets of fund variables. These regression results in Table 5 show that fundsize is positively related to the Cap ratio, while age and management fee arenegatively related to the Cap ratio. Most large funds have enough equity capital tosupport 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 isnegatively correlated with the Cap ratio. Younger funds may not be well established,

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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

LogðCapiÞ ¼ a0i þ a1i logðsizeÞ þ a2i logðageÞ þ a3iðmfeeÞ þ a4iðifeeÞ

þ a5iðleverageÞ þ a6iðwatermarkÞ þ a7iðlockupÞ þX10

j¼1

bjiðdummyjÞ,

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 dummyj ( 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)

Management fee �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)

Short bias �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)

Market neutral 0.491�� 0.470�� 0.481��

(0.006) (0.009) (0.008)

Event driven 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

��Significant at the 1% level.��Significant at the 5% level.�Significant at the 10% level.


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so they may be more cautious in their investment strategies in order to build up agood reputation in the early stages of their profession. These factors may result in anegative correlation between age and the Cap ratio. Across styles, the convertiblearbitrage and market neutral funds are better capitalized than the emerging markets,long/short equity hedge, and managed futures funds. These results are consistentwith those from the univariate analyses in Table 4.

Can the inferences from VaR-based capital measures be arrived at by justobserving the leverage ratios of these hedge funds? If leverage can capture risk theway VaR does, then there is no need for these calculations. However, that is not thecase. The median leverage ratio for all fund styles is one, since many hedge funds donot use borrowed funds.18 Therefore, it is unlikely that leverage ratios will conveyany relevant information about hedge funds’ risk profiles, and their true risk offailure. The correlation coefficient between VaR-based Cap ratios and the averageleverage 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 statisticallysignificant, the magnitudes are economically trivial. In addition, in the analysis inTable 5, after controlling for other factors, the leverage variable is insignificant.Therefore, there is not much information in the leverage ratios of hedge funds thatcan be related to their risk profiles and hence capital adequacy. This furtherreinforces the need to have capital estimation procedures based on VaR, instead ofleverage.

Table 6 reports the number and the percentage of funds that are undercapitalizedbased 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 morecomplex EVT-based measures.19 Comparing the results of Table 6 with those ofTable 3, we find that using the VaR based on normality leads to an underestimationof capital requirements, especially for dead funds. Standard deviation based VaR isable to detect undercapitalization in 2.4% of the live funds, and in only 3.0% of thedead funds, while the corresponding numbers are 3.7% and 10.9%, respectively,using the EVT VaR. This is not surprising, because assuming normality ignores thefat tails of the hedge fund return distribution, which in turn underestimates the riskof extremely low return realizations. This is especially true for dead funds, whichhave higher kurtosis and more negative skewness of returns. EVT captures theprobability of occurrence of extreme negative returns better, hence it provides amore accurate measure of VaR. Therefore, reliance on traditional standarddeviation-based risk measures alone could lead to funds keeping a lower capitalcushion than that dictated by their true risk profiles, thereby leading to a higherprobability of failure.

18As 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, as of March 2003. The rest of the funds have no direct borrowing or indirect

leverage through short selling or derivatives trading.19Note 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.

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Table 6

Undercapitalization based on VaR assuming normal distribution

This table presents the Cap ratios for hedge funds using VaR based on normality, for 942 live and 494 dead funds, as of March 2003, where Cap ¼

ðEactual � ErequiredÞ=Erequired (the Cap ratio) represents the degree of undercapitalization, Erequired is the required equity that is three times the 99% one month

VaR of the fund (assuming normality of returns), and Eactual is the actual equity which is taken from the data. 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.


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 0 0 12.2 8.3 14 0 0 8.0 4.9

Dedicated short bias 10 1 10.0 1.5 0.8 6 1 16.7 1.3 1.4

Emerging markets 62 7 11.3 1.2 0.9 40 1 2.5 2.0 1.2

Market neutral 36 0 0 9.0 4.8 10 0 0 5.3 5.5

Event driven 107 0 0 7.4 6.7 39 2 5.1 5.4 4.6

Fixed income arbitrage 32 0 0 11.3 6.8 18 0 0 6.6 5.1

Fund of funds 185 0 0 6.6 5.2 107 0 0 3.6 3.0

Global macro 40 0 0 4.1 2.9 40 1 2.5 2.9 1.8

Long/short equity hedge 288 11 3.8 2.2 1.6 114 7 6.1 2.1 1.4

Managed futures 94 3 3.2 2.1 1.5 105 3 2.9 2.2 1.6

Other 29 1 3.5 7.5 4.6 1 0 0 2.9 2.9

Total 942 23 2.4 5.0 2.9 494 15 3.0 3.2 2.1

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Table 7

Tail conditional losses (TCL) and 99.94% VaR ratios

This table presents the TCL/VaR and 99.94%/99% 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 ¼

½ð0 � R99%Þ � TNA�; where VaR is the 99% one month VaR, R99% is the cutoff return at 99% confidence

level estimated using EVT, TNA is the total net asset value (equity) of the fund, and TCL is estimated from

TCL ¼ ð0 � E½RjRoR99%�Þ � TNA: The 99.94% VaR is estimated using the same extreme value

distribution parameters as are used for estimating the 99% VaR.


No. TCL/VaR 99.94%/99% VaR No. TCL/VaR 99.94%/99% VaR

Mean Median Mean Median Mean Median Mean Median

Convertible arbitrage 59 1.55 1.38 2.28 1.95 14 1.61 1.42 2.65 2.07

Dedicated short bias 10 1.57 1.35 2.05 1.93 6 1.51 1.33 1.96 1.93

Emerging markets 62 1.51 1.34 2.14 1.96 40 1.57 1.39 2.46 2.11

Market neutral 36 1.46 1.35 2.03 1.91 10 1.49 1.41 2.11 2.11

Event driven 107 1.54 1.39 2.27 2.08 39 1.64 1.44 2.85 2.05

Fixed income arbitrage 32 1.63 1.44 2.79 2.42 18 1.73 1.47 2.87 2.61

Fund of funds 185 1.51 1.41 2.15 1.84 107 1.52 1.32 2.21 2.07

Global macro 40 1.45 1.32 1.99 1.82 40 1.48 1.33 2.14 2.05

Long/short equity hedge 288 1.49 1.33 2.07 1.85 114 1.55 1.38 2.41 2.01

Managed futures 94 1.48 1.35 2.04 1.92 105 1.51 1.35 1.99 1.94

Other 29 1.53 1.39 2.45 2.26 1 1.57 1.41 2.48 2.48

Total 942 1.51 1.35 2.15 1.87 494 1.54 1.37 2.31 2.06


5.4. Robustness tests

5.4.1. Tail conditional loss (TCL) and the 99.94% VaR

To examine the validity of using three as the safety multiplier for capitalrequirement, we estimate the tail conditional loss (TCL) for each fund. The ratio ofTCL to VaR is compared with the safety multiplier of three in order to test whetherit is adequate or not. Table 7 presents the TCL/VaR ratios by hedge fund styles. Themedian ratios for live and dead funds are 1.35 and 1.37, respectively. In fact, fornone of the funds is the TCL/VaR ratio greater than three. Hence, for all the funds inour sample, even if the VaR is breached, the expected loss is likely to be less thanthree times the VaR.

In addition to TCL, we also report the ratio of the 99.94% VaR to the 99% VaRfor the funds using EVT.20 This provides us with a further indication of whether theBasel ratio of three is safe. Table 7 presents the 99.94%/99% VaR ratios. Themedian ratios for live and dead funds are 1.87 and 2.06, respectively. Again, none ofthe mean or median ratios exceeds three, 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 three. For

20The 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.

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robustness, we recompute the Cap ratios using the maximum of the 99.94% VaR andthree times the 99% VaR, as the required equity capital. We find that 37 live funds(instead of 35 earlier) and 57 dead funds (instead of 54) are undercapitalized, whichis very similar to the percentage of funds under-capitalized using three times the 99%VaR as the required capital.

5.4.2. VaR confidence intervals

We construct (95%) confidence intervals on our (99%) VaR estimates, in order toestimate the tail quantile estimation errors. These confidence intervals provideimportant information about the reliability of the point estimates of VaR. Since weuse EVT to estimate the VaR, we use profile likelihood methods to estimate theconfidence intervals on these VaRs, as explained in Appendix A.3. Given the relativeuncertainty about quantiles in the tail of the distribution, the upper bound of theconfidence interval for the VaR provides a reasonable upper limit for how highthe VaR could be in reality. A ratio of the upper bound of the confidence interval tothe VaR itself can then be compared with the Basel multiplier of three to examinehow often the standard multiplier of three fails to capture the true risk of extremelylow returns.

Table 8 presents the upper and lower confidence intervals on VaR as a percentageof the estimated VaR. The median live fund has an upper (lower) confidence boundthat is 2.02 (0.52) times the VaR. The bounds are similar for dead funds. The upperbounds are much farther away from the VaRs than the lower bounds. This isconsistent with the intuition that there is much more uncertainty as we go deeper intothe tails of the return distribution. However, on average, these upper bounds areabout twice those of the VaR estimate, hence using the Basel multiplier of threeappears to be reasonably safe for estimating capital requirements. In fact, for only9.2% of the live funds (87 out of 942) and 12.6% of the dead funds (62 out of 494) isthe upper confidence bound greater than three times the VaR. When we re-estimatethe Cap ratios by defining the required equity to be the maximum of three times theVaR 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 undercapitalization using three timesthe VaR. Hence our primary capital adequacy results are not sensitive to the samplesize issue.

5.4.3. 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 non-normality in hedge fund returns. Leverage ratio is also ruledout as the correct measure since it does not proxy for asset risk in any meaningfulway. However, VaR can be used as an appropriate risk measure for hedge fundsonly if it can effectively detect the changing risk patterns over time. In general, therisk for dead funds increases toward the fund death date, either because of badperformance that leads to loss of capital and investor withdrawals, or because of theexcessive risk shifting incentives that occur when a fund has been performing poorlyin the past. If VaR is an appropriate risk measure for hedge funds, then it should

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Table 8

Estimation of confidence intervals for VaR based on EVT

This table presents the 95% confidence intervals for the 99% VaR, as a percentage of the VaR estimate. For example, a value of 2.0 for the upper confidence

interval implies that the upper confidence interval is 200% of the VaR estimate. The confidence intervals are estimated using profile likelihood methods for

Extreme Value Theory, for the 942 live and 494 dead funds in the sample as of March 2003.


No. Lower confidence interval Upper confidence interval No. Lower confidence interval Upper confidence interval

Mean Median Mean Median Mean Median Mean Median

Convertible arbitrage 59 0.51 0.55 2.01 1.90 14 0.35 0.40 2.32 2.49

Dedicated short bias 10 0.51 0.53 2.34 2.37 6 0.51 0.51 2.55 2.61

Emerging markets 62 0.46 0.52 2.16 1.97 40 0.37 0.40 1.93 1.83

Market neutral 36 0.49 0.54 2.12 2.05 10 0.42 0.44 2.14 2.00

Event driven 107 0.50 0.53 2.09 1.98 39 0.34 0.36 2.03 1.92

Fixed income arbitrage 32 0.50 0.54 2.28 2.21 18 0.25 0.22 1.73 1.71

Fund of funds 185 0.47 0.49 2.12 1.98 107 0.42 0.45 2.18 1.96

Global macro 40 0.51 0.54 2.11 2.07 40 0.44 0.47 2.19 2.04

Long/short equity hedge 288 0.48 0.52 2.09 2.03 114 0.43 0.45 2.13 1.98

Managed futures 94 0.52 0.57 2.15 2.00 105 0.45 0.46 2.17 2.05

Other 29 0.43 0.46 2.18 2.18 1 0.32 0.32 1.86 1.86

Total 942 0.49 0.52 2.12 2.02 494 0.41 0.45 2.13 1.97

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capture these risk characteristics, and we should be able to detect a significantincreasing trend for VaR over time for dead funds as the fund death dateapproaches. On the other hand, we should not expect such a systematic trend in theVaRs 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 fiveyears immediately preceding the fund death (window 3), the five years ending oneyear before the fund death date (window 2), and the five years ending two yearsbefore the fund death date (window 1). For comparison, we estimate the same threerolling VaRs for live funds as well, the only difference being that the time horizon forlive funds is the same for all funds, ending March 2003. Note that there may be sometrends in estimated fund VaRs due to the different magnitudes of non-fund-specificrisk over different periods.

In Fig. 2, we present the results for the 591 live and 272 dead funds that have atleast a seven-year return history. The box plots present the ratios for window 2 VaRto window 1 VaR, and the ratios for window 3 VaR to window 2 VaR, for live anddead funds, respectively. The mean (median) for dead funds is 1.27 (1.17) for theratios of window 2 VaR to window 1 VaR, and 1.37 (1.25) for the ratios of window 3VaR to window 2 VaR. These ratios show that the VaR of hedge funds, estimated

DEAD32DEAD21LIVE32LIVE21

VaR

Rat

io

2.5

2.0

1.5

1.0

0.5

0.0

Fig. 2. Ratios of Successive Rolling Window VaRs. This figure presents box plots of the ratios of

successive five-year rolling window VaRs for 591 live and 272 dead funds that have at least a seven-year

return history available. Window 3, window 2, and window 1 refer to the five-year windows immediately

preceding, one-year prior, and two-years prior, respectively, to the fund death date (or March 2003 for live

funds). Therefore, dead32 refers to the ratio of VaR in window 3 to that in window 2, for dead funds. The

other ratios are defined similarly. The plots show the median ratio, as well as the 5%, 25%, 75%, and 95%

quantiles of the distribution of the ratio.

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using 60 monthly returns, increases significantly in the two years prior to a fund’sdeath. In fact, over those two years, just prior to a fund’s death, the average increasein the VaR of the fund is 74%. VaR appears to be effective in capturing thecomponents of risk of hedge funds that lead to its death.

In contrast, the corresponding mean (median) ratios for live funds are much lowerat 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 aremany funds for which the recent VaRs are lower than before as well. The slightlyincreasing trend of VaR for live funds has been observed due to the increasingvolatility in financial markets in recent years. Hence, adjusting for the impact ofthese recent events, there should be no significant trend in the VaRs of live fundsover time, while the dead funds indicate a clear, sharp increase in VaR as the funddeath date approaches.

6. Monte Carlo simulation tests

In the previous section, even though we include a large number of dead funds inour analysis, some degree of survivorship and sample selection biases are inevitable.There may be some funds that have extremely high volatility, skewness and kurtosisof 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 toconduct robustness checks on the funds in our sample, it is important tofundamentally examine our methodology in a context that is free of these biasesand that can cover a wide range of market swings and rare events with largemagnitude and volatility.

We accomplish this using Monte Carlo simulation experiments, where we simulatereturns for a wide range of hedge fund profiles, and then examine the accuracy ofVaR estimates using the same methodology that is used for all hedge funds. Weanalyze 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 amedian fixed income dead fund, representing median funds from the style with themaximum skewness and kurtosis; and, an extreme live and an extreme dead fund,representing the most extreme skewness and kurtosis among live and dead funds(both of these extreme funds happen to be fixed income funds). Table 9 presents thefirst four moments of returns for these representative funds. The intuition behindpicking these fund profiles is to test our methodology on funds with moderateskewness and kurtosis in returns as well as on those with the most extreme skewnessand kurtosis.

6.1. The return generating process

We simulate two different processes—a Geometric Brownian Motion (GBM) forthe asset value process (implying normally distributed returns), and the GBM

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Table 9

Parameter estimates for jump diffusion process

This table presents the estimates of the parameters for the jump diffusion process, matched to the first

four moments for six representative hedge fund types. The jump diffusion process for fund returns is as

follows:

Z ¼ x þPKi¼0

yi ; where x Nða; s2Þ; K PoissonðlÞ;

yi NðmH;s

2HÞ

NðmL;s2LÞ

(w:p: pH;

w:p: pL ¼ 1 � pH:

This estimated process is later used in Monte Carlo simulation experiments. The six hedge fund

profiles selected for calibrating the parameters of the process represent median and extreme (in terms of

skewness and kurtosis of returns) funds from both the live and the dead funds categories. Two median

funds are chosen from the Fixed Income Arbitrage style since that style has the maximum skewness and

kurtosis of returns across all hedge fund styles in our sample of 942 live and 494 dead funds, as of

March 2003.

Hedge fund type Hedge fund return moments Jump diffusion process parameters

Mean Std. dev. Skew Kurt a s mH sH mL sL

Median live 0.72 3.70 0.06 4.85 0.35 3.11 �11.4 3.41 5.6 1.68

Median dead 0.62 4.57 �0.01 4.43 0.22 3.97 �13.5 4.05 6.2 1.86

Median live fixed income 0.64 1.83 �1.47 7.61 1.02 1.27 �6.4 1.92 �3.9 1.17

Median dead fixed income 0.62 2.34 �3.10 15.91 1.23 0.92 �11.0 3.29 �6.2 1.87

Extreme live 0.45 4.09 �6.16 54.12 1.47 0.09 �31.9 9.56 �9.1 2.73

Extreme dead 0.05 5.96 �6.29 56.20 1.53 0.08 �47.0 14.09 �13.1 3.92


process augmented by a jump process (jump diffusion model, JD), to generateskewness and kurtosis in returns (Z):

Z ¼ x þPKi¼0

yi; where x Nða;s2Þ; K PoissonðlÞ;

yi NðmH; s

2HÞ

NðmL; s2LÞ

(w:p: pH;

w:p: pL ¼ 1 � pH:

(5)

In a standard JD model, the jump frequency has to be unrealistically high tomatch the skewness and kurtosis observed in the data. In our process, the numberof jumps (K) is controlled by l; however, each jump can be one of two types—onejump (L) occurs with greater frequency, but has a smaller magnitude andvolatility. The other jump (H) occurs very rarely, but with a much higher magnitudeand volatility. This specification provides enough free parameters to match themoments of the process to actual fund moments, and still provide for large,infrequent jumps.

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The mean, variance, skewness, and kurtosis of this process, estimated using themethodology outlined in Das and Sundaram (1999), are as follows:

E½Z� ¼ aþ l½pHmH þ pLmL�;

Var½Z� ¼ s2 þ l½pHðm2H þ s2

HÞ þ pLðm2L þ s2

LÞ�;

Skew½Z� ¼l½pHðm

3H þ 3mHs

2HÞ þ pLðm

3L þ 3mLs

2LÞ�

fs2 þ l½pHðm2H þ s2


LÞ�g3=2

;

Kurt½Z� ¼ 3 þl½pHðm

4H þ 6m2

Hs2H þ 3s4

HÞ þ pLðm4L þ 6m2

Ls2L þ 3s4

LÞ�

fs2 þ l½pHðm2H þ s2


LÞ�g2

:

(6)

This JD process has eight free parameters. However, we only have four fundmoments to match the process moments for estimating the parameters.21 Therefore,we must specify four parameters, and then estimate the remaining four parameters tomatch the moments. We fix l; pH, and the coefficients of variation (c.v.) of the twonormal distributions from which jumps occur (sH=mH; and sL=mL) at realistic levels,and then vary them across a wide range to ensure that our results are robust. We fixlpH to be once every 10 years, and lpL at once every year. This gives us values ofl ¼ 11=120; and pH ¼ 1=11: The alternate l values we consider are:

l ¼ 11=360) extreme jump once every 30 years, moderate jump once every threeyears,l ¼ 11=240) extreme jump once every 20 years, moderate jump once every twoyears,l ¼ 11=60) extreme jump once every five years, moderate jump once every sixmonths,l ¼ 11=36) extreme jump once every three years, moderate jump once every 3.6months.

Similarly, we fix the coefficient of variation of both the jumps at 0.3, but run thesimulation experiments for c.v. values of 0.1, 0.2, 0.4, and 0.5 as well, for all l values.In summary, for each of the six hedge fund profiles, we run simulations for fivedifferent jump intensities, and five different c.v. for the jumps. Therefore, in all, weconduct simulation tests for 150 different sets of parameters for the JD process. Theestimated JD process parameter values for the six ‘‘base cases’’ (l ¼ 11=120; pH ¼

1=11; sH=mH ¼ 0:3; and sL=mL ¼ 0:3) are presented in Table 9. These parametervalues clearly indicate the type of process that is generating the returns for thatparticular hedge fund type. For example, for the extreme dead fund, the reportedparameters indicate that in order to generate the observed skewness (�6.29) andkurtosis (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 meansize of �13.1% occurring on average once every year.

21In theory, one could match higher moments (fifth and beyond) as well, but they do not make any

intuitive sense, and because doing so reduces the whole exercise to just mathematical fitting, we avoid it.

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6.2. Simulation results

Once the parameters of the two processes are estimated, we simulate 60 hedge fundreturns for each process, and estimate the 99-percentile return by applying EVT onthese 60 returns. The primary objective is to examine whether, based on 60 returnobservations, the EVT methodology is able to statistically distinguish the JD processfrom the GBM process.22 We repeat this experiment for the 150 sets of parametervalues for the JD process, to ensure that our results are not sensitive to anyparticular 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 ofthe 99-percentile return and its standard error are reported in the table.

Results show that the EVT methodology is able to clearly distinguish between theJD process and the GBM process, even when only 60 returns are available. Forexample, 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 thecorresponding GBM process is �6.43% (with a standard error of 0.01%), which issignificantly lower than the GBM return at any level of significance.

Table 10 also presents the several results for robustness checks on our EVT VaRapproach. The scaled confidence intervals show that for the median live fund, thesimulation-based 95% confidence interval on the 99% VaR is [0.62, 1.59]. For theextreme dead fund, which has the maximum skewness and kurtosis across all ourfunds, this confidence interval is [0.33, 1.89]. These confidence intervals are similarto those obtained for individual funds using profile likelihood methods in theprevious section.

In addition, all the TCL/VaR ratios are well below two, implying that the use ofthree as the safety multiplier is appropriate. The column VaR180 represents the ratioof 99% VaR using 60 return observations to the 99% VaR using 180 returnobservations. The objective of computing this ratio is to examine the bias, if any, inour VaR estimates due to reliance on a return history of five years. One may arguethat a five-year window may be too short for estimating the true VaR, and couldignore the impact of catastrophic events that may be observed over time periodslonger than five years.24 Although there is slight underestimation using a five-yearreturn history than a 15-year history, the difference is only marginal. The columnEmpVaR represents the ratio of EVT VaR to the empirical VaR, both from 180return observations. For the JD process, half of the six ratios are below one and theother half are above one. The errors are not systematic, which implies that our EVT

22In 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.23Increasing the number of simulation runs to 100,000 gives almost identical results.24However, 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.

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Table 10

Monte Carlo simulation results

This table presents the results for the Monte Carlo simulation of the jump diffusion process and a standard geometric Brownian motion process for the six

representative hedge fund profiles. Each of the figures presented is the average across 30,000 simulation runs. In each simulation, 60 return observations are

simulated, and then the 99-percentile return is estimated by applying Extreme Value Theory to these 60 returns. The mean and standard error of this 99-

percentile return are reported in the first column. Each of the figures in the remaining columns is the ratio w.r.t. the 99-percentile return. Confidence intervals

are the ratios of the lower and upper 95% confidence intervals to the mean estimate of the 99-percentile return, TCL is the ratio of the tail conditional loss to

the 99-percentile return, VaR180 is the ratio of the 99-percentile return using 60 return observations to the 99-percentile return using 180 return observations,

and EmpVaR is the ratio of the 99-percentile return using EVT on 180 observations to the empirical 99-percentile return from the same 180 observations.

Hedge

fund

type

Jump Diffusion Process Geometric Brownian Motion Process

99-Percentile

return (%)

95% Confidence

interval

TCL/VaR

ratio

VaR180

ratio

EmpVaR

ratio

99-Percentile

return (%)

95% Confidence

interval

TCL/VaR

ratio

VaR180

ratio

EmpVaR

ratio

1 �8.03 (0.01) [0.62, 1.59] 1.30 0.95 1.02 �6.43 (0.01) [0.68, 1.46] 1.17 0.97 0.85

2 �9.87 (0.01) [0.63, 1.58] 1.36 0.96 0.97 �8.11 (0.01) [0.68, 1.45] 1.17 0.97 0.85

3 �5.20 (0.01) [0.64, 1.56] 1.36 0.95 0.95 �3.00 (0.01) [0.65, 1.48] 1.17 0.96 0.86

4 �9.30 (0.02) [0.64, 1.78] 1.26 0.90 1.12 �4.00 (0.01) [0.66, 1.45] 1.16 0.96 0.86

5 �18.29 (0.05) [0.41, 1.93] 1.49 0.91 1.08 �7.42 (0.01) [0.69, 1.44] 1.16 0.97 0.85

6 �23.50 (0.06) [0.33, 1.89] 1.62 0.94 0.90 �11.03 (0.02) [0.69, 1.44] 1.17 0.96 0.85

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approach is appropriate and the estimated VaRs are very close to the true VaRsfrom the empirical distributions.

6.3. Backtesting the VaR estimation approach

Berkowitz and O’Brien (2002) question the statistical accuracy of the internal VaRmodels 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-percentiletrading P&L. How risk measures relate to actual performance over time is critical inevaluating the validity of the risk estimation model. Therefore, in this section, weexamine the out-of-sample accuracy of our VaR estimates, and explore how reliablethey are in forecasting future extreme returns.

We do backtesting in a formal statistical framework that consists of comparing thehistory of VaR estimates with actual realized returns in the next period, to determinewhether the frequency of exceedances is in line with the predicted confidence level ofthe VaR. For example, a 99% VaR should be exceeded 1% of the time, so we shouldexpect an extreme return lower than the 99% VaR in 1 out of 100 months, onaverage. If the actual number of VaR violations is significantly different from thepredicted number of violations, then the VaR estimation approach is not valid.

Statistically, the VaR model verification using failure rates is a sequence ofBernoulli trials, where the outcome of each trial is binary—either the actual lossexceeds the VaR level, or it does not. Let p be the predicted failure rate of the model(p ¼ 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 given by

f ðxÞ ¼T

x

� �pxð1 � pÞT�x. (7)

This binomial distribution is used to test the null hypothesis that the frequency offailures is equal to the predicted failure rate of the model. The 95% confidenceregions for a powerful test of this hypothesis can be estimated using the tail points ofthe log-likelihood ratio

LR ¼ �2 ln½ð1 � pÞT�NpN � þ 2 lnf½1 � ðN=TÞ�T�N ðN=TÞNg, (8)

where N is the actual number of exceedances in T trials.26 This likelihood ratio isasymptotically distributed chi-square with one degree of freedom under the nullhypothesis that p is the true probability. For example, for p ¼ 0:01; the nullhypothesis would be rejected if LR46.63, hence the 95% confidence interval forT ¼ 1; 000 is [4pNp19].

25It 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.26See Kupiec (1995) and Jorion (2001) for details. Also, 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.

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Since we need a large number of observations to reliably backtest a VaR model,we cannot do these tests on actual hedge fund data without imposing severesurvivorship bias. Therefore, we use simulated hedge fund returns for the six fundprofiles. In order to get 1,000 backtest trials, we need 1,060 simulated returns. We usethe first 60 months to estimate the 99% VaR, and then examine whether the actualloss in the 61st month is greater than the estimated VaR or not. Then we roll forwardby one month and repeat this exercise. This way, we conduct 1,000 trials for eachfund profile, and then calculate the number of times the actual loss exceeds theestimated VaR. We repeat this entire exercise 5,000 times and use the mean of thenumber of violations as our estimate for N.

Fig. 3 presents the plot of the number of violations for each fund profile (for eachof the 25 sets of parameters for that fund profile), the expected number of violationsbased on the confidence level, and a 95% confidence interval on the expected numberof violations (for the 99% VaR, if the number of violations is either less than four orgreater than 19, then the VaR model is not accurate). These plots show that thatthe number of violations for all fund profiles is not statistically different from theexpected number of violations for its confidence level. These backtesting results giveus additional confidence that our VaR estimation approach is accurate, since theestimated VaRs are able to forecast extreme negative hedge fund returns fairlyaccurately.

7. Conclusion

We examine the risk characteristics and capital adequacy of nearly 1,500 hedgefunds. Using Value-at-Risk-(estimated through Extreme Value Theory) basedcapital adequacy measures, we find that the majority of hedge funds in our sampleare adequately capitalized as of March 2003, with only a small proportion (3.7%) oflive funds being undercapitalized. Moreover, all the undercapitalized live funds arerelatively small, constituting a tiny fraction (1.2%) of the total fund assets in oursample. However, among dead funds, we find almost 11% to be undercapitalized,which is significantly higher than the percentage undercapitalized among live funds.This confirms that one of the reasons that funds die is lack of adequate capital.

Our study shows that VaR-based measures are superior to traditional riskmeasures like standard deviation of returns and leverage ratios, in capturing hedgefund risk. Normality-based standard deviation measures understate risk and areinappropriate for hedge funds, since their returns exhibit significantly high kurtosis.Leverage ratios do not effectively capture hedge fund risk either, since they are justnoisy indicators of credit risk to debt holders—they ignore the inherent riskiness ofthe assets in the portfolios completely. VaR based on historical returns, as estimatedin this paper, is reasonably effective in capturing the underlying risk trends in hedgefund returns that lead to a fund’s death, though VaR based on current positions islikely to be even more effective. This is evidenced by a significant upward trend inVaR for dead funds starting two years before their death, while no such trend isobserved for live funds.

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VaRViolations - Extreme Dead Fund

0

5

10

15

20

25

0 5 10 15 20 25Simulation Parameter Set

Nu

mb

er o

f V

iola

tio

nsVaR Violations - Extreme Live Fund

0

5

10

15

20

25


Nu

mb

er o

f V

iola

tio

ns

VaR Violations - MedianLive Fund (Fixed Income)

0

5

10

15

20

25


Nu

mb

er o

f V

iola

tio

ns

VaR Violations- Median Dead Fund (Fixed Income)

0

5

10

15

20

25


Nu

mb

er o

f V

iola

tio

ns

VaR Violations- Median Dead Fund

0

5

10

15

20

25


Nu

mb

er o

f V

iola

tio

nsVaR Violations- MedianLive Fund

0

5

10

15

20

25


Nu

mb

er o

f V

iola

tio

ns

Fig. 3. VaR Backtesting Results. This figure presents the plots of the number of violations (average across

5,000 simulation runs) of the 99% VaR for 1,060 simulated returns each for the six hedge fund profiles; 25

sets of results are presented for each hedge fund profile, corresponding to the various parameter sets for

the simulated jump diffusion process. The expected number of violations for each fund is indicated by the

solid horizontal line, while the dotted lines above and below the solid line represent the 95% confidence

intervals for the number of violations, to determine whether the number of actual violations for a fund is

significantly different from the predicted violations.


Monte Carlo simulation experiments and backtests indicate that our VaRestimates based on EVT are able to forecast extreme returns in hedge funds fairlyaccurately. However, in spite of these robustness tests, it is important to note thatthese VaR estimates are subject to several limitations. First, the historical data thatthey are based on may not include representative events for the future. Second, therisk profile of a hedge fund may change more rapidly than what a 60-month windowVaR can capture. Third, the liquidity aspect of hedge fund risk has been ignored inthis analysis.

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More importantly, due to dynamic trading strategies and changing marketconditions, hedge fund risk and capitalization show significant time variation.In the cross-section, the variability in fund capitalization is related to size, investmentstyle, age, and management fee. Market conditions appear to play a largerole in affecting the capital adequacy of hedge funds over time—they go aboveand below the threshold of adequate capital fairly often. We document a sharp dropin the level of fund capitalization just after the LTCM and the Russian debt crises in1998. Similarly, there is a significant increase in the fraction of undercapitalizedfunds over time. These findings should alert inventors, fund managers, and financialregulators regarding the overall trend of increasing hedge fund risk over the lastseveral years, and are relevant for the extensive debate currently going onbetween the regulatory agencies and market participants regarding hedge fund riskand regulation.27

Appendix A

A.1. Estimating VaR using Extreme Value Theory (EVT)

EVT models the limiting distribution of extreme returns (fitting only the tail of thedistribution), which is independent of the distribution of all returns. It can beimplemented using two different, but related, approaches. The first approachconsists of fitting one of the three standard extreme value distributions (Frechet,Weibull, or Gumbel) to block maxima values in a time series, while the secondapproach models the distribution of exceedances over a threshold as a generalizedPareto distribution (Peak Over Threshold (POT) method). The POT method ispreferred in applications in which there is less data available, since this method usesdata more efficiently. Hence, in this paper, we use the POT method to estimate the99-percentile return (R99%).

For random returns Ri, the POT method models the distribution Fu of values ofreturns above a certain threshold u. This conditional excess distribution function isdefined as

FuðyÞ ¼ P R � upyjR4uð Þ; where y ¼ r � u. (A.1)

As observed by Pickands (1975), the conditional excess distribution function,for large u, is well approximated by the generalized Pareto distribution (GPD)

27Market participants argue that, barring some exceptions, hedge funds in general operate within

prudent solvency norms, and that 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 could create enormous systemic risk. Therefore, regulators are pressing for increased

regulation of hedge funds.

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as follows:

FuðyÞ � GxsðyÞ; u ! 1,

GxsðyÞ ¼1 � 1 þ

xs

y

� ��1=x

if xa0;

1 � expð�y=sÞ if x ¼ 0;

8>><>>: ðA:2Þ

where x is the tail index and s is the scale parameter. This generalized distributionnests three standard distributions: The ordinary Pareto distribution (x40;polynomially decreasing tail), the Pareto type-II distribution (xo0; short tail), andthe exponential distribution (x ¼ 0; exponentially decreasing tail).

The conditional excess distribution function (Fu) can be written in terms of thecumulative distribution function (F) of all returns as follows:

FuðyÞ ¼F ðu þ yÞ � F ðuÞ

1 � F ðuÞ¼

F ðxÞ � F ðuÞ

1 � F ðuÞ; where y ¼ x � u,

) F ðxÞ ¼ ½1 � F ðuÞ�F uðxÞ þ F ðuÞ. ðA:3Þ

From this, an analytical expression can be derived for R99%. Note that Fu can bereplaced by the generalized Pareto distribution. The function F(u) is deterministic,since it is the fraction of observations below the threshold, hence F ðuÞ ¼ ðN � nÞ=N;where n is the number of observations above the threshold, while N is the totalnumber of observations. Therefore,

F ðxÞ ¼n

N

h i1 � 1 þ

xsðx � uÞ

� ��1=x" #

þ 1 �n

N

� �; for xa0

¼ 1 �n

N1 þ

xsðx � uÞ

� ��1=x

. ðA:4Þ

Thus, F(x) denotes the VaR confidence level. Let p denote the probability ofexceeding the VaR, which would be given by 1�F(x). For example, for the 99%VaR, p would be 0.01. Also, the x itself denotes R99%. Substituting for F(x) and x, weget the expression for R99% as

R99% ¼ u þsx

N

np

� ��x

� 1

" #; for xa0. (A.5)

When x ¼ 0; the form of the generalized Pareto distribution is different, and R99% isgiven by

R99% ¼ u þ s logn

Np

� �; for x ¼ 0. (A.6)

For a sample of return observations, the parameters of the generalized Paretodistribution can be estimated using maximum likelihood methods.28 Using the

28See Bali (2003), Longin (2000), Kellezi and Gilli (2000), McNeil (1999), Smith (1987), etc. for the

estimation procedure.

ARTICLE IN PRESS


estimated parameters, appropriate VaR estimates can be obtained at any confidencelevel based on (A.5) and (A.6). Please note that all of this EVT analysis assumes thatthe distributions are stationary.

A.2. Estimating tail conditional loss (TCL)

Using the mean excess function for the generalized Pareto distribution, withparameter xo1; we get the expression for the expected return conditional on it beingin the 1% tail as29

E½RjRoR99%� ¼R99%

1 � xþ

s� xu

1 � x. (A.7)

Depending on how heavy-tailed the return distribution is, any level of TCL can beobtained for a given level of VaR. Therefore, it is useful to quantify the expectedlosses in the tail of the return distribution, in addition to estimating the cutoff lossescorresponding to a certain quantile.

A.3. Estimating confidence intervals for VaR using profile likelihood methods

Following Azzalini (1996), McNeil (1998), and Kellezi and Gilli (2000), theconfidence interval for the return level R99% is constructed by reparametrizing theGPD log-likelihood function in terms of the parameters x and s and the EVT return(R99%). Then we test the null hypothesis that R99% ¼ r using an asymptoticlikelihood ratio test.

Let Lðx; s; rÞ be the maximum log-likelihood under the null hypothesis andLðx; s;R99%Þ be the unconstrained maximum log-likelihood. Asymptotically, underthe null hypothesis,

�2ðLðx; s; rÞ � Lðx; s;R99%ÞÞ X21; (A.8)

hence, the hypothesis is tested using a chi-squared distribution with one degree offreedom.

An a% confidence interval for R99% is the set of r for which the null hypothesis isnot rejected at the a% level. That is, the set

fr : Lðx; s; rÞXLðx; s;R99%Þ � 12X2

a;1g (A.9)

where X2a;1 is the a% point of the chi-squared distribution with one degree of

freedom. The curve ðr;Lðx; s; rÞÞ is known as the profile log-likelihood. In general,this curve is not symmetrical about its maximum, hence the confidence intervals arenot symmetrical about the maximum likelihood estimate R99%. This asymmetryreflects the asymmetry of our uncertainty concerning extreme events: It is easier tobound the confidence interval below than to bound it above.

29A general result concerning the existence of moments for generalized Pareto distributions is that for all

integers r such that ro1=x; the rth first moments exist. See Kellezi and Gilli (2000) for more information.

ARTICLE IN PRESS


References

Ackermann, C., McEnally, R., Ravenscraft, D., 1999. The performance of hedge funds: risk, return, and

incentives. Journal of Finance 54, 833–874.

Agarwal, V., Naik, N.Y., 2004. Risk and portfolio decisions involving hedge funds. Review of Financial

Studies 17 (1), 63–98.

Azzalini, A., 1996. Statistical Inference Based on the Likelihood. Chapman & Hall, London.

Bali, T.G., 2003. An extreme value approach to estimating volatility and value at risk. Journal of Business

76 (1), 83–108.

Berkowitz, J., O’Brien, J., 2002. How accurate are value-at-risk models at commercial banks? Journal of

Finance 57, 1093–1111.

Brown, S.J., Goetzmann, W.N., Ibbotson, R.G., 1999. Offshore hedge funds: survival & performance

1989–95. Journal of Business 72, 91–117.

Das, S., Sundaram, R., 1999. Of smiles and smirks: a term structure perspective. Journal of Financial and

Quantitative Analysis 34 (1), 60–72.

Fung, W., Hsieh, D.A., 1997. Empirical characteristics of dynamic trading strategies: the case of hedge

funds. The Review of Financial Studies 10, 275–302.

Fung, W., Hsieh, D.A., 2000. Measuring the market impact of hedge funds. Journal of Empirical Finance

7, 1–36.

Getmansky, M., Lo, A.W., Makarov, I., 2004. An econometric model of serial correlation and illiquidity

in hedge fund returns. Journal of Financial Economics 74 (3), 529–609.

Jorion, P., 2000. Risk management lessons from long term capital management. European Financial

Management 6, 277–300.

Jorion, P., 2001. Value At Risk: The New Benchmark For Managing Financial Risk, second ed. McGraw-

Hill Publications, New York.

Jorion, P., 2002. Fallacies about the effects of market risk management systems. Journal of Risk 5 (1),

75–96.

Kellezi, E., Gilli, M., 2000. Extreme value theory for tail related risk measures. Working paper, University

of Geneva.

Kupiec, P., 1995. Techniques for verifying the accuracy of risk measurement models. Journal of

Derivatives 3, 73–84.

Lhabitant, F.S., 2001. Assessing market risk for hedge funds and hedge fund portfolios. Working paper,

University of Lausanne.

Liang, B., 1999. On the performance of hedge funds. Financial Analysts Journal 55, 72–85.

Liang, B., 2000. Hedge funds: The living and the dead. Journal of Financial and Quantitative Analysis 35,

309–326.

Lo, A.W., 2001. Risk management for hedge funds: Introduction and overview. Financial Analysts

Journal 57 (6), 16–33.

Longin, F.M., 2000. From value at risk to stress testing: The extreme value approach. Journal of Banking

and Finance 24, 1097–1130.

McNeil, A.A., 1998. Calculating quantile risk measures for financial return series using extreme value

theory. Working paper, Department Methematik, ETH Zentrum, Zurich.

McNeil, A.A., 1999. Extreme value theory for risk managers. Working paper, Department Mathematik,

ETH Zentrum, Zurich.

Pickands, J., 1975. Statistical inference using extreme value order statistics. The Annals of Statistics 3,

119–131.

Sharpe, W.F., 1992. Asset allocation, management style and performance measurement. Journal of

Portfolio Management 18, 7–19.

Smith, R.L., 1987. Estimating tails of probability distributions. The Annals of Statistics 15 (3), 1174–1207.

Staff Report to the SEC, Implications of the Growth of Hedge Funds, September 2003.

Stahl, G., 1997. Three cheers. RISK 10, 67–69.

Tabachnick, B.G., Fidell, L.S., 1996. Using Multivariate Statistics, third ed. Harper Collins, New York.

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