Forthcoming: European Financial Management

Risk Measures for Hedge Funds: A Cross-Sectional Approach

by

Bing Liang and Hyuna Park*

This version: September 2006

Abstract

This paper analyzes the risk-return trade-off in the hedge fund industry. We compare semi-deviation, value-at-risk (VaR), Expected Shortfall (ES) and Tail Risk (TR) with standard deviation at the individual fund level as well as the portfolio level. Using the Fama and French (1992) methodology and the combined live and defunct hedge fund data from TASS, we find that the left-tail risk captured by Expected Shortfall (ES) and Tail Risk (TR) explains the cross-sectional variation in hedge fund returns very well, while the other risk measures provide statistically insignificant or marginally significant results. During the period between January 1995 and December 2004, hedge funds with high ES outperform those with low ES by an annual return difference of 7%. We provide empirical evidence on the theoretical argument by Artzner et al. (1999) that ES is superior to VaR as a downside risk measure. We also find the Cornish-Fisher (1937) expansion is superior to the nonparametric method in estimating ES and TR. Key words: hedge funds, expected shortfall, tail risk, conditional VaR, Cornish-Fisher expansion JEL classification: G11, G12, C31

* We thank Daniel Giamouridis, Hossein Kazemi, Bernard J. Morzuch, Mila Getmansky Sherman, and an anonymous referee for helpful comments and suggestions. We are responsible for any error. * Bing Liang is an associate professor of finance at the Department of Finance & Operations Management, Isenberg School of Management, University of Massachusetts, 121 Presidents Drive, Amherst, MA 01003-9310, Phone: (413) 545-3180, Fax: (413) 545-3858, E-mail: bliang@som.umass.edu. Hyuna Park is a Ph.D. candidate in finance at the Isenberg School of Management, University of Massachusetts, 121 Presidents Drive, Amherst, MA 01003-9310, Phone: (413) 348-9116, E-mail: hyuna@som.umass.edu.

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1. Introduction

One of the most important developments in modern finance theory is the ability to

model risk in a quantifiable fashion. This is important because if we know how to

measure and price financial risk correctly, we can properly value risky assets (Copeland

and Weston, 1992). Since the seminal work of Markowitz (1952), standard deviation of

returns has been one of the best-known measures for risk. The Markowitz model,

devised mainly for the long-only portfolios in the U.S. equity market, is based on the

assumption that investors’ utility curves are a function of the expected return and

standard deviation of returns only. Therefore, higher moments, if ever exist in the return

distribution, can be ignored.

Is this assumption still valid when investors include hedge funds in their portfolios?

Interest in alternative investments has grown rapidly since the recent downturn in the

U.S. equity market.1 There is increasing evidence that hedge funds offer higher mean

returns and lower standard deviations than traditional assets, but they also give investors

undesirable higher moment characteristics (Cremers, Kritzman, and Page (2005) and

Alexiev (2005)). After the demise of Long Term Capital Management (LTCM) in

August 1998, downside risk management and risk-adjusted performance measurement

have been strongly emphasized in this industry.

The source of this negatively skewed payoff is well documented in the literature.

Hedge funds implement dynamic, option-like strategies, trade derivative securities, and

have a fee structure that generates non-linear payoffs (Goetzmann, Ingersoll, Spiegel

and Welch (2002), Spurgin (2001), Mitchell and Pulvino (2001), Goetzmann, Ingersoll

3

and Ross (2003), Taleb (2004) and Chan, Getmansky, Haas and Lo (2005)). All these

facts make standard deviation an incomplete measure of a hedge fund’s risk.

Although most researchers agree that traditional risk management tools cannot

capture many of the risk exposures of hedge fund investments, an alternative framework

is not yet well established. Traditional risk measures are still dominant among

practitioners (see the survey result for the fund of hedge funds industry by Amenc,

Giraud, Martellini and Vaissie (2004)). However, academic research is beginning to

examine downside risk, asymmetric volatility, semi-deviation, extreme value analysis,

regime-switching, jump processes, and so on.

Semi-deviation considers standard deviation only over negative outcomes and is of

interest because investors only dislike downside volatility. Estrada (2001) argues that

semi-deviation combines information provided by two statistics: standard deviation and

skewness. Empirically, the semi-deviation has been reported to explain the cross-section

of returns of emerging markets and the cross-section of internet stocks returns (Harvey

(2000) and Estrada (2000, 2001)).

Another important alternative is Value-at-Risk (VaR). VaR is the worst loss that can

happen over a specified horizon at a specified confidence level. However, VaR is only

as accurate as its inputs. VaR based risk management has been criticized regarding the

failure of Long Term Capital Management (LTCM). Jorion (2000b) applies a VaR

approach to analyze the failure of LTCM and concludes that LCTM relied too much on

short-term history and significantly underestimated its risk profile.

Gupta and Liang (2005) adopt the extreme value theory (EVT) to estimate VaR and

address capital adequacy and risk estimation issues in the hedge fund industry. Using

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three times the 99% 1-month VaR as the required equity capital for hedge funds, they

find that the majority of hedge funds are adequately capitalized. They also compare

VaR with traditional risk measures in evaluating hedge fund risk. They conclude that

VaR is better than standard deviation due to negative skewness and high kurtosis in

hedge fund returns.

Bali and Gokcan (2004) estimate VaR for hedge fund portfolios using a normal

distribution, a fat-tailed generalized error distribution (GED), the Cornish-Fisher (CF)

expansion, and the extreme value theory (EVT). They use the HFR (Hedge Fund

Research) indexes and find that the EVT approach and the CF expansion capture tail

risk better than the other approaches.

Recently, Bali, Gokcan and Liang (2006) examine the cross-sectional relation

between hedge fund return and risk measured by VaR in an asset pricing framework.

They estimate VaR using both an empirical distribution and the Cornish-Fisher (CF)

expansion to incorporate the higher-order moments in fund returns. They find a

significant positive relation between VaR and the expected returns on live funds.

VaR, however, is subject to severe criticism. Lo (2001) points out that only several

years of historical data may not show the distribution of returns and questions the

usefulness of VaR based risk management. In addition to this empirical problem, VaR

has theoretical shortcomings. It does not provide the magnitude of the possible losses

below the threshold it identifies. A portfolio’s VaR is the maximum loss that the

investors might suffer during a time horizon at a specified confidence level. It should be

noted that there is still a small but nonzero probability that investors can experience a

loss more than VaR. In other words, VaR does not give any information on how big the

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loss can be when that level is breached. Expected Shortfall(ES) measures this amount. It

is the expected amount of loss conditional on the fact that VaR is exceeded. ES is also

known as tail conditional expectation, conditional loss or tail loss (see Jorion (2000a)).

ES is sometimes called conditional VaR (CVaR) (see Agarwal and Naik (2004) and

Alexander and Baptista (2004)).2

There is another theoretical reason why we prefer ES to VaR as an alternative risk

measure. VaR has some mathematical irregularity, such as lack of convexity and

monotonicity, as well as reasonable continuity (see Artzner, Delbaen, Eber and Heath

(1999), Uryasev (2000), and Alexander and Baptista (2004) for details).3 Recognizing

these shortcomings of VaR, Agarwal and Naik (2004) adopt ES in their empirical tests.

They use the empirical distributions of returns on the HFR hedge fund indexes to

estimate ES. They find that downside risk is significantly underestimated in the mean-

variance framework and suggest mean-ES optimization as an alternative.4

We use another downside risk measure called tail risk (TR) in this paper. It is

introduced by Bali, Demirtas and Levy (2005) to explain the time-series variation in

market returns. TR measures deviation from mean only when the return is lower than

VaR.5

Despite the debate on downside risk in hedge funds, empirical evidence is very

scarce. Academic literature on hedge funds has been mainly focused on performance

measurement. To the best of our knowledge, this is the first paper that empirically

compares alternative risk measures to explain the cross-section of hedge fund returns

using individual hedge fund data under the asset pricing framework.6 We recognize that

this is mainly due to the short history of the hedge fund industry and that the historical

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return distribution may not reveal the true risk in hedge funds. In addition, reporting to a

database is voluntary, which can cause biases in empirical research.7 This, however,

should not be a reason why researchers disregard available hedge fund data to measure

downside risk and stay away from analyzing cross-sectional variation in hedge fund

returns.8 In fact, the goal of this paper is to examine whether available data on hedge

funds can reveal the risk-return trade-off and, if so, which risk measure best captures the

cross-sectional variation in hedge fund returns.

We adopt research methodology from the traditional asset pricing literature. Similar

to Fama and French (1992), we first sort individual hedge funds by a risk measure at the

end of each period and form decile portfolios. Then we compare the rate of return on

the most risky portfolio with that from the least risky portfolio during the following

period. We use the empirical distribution of this return differential to test the presence

and significance of the relationship between risk and expected return. Using a

combined dataset from TASS live and defunct fund databases, we find that hedge funds

with high ES outperform those with low ES by an annual return difference of 7% during

the period January 1995 to December 2004, which is statistically significant at the 5%

level after adjusting for autocorrelation and heteroskedasticity. TR provides similar

results as ES. The other risk measures provide statistically insignificant or marginally

significant results.

In addition to the portfolio level analysis, we test the risk-return trade-off at the

individual fund level under the Fama and MacBeth (1973) framework using the same

dataset. That is, we use time series data to estimate risk for each fund and use the

estimate in a cross-sectional regression at each period. Using the empirical distribution

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of the parameter estimates from the cross-sectional regression, we find ES estimated

using the Cornish-Fisher expansion is positively related to the expected returns while

the relationship is not very significant when we used the other measures. The results are

consistent with the portfolio level analysis. When we use the monthly rate of return on

each hedge fund and adjust for style effects using dummy variables, the average

parameter estimate for Cornish-Fisher ES is 0.027, which is significantly different from

zero at the 5% level after correcting autocorrelation and heteroskedasticiy. The

model’s average adjusted R2 is 26.1%.

The rest of this paper is organized as follows. Section 2 describes the data and

Section 3 explains the methodology. We present the procedure used to estimate each

risk measure and explain how we test the risk-return trade-off at the portfolio level as

well as the individual fund level. Section 4 presents the empirical results and Section 5

concludes the paper.

2. Data

We obtain data on individual hedge funds from Lipper TASS (hereafter, TASS),

which is one of the most commonly utilized databases by academicians and

practitioners. The major databases used in academic literature are TASS, HFR (Hedge

Fund Research) and CISDM (Center for International Securities and Derivatives

Markets) databases. TASS supplies the CSFB Tremont indexes, HFR provides the HFR

family of indexes, and CISDM produces the CISDM family of indexes.9

The information provided by TASS includes net-of-fee monthly returns, investment

styles, assets under management (AUM), fee structure, high-water marks, minimum

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investment, subscription and redemption information, lockup provisions, and so on (see

Chan, Getmansky, Haas, and Lo (2005) for detailed description on TASS).

It is well known that hedge fund databases suffer from biases and we need to

mitigate them. To reduce survivorship bias, we include both live and defunct funds in

our analysis.10 TASS includes information on defunct funds as well as live funds, but

the graveyard database does not retain funds dropped out of the live fund database

before 1994. Thus our test period starts in January 1995 and ends in December 2004

(120 months).

We perform empirical tests for i) live funds, ii) defunct funds, as well as iii) all

funds that combine both the live and the defunct fund databases. As of January 2005,

there are 2,590 live funds and 1,726 defunct funds in TASS. These numbers are

obtained after we exclude those funds that report i) returns not in US dollars, ii)

quarterly (not monthly) returns, or iii) gross return (not net-of-fee returns) from the

original TASS database.

There are additional requirements for the funds to be included in our analysis. First,

funds with less than 24 months of return history are not included in our analysis because

we delete the first two years of return data to mitigate the instant history bias. In

addition, we need 60 months of returns to estimate the standard deviation, semi-

deviation, VaR, ES and TR of these funds. Where 60 months of data is not available, a

minimum of 24 months is used.

Second, we concentrate on the following investment styles: convertible arbitrage,

dedicated short bias, equity market neutral, event driven, fixed income arbitrage, global

macro, long/short equity hedge and multi strategies.11 We exclude funds of funds to

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avoid double counting and managed futures to focus on hedge funds.12 We do not

include emerging market funds to avoid making the highest risk decile dominated by a

specific style. Note that the average ES of emerging market funds is more than twice as

high as the average ES of the other funds. In addition, they point to a specific region,

not an investment strategy.13

Finally, we exclude funds with assets under management less than $10 million.

This is to reduce any bias that might be caused by very small funds. The recent rapid

growth in hedge fund assets is attributed mainly to the allocation of institutional

investors whose interest is not in small hedge funds. Note that TASS uses the same $10

million cut-off criterion to select hedge funds for the CSFB Tremont indexes. Also note

that when the SEC adopted a new rule under the Investment Advisers Act of 1940 for

hedge fund managers to be registered, small funds with AUM under $25 million were

exempted.

After all these requirements, we have 1,101 live funds and 429 defunct funds left in

our sample. Table 1 presents the statistical summary of our data and the Jarque-Bera

(JB) normality test. The average monthly rate of return of the 1,530 funds is 0.84 % and

the funds have negative skewness (-0.06) and high kurtosis (5.84) on average. The

average rejection rate for the normality test is 40.5%, but there is a large variation

across investment styles. As expected, high rejection rates are observed for fixed

income arbitrage funds (57.9%) and event driven funds (53.3%). Extreme returns are

least frequent in equity market-neutral funds, which have the lowest rejection rate

(23.3%) in the normality test. We do not find a large difference between live funds and

defunct funds in terms of the normality test on average, but there is also a large

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variation across styles. In the defunct category, fixed income arbitrage funds have the

highest rejection rate (71.8%), but there are lower rejection rates for long/short equity,

event driven and multi-strategy funds.

3. Methodology

3.1. Estimation of risk measures

Standard deviation (STD): At each month starting from January 1995, we use 60

month rolling windows of previous returns to estimate the standard deviation and other

risk measures of each hedge fund. That is, our five-year estimation window starts in

January 1990, and the test period spans 120 months from January 1995 to December

2004. For example, the standard deviation of returns as of January 1995 is calculated

based on the return history between January 1990 and December 1994. If the fund

started after January 1995, we use whatever return history is available during this time

period as long as the window is at least twenty-four months. For consistency, the same

cut-off rule is applied to estimating semi-deviation, VaR, ES and TR. Therefore, we use

monthly returns over the past 24 to 60 months (as available) to estimate risk for each

fund.

Semi-deviation (SEM): The estimation procedure for semi-deviation is similar to

that for the standard deviation except we consider the deviation from the mean only

when it is negative. Analytically, it can be formulated as follows:

2( ) [( ),0] Semi deviation SEM E Min R µ− ≡ − (1)

where µ is the average return.

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Estrada (2001) argues that the semi-deviation is more useful than the standard

deviation when the underlying distribution of returns is skewed and just as useful when

the underlying distribution is symmetric.

Value-at-Risk (VaR): VaR was originally developed to provide senior

management with a single number that summarizes the total risk in a portfolio. It

attempts to make the following statement: We are (1 )α− percent certain that we will

not lose more than ( , )VaR α τ dollars in τ days.14 That means we have three decision

variables to choose: the confidence level(1 )α− , the time horizon (τ ), and the

estimation model.

VaR can also be expressed in terms of the return on a portfolio instead of the dollar

value. Put more formally, let tR τ+ denote the portfolio return during the period between

t and t+ τ . Let ,R tF denote the cumulative distribution function (CDF) of tR τ+

conditional on the information available at time t. 1,R tF − denotes the inverse function of

,R tF . The portfolio VaR as of time t with a time horizon τ and a confidence level

1 α− can be formulated as follows:

1,( , ) ( )t R tVaR Fα τ α−= − (2)

In this paper, we use a 95% confidence level (0.05α = ) and a time horizon (τ ) of 1

month, which is the frequency of our hedge fund return data. As for the estimation

model, we use both nonparametric and parametric approaches. To compare these

approaches and find the best estimation model is one of the main goals of this paper.

Nonparametric VaR (VaR_NP): Nonparametric VaR does not impose any

parametric assumption on the distribution of a portfolio’s returns. It is based on the left

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tail of the actual empirical distribution. That is, we use monthly returns over the past

twenty-four to sixty months (as available) to estimate the nonparametric VaR from the

empirical distribution of individual hedge fund returns. For example, if we have sixty

observations for fund A in the first estimation window, we use the 5 percentile of all the

observations in the window for the 95% VaR_NP of fund A. If fund B starts at the

thirty-first month of the window, we use the 5 percentile of the thirty observations

available in the window. We use similar interpolation for other return histories between

twenty-four and sixty months. If fund C starts at the thirty-eighth month of the first

window, it is dropped from the window since it has less than twenty-four observations.

Cornish-Fisher VaR (VaR_CF): The Cornish-Fisher (1937) expansion is one of the

parametric approaches to estimating VaR. The traditional parametric method assumes

returns are normally distributed. That is, the VaR measure under this assumption

depends only on the mean and the standard deviation of returns. Hence, the 95% VaR

based on normality is calculated by the following formula:

_ ( 0.05) ( ( ) ) ( 1.645 )VaR Normal zα µ α σ µ σ= = − + × = − − × (3)

where µ and σ are, respectively, the sample mean and standard deviation of returns,

1 α− is the confidence level, and ( )z α is the critical value from the standard normal

distribution corresponding to the confidence level. However, it is inappropriate to use

equation (3) based on normality assumption to estimate VaR. As shown in Table 1, the

return distribution of hedge funds in our sample are left-skewed and leptokurtic on

average. The normality assumption is rejected in 40% of the funds.15

To account for the non-normality in the return distribution, we use the Cornish-

Fisher expansion. It was first introduced to estimating parametric VaR of option

13

portfolios by Zangari (1996). The rationale behind this expansion is that one can obtain

an approximate representation of any distribution with known moments in terms of any

known distribution. The Cornish-Fisher expansion uses the standard normal distribution

as the known distribution. That is, the explicit polynomial expansions for the α

percentile of a return distribution can be expressed in terms of the standardized

moments of the distribution and the corresponding percentile of the standard normal

distribution (see Johnson, Kotz and Balakrishnan (1994), Mina and Ulmer (1999),

Jaschke (2001) and Monteiro (2004) for details).

Equation (4) shows the first four terms of the Cornish-Fisher expansion for the α

percentile of R µ

σ−

. Hence, as shown in equations (4) and (5), the parametric VaR

approximated by the Cornish-Fisher expansion (VaR_CF) incorporates the skewness

and kurtosis of the empirical distribution.16

2 3 3 21 1 1( ) ( ) ( ( ) 1) ( ( ) 3 ( )) (2 ( ) 5 ( ))

6 24 36z z S z z K z z Sα α α α α α αΩ = + − + − − − (4)

_ ( ) ( ( ) )VaR CF α µ α σ= − + Ω × (5)

where µ is the average return, σ is the standard deviation, S is the skewness, and K is

the excess kurtosis of the past 24 to 60 (as available) monthly returns, 1 α− is the

confidence level, and ( )z α is the critical value from the standard normal distribution.

Note that if the portfolio return is normally distributed, skewness (S) and excess kurtosis

(K) are set equal to zero, which makes ( )αΩ equal z(α) and thus, _ ( )VaR CF α

equals _ (VaR Normal α ).

Expected Shortfall (ES): One of the shortcomings of VaR is that it does not

provide information on how big the loss could be once it is breached. ES measures this

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quantity. ES is the average loss greater or equal to VaR. ES can also be expressed in

terms of the portfolio return instead of a dollar amount. It is formulated as follows:

( , ) ( , )

, ,

,

( ) ( )

( , ) [ ( , )][ ( , )]

t tVaR t VaR t

R t R t

v vt t t t t

R t t

vf v dv vf v dv

ES E R R VaRF VaR

α α

τ τα τ α τα τ α

− −

=−∞ =−∞+ += − ≤ − = − = −

−

∫ ∫ (6)

where tR τ+ denotes the portfolio return during the period between t and

t+τt τ+

, ,R tf denotes the conditional probability density function (PDF) of tR τ+ , and ,R tF

denotes the conditional cumulative distribution function (CDF) of tR τ+ conditional on

the information available at time t. 1,R tF − denotes the inverse function of ,R tF and

1 α− is the confidence level.

Artzner et al. (1999) argue that ES has some mathematical properties such as

subadditivity and continuity that are desirable as a coherent measure of risk, while VaR

does not. To test this theoretical argument empirically using hedge funds data is one of

the goals of this paper.

Nonparametric Expected Shortfall (ES_NP): We use both parametric and

nonparametric approaches to estimate ES of individual hedge funds. To estimate

nonparametric ES, we use the left tail of the actual empirical distribution as in Agarwal

and Naik (2004), but we use individual hedge funds, not an index as the input.

Suppose Fund A has 60 observations in the first window. First, we estimate the 0.05

percentile of the return distribution using all 60 return values and set it as the 95%

VaR_NP. Then, we sort out all the return values less than or equal to the 95% VaR_NP

and take the average of them as the 95% ES_NP. We use the same methodology for

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other return histories between 24 and 60 months and any funds with less than 24 months

in the window are excluded in the analysis.

Expected Shortfall Based on the Cornish-Fisher Expansion (ES_CF): To find the

95% ES_CF, we first estimate the 95% Cornish-Fisher VaR (VaR_CF) using equations

(4) and (5). Then we search through the window and find the returns less than or equal

to the 95% VaR_CF. We take the average of those returns, and use the average as the

95% ES_CF of the fund in the window.17

Tail Risk (TR): While ES represents the mean of losses larger than VaR, TR

measures the deviation of losses larger than VaR from mean. Put more formally, TR is

defined as follows:

2( , ) [( ( )) ( , )]t t t t t t tTR E R E R R VaRτ τ τα τ α τ+ + += − ≤ − (7)

Note that among the downside risk measures used in this paper, TR is the best in

terms of capturing the impact of an extremely low return observation because the

deviations from mean are squared before being averaged. As in VaR and ES, we can

use both parametric and nonparametric approach to estimate TR. TR_NP denotes tail

risk based on nonparametric VaR (VaR_NP) and TR_CF denotes tail risk based on

Cornish-Fisher VaR (VaR_CF).

An illustrative example: We use a long/short equity fund as an example to provide

more information on our estimation procedure in Appendix. As illustrated in the

histogram, the normality assumption is a very poor approximation in this case. In this

left-skewed and leptokurtic distribution (Skewness: -1.61, Kurtosis: 11.38), standard

deviation is no longer a measure of total risk. In such a distribution with a high

frequency of extreme observations, downside risk measures depend highly on the

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estimation procedure. Note that the VaR_CF using the first four moments (22.47 %) is

more than twice as high as the VaR_NP that is based on the 5th percentile of the

empirical distribution (10.85 %).

3.2. Comparing risk measures at the portfolio level

Portfolios sorted by risk measures: Once we have the risk measures estimated for

each window as in Section 3.1, we use them to rank hedge fund returns and form 10

decile portfolios in the test periods. Our five-year estimation period starts in January

1990 and the test period is 120 months from January 1995 to December 2004. Our

methodology is very similar to Fama and French (1992), except that they update their

portfolios on an annual basis, while we do it on a monthly basis.

For example, when we use ES_CF as the risk measure, we rank all the hedge funds

in our combined dataset based on the ES_CF, which is estimated for January 1990 to

December 1994. Then we take the average of January 1995 returns of all the funds in

each decile. That is, we form 10 equally weighted portfolios based on ES_CF. Then we

roll over one month ahead and estimate ES_CF again. Now the second estimation

window is from February 1990 to January 1995 and the test month is February 1995.

We rebalance our portfolios based on these new estimates. We repeat the same

procedure 118 more times until the 120th estimation window becomes December 1999

to November 2004 and the test month is December 2004, which is the end of our

sample. Therefore, we have a 120 observation time series of returns on 10 equally

weighted portfolios formed on ES_CF. Following the standard asset pricing literature,

we use the difference between the returns on the most risky portfolio and those on the

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least risky portfolio to test the positive relationship between risk and expected return.

We repeat the same procedure for all the other risk measures and compare the test

results.

Overlapping windows and autocorrelation: Note that our sixty-month windows

overlap each other because only three non-overlapping five-year windows are available

during our sample period (January 1990 – December 2004), which makes statistical

tests infeasible. To deal with the autocorrelation problem that might arise from these

overlapping windows, we present the t-statistics based on the Newey-West (1987)

heteroskedasticity and autocorrelation consistent covariance matrix as well as the

traditional t-statistics.

Ex-post survival information and the risk-return trade-off: We implement the

portfolio level test for i) live funds, ii) defunct funds as well as iii) the combined

dataset, which includes both live and defunct funds. Note that the fund’s survival is ex-

post information which cannot be used in the portfolio decision. Therefore, the risk-

return trade-off should be tested using the combined dataset. High risk portfolios should

have higher expected returns than the less risky ones and the difference should be

statistically and economically significant. This should be true as long as the risk

measure can capture the true risk in the portfolio. Thus, we first implement the test

using the combined dataset. Then we move on to answer the following question: What

happens to the risk-return relationship if we divide the fund universe into two groups

using the ex-post survival information? Note that the realized returns might be different

from the expected ones. The riskier a fund, the more likely it has an extreme return. If

we divide the funds into two groups using the ex-post survival information, those lucky

18

funds with high realized returns belong to the live funds database will make the

relationship between risk and expected return more positive. On the other hand, unlucky

ones are likely to be moved to the graveyard, which makes the relationship negative or

insignificant.18

A caveat regarding the sample size: Even though we do not have a statistical test for

the minimum number of funds to be included in each portfolio, we should note that the

defunct fund sample is not large enough to provide empirical evidence as strong as

those from the live funds or all funds. In case of defunct funds, we have fewer than five

funds in each portfolio decile for the first and the last twelve months.19 Therefore, we

expect the power of the test will be the lowest when we use the defunct fund sample in

our analysis.

3.3. Comparing risk measures at the individual fund level

Univariate cross-sectional regressions: In addition to the portfolio level test

described in Section 3.2, we test the risk-return trade-off at the individual fund level.

Again we follow the traditional asset pricing literature, so the methodology is similar to

Fama and MacBeth (1973). That is, we first estimate each risk measure using the time

series data, and then we use the estimated parameters in the cross-sectional regression.

The estimation procedure is as described in Section 3.1. As in the portfolio level test

explained in Section 3.2, we have 120 time series estimates of the six risk measures

which are used in the corresponding 120 cross-sectional regressions. To test the risk-

return trade-off, we use the resulting 120 parameter estimates corresponding to each

risk measure. That is, if the regression coefficient is positive and statistically different

19

from zero, we conclude that the risk measure is positively related to the expected return.

Here again we use the Newey-West (1987) t-statistic to adjust for heteroskedasticity and

autocorrelation.

Adjustment for the style effect: Although we call all the funds in our sample hedge

funds as if they are a single asset class, the hedge fund universe is very heterogeneous.

It is more realistic to assume, for instance, that an event-driven fund investing in a

merger arbitrage spread has a different expected return from a global macro fund

betting on the dollar-yen exchange rate movement. Thus, we need to adjust for the style

effect before we test the risk-return trade-off by including the style dummy variables in

the regression.

Multivariate regressions: To find a clean relationship between the expected return

and a risk measure, we need to adjust for other fund characteristics that might affect the

expected return on the fund. Aragon (2004) argues that hedge funds with lockup

provision are more likely to hold illiquid assets and thus provide higher expected return.

Bali, Gokcan and Liang (2006) find that fund age and size also have some explanatory

power for a fund’s expected return. Hence we first test these variables in univariate

regressions. Then we use them as additional independent variables in our multivariate

analysis to separate liquidity, age, and size effects from the relationship between risk

and expected return.

4. Empirical results

4.1. Risk-return trade-off at the portfolio level

20

Portfolios sorted by standard deviation and semi-deviation: At the end of each

month from January 1995 to December 2004, ten equally weighted portfolios are

formed on the basis of the standard deviation (or semi-deviation) estimated using the

available (minimum 24) return history in the 60 months estimation window for each

fund. Table 2 shows the time series (120 months) average returns of the ten decile

portfolios for i) live funds, ii) defunct funds, and iii) all the funds in the combined

dataset. The reported standard deviation (semi-deviation) is the time series average of

the cross-sectional standard deviations (semi-deviations) of all the funds in each decile.

When we first look at the result from the combined dataset in Panel A, we find the

expected return on the portfolio is positively related to the risk measured by the average

standard deviation of all the funds in the portfolio. However, the return differential

between the riskiest portfolio (decile 10) and the least risky one (decile 1) is not

significantly different from zero when we adjust for autocorrelation and

heteroskedasticity. The result is not significant even when we implement the analysis

using the live funds only. We recall the JB test result presented in Table 1. Hedge fund

returns are on average left skewed and leptokurtic, and the JB test for normality is

rejected for 40% of the funds. Therefore, standard deviation is not a measure of total

risk for all moments. Panel A of Table 2 confirms our conjecture that standard

deviation, even though still widely used to measure the total risk in hedge funds, is

incapable of capturing the tail risk in alternative investments. We need better

alternatives.

As shown in Panel B of Table 2, part of the tail risk starts to be picked up by semi-

deviation. The return differential from the risk measured by semi-deviation is 0.68%,

21

which is significantly different from zero at the 10 percent level after we adjust for

autocorrelation and heteroskedasticity. The result becomes more significant when we

test the relationship using only live funds. As we know that semi-deviation (a variation

of the second moment) reflects only part of the information about the tail risk in hedge

fund returns, we move onto some more complete downside risk measures.

Portfolios sorted by 95% Value-at-Risk (VaR): Among the alternative risk

measures, VaR is the most widely used in practice.20 We use both parametric and

nonparametric approaches to estimate VaR and compare the results from the two

approaches. Panel A of Table 3 reports the results when we use the 95% VaR_NP

estimated using historical returns as a risk measure. The average return on the portfolio

mostly (but not always) increases as we move from the low risk portfolio (decile 1) to

the high risk portfolio (decile 10), but the difference in returns (0.39%) is not

significantly different from zero. In consistent with Bali, Gokcan and Liang (2006), we

find that the return differential (0.63%) is significantly different from zero at the 5%

level when we implement the test for the live fund sample.

We recognize that VaR_NP utilizes information contained only in the specific

observation corresponding to the 5th percentile and therefore, is more vulnerable to the

measurement error in a single observation. Note that VaR_CF utilizes all the

observations to calculate the four moments, and hence the impact of the error in a single

observation is largely mitigated. Therefore, we expect VaR_CF provides better results

than VaR_NP and Panel B of Table 3 confirms this conjecture. The return differential

based on VaR_CF (0.57%) is significantly different from zero at the 10 percent level.

However, the result is still not strong; this is not surprising when we recall the

22

shortcomings of VaR described in Section 1. Hence, we adopt expected shortfall (ES)

as an alternative risk measure and implement the test in the following.

Regarding the theoretical critiques on VaR raised by Altzner et al (1999), Mina and

Xiao (2001) argue that VaR does not always satisfy the subadditivity property but it is

difficult to come up with the cases in practice where the property is not actually

satisfied. If such a case ever exists in the real world, the hedge fund database would be

the place where we are most likely to find one. This paper aims at providing empirical

evidence on this debate. Table 3 and Table 4 show that ES explains the risk-return

trade-off much better than VaR. When we use the Cornish-Fisher expansion formula for

estimation, the hedge funds with high ES outperform those with low ES by an annual

return difference of 7%, and the return differential is significantly different from zero at

the 5 percent level even after we adjust for autocorrelation and heteroskedasticity.

When we drop defunct funds and test only live funds, the result becomes stronger as

expected and is significant at the 1 percent level.

Note that Panel A of Table 4 shows nonparametric ES fails to provide significant

result and it is attributable to the same reason as in the poor performance of

nonparametric VaR. If we live in an ideal world where we have no measurement error

and the realized return history reveals the true distribution, the Cornish-Fisher

expansion, a fourth order approximation of the true distribution using standard normal

distribution and the sample moments, would be inferior to the nonparametric method.

Recall the limitation of data collection in hedge fund research discussed in Section 2. If

all hedge funds are obliged to report their returns and there is no error in the reported

data, the nonparametric method might perform better than the Cornish Fisher

23

expansion. However, reporting to a hedge fund database is voluntary. We do not expect

that hedge funds’ data is perfect. Hence the parametric Cornish-Fisher ES performs

better than the nonparametric ES.

Table 4 presents only the time series average of the 10 portfolio returns and the

statistical test results. To illustrate the difference in the portfolio risk as we move from

decile 1 through decile 10, we provide a histogram for each decile as well as the plot of

average returns in Figure 1. The plot in Figure 1 shows the positive relationship

between the expected return and risk measured by the ES_CF. The histograms show

how portfolio risk increases as we move from decile 1 to decile 10. In case of decile 1

(portfolio with the lowest ES_CF), returns are clustered around the mean, while extreme

returns are observed more often in decile 10, which have the highest ES_CF.

As already pointed out in Section 3.1, the impact of an extremely low return is

amplified when we use TR as a risk measure because the deviations form mean are

squared before being averaged. If those extremely low returns are mainly outliers to be

ignored in statistical analyses, TR would not be able to explain the cross-sectional

variation in hedge fund returns. However, as shown in Table 5, TR is positively related

to the expected return on hedge funds. The return differential between high TR_CF and

low TR_CF portfolios is significant at the 5 percent level. That is why we call the most

extreme returns (mainly observed in August 1998) ‘informative outliers’. Table 5 also

shows that the nonparametric method performs worse than the parametric method in

estimating TR due to the same reason as in estimating VaR and ES.

So far we have compared the risk measures at the portfolio level and found

downsided risk measures such as ES or TR are better than standard deviation in terms

24

of explaining the cross-sectional variation in hedge funds returns. Now we move onto

the cross-sectional regressions to test the risk-return relationship at the individual fund

level.

4.2. Cross-sectional regressions

Univariate analysis: Table 6 shows the univariate cross-sectional regression results

for i) live funds, ii) defunct funds, and iii) the combined dataset. Following Fama and

MacBeth (1973), we use the time series of the parameter estimates from the cross-

sectional regressions to test the risk-return trade-off. As explained in the portfolio level

analysis in the previous section, we use the autocorrelation and heteroskedasticity

adjusted result from the combined dataset to compare the explanatory power of the risk

measures and other independent variables. Consistent with the portfolio level analysis,

ES_CF and TR_CF provide more significant results than the other risk measures. The

coefficient of ES_CF is 0.034, which is positive and significantly different from zero at

the 1 percent level.21

Other independent variables (fund age, size and liquidity): Table 6 also shows that

lockup provisions explain the liquidity premium in hedge fund returns as argued by

Aragon (2004). Age and size also have some explanatory power. Consistent with the

literature, we find the younger and the smaller a fund is, the higher the expected return

on the fund. Note that the coefficient on the size variable is negative but significantly

different from zero only at the 10 percent level for the combined dataset. The expected

return-size relationship is weakened because we exclude those small funds with less

than US $10 million assets under management (AUM) in our analysis to avoid noise

25

caused by very small funds. Note that the adjusted R2 is much smaller when we use age,

size or lockup as the independent variable, which means risk measures have much

higher explanatory power than others.

Style dummy variables: Recognizing that hedge funds are a very heterogeneous

asset class and each investment style is likely to provide different expected returns, we

repeat the univariate regressions by including style dummy variables. As shown in

Panel B of Table 6, this adjustment makes the distinction between standard deviation

and downside risk measures more prominent. While most of the other risk measures

lose explanatory power due to the style dummy variables, the coefficients on ES_CF

and TR_CF are still significantly different from zero at the 5 percent level and the

adjusted R2 is over 25%. The explanatory power of fund age and liquidity remains after

adjusting for the style effect, while the size effect becomes insignificant when we

include style dummy variables.

Multivariate regressions: To compare alternative risk measures after controlling for

all the other effects such as style, liquidity, age and size, we provide the multivariate

cross-sectional regressions results in Table 7. Consistent with the portfolio level tests

and the univariate regressions, we find expected shortfall (ES) and tail risk(TR) are

superior to the other risk measures in terms of explaining the cross-section of hedge

fund returns. When we use the combined dataset, the coefficient on the 95% ES_CF is

positive (0.0351) and significantly different from zero at the 5 percent level. The

adjusted R2 of this model is 27.4%. TR_CF provides similar results as ES_CF. The

coefficient on standard deviation is significantly different from zero only at the 10

percent level.

26

5. Conclusions

Risk management for alternative investments has been under the spotlight since the

aftermath of Long Term Capital Management (LTCM). The flexibility entitled to hedge

fund managers regarding short selling, leverage and trading derivatives is challenging

the usefulness of traditional risk management tools such as mean-variance analysis, beta

and VaR (Lo (2001), Jorion (2000b) and Agarwal and Naik (2004)). This paper focuses

on alternative risk measures such as semi-deviation, VaR, expected shortfall (ES) and

tail risk (TR) and compares them with standard deviation in terms of their explanatory

power of the cross-sectional variation in expected returns of hedge funds.

Using 1,530 live and defunct funds in the TASS database, we make three main

points about the risk profile of hedge funds. First, we confirm and extend the finding in

the literature that skewness and kurtosis should not be ignored when we analyze the risk

of hedge funds. We provide evidence that the cross-sectional variation in expected

returns of hedge funds can be explained better when we take higher moments into

consideration.

Second, we show that expected shortfall (ES) is superior to VaR as a risk measure

of hedge funds. Our empirical finding confirms the theoretical argument in the literature

that ES is a coherent measure of risk while VaR is not. Finally, we find that the

Cornish-Fisher expansion is better than the nonparametric method when we estimate

downside risk measures. We show that ES and TR based on the Cornish-Fisher formula

(ES_CF) are better than the other risk measures in capturing the risk-return trade-off.

27

During the period between January 1995 and December 2004, the hedge funds with

high ES_CF outperform those with low ES_CF by an annual return difference of 7%.

One caveat here is that we exclude very small and young funds from our analysis to

focus on the risk profile of funds that are more likely to be of interest to large

institutional investors. The hedge fund industry is a very heterogeneous asset class with

light regulation. Risk profiles of nascent funds with small AUM may be very different

from large funds with longer track records. There may be many small funds that take

excess risk that are not reasonably priced, use up their capital and thus disappear

without being known to the public through any database. Another caveat is that there

are other risks that are hardly detected by the six risk measures covered here. We use

lock-up provisions as a proxy for liquidity risk, but credit risk or operational risk in

hedge funds is beyond the scope of this paper.

We should also note that this paper uses a cross-sectional approach to compare

alternative risk measures with regard to the risk-return trade-off, but there are other

ways to find the best measure. For example, a desirable risk measure for alternative

investments should be able to predict the survival of hedge funds (see Park (2006) for

details).

Our future research agenda includes the aggregation property of alternative risk

measures. When we increase the number of assets in our portfolio, standard deviation of

the portfolio always decreases. This diversification benefit may not be as large as we

think when standard deviation is not a measure of total risk. For example, left-tail risk

measured by skewness may increase as we ‘diversify’ our portfolio (see Harvey,

28

Liechty, Liechty, and Müller (2002) for an intuitive example). This implies that the risk

profile of hedge fund indexes may not reveal the true risk of individual hedge funds.

The inaccuracy of estimation due to short history is another issue we would like to

address in our future research. We require at least two years of return data to estimate

the risk measures, and we recognize that the available return history may not show the

true return distribution. We find that VaR, ES and TR estimates are uniformly much

higher in funds with return data in August 1998 (the Russian Debt Crisis), which we

call ‘informative’ outliers. Note that there are many firms that manage several hedge

funds within the same investment style simultaneously and usually the fund returns are

highly correlated. One of our future research topics is to test the validity of using style

or fund family information to overcome the short history problem in the estimation of

downside risk measures.

29

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35

Table 1 Statistical summary of hedge fund returns and the test for normality

This table shows the number of hedge funds (N) in each investment style, the mean and median values of the sample average, standard deviation, skewness and kurtosis of individual hedge fund returns. It also shows how many funds in each style fail the Jarque-Bera (JB) test for normality1. Panel A shows the statistics and the JB test result for all the funds in the combined dataset, while Panel B and C show the results for the live funds and the defunct funds database, respectively. The data is from TASS database, and the sample period is from January 1995 to December 2004.

Panel A. All funds

Average Return (%)

Standard Deviation (%) Skewness Kurtosis

Minimum Return (%)

Maximum Return (%) Investment Style

Number of Funds

Mean Median Mean Median Mean Median Mean Median Mean Median Mean Median

% Rejection in the JB test for

Normality

Convertible Arbitrage 126 0.63 0.70 1.86 1.29 -0.25 -0.23 6.64 4.05 -4.57 -2.26 5.08 3.11 34.9

Dedicated Shortseller 15 0.25 0.27 6.77 7.27 0.11 0.14 4.44 3.80 -18.17 -15.58 20.22 18.50 26.7

Equity Market Neutral 116 0.62 0.44 1.73 1.50 0.06 0.01 4.27 3.47 -3.29 -2.59 4.58 3.48 23.3

Event Driven 246 0.99 0.85 2.15 1.62 -0.09 -0.06 5.82 5.10 -4.84 -3.09 6.63 4.92 53.3

Fixed Income Arbitrage 114 0.54 0.55 2.07 1.54 -1.23 -0.73 11.64 6.24 -7.64 -3.66 4.41 2.89 57.9

Global Macro 109 0.52 0.74 3.72 3.03 0.33 0.37 5.46 4.48 -7.37 -5.57 10.34 7.95 35.8

Long/Short Equity Hedge 730 0.96 0.98 4.49 3.62 0.13 0.10 4.90 4.06 -9.95 -7.33 13.13 9.86 37.3

Multi-Strategy 74 0.85 0.75 2.54 1.42 -0.43 -0.23 8.38 4.99 -5.93 -3.21 8.24 3.81 50.0

All Funds 1530 0.84 0.79 3.38 2.56 -0.06 0.03 5.84 4.31 -7.71 -5.10 9.76 6.15 40.5

1 [ ]24/)3()6/( 22 −+= KSnJB , where S is skewness, K is kurtosis, and n is number of observations. This is a joint test of S = 0 and K = 3. The JB statistic has a Chi-square distribution with two degrees of freedom.

36

Panel B. Live funds

Average Return (%)

Standard Deviation (%)

Skewness Kurtosis Minimum Return (%)

Maximum Return (%) Investment Style

Sample Size

Mean Median Mean Median Mean Median Mean Median Mean Median Mean Median

% Rejection in the JB test for

Normality

Convertible Arbitrage 96 0.63 0.70 1.58 1.21 -0.08 -0.23 5.25 3.68 -3.44 -2.26 4.83 2.53 30.2

Dedicated Shortseller 9 -0.01 0.03 7.01 7.27 0.03 0.13 4.73 3.76 -22.58 -19.80 22.27 20.07 22.2

Equity Market Neutral 78 0.46 0.43 1.56 1.23 0.04 0.01 4.19 3.47 -3.36 -2.14 4.20 3.38 23.1

Event Driven 184 1.11 0.92 2.12 1.62 0.02 0.05 5.81 5.11 -4.49 -2.94 6.95 5.11 55.4

Fixed Income Arbitrage 75 0.68 0.68 1.53 1.05 -0.92 -0.31 10.44 5.43 -5.12 -2.25 3.81 2.94 50.7

Global Macro 73 0.77 0.72 3.12 2.64 0.36 0.47 5.29 4.51 -6.06 -4.76 9.35 6.85 32.9

Long/Short Equity Hedge 529 1.11 1.05 3.97 3.39 0.18 0.13 4.94 4.11 -8.74 -6.98 12.54 9.60 40.1

Multi-Strategy 57 0.83 0.73 2.10 1.34 -0.53 -0.31 8.21 5.34 -5.62 -2.75 6.50 3.46 54.4

Live Funds 1101 0.94 0.84 2.99 2.25 0.02 0.06 5.60 4.30 -6.71 -4.47 9.30 5.81 41.4

Panel C. Defunct funds

Average Return (%)

Standard Deviation (%) Skewness Kurtosis Minimum Return

(%) Maximum Return

(%) Investment Style Sample Size

Mean Median Mean Median Mean Median Mean Median Mean Median Mean Median

% Rejection in the JB test for

Normality

Convertible Arbitrage 30 0.63 0.68 2.74 1.49 -0.77 -0.25 11.01 5.39 -8.19 -2.24 5.85 4.19 50.0

Dedicated Shortseller 6 0.63 0.67 6.40 5.68 0.22 0.50 4.00 3.93 -11.54 -10.08 17.29 15.93 33.3

Equity Market Neutral 38 0.96 0.47 2.11 2.00 0.09 0.00 4.46 3.52 -3.14 -3.17 5.37 3.79 23.7

Event Driven 62 0.63 0.57 2.22 1.64 -0.41 -0.14 5.84 4.83 -5.86 -3.93 5.67 4.21 46.8

Fixed Income Arbitrage 39 0.25 0.42 3.11 2.39 -1.81 -1.61 13.78 7.28 -12.50 -11.13 5.57 2.84 71.8

Global Macro 36 0.02 0.84 4.92 4.65 0.28 0.37 5.79 4.38 -10.03 -7.34 12.36 10.94 41.7

Long/Short Equity Hedge 201 0.59 0.63 5.88 4.78 -0.01 0.00 4.81 3.95 -13.15 -9.56 14.66 10.40 29.9

Multi-Strategy 17 0.93 0.79 4.01 2.63 -0.08 0.14 8.97 4.81 -6.97 -3.86 14.10 5.93 35.3

Defunct Funds 429 0.57 0.58 4.40 3.43 -0.26 -0.05 6.44 4.33 -10.27 -7.10 10.92 7.28 38.2

37

Table 2 Average returns of the hedge fund portfolios sorted by

standard deviation (Panel A) and semi-deviation (Panel B): January 1995 to December 2004 (120 months)

At the end of each month, 10 equally weighted portfolios are formed on the basis of the standard deviation (or semi-deviation) calculated using the previous 24 to 60 months return history (as available) for each fund. The first part of each panel shows the time series (120 months) average returns of these portfolios for Deciles 1 to 10 for the live funds, the defunct funds as well as the combined datasets. The reported standard deviation (or semi-deviation) is the time series (120 months) average of the average standard deviation (or semi-deviation) of all the funds in each decile. The second part of each panel presents the average return differential between the decile 10 (riskiest portfolio) and the decile 1 (least risky one), the standard t-statistics for the average return differential and the Newey-West (1987) adjusted t-statistics for the live funds, the defunct funds and the combined dataset. ***, **, and * denote the return differential is significantly different from zero at the 1%, 5% and 10% level, respectively.

Panel A. Standard Deviation (STD)

Live Defunct All

Deciles STD Return Deciles STD Return Deciles STD Return

Low STD 0.66 0.80 Low STD 0.77 0.61 Low STD 0.69 0.77

2 1.12 0.85 2 1.30 0.61 2 1.19 0.79

3 1.60 0.88 3 1.81 0.75 3 1.69 0.82

4 2.17 1.01 4 2.35 0.69 4 2.23 0.97

5 2.82 1.19 5 2.81 0.85 5 2.86 1.11

6 3.48 1.18 6 3.43 0.65 6 3.50 1.07

7 4.07 1.27 7 4.29 0.76 7 4.17 1.21

8 4.97 1.41 8 5.27 1.04 8 5.10 1.23

9 6.16 1.23 9 6.68 1.20 9 6.40 1.19

High STD 8.80 1.27 High STD 10.40 0.71 High STD 9.60 1.40

Average Return Differential for STD Average Return Differential for STD Average Return Differential for STD

High STD - Low STD 0.47% High STD - Low STD 0.10% High STD - Low STD 0.63%

standard t-statistic 1.67* standard t-statistic 0.16 standard t-statistic 1.66*

Newey-West t-statistic 1.53 Newey-West t-statistic 0.15 Newey-West t-statistic 1.57

38

Panel B. Semi-deviation (SEM)

Live Defunct All

Deciles SEM Return Deciles SEM Return Deciles SEM Return

Low SEM 0.46 0.81 Low SEM 0.53 0.63 Low SEM 0.47 0.78

2 0.79 0.85 2 0.90 0.58 2 0.83 0.78

3 1.15 0.87 3 1.23 0.82 3 1.18 0.90

4 1.54 1.02 4 1.60 0.85 4 1.57 0.96

5 1.94 1.07 5 1.98 0.75 5 1.97 1.00

6 2.38 1.25 6 2.43 0.75 6 2.42 1.09

7 2.80 1.10 7 2.96 0.73 7 2.87 1.09

8 3.41 1.51 8 3.53 1.08 8 3.51 1.39

9 4.24 1.26 9 4.57 0.88 9 4.38 1.10

High SEM 5.87 1.37 High SEM 7.17 0.76 High SEM 6.44 1.46

Average Return Differential for SEM Average Return Differential for SEM Average Return Differential for SEM

High SEM - Low SEM 0.57% High SEM - Low SEM 0.14% High SEM - Low SEM 0.68%

standard t-statistic 2.08** standard t-statistic 0.23 standard t-statistic 1.85*

Newey-West t-statistic 1.89* Newey-West t-statistic 0.23 Newey-West t-statistic 1.78*

39

Table 3

Average returns of the hedge fund portfolios sorted by 95% nonparametric VaR (Panel A) and the Cornish-Fisher VaR (Panel B):

January 1995 to December 2004

At the end of each month, 10 equally weighted portfolios are formed on the basis of the 95% nonparametric VaR (or Cornish-Fisher VaR) calculated using the previous 24 to 60 months return history (as available) for each fund. The first part of each panel shows the time series (120 months) average returns of these portfolios for Deciles 1 to 10 for the live funds, the defunct funds as well as the combined datasets. The reported 95% VaR is the time series (120 months) average of the average VaR of all the funds in each decile. The second part of each panel presents the average return differential between the decile 10 (riskiest portfolio) and the decile 1 (least risky one), the standard t-statistics for the average return differential and the Newey-West (1987) adjusted t-statistics for the live funds, the defunct funds and the combined dataset. ***, **, and * denote the return differential is significantly different from zero at the 1%, 5% and 10% level, respectively.

Panel A. 95% Nonparametric VaR (VaR_NP)

Live Defunct All

Deciles VAR_NP Return Deciles VAR_NP Return Deciles VAR_NP Return

Low VAR -0.06 0.87 Low VAR -0.08 0.86 Low VAR -0.08 0.86

2 0.60 0.80 2 0.84 0.83 2 0.67 0.81

3 1.29 1.00 3 1.51 0.59 3 1.40 0.91

4 2.05 0.99 4 2.33 0.68 4 2.14 0.95

5 2.85 1.21 5 3.11 0.67 5 2.96 1.05

6 3.72 1.13 6 4.00 0.76 6 3.83 1.09

7 4.65 1.16 7 5.00 0.88 7 4.79 1.07

8 5.85 1.24 8 6.22 0.90 8 5.99 1.11

9 7.57 1.20 9 8.09 1.17 9 7.87 1.44

High VAR 11.38 1.50 High VAR 13.71 0.50 High VAR 12.46 1.26

Average Return Differential for VAR Average Return Differential for VAR Average Return Differential for VAR

High VAR - Low VAR 0.63% High VAR - Low VAR -0.35% High VAR - Low VAR 0.39%

standard t-statistic 2.37** standard t-statistic -0.54 standard t-statistic 1.18

Newey-West t-statistic 2.10** Newey-West t-statistic -0.49 Newey-West t-statistic 1.11

40

Panel B. 95% Cornish-Fisher VaR (VaR_CF)

Live Defunct All

Deciles VAR_CF Return Deciles VAR_CF Return Deciles VAR_CF Return

Low VAR -0.06 0.88 Low VAR -0.16 0.86 Low VAR -0.06 0.88

2 0.77 0.82 2 0.93 0.64 2 0.85 0.80

3 1.50 0.88 3 1.68 0.71 3 1.59 0.82

4 2.30 1.01 4 2.47 0.73 4 2.37 0.96

5 3.17 1.04 5 3.42 0.70 5 3.27 0.99

6 4.10 1.12 6 4.42 0.74 6 4.23 1.08

7 5.10 1.18 7 5.53 0.94 7 5.25 1.09

8 6.35 1.44 8 6.76 0.64 8 6.55 1.28

9 8.19 1.36 9 8.93 1.13 9 8.52 1.20

High VAR 12.15 1.38 High VAR 15.03 0.72 High VAR 13.32 1.45

Average Return Differential for VAR Average Return Differential for VAR Average Return Differential for VAR

High VAR - Low VAR 0.50% High VAR - Low VAR -0.13% High VAR - Low VAR 0.57%

standard t-statistic 2.09** standard t-statistic -0.21 standard t-statistic 1.85*

Newey-West t-statistic 1.93* Newey-West t-statistic -0.19 Newey-West t-statistic 1.77*

41

Table 4 Average returns of the hedge fund portfolios sorted by

95% nonparametric expected shortfall (Panel A) or Cornish-Fisher expected shortfall (Panel B):

January 1995 to December 2004

At the end of each month, 10 equally weighted portfolios are formed on the basis of the 95% nonparametric expected shortfall (or Cornish-Fisher expected shortfall) calculated using the previous 24 to 60 months return history (as available) for each fund in the TASS database. The first part of each panel shows the time series (120 months) average returns of these portfolios for Deciles 1 to 10 for the live funds, the defunct funds as well as the combined datasets. The reported 95% ES is the time series (120 months) average of the average ES of all the funds in each decile. The second part of each panel presents the average return differential between the decile 10 (riskiest portfolio) and the decile 1 (least risky one), the standard t-statistics for the average return differential and the Newey-West (1987) adjusted t-statistics for the live funds, the defunct funds and the combined dataset. ***, **, and * denote the return differential is significantly different from zero at the 1%, 5% and 10% level, respectively. Panel A. 95% nonparametric expected shortfall (ES_NP)

Live Defunct All

Deciles ES_NP Return Deciles ES_NP Return Deciles ES_NP Return

Low ES 0.32 0.85 Low ES 0.38 0.94 Low ES 0.34 0.88

2 1.42 0.87 2 1.56 0.71 2 1.49 0.83

3 2.35 0.84 3 2.54 0.68 3 2.45 0.82

4 3.43 0.96 4 3.56 0.71 4 3.50 0.91

5 4.52 1.20 5 4.63 0.60 5 4.60 1.11

6 5.59 1.14 6 6.03 0.80 6 5.77 1.05

7 6.88 1.09 7 7.41 1.08 7 7.11 1.05

8 8.54 1.46 8 8.90 0.77 8 8.80 1.34

9 11.08 1.28 9 11.87 0.74 9 11.55 1.23

High ES 16.35 1.40 High ES 20.13 0.80 High ES 17.86 1.33

Average Return Differential for ES Average Return Differential for ES Average Return Differential for ES

High ES - Low ES 0.55% High ES - Low ES -0.13% High ES - Low ES 0.46%

standard t-statistic 2.33** standard t-statistic -0.23 standard t-statistic 1.50

Newey-West t-statistic 2.19** Newey-West t-statistic -0.22 Newey-West t-statistic 1.55

42

Panel B. 95% Cornish-Fisher expected shortfall (ES_CF)

Live Defunct All

Deciles ES_CF Return Deciles ES_CF Return Deciles ES_CF Return

Low ES 0.49 0.86 Low ES 0.49 1.01 Low ES 0.52 0.89

2 1.63 0.89 2 1.74 0.52 2 1.71 0.78

3 2.69 0.78 3 2.77 0.62 3 2.77 0.78

4 3.79 1.02 4 3.77 0.95 4 3.84 1.00

5 4.81 1.18 5 5.02 0.63 5 4.95 1.05

6 6.10 1.11 6 6.54 0.79 6 6.31 1.06

7 7.58 1.11 7 8.15 1.08 7 7.84 1.13

8 9.54 1.37 8 9.93 0.72 8 9.90 1.25

9 12.30 1.16 9 13.20 0.51 9 12.76 1.18

High ES 19.65 1.65 High ES 24.92 1.04 High ES 21.42 1.44

Average Return Differential for ES Average Return Differential for ES Average Return Differential for ES

High ES - Low ES 0.78% High ES - Low ES 0.04% High ES - Low ES 0.55%

standard t-statistic 3.17*** standard t-statistic 0.07 standard t-statistic 1.84*

Newey-West t-statistic 3.12*** Newey-West t-statistic 0.07 Newey-West t-statistic 2.04**

43

Table 5 Average returns of the hedge fund portfolios sorted by

5% nonparametric tail risk (Panel A) or Cornish-Fisher tail risk (Panel B): January 1995 to December 2004

At the end of each month, 10 equally weighted portfolios are formed on the basis of the 5% nonparametric tail risk (or Cornish-Fisher tail risk) calculated using the previous 24 to 60 months return history (as available) for each fund in the TASS database. The first part of each panel shows the time series (120 months) average returns of these portfolios for deciles 1 to 10 for the live funds, the defunct funds as well as the combined datasets. The reported 5% TR is the time series (120 months) average of the average TR of all the funds in each decile. The second part of each panel presents the average return differential between the decile 10 (riskiest portfolio) and the decile 1 (least risky one), the standard t-statistics for the average return differential and the Newey-West (1987) adjusted t-statistics for the live funds, the defunct funds and the combined dataset. ***, **, and * denote the return differential is significantly different from zero at the 1%, 5% and 10% level, respectively. Panel A. 5% Nonparametric Tail Risk (TR_NP)

Live Defunct All

Deciles TR Return Deciles TR Return Deciles TR Return

Low TR 1.30 0.80 Low TR 1.50 0.67 Low TR 1.36 0.79

2 2.48 0.88 2 2.70 0.81 2 2.60 0.85

3 3.51 0.86 3 3.66 0.77 3 3.59 0.80

4 4.70 0.96 4 4.77 0.64 4 4.76 0.89

5 5.88 1.14 5 5.91 0.66 5 5.94 1.09

6 7.05 1.22 6 7.38 0.86 6 7.22 1.04

7 8.44 1.16 7 8.82 0.91 7 8.66 1.28

8 10.21 1.37 8 10.46 1.04 8 10.51 1.25

9 12.90 1.27 9 13.68 0.56 9 13.34 1.19

High TR 18.40 1.45 High TR 22.71 0.92 High TR 20.11 1.36

Average Return Differential for TR Average Return Differential for TR Average Return Differential for TR

High TR - Low TR 0.65% High TR - Low TR 0.26% High TR - Low TR 0.57%

standard t-statistic 2.52** standard t-statistic 0.48 standard t-statistic 1.74*

Newey-West t-statistic 2.50** Newey-West t-statistic 0.47 Newey-West t-statistic 1.82*

44

Panel B. 5% Cornish-Fisher Tail Risk (TR_CF)

Live Defunct All

Deciles TR_CF Return Deciles TR_CF Return Deciles TR_CF Return

Low TR 1.50 0.82 Low TR 1.61 0.86 Low TR 1.56 0.84

2 2.67 0.86 2 2.83 0.54 2 2.78 0.81

3 3.77 0.86 3 3.84 0.72 3 3.84 0.79

4 5.00 0.89 4 4.96 0.73 4 5.03 0.90

5 6.15 1.22 5 6.27 0.76 5 6.26 1.15

6 7.46 1.17 6 7.84 0.57 6 7.66 1.08

7 8.96 1.15 7 9.42 1.39 7 9.25 1.04

8 11.07 1.38 8 11.37 0.80 8 11.43 1.31

9 13.79 1.19 9 14.62 0.32 9 14.27 1.15

High TR 21.09 1.57 High TR 26.52 1.18 High TR 22.98 1.49

Average Return Differential for TR Average Return Differential for TR Average Return Differential for TR

High TR - Low TR 0.74% High TR - Low TR 0.33% High TR - Low TR 0.65%

standard t-statistic 2.68*** standard t-statistic 0.61 standard t-statistic 2.02**

Newey-West t-statistic 2.57** Newey-West t-statistic 0.56 Newey-West t-statistic 2.08**

45

Table 6 Univariate cross-sectional regression of hedge fund returns on risk measures estimated from the return history: average

parameter estimates and adjusted R2 for the 120 months from January 1995 to December 2004

This table presents the time series (120 months) average of the slope coefficient and the adjusted R2 obtained from the univariate cross-sectional regressions. The standard t-statistics given in parentheses are the average slope divided by its time-series standard error. The Newey-West (1987) adjusted t-statistics are given in square brackets. ***, ** and * denote significant results at the 1%, 5%, and 10% level, respectively.

Panel A. Without adjusting for style effects

All Funds Live Funds Defunct Funds Regression Models

tβ Adj-R2 tβ Adj-R2

tβ Adj-R2

, , 1 ,i t t t i t i tR Stdγ β ε−= + + 0.0902 (2.19)** [2.20]**

5.99% 0.0816 (2.07)** [2.14]**

5.28% 0.0791 (1.59) [1.44]

6.76%

, , 1 ,i t t t i t i tR SEMγ β ε−= + + 0.1383

(2.28)** [2.30]**

5.83% 0.1312

(2.30)** [2.34]**

5.11% 0.0873 (1.16) [1.05]

6.75%

0.05, , 1 ,_i t t t i t i tR VaR NPαγ β ε=

−= + + 0.0515 (1.95)* [1.87]*

5.18% 0.0520

(2.21)** [2.17]**

4.21% 0.0253 (0.68) [0.57]

7.20%

0.05, , 1 ,_i t t t i t i tR VaR CF αγ β ε=

−= + + 0.0542

(2.23)** [2.19]**

4.84% 0.0543

(2.51)** [2.43]**

3.93% 0.0238 (0.69) [0.61]

6.95%

0.05, , 1 ,_i t t t i t i tR ES NPαγ β ε=

−= + + 0.03l88

(2.13)** [2.20]**

4.44% 0.0377

(2.31)** [2.41]**

3.65% 0.0194 (0.78) [0.71]

6.24%

0.05, , 1 ,_i t t t i t i tR ES CFαγ β ε=

−= + + 0.0340

(2.34)** [2.60]***

3.42% 0.0339

(2.60)** [2.80]***

2.77% 0.0214 (1.04) [1.05]

4.84%

0.05, , 1 ,_i t t t i t i tR TR NPαγ β ε=

−= + + 0.0415

(2.32)** [2.46]**

4.34% 0.0372

(2.36)** [2.41]**

3.23% 0.0275 (1.16) [1.14]

6.21%

0.05, , 1 ,_i t t t i t i tR TR CFαγ β ε=

−= + + 0.0366

(2.46)** [2.76]***

3.57% 0.0325

(2.43)** [2.57]**

2.62% 0.0277 (1.36) [1.43]

5.27%

, , 1 ,i t t t i t i tR Ageγ β ε−= + + -0.0016

(-2.62)*** [-2.01]**

0.13% -0.0017

(-2.84)*** [-2.32]**

0.10% -0.0030

(-2.45)** [-2.15]**

-0.24%

46

, , 1 ,i t t t i t i tR LnAγ β ε−= + + -0.0550

(-1.77)* [-1.79]*

0.69% -0.0843

(-3.12)*** [-3.43]***

0.64% 0.0248 (0.42) [0.40]

0.89%

, ,i t t t i i tR dlockγ β ε= + + 0.1966

(3.25)*** [3.61]***

0.30% 0.1663

(2.58)** [2.54]**

0.38% 0.2779

(2.50)** [2.39]**

0.11%

Panel B. With style dummy variables

All Funds Live Funds Defunct Funds Regression Models

tβ Adj-R2 tβ Adj-R2

tβ Adj-R2

8

, , , 1 ,1

i t s s t t i t i ts

R D Stdγ β ε−=

= + +∑

0.0732 (1.88)* [1.76]*

27.2% 0.0713 (1.77)* [1.65]

27.6% 0.0613 (1.27) [1.17]

25.2%

8

, , , 1 ,1

i t s s t t i t i ts

R D SEMγ β ε−=

= + +∑

0.1103 (1.91)* [1.82]*

27.2% 0.1160 (1.97)* [1.84]*

27.9% 0.0431 (0.57) [0.53]

25.4%

80.05

, , , 1 ,1

_i t s s t t i t i ts

R D VaR NPαγ β ε=−

=

= + +∑

0.0377 (1.44) [1.33]

27.1% 0.0452 (1.84)* [1.66]*

27.0% 0.0139 (0.34) [0.30]

26.6%

80.05

, , , 1 ,1

_i t s s t t i t i ts

R D VaR CFαγ β ε=−

=

= + +∑

0.0419 (1.71)* [1.65]

27.0% 0.0513

(2.23)** [2.02]**

27.1% -0.0039 (-0.10) [-0.11]

26.5%

80.05

, , , 1 ,1

_i t s s t t i t i ts

R D ES NPαγ β ε=−

== + +∑

0.0288 (1.60) [1.61]

26.7% 0.0337 (1.95)* [1.94]*

27.2% 0.0103 (0.37) [0.36]

25.9%

80.05

, , , 1 ,1

_i t s s t t i t i ts

R D ES CFαγ β ε=−

== + +∑

0.0270 (1.89)* [2.00]**

26.1% 0.0332

(2.40)** [2.39]**

26.8% -0.0045 (-0.19) [-0.22]

25.2%

80.05

, , , 1 ,1

_i t s s t t i t i ts

R D TR NPαγ β ε=−

== + +∑

0.0316 (1.86)* [1.89]*

26.7% 0.0330

(2.01)** [1.90]*

27.3% 0.0138 (0.57) [0.59]

25.3%

80.05

, , , 1 ,1

_i t s s t t i t i ts

R D TR CFαγ β ε=−

== + +∑

0.0293 (2.08)** [2.19]**

26.2% 0.0311

(2.23)** [2.15]**

27.2% 0.0566 (1.08) [1.02]

24.9%

8

, , , 1 ,1

i t s s t t i t i ts

R D Ageγ β ε−=

= + +∑

-0.00178 (-2.87)*** [-2.40]**

22.2% -0.00171

(-2.64)*** [-2.41]**

23.3% -0.00386

(-2.83)*** [-2.68]***

19.7%

47

8

, , , 1 ,1

i t s s t t i t i ts

R D LnAγ β ε−=

= + +∑

-0.0403 (-1.36)

[-1.40] 22.9%

-0.0692 (-2.73)*** [-3.19]***

23.5% -0.1075 (-0.59) [-0.60]

20.8%

8

, , ,1

i t s s t t i i ts

R D dlockγ β ε=

= + +∑

0.1920 (3.62)*** [3.43]***

22.3% 0.1339

(2.13)** [1.87]*

23.4% 0.4389

(4.05)*** [4.19]***

21.0%

48

Table 7 Multivariate cross-sectional regression of hedge fund returns on risk measures and other fund characteristics:

average parameter estimates and adjusted R2 for the 120 months from January 1995 to December 2004 This table presents the time series (120 months) average of the slope coefficient and the adjusted R2 obtained from the multivariate cross-sectional regressions. The standard t-statistics given in parentheses are the average slope divided by its time-series standard error. The Newey-West (1987) adjusted t-statistics are given in square brackets. ***, ** and * denote significant results at the 1%, 5%, and 10 level, respectively.

Panel A. All funds

Multivariate Cross-Sectional Regressions with Style Dummy Variables 1β 2β 3β 4β Adj-R2

8

, , 1, , 1 2, , 3, , 4, ,1

i t s s t t i t t i t t i t t i i ts

R D Std Age LnA dlockγ β β β β ε−=

= + + + + +∑ 0.0870 (1.97)* [1.93]*

-0.0014 (-2.24)** [-1.93]*

0.0099 (0.27) [0.28]

0.2369 (3.40)*** [2.54]**

28.1%

8

, , 1, , 1 2, , 3, , 4, ,1

i t s s t t i t t i t t i t t i i ts

R D SEM Age LnA dlockγ β β β β ε−=

= + + + + +∑ 0.1306 (1.98)** [1.98]**

-0.0011 (-1.73) [-1.45]

0.0055 (0.15) [0.16]

0.2351 (3.36)*** [2.50]**

28.6%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D VaR NP Age LnA dlockαγ β β β β ε=−

=

= + + + + +∑ 0.0517 (1.72)* [1.71]*

-0.0013 (-2.23)** [-1.78]*

0.0072 (0.19) [0.20]

0.2361 (3.37)*** [2.50]**

28.0%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D VaR CF Age LnA dlockαγ β β β β ε=−

=

= + + + + +∑ 0.0566

(2.03)** [2.06]**

-0.0010 (-1.69)* [-1.43]

0.0052 (0.14) [0.15]

0.2422 (3.44)*** [2.56]**

27.9%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D ES NP Age LnA dlockαγ β β β β ε=−

== + + + + +∑

0.0391 (1.90)* [2.02]**

-0.0012 (-2.12)**

[-1.62]

0.0080 (0.21) [0.23]

0.2344 (3.35)*** [2.51]**

28.0%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D ES CF Age LnA dlockαγ β β β β ε=−

== + + + + +∑

0.0351 (2.09)** [2.26]**

-0.0009 (-1.49) [-1.13]

0.0040 (0.09) [0.10]

0.2428 (3.58)*** [2.75]***

27.4%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D TR NP Age LnA dlockαγ β β β β ε=−

== + + + + +∑

0.0394 (2.01)** [2.11]**

-0.0018 (-2.50)** [-2.32]**

0.0091 (0.23) [0.24]

0.1671 (2.69)*** [2.54]**

28.0%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D TR CF Age LnA dlockαγ β β β β ε=−

== + + + + +∑

0.0366 (2.17)** [2.31]**

-0.0014 (-1.94)* [-1.81]*

0.0063 (0.16) [0.17]

0.1759 (2.84)*** [2.82]***

27.5%

49

Panel B. Live funds

Multivariate Cross-Sectional Regressions with Style Dummy Variables 1β 2β 3β 4β Adj-R2

8

, , 1, , 1 2, , 3, , 4, ,1

i t s s t t i t t i t t i t t i i ts

R D Std Age LnA dlockγ β β β β ε−=

= + + + + +∑ 0.0810 (1.59) [1.50]

-0.0003 (-0.36) [-0.31]

-0.020 (-0.68) [-0.74]

0.1867 (2.25)**

[1.62] 28.0%

8

, , 1, , 1 2, , 3, , 4, ,1

i t s s t t i t t i t t i t t i i ts

R D SEM Age LnA dlockγ β β β β ε−=

= + + + + +∑ 0.1309 (1.76)* [1.64]

-0.00011 (-0.15) [-0.13]

-0.0182 (-0.62) [-0.69]

0.179 (2.14)**

[1.54] 28.3%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D VaR NP Age LnA dlockαγ β β β β ε=−

=

= + + + + +∑ 0.0534 (1.65) [1.57]

-0.00025 (-0.37) [-0.30]

-0.0199 (-0.67) [-0.78]

0.1863 (2.19)**

[1.56] 27.3%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D VaR CF Age LnA dlockαγ β β β β ε=−

=

= + + + + +∑ 0.0591

(2.04)** [1.94]*

-0.00012 (-0.17) [-0.15]

-0.0168 (-0.59) [-0.67]

0.1743 (2.05)**

[1.50] 27.2%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D ES NP Age LnA dlockαγ β β β β ε=−

== + + + + +∑

0.0382 (1.73)* [1.70]*

-0.00027 (-0.39) [-0.32]

-0.0147 (-0.50) [-0.61]

0.1669 (1.97)**

[1.43] 27.3%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D ES CF Age LnA dlockαγ β β β β ε=−

== + + + + +∑

0.0369 (2.09)** [2.02]**

-0.00008 (-0.12) [-0.10]

-0.0151 (-0.53) [-0.62]

0.1628 (2.01)**

[1.46] 26.9%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D TR NP Age LnA dlockαγ β β β β ε=−

== + + + + +∑

0.0367 (1.72)* [1.59]

-0.0011 (-1.13) [-0.96]

-0.0078 (-0.25) [-0.29]

0.0727 (0.91) [0.78]

27.1%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D TR CF Age LnA dlockαγ β β β β ε=−

== + + + + +∑

0.0364 (2.01)** [1.84]*

-0.0008 (-0.84) [-0.77]

-0.0073 (-0.24) [-0.27]

0.0728 (0.93) [0.79]

26.8%

50

Panel C. Defunct funds

Multivariate Cross-Sectional Regressions with Style Dummy Variables 1β 2β 3β 4β Adj-R2

8

, , 1, , 1 2, , 3, , 4, ,1

i t s s t t i t t i t t i t t i i ts

R D Std Age LnA dlockγ β β β β ε−=

= + + + + +∑ -1.1709 (-1.01) [-0.96]

0.0104 (0.70) [0.71]

-0.5097 (-0.89) [-0.89]

5.6987 (1.07) [1.08]

28.6%

8

, , 1, , 1 2, , 3, , 4, ,1

i t s s t t i t t i t t i t t i i ts

R D SEM Age LnA dlockγ β β β β ε−=

= + + + + +∑ -0.1902 (-0.81) [-0.79]

-0.0036 (-1.98)** [-2.21]**

0.0307 (0.33) [0.36]

0.4682 (3.13)*** [2.53]**

30.8%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D VaR NP Age LnA dlockαγ β β β β ε=−

=

= + + + + +∑ -0.0530 (-0.85) [-0.67]

-0.0040 (-2.04)** [-2.21]**

-0.0006 (-0.006)

-0.01]

0.6554 (2.66)*** [2.32]**

29.7%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D VaR CF Age LnA dlockαγ β β β β ε=−

=

= + + + + +∑ -0.0537 (-0.59) [-0.75]

-0.0041 (-2.25)** [-2.44]**

0.0504 (0.52) [0.58]

0.2614 (1.13) [1.14]

29.6%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D ES NP Age LnA dlockαγ β β β β ε=−

== + + + + +∑

-0.0387 (-0.78) [-0.79]

-0.0040 (-2.29)** [-2.19]**

0.0266 (0.28) [0.30]

0.4714 (3.18)*** [2.56]**

30.8%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D ES CF Age LnA dlockαγ β β β β ε=−

== + + + + +∑

-0.0674 (-1.07) [-0.98]

-0.0037 (-2.11)** [-2.01]**

0.0511 (0.53) [0.54]

0.4076 (2.96)*** [2.33]**

30.0%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D TR NP Age LnA dlockαγ β β β β ε=−

== + + + + +∑

-0.0643 (-0.90) [-0.87]

-0.0037 (-2.16)** [-2.10]**

0.0327 (0.34) [0.38]

0.4614 (3.19)*** [2.60]**

30.7%

80.05

, , 1, , 1 2, , 3, , 4, ,1

_i t s s t t i t t i t t i t t i i ts

R D TR CF Age LnA dlockαγ β β β β ε=−

== + + + + +∑

-0.1412 (-1.05) [-1.00]

-0.0034 (-1.88)* [-1.77]*

0.0527 (0.54) [0.57]

0.4128 (2.99)*** [2.38]**

30.1%

51

Figure 1. Returns on the portfolios sorted by 95% ES_CF : January 1994 to December 2004 (All funds)

Average Return of the 120 Months

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

Low 2 3 4 5 6 7 8 9 High

Deciles

Re

turn

s (%

)

Histogram of Returns for Each Decile (120 Months)

52

Appendix. An example to show how to estimate the risk measures

- Investment Style: Long/Short Equity Hedge - Size: US $87.6 million (as of January 1995) - Estimation Window: January 1995 – December 1999 (60 Months) - Return History in the Estimation Window (Sorted in ascending order)

Min 2 3 4 5 6 7 8 ………… 56 57 58 59 Max

-62.7 % -24.0 -14.4 -10.7 -9.4 -8.8 -8.2 -6.1 ………… 21.0 22.5 23.8 32.4 32.9 %

- The Distribution of Returns

- Summary Statistics Number of Months 60 Skewness -1.61 Mean 3.52 % Kurtosis 11.38 Standard Deviation (STD) 13.79 % Minimum -62.74 % Semi-deviation (SEM) 10.67 % Maximum 32.88 % - Calculation of the 95% VaR, 95% expected shortfall and 5% tail risk

VaR_NP The 5th percentile of the distribution = 10.85 %

VaR_CF 2 3 3 21 1 1

[3.52 1.645 (( 1.645) 1)( 1.61) (( 1.645) 3( 1.645))(11.38) (2( 1.645) 5(1.645))( 1.61) (13.79)]6 24 36

− + − + − − − + − − − − − + − = 22.47 %

ES_NP ( 62.7 24.0 14.4)

33.70%3

− − −= −

ES_CF ( 62.7 24.0)

43.35 %2

− −= −

TR_NP 2 2 2( 62.7 3.52) ( 24.0 3.52) ( 14.4 3.52)

42.7 %3

− − + − − + − −=

TR_CF 2 2( 62.7 3.52) ( 24.0 3.52)

50.7 %2

− − + − −=

With Normality Assumption

With Nonparametric Kernel Method

53

1 The Economist (Feb 17th, 2005, ‘The New Money Men’) reports that in recent years hedge funds have doubled in size and number while the size of the mutual-fund industry in terms of assets and offerings has merely returned to its level of 2000. Fung, Hsieh, Naik and Ramadorai (2005) report that assets under management in the hedge fund industry have grown from U.S. $50 billion at the end of 1993 to over $670 billion by the end of 2004 using TASS Asset Flows report. 2 CVaR, however, means a very different concept in other papers (see Engle and Manganelli (2004) and Bali and Theodossiou (2005)). That is, CVaR is related to some GARCH or stochastic volatility estimation. Therefore, we use the term ‘expected shortfall(ES)’, not ‘conditional VaR (CVaR)’, in this paper. ES is not really a conditional measure in the sense that it does not incorporate time-variation or stochastic movements in the return distribution. ES is unconditional, but it is calculated based on (or conditioned on) the VaR threshold. We thank the anonymous referee for pointing this out. 3 For example, Artzner et al. (1999) show that the VaR of a portfolio with two securities can be larger than the sum of the VaR of each security in the portfolio. That is, VaR is not a ‘coherent’ measure of risk, because it fails to satisfy the ‘subadditivity property’. 4 Note that Agarwal and Naik (2004) utilize index data to analyze downside risk in hedge funds while we use individual fund data. Fung and Hsieh (2002a) report that hedge fund indexes behave very differently from individual hedge funds. While standard deviation is always reduced when we form a portfolio, the aggregation property of the alternative risk measures is not very well studied. This is one of our future research topics. 5 See Section 3.1 for detailed explanation on TR. 6 Bali and Cakici (2004) examine the cross-sectional relation between VaR and expected returns on individual stocks trading at NYSE, AMEX and Nasdaq. Bali,Gokcan and Liang (2006) use VaR to explain the cross-section of hedge fund returns but they do not compare alternative risk measures. 7 See Ackermann, McEnally, and Ravenscraft (1999), Brown, Goetzmann, and Ibbotson (1999), Liang (2001), Fung and Hsieh (2000a, 2002a), Fung, Hsieh, Naik and Ramadorai (2005) and Baquero, Horst, and Verbeek (2002) for more information on biases in hedge fund databases. 8 As pointed out by Joseph Nocera in his New York Times article (Oct. 1, 2005, ‘A Skeptic Who Merits Skepticism’), even though hedge fund managers recognize that there is always the possibility of huge events that their risk management system cannot capture, the only thing worse than having a risk management system is having no risk-management system. 9 TASS is used by Fung and Hsieh (1997, 2000), Liang (2000), Lo (2001), Brown, Goetzmann, and Park (2001), Brown and Goetzmann (2003), Getmansky, Lo, and Makarov (2004), Bali, Gokcan and Liang (2006) and Fung, Hsieh, Naik and Ramadorai (2005). HFR is used by Ackermann, McEnally, and Ravenscraft (1999), Liang (2000), Bali, Gokcan and Liang (2006) and Fung, Hsieh, Naik and Ramadorai (2005). CISDM (formerly known as MAR) is used by Ackermann, McEnally, and Ravenscraft (1999), Cremers, Kritzman and Page (2005) and Fung, Hsieh, Naik and Ramadorai (2005). 10 For the details on biases in hedge fund databases, see Fung and Hsieh (2000a and 2002a). For the reasons why funds drop out of the live fund database and move to the graveyard, see Getmansky, Lo and Mei (2004), Rouah (2005) and Park (2006).

54

11 TASS classifies hedge funds into ten investment styles, which are mutually exclusive and collectively exhaustive. See the Credit Swiss First Boston/ Tremont HedgeIndex website (www.hedgeindex.com) and Hedge Fund Research, Inc. website (www.hedgefundresearch.com) for detailed description on each investment style. 12 See Liang (2004) for the difference between funds of funds, managed futures (or commodity trading advisors, CTAs) and hedge funds. 13 We find that including emerging market funds does not change our main results. 14 See Hull (1999) and Mina and Xiao (2001) for details. 15 Although we recognize the difference in non-normality across styles, we do not have enough observations to implement the portfolio level tests by style. We make a sub-sample only for live funds and defunct funds due to the data restriction. However, we adjust for the style effect in the individual fund level tests by including the style dummy variables in the cross-sectional regressions. 16 In equation (2) and (5), we multiply the original VaR number by -1 to avoid confusion. Note that standard deviation and semi-deviation are always positive while the original VaR and ES numbers are usually negative. Therefore, VaR and ES in this paper are usually positive and we always expect positive relationship between risk and expected return for all the risk measures tested in this study. 17 An alternative way to estimate ES_CF is to use the analytical solution provided by Giamouridis (2006). His formula is based on Gram-Charlier approximation, which is known to perform well in cases of moderate skewness within the range of -1.2 to +1.2. We test the formula empirically and find that the formula is not applicable to our data because many hedge funds in our dataset have extreme skewness. The formula provides estimates that are too large to be considered as reasonable in case of the hedge funds in our data set. Note that the Gram-Charlier series is not guaranteed to be positive and therefore is not a valid probability distribution in case of extreme skewness. Hence we do not use this approach in this paper. 18 We, however, recognize that the ‘graveyard (defunct fund database)’ may be a misnomer. Although most funds in the graveyard perform worse than the funds in the live fund database, the graveyard may contain successful hedge funds. See Park (2006) for the tail risk in graveyard funds and a survival analysis of hedge funds. 19 In Fama and French (1992), the average number of stocks per month for the size-beta portfolio varies from 11 to 177 depending on the decile. Note that the stock price data is the most transparent information in the world, and we do not expect the same level of transparency in hedge funds return data. The more funds we have in each decile, the more powerful the test could be as the errors in the individual fund data would hopefully cancel out. 20 The VaR-based risk management is adopted by the Basel Committee for Banking Supervision (BCBS). RiskMetrics group, which is a spin-off from JP Morgan, has been developing and releasing software and documents on the use of VaR as a market risk measure since 1994.

21 Although in Panel A the regression model using ES_CF as a risk measure provides the lowest adjusted R2, Panel B shows that most of the difference in the adjusted R2 disappears when we adjust for the style effect.