dynamic risk exposures in hedge funds

Computational Statistics and Data Analysis 56 (2012) 3517–3532

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Computational Statistics and Data Analysis

journal homepage: www.elsevier.com/locate/csda

Dynamic risk exposures in hedge fundsMonica Billio a,1, Mila Getmansky b,∗, Loriana Pelizzon a,2

a University of Venice, Department of Economics, Fondamenta San Giobbe 873, 30100 Venice, Italyb Isenberg School of Management, University of Massachusetts, 121 Presidents Drive, Room 308C, Amherst, MA 01003, United States

a r t i c l e i n f o

Article history:Received 21 January 2010Received in revised form 17 August 2010Accepted 28 August 2010Available online 9 September 2010

Keywords:Hedge fundsRegime-switching modelsRisk managementLiquidityFinancial crises

a b s t r a c t

A regime-switching beta model is proposed to measure dynamic risk exposures of hedgefunds to various risk factors during different market volatility conditions. Hedge fundexposures strongly depend on whether the equity market (S&P 500) is in the up, down, ortranquil regime. In the down-state of themarket,whenmarket volatility is high and returnsare very low, S&P 500, Small–Large, Credit Spread, and VIX are common risk factors formost of the hedge fund strategies. This suggests that hedge fund exposures to the market,liquidity, credit, and volatility risks change depending on market conditions, and theserisks are potentially common factors for the hedge fund industry in the down-state of themarket.

© 2010 Elsevier B.V. All rights reserved.

1. Introduction

The last decade has seen an increase in the number of hedge funds and the availability of hedge fund data bothon individual hedge funds and on hedge fund indexes. However, the recent financial crisis of 2007–2009 uncoveredvulnerabilities of the industry to many risk factors including the equity market, liquidity, volatility, and credit. Unliketraditional investments like mutual funds, hedge funds engage in dynamic strategies, use leverage, take concentrated andstate-dependent bets, bet on volatility, and have non-linear payoffs. The tremendous increase in the number of hedge fundsand the availability of hedge funddata has attracted a lot of attention in the academic literature,which has been concentratedon analyzing hedge fund styles (Fung and Hsieh (2001) and Mitchell and Pulvino (2001)), performance and risk exposures(Fung and Hsieh (1997); Brealey and Kaplanis (2001); Edwards and Caglayan (2001); Agarwal and Naik (2004); Bali et al.(2007); Gupta and Liang (2005) and Schneeweis et al. (2002)), and liquidity, systemic risk and contagion issues (Getmanskyet al. (2004); Chan et al. (2006) and Boyson et al. (forthcoming)).

The aim of this paper is to analyze time-varying and state-dependent risk exposures for various hedge fund strategiesand to obtain reliable estimates for predicted risk exposures of hedge fund returns. The innovative aspect of this paper isthat we study hedge fund risk exposures conditional on different levels of mean and volatility of the equity market riskfactor, characterized by the S&P 500. Understanding andmodeling hedge fund risk exposures is fundamental for both hedgefund investors and regulators. For investors, the knowledge of hedge fund risk exposures is essential in order to analyzerisk-adjusted hedge fund performance and to perform optimal asset allocations. Regulators are concerned about risks thatare common across different hedge fund strategies. These risks can be responsible for financial trouble in the hedge fundindustry and can act as catalysts for a spillover to other financial sectors, i.e., systemic risk. Moreover, regulators are worried

∗ Corresponding author. Tel.: +1 413 577 3308; fax: +1 413 545 3858.E-mail addresses: [email protected] (M. Billio), [email protected] (M. Getmansky), [email protected] (L. Pelizzon).

1 Tel.: +39 041 234 9170; fax: +39 041 234 9176.2 Tel.: +39 041 234 9164; fax: +39 041 234 9176.

0167-9473/$ – see front matter© 2010 Elsevier B.V. All rights reserved.doi:10.1016/j.csda.2010.08.015

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3518 M. Billio et al. / Computational Statistics and Data Analysis 56 (2012) 3517–3532

if this joint risk across hedge fund strategies is associated with a particular state or regime of the equity market. In addition,regulators are concerned about an event where all different hedge fund strategies move to a high volatility state due toliquidity or non-market related shocks. This can lead to the downfall of all hedge fund strategies at the same time andpotential spillovers to financialmarkets. The recent global financial crisis of 2007–2009 exposed new types of risks for hedgefunds such as credit, volatility, and liquidity, and currently there are discussions of potential regulations that are aimed atmore transparent risk exposures and reducing large risk exposures during volatile times.

Our approach provides a framework that can be used to address these issues and can be applied in stress-testing analysis.Specifically, we provide a framework for analyzing time-varying and state-dependent risk exposures for various hedge fundstrategies and identify common risk factors for these strategies. Hedge fundsmay exhibit time-varying and state-dependentrisk exposures for various reasons such as their use of options, or more generally dynamic trading strategies. Unlike mostmutual funds (Koski and Pontiff (1999) and Almazan et al. (2004)), hedge funds frequently trade in derivatives. Further,hedge funds are known for the ‘‘opportunistic’’ nature of their trading strategies and a significant part of their returns arisefrom taking state-contingent and volatility bets.

There is currently a limited understanding of non-linear risk factor exposures of individual hedge fund strategies. Inthe hedge fund literature, the analysis of risk exposures is based on three main approaches. The first approach is based onthe classical linear factor model applied to mutual funds. The second, introduced by Fung and Hsieh (1997), is based on apredetermined structure of risk factors (quintile analysis or extreme event analysis). The third approach is based on option-like payoffs, also called Asset-Based Style Factors (ABS Factors), introduced by Fung and Hsieh (2001) and Agarwal and Naik(2004). We add to the literature by proposing a new way of capturing dynamic risk exposures in hedge funds based onvolatility changes of the market risk factor.

In this paper, consistent with the asset pricing perspective proposed by Bekaert and Harvey (1995), we suggest analyzingthe risk exposures of hedge fund indexes with a factor model based on regime-switching volatility of the market risk factor,where non-linearities in the exposures are captured by factor loadings that are state-dependent. The regime-switchingapproach is able to identify when themarket risk factor is characterized by tranquil, down-market or up-market conditions,and the state-dependent factor loadings are able to capture the risk factor exposures of hedge funds in these differentmarketvolatility states. To our knowledge this is the first attempt to analyze hedge fund risk exposures considering that themarketrisk factor is characterized by stochastic volatility, i.e., calculating hedge fund exposures to various risk factors by explicitlyaccounting for the change in volatility of the market risk factor. This feature is relevant because hedge funds bet on marketvolatility, and factor loadings are affected by this volatility.

The importance of using regime-switchingmodels is well established in the financial economics literature and examplesare found in Bekaert and Harvey’s (1995) regime-switching asset pricing model, Ang and Bekaert’s (2002) and Guidolinand Timmermann’s (2008) regime-switching asset allocation models, and Billio and Pelizzon’s (2000; 2003) analysis of VaRcalculation, volatility spillover and contagion among markets. Moreover, regime-switching models have been successfullyapplied to constructing trading rules in equity markets (Hwang and Satchell (2007)), equity and bond markets (Brooksand Persand (2001)), and foreign exchange markets (Dueker and Neely (2004)). Chan et al. (2006) apply regime-switchingmodels to CS/Tremont hedge fund indexes to analyze the possibility of switching from a tranquil to a distressed regime inthe hedge fund industry. The implementation of the regime-switching methodology is similar in spirit to ours; however, inour paper we propose a regime-switching betamodel to measure the exposures of hedge fund index returns to various riskfactors during different market risk factor regimes. Such exposures cannot be measured with the simple regime-switchingmodel used by Chan et al. (2006) because their model does not account for market risk factor regimes.

Our approach maintains the spirit of Fung and Hsieh (1997) and Agarwal and Naik (2004), but we differ from thesestudies in the focus of our investigation. Specifically, rather than using ABS factors to capture dynamic strategies, we allowfor dynamic risk factor loadings, where factor loadings are endogenously determined. In this way we capture, with a formalmodel, the idea of Fung and Hsieh (1997) to separate factors into different quintiles based on historical performance and tryto access the exposures of hedge fund returns to factors in each of the quintiles. However, the use of quintiles implies theexogenous definition of states. Rather, we let themodel determine the states. Our analysis shows that hedge fund exposuresto risk factors are related to a mixture of strategies based on options. The framework is also flexible, as we do not need todefine a priori a strategy that hedge funds may follow, but in line with the classical Sharpe-style analysis approach (Sharpe,1992), the data highlight the dynamic exposures to risk factors. Therefore, we observe different risk factor loadings for hedgefund strategies during different market regimes.

Our analysis confirms that hedge funds change their exposures based on different market conditions. Bollen andWhaley(2009) show that allowing for switching in risk exposures is essential when analyzing hedge fund performance. We showthat the exposures change over time for all strategies, confirming the time-varying risk exposures of hedge funds. Factorloadings with respect to systematic risk factors vary in different regimes for almost all hedge fund indexes, validatingthe non-linear exposure to the market risk factor. Moreover, our framework can capture the phase-locking property(i.e. uncorrelated actions suddenly become synchronized, as defined in natural sciences) of hedge funds introduced by Chanet al. (2006). For example, we observe that for many strategies in the tranquil market regime, factor loadings are very lowor zero for some particular risk factors; however, factor loadings become very large when volatility increases. These resultssuggest that by using regime-switching models investors can identify and select hedge funds and hedge fund indexes withfavorable market exposures in each regime and in particular during market downturns. They will also be more informedabout transition probabilities between different regimes and probabilities of being exposed to several market factors.

M. Billio et al. / Computational Statistics and Data Analysis 56 (2012) 3517–3532 3519

We find that hedge fund exposures during market downturns decreased for the S&P 500 (market proxy), became largerfor the Small–Large risk factor (which may potentially capture liquidity risk in line with Acharya and Pedersen (2005)), andbecamemore negative for Credit Spread (credit risk proxy) and1VIX (volatility proxy) risk factors. This suggests thatmarket,liquidity, credit, and volatility factors are commonhedge fund risk factors during crisis periods. Robustness analysis confirmsthe economic importance of accounting for the presence of market regimes for determining hedge fund risk exposures.

The rest of the paper is organized as follows. In Section 2 we develop a theoretical framework and define a series ofbeta regime-switching models that can be used to analyze different hedge fund style indexes. Section 3 describes data andpresents results for the one-factor and multi-factor beta regime-switching models. Section 4 provides robustness checks.We compare our approach to OLS and also adjust for potential illiquidity and smoothing in the data. Section 5 concludes.

2. Theoretical framework

Linear factor models such as the capital asset pricing model (CAPM) and the arbitrage pricing theory (APT) have been thefoundation of most of the theoretical and empirical asset pricing literature. Formally, a simple one-factor model applied tohedge fund index returns could be represented as:

Rt = α + βIt + ωut , (1)where Rt is the return of a hedge-fund index in period t , It is a market factor, for example, the S&P 500 in period t , and ut isIID.

In this model, we can identify the exposure of hedge fund returns to factor I . Unfortunately this theory constrains therelation between risk factors and returns to be linear. Therefore it cannot price securities whose payoffs are nonlinearfunctions of risk factors, i.e., hedge fund returns that are characterized by the implementation of dynamic strategies. Forthis reason we propose a more flexible and complete model for capturing this feature: a regime-switching model.

AMarkov regime-switchingmodel is one in which systematic and un-systematic events may affect the output due to thepresence of discontinuous shifts in average return and volatility. The change in regime should not be regarded as predictablebut rather as a random event.

More formally, the model could be represented as:Rt = α + β(St)It + ωut , (2)

It = µ(St) + σ(St)ϵt , (3)where St is a Markov chain with n states and transition probability matrix P. ut and ϵt are independent and both normallydistributed with zero mean and unit variance.

Each state of the market index I has its ownmean and variance. Hedge fund mean returns are related to the states of themarket index and are defined by a parameter α plus a factor loading, β , on the conditional mean of the factor. Hedge fundvolatilities are also related to the states of the market index I and are defined by the factor loading, β , on the conditionalvolatility of the factor plus the volatility of the idiosyncratic risk factor ω. In both cases β could be different conditional onthe state of the risk factor I .

For example, if n = 3 (state labels are denoted as 0, 1 or 2), the model can be represented as follows:

Rt =

α + β0It + ωut if St = 0,α + β1It + ωut if St = 1,α + β2It + ωut if St = 2,

(4)

where the state variable S depends on time t , and β depends on the state variable:

β(St) =

β0 if St = 0,β1 if St = 1,β2 if St = 2,

(5)

and the Markov chain St (the regime-switching process) is described by the following transition probability matrix P:

P =

p00 p01 p02p10 p11 p12p20 p21 p22

, (6)

where pij is the transition probability of moving from regime i to regime j and with p02 = 1− p00 − p01, p12 = 1− p10 − p11,and p22 = 1 − p20 − p21. The parameters p00, p11 and p22 determine the probability of remaining in the same regime.Moreover, in all Markov states, we assume normality of the error terms and homoskedasticity within regimes. This modelallows for a change in variance of returns only in response to occasional, discrete events. Despite the fact that the state St isunobservable, it can be statistically estimated (see for example Hamilton (1989, 1990)).

Our specification is similar to the well-known ‘‘mixture of distributions’’ model. However, unlike standard mixturemodels, the regime is not independently distributed over time unless transition probabilities pij are equal to 1/n, where nis the number of states. The advantage of using a Markov chain as opposed to a ‘‘mixture of distributions’’ is that the formerallows for conditional information to be used in the forecasting process. It allows us to: (i) fit and explain the time series,(ii) capture thewell known cluster effect, underwhich high volatility is usually followed by high volatility (in the presence of


persistent regimes), (iii) generate better forecasts compared to the mixture of distributions model, since regime-switchingmodels generate a time-conditional forecast distribution rather than an unconditional forecast distribution, and (iv) providean accurate representation of the left-hand tail of the return distribution, as the regime-switching approach can account for‘‘short-lived’’ and ‘‘infrequent’’ events.

More precisely, the Markov switching model is more flexible than simply using a truncated distribution approach, as ateach time t , we have amixture of one ormore normal distributions, and thismixture changes every time. Using the truncateddistribution will lead to a non-parametric estimation, where the down-state of the market is exogenously imposed, and itis hard to make inferences about beta forecast and conditional expectations. Instead, we use a parametric model to help usseparate the states of the world. We are able to infer time-varying risk exposures of hedge funds, make forecasts, calculatetransition probabilities from one state to another, and calculate conditional expectations.

Moreover, our formal model allows us to make dynamic forecasts. More specifically, once parameters are estimated,the likelihood of regime changes can be readily obtained, as well as forecasts of βt itself. In particular, because thek-step transition matrix of a Markov chain is given by Pk, the conditional probability of the regime St+k given date-t dataRt ≡ (Rt , Rt−1, . . . , R1) takes on a particularly simple form when the number of regimes is 2 (regime 0 and 1):

Prob(St+k = 0|Rt) = π1 + (p00 − (1 − p11))kProb(St = 0|Rt) − π1

, (7)

π1 ≡(1 − p11)

(2 − p00 − p11), (8)

where Prob (St = 0|Rt) is the probability that the date-t regime is 0 given the historical data up to and including date t(this is the filtered probability and is a by-product of the maximum-likelihood estimation procedure). More generally, theconditional probability of the regime St+k given date-t data is:

Prob(St+k = 0|Rt) = Pk′at , (9)

at =Prob(St = 0|Rt) Prob(St = 1|Rt) . . . Prob(St = n|Rt)

′. (10)

The model described in Eqs. (2) and (3) could be extended in several ways. For example, we propose a regime-switchingmodel with non-linearity in the volatility of residuals and in the intercept coefficient:

Rt = α(Zt) + β(St)It + ω(Zt)ut , (11)

It = µ(St) + σ(St)ϵt . (12)

In this model, additional non-linearities are captured by the intercept and residuals. Zt is another Markov chain whichproxies for all other non-linearities not captured by the non-linear relationship between a particular hedge fund (index)and the risk factor I . In this case we are interested in analyzing the possibility for residuals to switch from a tranquil to adistressed state and thus a two-regimemodel is assumed for Zt , which combines in a six-regime (3×2)model with St . Moreprecisely, given the idiosyncratic nature of residuals, Zt and St are assumed to be independent, and ut and ϵt are independentand both normally distributed with zero mean and unit variance.

Moreover, the model (11) allows us also to relax the assumption of homoskedasticity and normality for the idiosyncraticerror term, since at each time t , we have a mixture of normal distributions. The residuals are only homoskedastic andnormally distributed when conditioned on the Markov chain Zt . Usually more than one factor affects hedge fund returns.Our regime-switching beta model can be easily extended to a multi-factor model.

The first extension is a model in the same spirit as the model developed by Agarwal and Naik (2004) with a non-linearexposure to the S&P 500 and a linear exposure to other risk factors. More formally:

Rt = α(Zt) + β(St)It +

Kk=1

θkFkt + ω(Zt)ut , (13)

It = µ(St) + σ(St)ϵt , (14)

where θk is the linear factor loading of the hedge fund index on the k-th risk factor and Fkt is the return of the k-th risk factorat time t.

However, this model does not consider the possibility that the exposure to other risk factors could be affected by theregime that characterizes the S&P 500. To capture this feature, we propose a multi-factor beta switching model with non-linearity in residuals:

Rt = α(Zt) + β(St)It +

Kk=1

θk(St)Fkt + ω(Zt)ut , (15)

It = µ(St) + σ(St)ϵt . (16)

This model allows us to detect the exposures of hedge fund indexes to different factors conditional on the state of themarket risk factor, I . From an econometric point of view, as Hamilton (1996), among others, has pointed out, conductingproper inference in regime-switching models is particularly challenging. This challenge arises due to the fact that when


Table 1Summary statistics.

Strategy N βS&P500 Ann. meanreturn (%)

Ann. SD(%)

Min. return(%)

Med. return(%)

Max. return(%)

Skew Kurt JB p-value

Convertible bond arbitrage 189 0.16 3.92 7.20 −12.63 0.63 5.80 −2.55 17.68 0.00Dedicated short bias 189 −0.82 −4.55 16.95 −9.58 −0.83 22.31 0.74 4.56 0.00Emerging markets 189 0.53 5.28 15.69 −23.43 1.09 16.04 −0.78 7.53 0.00Equity market neutral 189 0.09 5.30 3.28 −5.63 0.43 3.62 −1.16 11.64 0.00Long/short equity 189 0.41 6.80 10.05 −11.83 0.67 12.58 −0.05 6.41 0.00Distressed 189 0.25 7.17 6.68 −12.85 0.84 4.14 −2.31 15.23 0.00Event driven MS 189 0.22 5.88 6.42 −11.92 0.61 4.30 −2.04 13.72 0.00Risk arbitrage 189 0.13 3.66 4.17 −6.55 0.34 3.38 −1.19 8.54 0.00S&P 500 189 1.00 5.89 15.66 −16.69 0.96 9.50 −0.70 4.00 0.00

This table presents summary statistics for monthly CS/Tremont hedge fund index excess returns and the S&P 500 excess returns from January 1994 toSeptember 2009. All returns are in excess of three-month Treasury Bill rates. N is the number of observations, βS&P500 is contemporaneous market beta,Ann. Mean Return is annualized mean return, and Ann. SD is annualized standard deviation. Min. Return, Med. Return, and Max. Return are minimum,median, and maximum monthly returns, respectively. The returns are in percentage terms. Skew measures skewness and Kurt measures excess kurtosis.JB p-value is p-value of the Jarque–Bera test. The JB statistic has an asymptotic chi-squared distribution with two degrees of freedom and can be used totest the null hypothesis that the data are from a normal distribution. In the Equity Market Neutral category all Madoff funds are excluded.

formulated in the natural way, testing the null hypothesis that there is a single regime (versus the alternative of, say, tworegimes) can involve a nuisance parameter identified only under the alternative, as well as a parameter on the boundary ofthe parameter space. Standard likelihood ratio (LR) tests (and related Lagrangemultiplier orWald tests) cannot be conductedin the usual manner (see Hansen (1992, 1996), Cho and White (2007) and recently Dannemann and Holzmann (2010)).

Regarding the specification of the number of regimes, procedures for the derivation of the asymptotic null distributionproposed in the literature are particularly computationally demanding. As a resultwe combined several specification/testingapproaches to define the appropriate number of regimes, as detailed in the empirical analysis below. More precisely, weconsidered the specification procedure defined by Hamilton (1996), the model selection procedure suggested by Krolzig(1997), based on ARMA representations, and a simulated likelihood ratio test.

Moreover, comparisons among different models have been performed through a goodness-of-fit measure, usingMcFadden’s (1974) pseudo-R2 approach. In this approach, the unrestricted (full model) likelihood LUR is compared to therestricted (constant only) likelihood LR as follows:

Pseudo-R2= 1 −

log LURlog LR

. (17)

The pseudo-R2 measure has also been used by Boyson et al. (forthcoming) to compare different hedge fund risk models.The ratio of the likelihoodsmeasures the level of improvementmade by the unrestrictedmodelwith respect to the restrictedone. A likelihood falls between 0 and 1, so the log of a likelihood is less than or equal to zero. If a model has a very lowlikelihood, then the log of the likelihood will have a larger magnitude than the log of a model with high likelihood. A smallratio of log likelihoods indicates that the unrestricted model has a far better fit than the restricted model. The pseudo-R2 measures an improvement of the unrestricted model with respect to the restricted model. Thus, when comparing twomodels on the same data, McFadden’s pseudo-R2 is higher for the model with the greater likelihood. However, even thoughit ranges from 0 to 1 with higher values indicating a better model fit, pseudo-R2 is not a measure of explained variability, asis the classical OLS R2.

3. Empirical analysis

3.1. Data

For the empirical analysis in this paper, we use aggregate hedge-fund index returns from the Credit Suisse/Tremontdatabase from January 1994 to September 2009. The CS/Tremont indexes are asset-weighted indexes of funds with aminimum of $10 million of assets under management, a minimum one-year track record, and current audited financialstatements. An aggregate index is computed from this universe, and 10 sub-indexes based on investment style are alsocomputed using a similar method.3 Indexes are computed and rebalanced on amonthly frequency and the universe of fundsis redefined on a quarterly basis. We use net-of-fee monthly excess return (in excess of T-Bill rate). This database accountsfor survivorship bias in hedge funds (Fung and Hsieh (2000)). Table 1 describes the sample size, β with respect to the S&P500, annualized mean, annualized standard deviation, minimum, median, maximum, skewness and kurtosis for monthlyCS/Tremont hedge-fund index returns as well as for the S&P 500.

For our empirical analysis, we evaluate the exposures of hedge fund indexes to the market index, the S&P 500; therefore,we concentrate only on hedge fund styles that either directly or indirectly have S&P 500 exposures. For example, weconcentrate on directional strategies such as Dedicated Short Bias, Long/Short Equity, and Emerging Markets as well as

3 The description of the indexes is obtained from the Credit Suisse website: http://www.hedgeindex.com.

http://www.hedgeindex.com


non-directional strategies such as Distressed, Event Driven Multi-Strategy, Equity Market Neutral, Convertible Bond, andRisk Arbitrage.4

Categories greatly differ. For example, annualized mean of excess return for the Dedicated Short Bias category is thelowest: −4.55%, and the annualized standard deviation is the highest at 16.95%. Distressed has the highest mean, 7.17%.The lowest annualized standard deviation is reported for the Equity Market Neutral strategy at 3.28% with an annualizedmean of 5.30%. Hedge fund strategies also show different third and fourthmoments. Specifically, non-directional funds suchas Event Driven Multi-Strategy, Risk Arbitrage, Distressed, Equity Market Neutral, and Convertible Bond Arbitrage all havenegative skewness and high kurtosis. The market risk factor is characterized by high annualized excess return of 5.89% andhigh standard deviation of 15.66% during our sample period. Moreover, the distribution of the market factor is far fromnormal and is characterized by negative skewness.

3.2. Beta regime-switching models

In the following sub-sections, all switching regime models have been estimated by maximum likelihood using theHamilton’s filter and the econometric software GAUSS. Because of the limited dataset, we prefer to adopt a two-stepprocedure. We first characterize the S&P 500 behavior by a switching-regime model and then, conditional on this result,we estimate our one-factor and multi-factor models.

Finally, in all our estimations we compute the robust covariance matrix estimator (often known as the sandwichestimator) to calculate the standard errors (see Huber (1981) and White (1982)). The estimator’s virtue is that it providesconsistent estimates of the covariance matrix for parameter estimates even when a parametric model fails to hold, or is noteven specified. In all tables we present the t-statistics obtained with the robust covariance matrix estimators, which allowsus to take into account a possibility that data may deviate to some extent from the specified model.

3.2.1. S&P 500 regimesTo determine the number of regimes to appropriately capture the S&P 500 return dynamics, we estimated and tested

models with a different number of regimes. In particular we estimated models with two and three regimes and consideredboth the model selection procedure suggested by Krolzig (1997) and a simulated likelihood ratio test to finalize the numberof regimes. The model selection procedure suggested by Krolzig (1997) is based on the ARMA representation theoremsfor Markov switching models in the mean. Specifically, an ARMA structure in the autocovariance function may reveal thecharacteristics of a data generating Markov switching process. In order to apply this model selection strategy to the S&P500 return data, we performed an univariate ARMA analysis. The Akaike information criterion (AIC, Akaike (1974)) has beenemployed to assist in choosing the appropriate order of the ARMA process. The AIC is defined as follows:

AIC = 2k − 2 log L, (18)where k is the number of parameters in the statistical model, and L is the maximized value of the likelihood function for theestimated model. We obtained an AIC of 1421 for the ARMA representation of the two-regime model and an AIC of 1419 forthe ARMA representation of the three-regime model. These results suggest that a three-regime model is more appropriateto capture the S&P 500 return dynamics.

We also considered a simulated likelihood ratio test. In this case we simulated the data under the null hypothesis oftwo regimes (parameters have been set to the estimated values), estimated both the two and three regime models on thesimulated data, and computed the likelihood ratio test as follows:

LRi= 2

log Li3R − log Li2R

, i = 1, 2, . . . , 3000. (19)

We obtained the simulated distribution of the likelihood ratio using 3000 simulations. The empirical likelihood ratio testis 14.37, and according to the simulation results its p-value is 0.0924. Therefore, we reject the null of two regimes at the 10%confidence level in favor of the three-regime models. Finally, we checked the two-regime specification through the testingprocedure suggested by Hamilton (1996), both performing the dynamic specification test for ARCH effects and the dynamicspecification test for validity of Markov assumptions, in their small sample version. The two-regime model fails to pass thedynamic specification test for ARCH effects with a small sample p-value of 0.000, clearly suggesting that a model able tocapture the full volatility dynamics is required. As a result a three-regime model is better able to describe S&P 500 returndynamics.

According to these results, we ultimately decided that using three regimes is optimal for our analysis. Using three regimesis also consistent with the literature that well recognizes the presence of tranquil, up-market, and down-market regions inthe returns of the equity market. In fact, Goetzmann et al. (2007) show that an optimal strategy for hedge funds might beselling out-of-the-money puts and calls, ensuring that during tranquil regimes, hedge fund managers obtain a positive cashflow, and have a large exposure in extreme events. We estimate the model for three states of the S&P 500 based on differentmeans and volatilities described in Eq. (3). The results of the estimation are shown in Table 2.

Table 2 shows that the return pattern of the S&P 500 could be easily captured with three regimes, where regime 0 hasa mean of 6.48% and a relatively low monthly volatility of 1.61%. We denote this regime as an up-market regime, which is

4 Details on the different strategies are obtained from http://www.hedgeindex.com.

http://www.hedgeindex.com


Table 2Regime-switching model for the market risk factor, S&P 500.

Mean µ (%)

Regime 0 (Up-market) µ0 Regime 1 (Tranquil) µ1 Regime 2 (Down-market) µ2

Estimate t-stat Estimate t-stat Estimate t-stat

6.48 171.04 0.94 24.78 −1.75 −46.21

Standard deviation σ (%)

Regime 0 (Up-market) σ0 Regime 1 (Tranquil) σ1 Regime 2 (Down-market) σ2

Estimate t-stat Estimate t-stat Estimate t-stat

1.61 42.44 2.31 61.04 5.04 133.08

Frequency of S&P 500 regimes from 1994–2009 (%)

Regime 0 (Up-market) Regime 1 (Tranquil) Regime 2 (Down-market)

14% 44% 42%

Transition probabilities

Regime 0 Regime 1 Regime 2

Regime 0 0.34 0.08 0.57Regime 1 0.00 0.98 0.02Regime 2 0.20 0.00 0.80

This table presents results for the regime-switching model for the market risk factor, S&P 500, labeled as I . S&P 500 returns are in excess of three-monthTreasury Bill rates. The following model is estimated: It = µ(St ) + σ(St )ϵt . The state of the market index I is described by the Markov chain St . Each stateof the market index has its own mean µ(St ) and standard deviation σ(St ). There are three regimes that are estimated: regime 0 (up-market), regime 1(tranquil), and regime 2 (down-market). The frequency of the S&P 500 regimes from January 1994 to September 2009 is calculated. The 3 × 3 matrix oftransition probabilities is estimated (Pij is the transition probability of moving from regime i to regime j). Parameters that are significant at the 10% levelare shown in bold type.

not persistent (the probability of remaining in this regime in the following month is 34%). Regime 1 has a mean statisticallydifferent from zero and equal to 0.94% and a volatility of 2.31%, and we call it a tranquil state. This is a persistent regime, andthe probability of remaining in it is 98%. The last regime, Regime 2, which is often associated with financial crises, capturesmarket downturns and has a mean of −1.75% and a volatility of 5.04%. The probability of remaining in this down-marketregime is 80%. As previously noted, in all our estimations we compute the robust covariance matrix estimator to calculatethe standard errors. For the regime-switching models the standard deviations obtained with the usual covariance matrixestimator and the robust covariance matrix estimator are similar.

Themodel estimation also allows us to inferwhen the S&P 500was in one of the three regimes for each date of the sampleusing the Hamilton filter and smoothing algorithms (Hamilton (1994)). From Fig. 1 we observe that in the first part of thesample, the S&P 500 returns are frequently characterized by the tranquil regime 1. Specifically, from July 1994 to December1996, 91.7% of time themarketwas in tranquil regime and 8.3% in themarket downturn. The period from1997 through 2003is characterized primarily by two other regimes: up-market (30.4%) and down-market (64.6%). This outcome is generatedmainly by high instability of the financial markets starting from the Asian down-market in 1997, well captured by regime2; the technology and Internet boom, well captured by regime 0; the Japanese down-market of March 2001, September 11,2001, and the market downturns of 2002 and 2003, capturedmostly by regime 2. The part of the sample from 2003 throughthe third quarter of 2007 is characterized by the tranquil regime 1 (100%). Finally, starting August 2007 the S&P 500 returnsare again almost exclusively in regime 2, themarket down-turn. This clearly captures the effects of the recent subprime crisisand the current economic downturn of 2007–2009. It is important to note that the three-regime approach does not simplyimply splitting the data sample into large negatives, large positives, or close to the mean returns. The regime-switchingapproach allows us to capture periods when the market return distribution belongs to large volatility periods characterizedby substantial return downturns, or more tranquil periods. In all these different regimes we may face positive or negativereturns.

Overall, the results confirm that during the period of January 1994 to December 2008, the S&P 500 was clearlycharacterized by three separate regimes based on different means and volatilities of the S&P 500: 44% of the sample was inthe tranquil regime, 14% was in the up-market regime, and 42% was in the down-regime.

3.2.2. One-factor modelAfter having characterized the process for the S&P 500, we analyze the exposures of different hedge fund strategies to

the different states of the S&P 500 market index. The analysis is based on the model presented in Eq. (11) and results areshown in Table 3.

We find different factor loadings with respect to the S&P 500 regimes for almost all hedge funds indexes. For most ofthe strategies, the exposure to the S&P 500 during crisis periods is smaller or negative compared to tranquil periods. Thissuggests that hedge fund managers are able to hedge market exposures, especially during financial crises. For example,


Fig. 1. Market regime probability. These figures depict the probability of being in regime 1 (tranquil market) and regime 2 (down-market) from January1994 to December 2008. The market is characterized by the S&P 500. S&P 500 returns are in excess of three-month Treasury Bill rates. The tranquil regimehas a mean of 0.94% and a volatility of 2.31%. The down-market regime captures market downturns and has a mean of −1.75% and a volatility of 5.04%.

Table 3One-factor model.

Convertible bond arb Dedicated short bias Emerging markets Equity market neutralEstimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat

α0 0.50 4.09 0.61 1.38 1.06 5.15 0.46 6.73α1 −0.72 −1.20 0.10 0.35 −0.39 −0.64 0.17 0.81β0 (S&P 500) 0.11 2.04 −0.98 −5.58 0.14 0.85 0.11 2.84β1 (S&P 500) 0.12 2.01 −1.28 −9.08 0.45 3.79 0.04 1.02β2 (S&P 500) 0.00 −0.09 −0.65 −9.12 0.45 7.65 0.05 2.93ω0 0.79 10.26 2.35 8.93 1.83 12.38 0.58 13.06ω1 3.85 6.87 3.99 13.02 5.22 11.52 1.36 7.07pZ00 0.94 0.97 0.97 0.98pZ11 0.79 0.94 0.98 0.95Pseudo-R2 0.14 0.12 0.09 0.10Log-likelihood −498.49 −652.82 −658.08 −389.78

Long/short equity Distressed Event driven multi-strategy Risk arb

Estimate t-stat Estimate t-stat Estimate t-stat Estimate t-stat

α0 0.17 1.18 0.70 6.26 0.75 6.68 0.32 4.13α1 0.00 0.00 −2.71 −2.35 −0.68 −1.26 −0.10 −0.31β0 (S&P 500) 0.51 5.40 0.13 1.90 0.10 1.46 0.11 2.06β1 (S&P 500) 0.61 7.60 0.33 5.54 0.28 4.20 0.16 3.88β2 (S&P 500) 0.27 5.47 0.14 4.03 0.18 4.94 0.07 2.77ω0 1.54 16.31 1.14 16.92 1.01 9.84 0.68 10.31ω1 5.84 4.63 2.74 4.59 2.32 8.53 1.74 6.67pZ00 0.98 0.98 0.96 0.89pZ11 0.74 0.68 0.88 0.67Pseudo-R2 0.13 0.12 0.11 0.08Log-likelihood −558.19 −495.04 −498.05 −438.24

This table presents the exposures of the CS/Tremont hedge-fund index strategies to different S&P 500 regimes. The following model is estimated:Rt = α(Zt ) + β(St )It + ω(Zt )ut . It is the market factor, S&P 500. St is the Markov Chain for the S&P 500. It is characterized by 3 states (regime 0: up-market, regime 1: tranquil and regime 2: down-market). Each state of the market index I has its own mean and variance: It = µ(St ) + σ(St )ϵt . ut andϵt are independent and both normally distributed with zero mean and unit variance; ω is volatility of the idiosyncratic risk factor, which is characterizedby the Markov Chain Zt . The Zt Markov Chain has two states (state 0: low volatility and state 1: high volatility of idiosyncratic risk factor). pz00 and pz11 aretransition probabilities of staying in state 0 (1) given state 0 (1) of the idiosyncratic risk factor. Parameters that are significant at the 10% level are shownin bold type.


Table 4Variable definitions.

Variable Definition

S&P500 Monthly return of the S&P 500 index including dividendsSmall–Large Monthly return difference between Russell 2000 and Russell 1000 indexesValue-Growth Monthly return difference between Russell 1000 Value and Growth indexesTerm spread 10-year T Bond minus 6-month LIBOR1VIX Monthly change in implied volatility based on the CBOE’s OEX options.Credit spread The difference between BAA and AAA indexes provide by Moody’sGold Monthly return using gold bullion $/Troy Oz. PriceMomentum factor Momentum factor

This table presents definitions ofmarket and other risk factors used inmulti-factormodels. All variables except1VIX and theMomentum factor are obtainedusing Datastream. 1VIX is obtained from the CBOE. The Momentum factor is obtained from Ken French’s website.

the exposure of hedge fund returns of the Long/Short Equity strategy to the S&P 500 during tranquil periods is 0.61, andis reduced by more than half to 0.27 during market down-turns, as shown in Table 3. Long/Short Equity strategy aims togo both long and short on the market during the tranquil regime. The Dedicated Short Bias strategy shows a large negativeexposure of −1.28 to the S&P 500 in tranquil times. This relationship is reduced by more than half for both the up-marketand down-market states.

The Emerging Markets strategy has a positive market exposure both when the market is characterized by the down-market state and in tranquil time. The exposure of the strategy to the S&P 500 during the up-market is not significant fromzero. The Equity Market Neutral strategy is market neutral during tranquil times, i.e., the exposure of the strategy returnsto the market is zero in tranquil times. However, the exposures are positive during up and down-market conditions. Theseresults are in line with the fact that the Market Neutral strategy can neutralize the effects of normal movements of themarket, but when the market is suddenly moving to another regime facing a phase-locking phenomenon, the exposurebecomes positive.

During both tranquil and up-market conditions, the market exposures of the Convertible Bond Arbitrage strategy arepositive. As expected, during the market downturn, the exposure is reduced and is not significant from zero. The otherthree strategies are related to the Event Driven categories. The exposures to the S&P 500 are positive in different states ofthe market for Distressed, Event Driven Multi-Strategy, and Risk Arbitrage strategies. Distressed strategy presents a largerexposure in tranquil times. The Event Driven Multi-strategy presents a positive exposure in the tranquil and down-marketregimes, and an almost zero exposure in the up-market regime. Nevertheless, other risk factors play a role as important asthe S&P 500 in characterizing the time-varying hedge fund exposures. This aspect is investigated in the next section with amulti-factor model.

3.2.3. Multi-factor modelMulti-factor model with non-linear exposures only to the S&P 500

As discussed above, other risk factors affect hedge fund index returns, and this calls for the use of a multi-factorframework. We begin with a comprehensive set of risk factors that will be candidates for each of the risk models, coveringstocks, bonds, commodities, emerging markets, momentum factor, and volatility. These factors are presented in Table 4.They are also described by Chan et al. (2006) as relevant factors to be used for each hedge fund strategy. In all our analyses,hedge fund returns, S&P 500, Gold, and the Momentum factor are used in excess of T-Bill returns.

Given the limited dataset, we use a step-wise approach to limit the final list of factors for our analysis. Employing acombination of statistical methods and empirical judgement, we use these factors to estimate risk models for the 8 hedgefund indexes. Other approaches could also be considered to specify the relevant risk factors, see for exampleMaringer (2004).

We first consider themodel presented in Eq. (13) and the results for thismodel are contained in Table 5. Herewe considerthe exposures of hedge fund strategies to the following linear factors: Small–Large, 1VIX , Credit Spread, Value-Growth,TermSpread, Gold, Momentum factor, and non-linear exposures to different states of the S&P 500.

The number of factors F selected for each risk model varies from a minimum of 1 for Equity Market Neutral andConvertible Bond Arbitrage to a maximum of 5 for the Long/Short Equity strategy, not including the S&P 500 index. Thestatistical significance of the factor loadings on the S&P 500 conditional on the different regimes is almost the same as theone obtained in the previous analysis with only the S&P 500 risk factor. This indicates that the analysis performed above isrobust to the inclusion of other factors that may affect hedge index returns.

Regarding the Small–Large factor, we observe that this factor is relevant for almost all hedge fund strategies, the onlyexception is the EquityMarket Neutral strategy. The exposure to the Small–Large factor is positive for almost all hedge fundsindexes (the only exception is the Dedicated Short Bias) suggesting that returns of these hedge indexes resemble thoseachieved by going long on small stocks and short on large stocks (as shown previously by Agarwal and Naik (2004) andChan et al. (2006)). Another potential explanation is that this factor is capturing liquidity risk as highlighted by Amihud(2002) and Acharya and Pedersen (2005). We consider this aspect later.

The hedge fund exposure to Value-Growth factor is positive for Dedicated Short Bias and Risk Arbitrage, and is negativefor the Long/Short Equity strategy. We also consider Fama and French ‘‘size’’ and ‘‘book-to-market’’ risk factors (Fama andFrench, 1993) and the results are similar. We prefer to present the results with the Small–Large and Value-Growth Russell


Table5

Multi-

factor

mod

elwith

non-

linea

rexp

osur

eson

lyto

S&P50

0.Co

nvertib

lebo

ndarbitrag

eDed

icated

shortb

ias

Emerging

marke

tsEq

uity

marke

tne

utral

Long

/sho

rteq

uity

Distressed

Even

tdrive

nmulti-

strategy

Risk

arb

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

α0

0.48

4.65

1.41

3.71

0.75

4.07

0.42

6.48

0.35

2.12

1.37

4.96

0.72

8.00

−0.03

−0.30

α1

−0.74

−1.19

1.10

0.93

0.01

0.01

0.24

0.88

0.88

2.59

0.59

1.43

−0.53

−1.36

0.37

3.42

β0(S&P50

0)0.12

2.34

−0.98

−7.02

0.14

0.94

0.12

2.94

0.43

5.03

0.10

1.48

0.15

2.83

0.08

1.72

β1(S&P50

0)0.11

2.17

−1.22

−10

.97

0.30

2.86

0.03

0.72

0.48

8.52

0.21

4.01

0.27

4.67

0.12

2.86

β2(S&P50

0)0.00

0.07

−0.66

−9.08

0.49

9.37

0.06

2.99

0.38

9.18

0.21

5.23

0.23

6.87

0.11

4.33

θ 1(Small–La

rge)

0.07

3.26

−0.50

−9.18

0.16

2.95

0.25

7.56

0.13

4.47

0.14

6.79

0.11

5.59

θ 2(1

Vix)

−0.10

−1.68

0.06

2.06

−0.04

−1.94

θ 3(Credits

prea

d)−1.04

−3.04

−0.49

−1.76

θ 4(V

alue

-Growth)

0.28

5.33

−0.07

−2.24

0.04

2.09

θ 5(Term

spread

)−0.19

−2.53

θ 6(G

old)

−0.08

−1.90

0.21

5.50

0.02

1.95

0.09

4.04

0.04

3.16

θ 7(M

omen

tum)

0.15

7.11

ω0

0.78

11.85

2.01

15.17

1.59

12.15

0.60

12.67

1.05

14.79

0.74

7.47

0.84

14.19

0.53

8.95

ω1

3.92

7.17

4.84

4.38

5.10

11.82

1.47

5.93

2.29

9.81

1.92

7.63

2.40

8.96

1.05

15.86

pZ 000.94

0.97

0.97

0.98

0.99

0.90

0.97

0.97

pZ 110.76

0.74

0.98

0.89

0.96

0.90

0.90

0.99

Pseu

do-R

20.15

0.18

0.12

0.10

0.22

0.13

0.14

0.12

Log-lik

elihoo

d−49

3.78

−60

5.87

−64

0.19

−38

8.04

−50

4.42

−48

9.69

−48

0.11

−41

8.27

Thistablepr

esen

tstheex

posu

reso

fthe

CS/Tremon

thed

ge-fun

dinde

xstrategies

totheS&

P50

0in

diffe

rent

S&P50

0regimes

andothe

rriskfactors:

Small–La

rge,

1VIX,C

reditS

prea

d,Va

lue-Gr

owth,Term

Spread

,Go

ld,and

Mom

entum

factor.T

hefollo

wingmod

elisestim

ated

:Rt=

α(Z

t)+

β(S

t)I t

+ K k=

1θ kF k

t+

ω(Z

t)u t

.I tisthemarke

tfactor,S&

P50

0an

dF k

tareothe

rriskfactors.S t

istheMarko

vCh

ainforthe

S&P50

0.I t

isch

aracterize

dby

3states

(reg

ime0:

up-m

arke

t,regime1:

tran

quilan

dregime2:

down-

marke

t).E

achstateof

themarke

tind

exIh

asits

ownmea

nan

dva

rian

ce:I

t=µ

(St)

+σ(S

t)ϵ t.u

tan

dϵ t

areinde

pend

ent

andbo

thno

rmally

distribu

tedwith

zero

mea

nan

dun

itva

rian

ce;ω

isvo

latility

oftheidiosync

ratic

risk

factor,w

hich

isch

aracterize

dby

theMarko

vCh

ainZ t.T

heZ t

Marko

vCh

ainha

stw

ostates

(state

0:low

volatility

andstate1:

high

volatility

ofidiosync

ratic

risk

factor).pz 00

andpz 11

aretran

sitio

npr

obab

ilitie

sof

stay

ingin

state0(1)g

iven

state0(1)o

fthe

idiosync

ratic

risk

factor.H

edge

fund

return

s,S&

P50

0an

dGo

ldareus

edin

exce

ssof

LIBO

Rreturn

s.Pa

rametersthat

aresign

ificant

atthe10

%leve

lare

show

nin

bold

type

.


indexes because they are investable portfolios, following Chan et al. (2006). Credit Spread is a common negative factor fortwo out of eight strategies.

1VIX is only significant for three strategies: negative for Dedicated Short Bias and Risk Arbitrage strategies and positivefor the Event Driven Multi-Strategy. This is surprising given that hedge funds take bets on volatility. There could be twocomplementary reasons for this: (1) Switching in S&P 500 regimes based on mean and volatility already captures thisexposure, and (2) Hedge funds take non-linear bets in volatility and thus the current linear 1VIX exposure does not capturethe true underlying non-linear exposure. If the first reason were the only true reason, then 1VIX will be captured by OLS;however, it is not as will be shown in the robustness analysis.Multi-factor model with non-linear exposures to all factors

Finally, we estimate the multi-factor model specified in Eq. (15) and the results are contained in Table 6. Here weconsider non-linear exposures to all factors: S&P 500, Small–Large, 1VIX , Credit Spread, Value-Growth, TermSpread, Gold,and Momentum factor. For each risk factor, we estimate three exposures that are based on the regime of the S&P 500.

We find that all strategies have exposures to the S&P 500 in at least one regime even after accounting for conditionalexposures to other risk factors, as shown by Table 6. Generally, we find that themodel that accounts for different risk factorsconditional on the state of the market is richer and captures more exposures compared to previous models. Moreover, themodel shows that factor exposures are changing conditional on the state of the market. Finally, this model captures moreof the hedge fund return variation as is evidenced by a higher pseudo-R2 for all hedge fund strategies compared to theone-factor model and the multi-factor model with non-linear exposures only to the S&P 500.

We further examine whether hedge fund managers are able to reduce hedge fund exposures to other risk factors duringfinancial market distress. We find that Small–Large is a common factor during down-market for six out of eight strategies(Convertible Bond Arbitrage, Dedicated Short Bias, Emerging Markets, Long/Short Equity, Event Driven Multi-Strategy, andRisk Arbitrage) and for five out of eight it has the same sign (note that for Dedicated Short Bias the exposure to Small–Large ispositive: this strategymakesmoney in thedown-state of themarket; therefore, the shock in the down-statewill be beneficialfor this strategy). This result suggests that the Small–Large risk factor may potentially capture a common factor in the hedgefund industry. Small–Large can serve as a market liquidity proxy (Amihud (2002) and Acharya and Pedersen (2005)). Smallstocks have greater sensitivity to market illiquidity than large stocks, meaning that they have greater liquidity risk. Wefind that liquidity is highly relevant for hedge funds. This result is in line with the potential interpretation of Acharya andSchaefer (2006) that the ‘‘illiquidity’’ prices in capital markets exhibit different regimes. Specifically, in a tranquil regime,hedge funds are well capitalized and liquidity effects are minimal. However, in the ‘‘illiquidity’’ regime usually related tocrises, hedge funds are close to their collateral constraints and there is ‘‘cash-in-the-market’’ pricing (Allen and Gale (1994,1998)).

Moreover, we find that another common risk factor for hedge funds is Credit Spread, especially the effect of theCredit Spread in the negative states of themarket. For Dedicated Short Bias, Equity Market Neutral, and Distressed strategiesthe impact of the Credit Spread in the down-market regime on hedge fund index returns is negative. For Dedicated ShortBias and Distressed strategies the exposures during tranquil times to the Credit Spread are zero, but become negative duringmarket down-turns. Credit Spread variable is a proxy for credit risk (Longstaff et al. (2005)) and funding liquidity risk (Boysonet al. (forthcoming) and Brunnermeier (2009)). In times of uncertainty the rate on low-credit illiquid investments such asBAA corporate bonds increases. At the same time, the demand for high-credit liquid investments such as AAA corporatebonds increases, leading to the increase in Credit Spread. Adverse shocks to funding liquidity accompanied by an increasein Credit Spreads lead to an increase in margins, de-leveraging and margin calls, causing the unwinding of illiquid positions,generating further losses and margin calls, and finally culminating in hedge funds’ collapse. During crisis periods, hedgefunds are faced with sudden liquidation and margin calls (Khandani and Lo (2007)). The credit risk in the down-state of themarket is the most important risk factor that should be controlled by regulators.

Also, 1VIX is a common risk factor for the hedge fund industry. Five out of eight strategies show a negative exposureto this variable in the down-market. The comparison with the results in Table 5 underlies the importance of measuringhedge fund exposures to VIX conditional on the S&P 500. Higher volatility is often associated with lower liquidity, higherCredit Spreads, higher correlations, and ‘‘flights to quality’’ (Bondarenko (2007) and Brunnermeier and Pedersen (2009)).After observing sharp price drops due to an increase in volatility, prime brokers are likely to increase margins and financiersmight be reluctant to roll over short-term asset-backed commercial paper. Volatility also tends to ‘‘spill over’’ across assetsand regions. During crisis periods, an increase in volatility is more likely to lead to hedge fund losses compared to other timeperiods (tranquil or up-market). In fact, we found that this result is not observed for either up-market or tranquil periods.

1VIX is a variable that needs to be interpreted jointly with different regimes of the S&P 500. For the Convertible BondArbitrage strategy, the effect of 1VIX is negative in crises markets (−0.10). The relationship between a convertible bondprice and stock price is concave when stock price is low (down-market) and highly convex when the stock price is high (up-market). Therefore, in the up-market, we expect change in volatility to attribute to additional returns of the strategy, and indown-markets, the change in volatility negatively affects the returns of the strategy. For Risk Arbitrage, the exposure to1VIXis positive and significant during tranquil periods (0.06), but negative during down-market periods (−0.07). Risk Arbitragestrategy is concerned with the success of a merger, and increase in volatility in down-times often signals an increase inthe probability of failure. The same applies to the Distressed strategies. For most of the strategies considered, exposuresto the 1VIX have opposite signs for down and up markets. In conclusion, considering only linear factor exposures willunderestimate true non-linear state-dependent exposures.


Table6

Multi-

factor

regime-sw

itching

mod

el.

Conv

ertib

lebo

ndarbitrag

eDed

icated

short

bias

Emerging

marke

tsEq

uity

marke

tne

utral

Long

/sho

rteq

uity

Distressed

Even

tdrive

nmulti-

strategy

Risk

arb

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

α0

1.01

5.75

0.87

1.68

0.33

0.62

0.90

5.14

−0.20

−0.59

0.97

3.67

0.67

2.23

0.58

2.62

α1

−0.33

−0.82

1.64

1.07

−0.79

−1.12

0.53

1.95

−0.02

−0.04

1.35

4.24

−0.48

−0.82

0.36

1.40

β0(S&P50

0)0.00

−0.05

−1.24

−2.87

0.14

0.44

0.41

3.02

1.24

4.72

−0.35

−1.49

0.37

2.44

−0.10

−0.48

β1(S&P50

0)0.09

2.40

−1.18

−8.85

0.26

2.16

0.08

1.81

0.56

8.89

0.22

4.33

0.26

4.00

0.15

3.96

β2(S&P50

0)−0.07

−2.41

−0.71

−8.34

0.32

3.71

0.02

0.55

0.32

5.17

0.19

4.31

0.16

3.97

0.10

3.32

θ 1,0(Small–La

rge)

−0.04

−0.70

−0.27

−0.98

0.73

2.82

0.01

0.11

0.85

4.75

0.37

3.79

0.15

1.85

0.07

0.67

θ 1,1(Small–La

rge)

0.02

0.56

−0.77

−6.70

0.09

0.99

0.01

0.37

0.28

4.68

0.12

3.03

0.12

2.34

0.10

2.71

θ 1,2(Small–La

rge)

0.10

4.47

−0.34

−4.72

0.28

3.64

−0.01

−0.62

0.25

5.82

0.01

0.46

0.14

4.80

0.14

6.11

θ 2,0(1

Vix)

0.23

3.43

−0.04

−0.16

0.42

1.82

0.13

2.13

0.30

2.18

0.04

0.27

0.34

3.96

0.06

0.42

θ 2,1(1

Vix)

0.00

−0.04

−0.15

−1.13

0.17

1.49

0.06

1.50

0.09

1.42

0.07

1.37

0.07

1.03

0.06

1.76

θ 2,2(1

Vix)

−0.10

−3.19

−0.13

−1.49

−0.26

−2.93

−0.06

−2.21

−0.02

−0.35

−0.09

−1.99

−0.04

−0.96

−0.07

−1.82

θ 3,0(Credits

prea

d)0.78

1.18

1.01

0.34

2.54

0.86

−1.81

−2.39

−0.33

−0.16

2.24

1.26

−0.49

−0.48

2.50

2.18

θ 3,1(Credits

prea

d)−0.50

−2.07

−0.30

−0.40

0.80

1.05

−0.72

−3.12

0.70

1.55

−0.32

−0.95

0.13

0.31

−0.49

−1.59

θ 3,2(Credits

prea

d)−0.06

−0.37

−0.92

−2.54

0.24

0.71

−0.32

−1.92

0.65

1.55

−0.89

−3.50

−0.05

−0.18

0.33

1.46

θ 4,0(V

alue

-Growth)

−0.10

−0.38

0.15

2.11

0.43

2.50

0.23

2.36

θ 4,1(V

alue

-Growth)

0.33

2.00

0.08

1.82

0.16

1.77

−0.07

−1.31

θ 4,2(V

alue

-Growth)

0.29

4.64

−0.01

−0.68

−0.13

−3.23

0.03

1.21

θ 5,0(Term

spread

)−1.09

−1.61

−0.20

−0.43

θ 5,1(Term

spread

)−0.16

−1.55

0.01

0.23

θ 5,2(Term

spread

)−0.30

−1.80

−0.30

−2.81

θ 6,0(G

old)

−0.04

−0.51

−0.12

−0.37

−1.85

−5.29

−0.07

−0.91

−0.11

−0.57

−0.35

−1.72

−0.12

−1.34

θ 6,1(G

old)

0.02

0.69

−0.05

−0.76

0.21

4.30

0.04

2.45

0.06

1.47

0.02

0.84

0.04

1.66

θ 6,2(G

old)

0.01

0.68

−0.10

−1.56

0.19

2.70

0.00

0.25

0.09

2.18

0.08

2.82

0.07

3.36

θ 7,0(M

omen

tum)

−0.07

−2.31

0.13

0.94

0.29

2.82

θ 7,1(M

omen

tum)

0.04

1.14

0.09

1.01

0.03

0.57

θ 7,2(M

omen

tum)

−0.03

−2.12

−0.01

−0.13

0.15

5.56

ω0

0.44

10.14

2.05

17.12

1.08

5.95

0.52

15.18

0.94

14.13

0.52

9.64

0.81

13.79

0.39

4.96

ω1

2.79

11.01

5.07

4.12

4.86

11.85

1.33

7.50

2.30

8.13

1.66

15.30

2.23

9.07

1.18

9.02

pZ 000.86

0.98

0.85

0.98

0.99

0.95

0.96

0.79

pZ 110.78

0.76

0.86

0.96

0.95

0.98

0.91

0.82

Pseu

do-R

20.17

0.19

0.13

0.13

0.25

0.15

0.15

0.15

Log-lik

elihoo

d−48

0.87

−60

0.77

−63

3.86

−37

4.69

−48

4.45

−48

0.74

−47

2.99

−40

5.40

Thistablepr

esen

tstheno

nlinea

rexp

osur

esof

theCS

/Tremon

thed

gefund

inde

xstrategies

totheS&

P50

0,Sm

all–La

rge,

1VIX,C

reditSp

read

,Value

-Gro

wth,T

ermSp

read

,Gold,

andMom

entum

factor

ford

ifferen

tS&

P50

0regimes.The

follo

wingmod

elisestim

ated

:Ri,t

=αi(Z t

)+

βi(S t

)It+

K k=1θ ik(S t

)Fkt

+ω

i(Z i

,t)u

i,t.I tisthemarke

tfactor,S&

P50

0,F k

tareothe

rriskfactors,an

dω

i(Z i

,t)isthevo

latility

oftheidiosync

ratic

risk

factor.Z

i,tis

aMarko

vch

ainwith

2states

(0=

low

idiosync

ratic

volatility

statean

d1

=high

idiosync

ratic

volatility

state)

Pz 00(P

z 11)isthetran

sitio

npr

obab

ility

ofstay

ingin

thelow

(high)-idios

yncratic

volatility

state.

Regime0:

up-m

arke

t,regime1:

tran

quil,

andregime2:

down-

marke

t.Pa

rametersthat

aresign

ificant

atthe10

%leve

lare

show

nin

bold

type

.


It is important to underline that our results may suffer due to data limitation. However, we still find convincing evidencethat factor exposures are different for various factors conditional on the state of the market and for different hedge fundindexes. Moreover, the model shows that factor exposures are changing conditional on the volatility of the market riskfactor. This confirms our initial hypothesis that the exposures to different risk factors are time-varying and conditional onthe state of the market risk factor. Indeed, for many factors we observe that the risk exposure is zero during tranquil times,and suddenly becomes positive or of opposite sign during market downturns characterized by high volatility.

4. Robustness analysis

4.1. Comparison with OLS regression

In amulti-factor setting,we consider themodel presented in Eq. (13). The naturalway to test the regime-switchingmodelis to compare its results to those obtained using OLS regression. The results for the OLS regression are presented in Table 7.They are consistent, meaning, that factor loadings have the same sign in bothmodels; however, the regime-switchingmodelis clearly superior based on pseudo-R2 metric. As already underlined, it is important to note that a pseudo-R2 only hasmeaning when compared to another pseudo-R2 of the same type, on the same data, and predicting the same outcome. Inthis situation, a higher pseudo-R2 indicates which model is preferable.

For each hedge fund index, pseudo-R2 is larger for regime-switching models specified in Eqs. (13) and (15) comparedto OLS models. Moreover, several estimates that are significant in the regime-switching model are not significant for theOLS model. The OLS model is missing several factor exposures and does not take into account time-variability of risk factorsbased on market conditions.

4.2. Data smoothing and illiquidity effects

As shown by Getmansky et al. (2004), observed hedge fund returns are biased by performance smoothing and illiquidity,leading to autocorrelation of hedge fund returns on a monthly basis. Following the approach of Getmansky et al. (2004), wede-smooth returns using the following procedure:

Denote by Rt the true economic return of a hedge fund in period t , and let Rt satisfy the following single linear factormodel:

Rt = µ + βΛt + ϵt , E[Λt ] = E[ϵt ] = 0, ϵt , Λt ∼ IID (20)

Var[Rt ] ≡ σ 2. (21)True returns represent the flow of information that would determine the equilibrium value of the fund’s securities in africtionless market. However, true economic returns are not observed. Instead, Ro

t denotes the reported or observed returnin period t , and let

Rot = θ0Rt + θ1Rt−1 + · · · + θkRt−k, (22)

θj ∈ [0, 1], j = 0, . . . , k, (23)

1 = θ0 + θ1 + · · · + θk, (24)which is a weighted average of the fund’s true returns over the most recent k+1 periods, including the current period.Similar to the Getmansky et al. (2004) model, we estimate the MA(2) model where k = 2 using the maximum likelihoodmethod.

In line with this approach we determine Rt , i.e., ‘‘real returns’’ and estimate our models on the real returns. The resultsshow that indeed there is evidence of data smoothing, but the estimated exposures to the different factors conditional onthe states of the market are largely unaffected by the smoothing phenomenon. Results are available on request.

5. Conclusion

In this paper we characterize the exposures of hedge fund indexes to risk factors using regime-switching beta models.This approach allows us to capture common risk factors and analyze time-varying risk exposures and the phase-lockingphenomenon for hedge funds. In particular, the changes in hedge fund exposures to various risk factors explicitly accountfor the changes in volatility of the equity market risk factor.

We have two main results. First, hedge funds exhibit significant non-linear exposures not only to the equity marketrisk factor, but also to liquidity risk factor, commodities, volatility, credit and term spreads. In particular, we show thatexposures can be strongly different in the down-market and up-market regimes compared to tranquil times, suggestingthat risk exposures of hedge funds in the down-market regimes are quite different than those faced during tranquil regimes.Moreover, many risk factor exposures can only be captured with the regime-switching analysis because for many riskfactors the exposures exhibit a phase-locking characteristic where in the tranquil regime the exposure is zero and inmarketdownturns it is statistically different from zero or there is a change in the sign of the exposure.

Second, we find that the S&P 500, Credit Spread, Small–Large, and 1VIX are common hedge fund factors in the down-state of the market, suggesting that these factors are important in accessing hedge fund risk especially in the down-state of


Table7

Multi-

factor

OLS

mod

el. Co

nvertib

lebo

ndarbitrag

eDed

icated

short

bias

Emerging

marke

tsEq

uity

marke

tne

utral

Long

/sho

rteq

uity

Distressed

Even

tdrive

nmulti-

strategy

Risk

arb

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

Estim

ate

t-stat

α0

0.75

2.31

1.38

3.20

−0.37

−0.59

0.43

3.08

0.00

0.00

1.10

4.57

0.54

2.09

0.35

2.12

β(S&P50

0)0.07

1.56

−0.91

−14

.90

0.57

6.52

0.12

5.86

0.50

13.15

0.22

6.57

0.20

5.56

0.10

4.39

θ 1(Small–La

rge)

0.05

1.17

−0.46

−7.92

0.25

3.11

−0.01

−0.37

0.30

8.79

0.12

3.69

0.13

4.01

0.10

4.79

θ 2(dVix)

−0.09

−1.93

−0.17

−2.75

−0.03

−0.35

0.02

1.12

0.04

1.17

−0.03

−0.85

−0.04

−1.25

−0.06

−2.62

θ 3(Credits

prea

d)−0.48

−1.60

−1.34

−3.40

0.46

0.80

−0.06

−0.48

0.60

2.34

−0.66

−2.97

−0.17

−0.73

0.06

0.40

θ 4(V

alue

-Growth)

0.21

3.61

0.03

1.77

−0.11

−3.00

0.04

1.86

θ 5(Term

spread

)−0.23

−2.22

−0.10

−1.65

θ 6(G

old)

0.08

2.51

−0.09

−1.98

0.16

2.61

0.03

2.37

0.08

3.10

0.05

2.04

0.04

2.48

θ 7(M

omen

tum)

−0.05

−2.06

0.09

1.88

0.18

8.04

0.03

1.70

σ0

1.86

5.62

2.56

3.86

3.57

1.96

0.82

7.03

1.49

6.52

1.44

6.64

1.47

6.57

0.93

7.13

Adjusted

-R2

0.19

0.72

0.36

0.36

0.73

0.48

0.39

0.41

Pseu

do-R

20.03

0.16

0.06

0.06

0.19

0.10

0.08

0.10

Log-lik

elihoo

d55

9.39

619.16

682.30

405.08

517.41

510.57

514.98

428.09

Thistablepr

esen

tstheresu

ltsfort

heOLS

regression

oftheCS

/Tremon

thed

ge-fun

dinde

xstrategies

onS&

P50

0,Sm

all–La

rge,

1VIX,C

reditS

prea

d,Va

lue-Gr

owth,Term

Spread

,Gold,

andMom

entum

factor.H

edge

fund

return

s,S&

P50

0an

dGo

ldareus

edin

exce

ssof

LIBO

Rreturn

s.Pa

rametersthat

aresign

ificant

atthe10

%leve

lare

show

nin

bold

type

.


the market. Specifically, in the market downturn regime six out of eight strategies are all significantly exposed to theSmall–Large risk factor (this represents 84% of hedge funds in the sample). This feature is important in the light of the resultsof Acharya and Pedersen (2005) that the size risk factor is capturing liquidity risk.Moreover, considering that liquidity shocksare highly episodic and tend to be preceded by or associated with large and negative asset return shocks, our results indeedsuggest that liquidity is a risk factor for hedge fund returns and needs further investigation.

Themain goal of the paper is to analyze hedge fund risk.We provide a robust framework that can be used by investors andregulators to assess this risk. The framework can be used by investors for portfolio allocation and risk assessment, portfolioconstruction, risk management, and benchmark design. Regulators, on the other hand, can use this framework for stresstesting and endogenously consider the effects of switching volatility of the market factor on the overall risk hedge fundindustry may face. Regulators are particularly interested in identifying common risk factors, especially in the down-state ofthemarket. Our analysis suggests that they can use this framework to create ‘‘earlywarning indicators’’ for potential changesin risk exposures of hedge funds and increases in volatility of the hedge fund industry. This can help address regulators’concerns regarding the potential risk hedge funds may pose for stability of financial markets.

Acknowledgements

A co-editor and three anonymous referees provided very helpful comments for which we are most grateful. We alsothank Vikas Agarwal, Ben Branch, Stephen Brown, David Hsieh,Will Goetzmann, Ravi Jagannathan, Nikunj Kapadia, HosseinKazemi, Martin Lettau, Bing Liang, Andrew Lo, Narayan Naik, Geert Rouwenhorst, Tom Schneeweis, Matthew Spiegel,Heather Tookes, Marno Verbeek, and seminar participants at the NYU Stern School of Business, Cornell, Rutgers BusinessSchool, Brandeis University, and European Finance Association Conference (EFA), for valuable comments and suggestions.Finally, we thank Lorenzo Frattarolo and Mirco Rubin for excellent research assistance.

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dynamic risk exposures in hedge funds

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