The response of hedge fund tail risk to macroeconomic shocks: A nonlinear VAR approach

Journal Pre-proof The response of hedge fund tail risk to macroeconomic shocks: A nonlinear VAR approach Greg N. Gregoriou, François-Éric Racicot, Raymond Théoret PII:

S0264-9993(18)31613-4

DOI:

https://doi.org/10.1016/j.econmod.2020.02.025

Reference:

ECMODE 5167

To appear in:

Economic Modelling

Received Date: 6 November 2018 Revised Date:

6 January 2020

Accepted Date: 12 February 2020

Please cite this article as: Gregoriou, G.N., Racicot, Franç.-É., Théoret, R., The response of hedge fund tail risk to macroeconomic shocks: A nonlinear VAR approach, Economic Modelling (2020), doi: https:// doi.org/10.1016/j.econmod.2020.02.025. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. © 2020 Published by Elsevier B.V.

The response of hedge fund tail risk to macroeconomic shocks: A nonlinear VAR approach† (This version January 2020)

Greg N. Gregorioua, François-Éric Racicotb*, Raymond Théoretc a

Formerly Professor at State University of New York (Plattsburgh), 101 Broad Street, Plattsburgh, NY. Telfer School of Management, University of Ottawa, 55 Laurier Avenue East, Ottawa, Ontario; Affiliate Research Fellow, IPAG Business School, Paris, France. E-mail: [email protected]. Tel. number: (613) 562-5800 ext. 4757. c Université du Québec (Montréal), École des sciences de la gestion, 315 est Ste-Catherine, R2915, Montréal, Québec; Chaire d’information financière et organisationnelle, ESG-UQAM; Université du Québec (Outaouais). E-mail: [email protected]. Tel. number: (514) 987-3000 ext. 4417. b

______________________________________________________________________________________________________

Abstract We study how downside risk taken by hedge fund strategies responds to macroeconomic and financial shocks. Using new empirical measures of systematic tail risk, we find that the impulse response functions of strategies’ multi-moment risk display an asymmetric behavior, being much more significant and nonlinear during the subprime crisis than during the recent expansion period. Our results indicate that managers of hedge fund strategies seek to monitor their tail risk during crises, in the sense that their portfolios seem to behave as puts, which are kurtosis reducers in times of turmoil. However, while some strategies (e.g., futures) succeed quite well in controlling their tail risk during crises, others (e.g., equity market neutral) have difficulties to do so. This is an important result for institutional investors who think that hedge funds are “hedged” and thus immunized against adverse macroeconomic shocks—especially market volatility shocks. Keywords: Hedge fund; Tail risk; Local projection; Business cycle; Systemic risk. JEL classification: C13; C58; G11; G23.

______________________________________ *Corresponding author.

† Aknowledgements. We thank Professor Alain Guay and Brian Lucey for their technical suggestions and comments. We also benefited from the skills of Vincent Beauséjour who assisted us in the writing of the MATLAB® codes of the nonlinear projection algorithm. We acknowledge the financial support of the Social Sciences and Humanities Research Council (SSHRC)—grants no. 435-2015-0106 and 435-2019-0078—which helped us to finance the works of our research assistants. We thank the seminar participants of the November 2018 Southern Finance Association meeting held in Asheville, NC. Financial support from the IPAG Business School is gratefully acknowledged.

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The response of hedge fund tail risk to macroeconomic shocks: A nonlinear VAR approach Abstract We study how downside risk taken by hedge fund strategies responds to macroeconomic and financial shocks. Using new empirical measures of systematic tail risk, we find that the impulse response functions of strategies’ multimoment risk display an asymmetric behavior, being much more significant and nonlinear during the subprime crisis than during the recent expansion period. Our results indicate that managers of hedge fund strategies seek to monitor their tail risk during crises, in the sense that their portfolios seem to behave as puts, which are kurtosis reducers in times of turmoil. However, while some strategies (e.g., futures) succeed quite well in controlling their tail risk during crises, others (e.g., equity market neutral) have difficulties to do so. This is an important result for institutional investors who think that hedge funds are “hedged” and thus immunized against adverse macroeconomic shocks— especially market volatility shocks.

1. Introduction Prudent investors dislike downside risk (Geiss et al., 1980; Eeckhoudt et al., 2005). However, it is well-known that the standard deviation of stock returns or a stock beta does not adequately measure this kind of risk1. To really account for downside risk, we must rely on measures of tail risk2. Moreover, the indicators of financial risk ought to be analyzed in a dynamic setting. Indeed, risk-averse investors dislike a transfer of risk (or probability mass) from wealthy or good states—e.g., economic expansions—to poorer or bad states—e.g., crises or recessions3. These transfers are at the roots of the stochastic dominance theory (Hanoch and Levy, 1969; Rothchild and Stiglitz, 1970, 1971). This is an obvious source of asymmetry in the risk preferences of investors according to the phase of the business cycle.

In finance, portfolio risk is usually measured by its return standard deviation. This measure was introduced by Markowitz (1952) who recently reiterated his faith in this gauge of risk (Markowitz, 2012; Kritzman and Markowitz, 2017). Markowitz’s theory is developed for portfolios that generate linear payoffs with respect to investors’ final wealth. However, for portfolios generating nonlinear payoffs with respect to their underlying assets—e.g., portfolios of structured products—the standard deviation stands as an imperfect measure of risk. More precisely, investing in traditional securities entails a linear payoff function—i.e., a linear risk sharing rule. This payoff function may be written as: g ( x ) = w1 + α x , where w1 is the investor’s risk-free wealth at time 1 and x is the market’s portfolio payoff or any other relevant random variable, whereas α is the investor’s exposure to x, a decision variable (Gollier et al., 1995; Franke et al., 1998). Introducing Arrow-Debreu contingent claims—i.e., derivatives—makes the payoff function 1

non-linear. This function may be written as: g ( x ) = w1 + α x + θ ω ( x )  , where θ ω ( x ) is the investor’s payoff for non-linear

claims. The payoff g(x) is convex with respect to x for investors who buy insurance—i.e., who buy put options, among others. This function is concave for agents who sell insurance (Franke et al, 1998). Agents who buy insurance may be agents with high background risk—i.e., non-insurable risk like risk related to labor income (Franke et al., 1998). Or they may be investors with average return expectations but whose risk tolerance increases more quickly than average with wealth, like pension or endowment funds (Leland, 1980; Gollier, 2001). Finally, investors buying insurance may be investors whose expectations of returns are more optimistic than average and who are in search of high α like hedge funds (Leland, 1980; Gollier, 2001). 2 i.e., co-skewness and co-kurtosis in this paper. 3 i.e., risk-averse investors prefer a smooth consumption plan over time (Cochrane, 2005).

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Institutional investors—like pension funds and university endowments—are particularly concerned about tail risk in the occurrence of an adverse macroeconomic event. By loose arguments to the effect that they are somehow “hedged” and adopt market neutral strategies, hedge funds were made attractive to institutional investors who were concerned by such an event. Hedge funds could act as portfolio insurers for investors during crises if their payoffs were similar to put options. This would be the case if hedge funds were successfully timing tail risk. As shown by Mitchel and Pulvino (2001), the returns of many hedge fund strategies usually behave as portfolios of short puts. During crises, if they act as portfolio insurers of their investors, managers of these strategies should transform their portfolios in long puts (e.g., Agarwal et al., 2017b). As shown by Bali et al. (2017), puts behave as kurtosis reducers during crises like the Lehman Brothers bankruptcy in September 2008. Actually, trailing tail risk is ideally associated with an increase in skewness and a reduction in kurtosis during crises, a target difficult to attain according the Wilkins’ lower bound (1944)4. In fact, the kurtosis of the distribution of put returns in a risk neutral universe decreases during a crisis but its skewness is also reduced (Bali et al., 2017). However, kurtosis weights much more in the utility function of an investor during crises—i.e., when a high kurtosis may generate a substantial negative payout. Previous studies have found many asymmetries or nonlinearities in the behavior of financial institutions dependent on the stance of the business cycle (e.g., Marcucci and Quagliariello, 2006; Jawadi and Khanniche, 2012; Puddu, 2012; Bali et al., 2014; Calmès and Théoret, 2014; Namvar et al., 2016; Racicot and Théoret, 2016; Bali et al., 2017). These nonlinearities arise because endogenous risk5 is high during crises while it is very low in normal times (Brunnermeier and Sannikov, 2014). Endogenous risk is driven by the illiquidity of financial markets —a friction—and is amplified by the leverage of investors—especially financial institutions (Shleifer and Vishny, 2010; Brunnermeier and Sannikov, 2014). We must thus account for nonlinearities when building an empirical model of risk management because ignoring them might result in a serious underestimation of the response of hedge funds risk to macroeconomic and financial shocks (Marcucci and Quagliariello, 2006; Puddu, 2012). For instance, this may explain why several studies on market timing in the hedge fund industry obtain non-significant results (Namvar et al., 2016). In this study, we analyze how hedge funds track their risk—especially tail risk— over the business cycle. More precisely, we aim at examining how hedge fund strategies monitor

i.e., kurtosis ≥ 1 + skewness 2 . In contrast to exogenous risk which is associated with changes in fundamentals, endogenous risk—i.e., risk self-generated by the economic system—refers to changes in asset prices attributable to portfolio adjustments in response to constraints or precautionary motives (Brunnermeier and Sannikov, 2014). 4 5

3

their downside risk over the cycle to meet the preferences of their investors. To do so, we rely on a nonlinear local projection method (Jordà 2004, 2005) for analyzing the response of multimoment risk of hedge fund strategies—as measured by beta, co-skewness and co-kurtosis—to macroeconomic and financial shocks. This nonlinear specification allows us accounting for the nonlinearities driving the interactions between the explanatory variables. Moreover, being based on a direct forecast setting, the local projection approach is preferable to the standard VAR when the data generating process is nonlinear (Jordà, 2005; Puddu, 2012). We perform this local projection first on a recent expansion period—i.e., from 2013 to 2016—and then over the subprime crisis—i.e., the most important crisis since the end of the Second World War—in order to decrypt the asymmetric pattern of hedge fund behavior towards risk in expansion and recession (or crisis). Our nonlinear VAR device is particularly relevant owing to the nonlinear behavior of the economic system during crises and to the endogeneity of risk. Indeed, all variables are endogenous in a VAR approach. To the best of our knowledge, we are the first to study market-timing in the hedge industry by relying on a nonlinear VAR method. Similarly to Bali et al. (2014) and Racicot and Théoret (2016) who focus on the impact of macroeconomic uncertainty on hedge fund returns and betas, our main contribution is to show that this kind of uncertainty has also an important role to play in the dynamics of risk associated with the higher moments of strategies’ returns—i.e., co-skewness and co-kurtosis. More precisely, hedge funds tend to track beta and higher moment risk either to hedge it (reduce it)— especially in recession—or to capture positive payoffs associated with risk premia—particularly in expansion. Our study further investigates the market timing abilities of hedge funds regarding their risk level—especially higher moment risk—in a nonlinear macroeconomic dynamic setting. Some researchers find that hedge funds tend to sell insurance in good times but adopt the opposite behavior in times of turmoil by buying protection to reduce their tail risk (Agarwal et al., 2017b; Bali et al., 2017)6. However, these studies are performed using cross-sectional approaches engineered to analyze the microstructure of tail-risk timing in the hedge fund industry while our paper presents a business cycle study of the risk dimensions of this industry7. We find that the response of hedge funds is actually asymmetric dependent on the stance of the business cycle—recession or expansion. During recessions, the reaction of hedge fund risk to shocks, regardless of their provenance, is less smooth, stronger and more significant than during expansion. More importantly, we find that risk is more difficult to monitor or hedge

6 i.e., the payoffs of many hedge fund strategies are similar to those of a short put in expansion and to those of a (long) put in recession. There is thus an obvious asymmetry or change in regime regarding their behavior. 7 Note that Agarwal et al. (2017b) have correlated their tail risk measure with the global uncertainty indicator developed by Bali et al. (2014), which is the first principal component of well-known macroeconomic and financial uncertainty indicators. They find a negative correlation. Although these authors do not discuss this result, it may be further evidence that hedge funds do track their tail risk.

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during the subprime crisis than in a normal expansion period. During crises, hedge funds may lose temporarily the control over their risk measures— especially following VIX or credit shocks to which they are particularly exposed (Gregoriou, 2004). The co-skewness of hedge fund strategies’ returns is markedly difficult to track during a crisis. Nevertheless, some strategies succeed quite well in fine-tuning their risk during a crisis. This is particularly the case for the growth strategy—which displays hedging properties with respect to market reversals (Campbell et al., 2010)—and for futures8 or short-sellers strategies owing, in large part, to their short sales. The high degree of liquidity9 of these strategies’ portfolios also fosters their capacity to respond optimally to macroeconomic and financial shocks. One of our contributions is to show that nonlinearities, and especially those related to VIX—i.e., a measure of the implied volatility of US financial markets—are crucial for the dynamics of our nonlinear local projection model10. In this respect, a comparison of the results obtained with linear and nonlinear impulse response functions (IRF) indicates that squared endogenous variables—e.g., VIX2—, which account for their volatility, are usually much more important than their levels, particularly during crises (Agarwal et al., 2017a). Another contribution of our paper is to shed more light on contagion in the hedge fund industry (e.g., Boysin et al., 2008; Dudley and Nimalendran, 2011; Jiang and Kelly, 2012). In this respect, we study the asymmetries related to systemic risk in the hedge fund industry using an underpinning developed by Beaudry et al. (2001) and transposed to the analysis of risk in the financial sector by Baum et al. (2002, 2004, 2009), Calmès and Théoret (2014), Caglayan and Xu (2016a, 2016b), and Racicot and Théoret (2016). We find that tail risk may be an important source of systemic risk for hedge funds during crises. Monitoring only the beta may then result in a serious underestimation of risk borne by hedge fund strategies11. This paper is organized as follows. Section 2 presents a survey of the literature. Section 3 provides the time-varying risk measures we use—i.e., beta, co-skewness and co-kurtosis—as well as the theoretical and empirical background, particularly the nonlinear local projection method we rely on to implement our experiments. Section 4 presents the database and the stylized facts. In Section 5, we analyze the impulse response functions of our risk measures to shocks computed with a nonlinear local projection model over the 2013-2016 expansion period and over the subprime crisis. We also compare these nonlinear impulse response functions to

This result is consistent with Asness et al. (2001) who find that the managed futures strategy displays market-timing skills. As measured by the coefficient of first-order autocorrelation of the returns of these portfolios (Getmansky et al., 2004). 10 Hasanhodzic and Lo (2007) show that using a linear asset pricing model estimated with OLS may result in spurious results for hedge funds whose payoffs are similar to those of derivatives. 11 In this respect, Agarwal and Naik (2004) argue that the mean-variance approach may underestimate portfolio losses and that this underestimation may be substantial for portfolios with low volatility. In our study, we find that the equity market neutral strategy— that displays low volatility—may be quite risky during crises when we account for higher moment risk. It is not so neutral after all (Duarte et al., 2007; Patton, 2009). See also Agarwal et al. (2017b). 8 9

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standard linear ones. Section 6 focuses on systemic risk in the hedge fund industry. As a robustness check, Section 7 applies our nonlinear VAR approach to monthly data. Section 8 concludes.

2. Literature review on the dynamics of hedge funds’ strategies Treynor and Mazuy (1966) and Henriksson and Merton (1981) opened the way to a dynamic analysis of systematic risk in portfolio construction. To account for market timing, Treynor and Mazuy (1966) introduces a quadratic term in the CAPM empirical model, i.e.,

(

)

(

R pt − R ft = α p + γ 1 Rmt − R ft + γ 2 Rmt − R ft

)

2

+ ε t , where α P is the alpha of Jensen, Rpt is

the return on a managed portfolio, Rft is the risk-free rate, Rmt is the return on the market portfolio and ε t is the innovation. A good market timing is associated with a positive sign for γ 2 . This equation is also the empirical version of the three-moment CAPM that will be formulated later (Kraus and Litzenberger, 1976). Henriksson and Merton (1981) have tested market timing by

relying

on

the

(

following

)

transformation

(

of

the

CAPM

empirical

model:

)

R pt − R ft = α p + π1 Rmt − R ft + π 2 Rmt − R ft D + ξt , where D is a binary variable which

takes the value of one if Rmt − R ft > 0 and 0 otherwise. A good market timing is associated with a

positive

sign

for

(

π2 .

)

Note

that

this

(

equation

may

be

rewritten

as:

)

R pt − R ft = α p + π1 Rmt − R ft + π 2 MAX Rmt − R ft ,0 + ξt , which shows that market timing is

also a call option on the market portfolio return (Glosten and Jagannathan, 1994; Agarwal and Naik, 2004; Stafylas et al., 2017). For Fung and Hsieh (1997, 2001, 2002, 2004), the transactions of hedge fund are essentially dynamic—i.e., option-like. Indeed, the return they deliver is nonlinear with respect to the underlying assets. Moreover, most hedge fund strategies’ managers aim at “decorrelating” their return from the financial markets when they are bearish. To introduce dynamics in the analysis of hedge fund payoffs, Mitchel and Pulvino (2001), Fung and Hsieh (1997, 2001, 2002, 2004) and Agarwal and Naik (2004) have established that many hedge fund strategies—and especially those involved in arbitrage activities less captured by the CAPM model, which may be considered as non-directional—display payoffs that are similar to those of a short put12. These studies are in line with others which also relate the payoffs of hedge fund strategies to those of derivatives or structured products (Mitchell and Pulvino, 2001; Stafylas et al., 2017, 2018). For in12 According to this view, hedge funds “generate small positive returns most of the times before incurring a substantial loss” (Agarwal et al., 2017b). However, this pattern may be considered as somewhat “reductivist” in the sense that hedge funds are often associated with high positive absolute returns (alphas) and that their payoffs are usually procyclical. Moreover, hedge funds strategies with payoffs similar to straddles—like CTA, managed futures or macro—can produce substantial positive payoffs when stock markets are volatile—especially during crises. Short-sellers also benefit from falling stock markets.

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stance, the payoffs of the trend followers and market timers—particularly managed futures or Commodities Trading Advisors (CTAs)—may be associated with the payoffs of straight straddles or lookback straddles13 (Fung and Hsieh, 1997, 2001, 2002, 2004; Stafylas et al., 2017). However, even if they consider that the payoffs of hedge funds are option-like or nonlinear (dynamic), these early analyses are quite static since they do not envision time-varying measures of market risk and tail risk. Recent studies feature such measures. We can distinguish two groups of papers: (i) those which adopt a cross-sectional approach and are more interested in the relationship between the impact of their gauges of risk on risk premia embedded in hedge fund returns (Jiang and Kelly, 2012; Bali et al., 2014; Namvar et al., 2016; Agarwal et al., 2017b); (ii) those which are more concerned with the asymmetric behavior of hedge funds dependent on the phase of the business cycle and with market timing of risk over the cycle (Jawadi and Kanniche, 2012, Racicot and Théoret, 2016, 2018, 2019; Stafylas et al., 2018). Regarding the first set of studies, Bali et al. (2014) find that hedge funds with higher time-varying uncertainty betas—i.e., exposures of hedge funds to macroeconomic and financial risk computed with rolling regressions—generate greater future returns, which suggests that macroeconomic risk is rewarded. Directional funds—i.e., funds with higher betas—support more macroeconomic risk than non-directional funds. Their results also show that fund managers have heterogeneous expectations and different reactions to changes in the economy. In the same vein, Namvar et al. (2016) find that funds in the highest systematic risk quintile generate 6% higher average returns than funds in the lowest quintile. The former funds tend to be involved in market timing, delivering more systematic risk in strong market states and less in weak market states. Jiang and Kelly (2012) develop a non-parametric time-varying tail risk measure based on the distribution of hedge fund returns. In their setting, a tail event tends to reduce returns but hedge funds more exposed to tail risk provide higher expected (average) returns. These authors also contribute to the literature on contagion in the hedge fund industry by showing that common exposures to tail risk leads to co-movements of hedge fund returns in times of distress. In line with this paper, Agarwal et al. (2017b) also build a non-parametric time-varying measure of tail risk based on return distributions and find that tail risk affects the cross-sectional variation of hedge fund returns. Their results suggest that hedge funds managed to reduce their tail risk exposure prior to the onset of the subprime crisis. The second set of papers, mainly published after the subprime crisis, have been especially devoted to the asymmetric behavior of hedge funds dependent on the phase of the business cycle (Holmes and Faff, 2008; Jawadi and Khanniche, 2012; Namvar et al., 2016; Racicot and

13 As previously argued, the payoffs of a perfect market timer who is not involved in short sales are similar to the ones of a long call. Being also involved in short sales transforms these payoffs in straddles or lookback straddles (Stafylas et al., 2017).

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Théoret, 2016, 2018, 2019; Stafylas et al., 2018; Thomson and van Vuuren, 2018). Using the Kalman filter to build their time-varying betas, Holmes and Faff (2008) show that hedge funds decrease their beta when volatility increases instead of increasing their exposure to volatility. Indeed, market volatility tends to persist and it is therefore to some extent predictable (Busse, 1999). Thomson and van Vuuren (2018) also resort to the Kalman filter but they define two kinds of beta: a long-run beta and an instantaneous time-varying beta—i.e., the beta related to market timing—which oscillates around the long-run beta according to a mean-reverting process. They find that macro and event driven funds outperform managed futures and long/short equity strategies. Relying on a Markov switching approach to investigate the asymmetric behavior of hedge funds in high and low regimes, Stafylas et al. (2018) find that hedge fund managers are concerned with minimizing their systematic risk during bad times whereas they increase their risk during good times in order to deliver high returns. According to their results, market volatility impacts more hedge fund performance than business cycle volatility. Finally, using tail risk measures based on time-varying covariances built with MGARCH procedures, Racicot and Théoret (2016, 2018, 2019) show that the behavior of hedge funds vis-à-vis risk is asymmetric dependent on the phase of the business cycle. Hedge funds are more concerned with tracking risk during recessions or crises than during expansions. While some strategies succeed in timing risk during crises—especially futures and short-sellers—others ones have difficulties in doing so—i.e., equity market neutral and fixed income.

3. Methodology 3.1 The empirical measures of hedge fund conditional (time-varying) return moments Similarly to the beta which is the keystone of the two-moment CAPM, the concepts of co-skewness and co-kurtosis originate from the four-moment CAPM, which takes the following form (Fang and Lai, 1997): E ( Ri ) − rf = β1Cov ( Rm , Ri ) + β 2 Cov ( Rm2 , Ri ) + β 3Cov ( Rm3 , Ri )

(1)

where E(Ri) is the expected value of return i; rf is the risk-free rate and Cov(.) is the operator of covariance. The unscaled beta, co-skewness and co-kurtosis of the asset being priced are defined as Cov ( Rm , Ri ) , Cov ( Rm2 , Ri ) , and Cov ( Rm3 , Ri ) , respectively. As usual, the risk associated with Ri is thus seen as co-movements between this return and the stock market return (unscaled beta), its square (unscaled co-skewness) and its cube (unscaled co-kurtosis), respectively.

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Let us provide some intuition about these measures of risk. In the classical CAPM model, the only priced risk related to an asset is the covariance of the return of this asset with the market

return

or

its

systematic

standard

deviation—i.e.,

momim = E ( Ri − Ri )( Rm − Rm )  = Cov ( Ri , Rm ) , where E(.) and Cov(.) are the operators of expected

value and covariance, respectively, and mom stands for moment. Ri is the return on asset i and

Ri , its mean value; Rm is the market return and Rm , its mean value. However, as shown by Kraus and Litzenberger (1976), Scott and Hovarth (1980) and Dittmar (2002), investors may be concerned about skewness—i.e., a measure of the asymmetry of a distribution— and kurtosis— i.e., a measure of the tails of a distribution—in addition to mean and variance. In line with systematic standard deviation, only systematic skewness and kurtosis are priced on financial markets. Consistent with the definition of systematic standard deviation, co-skewness is defined as 2 (Kraus and Litzenberger, 1976) momimm = E ( Ri − Ri )( Rm − Rm )  = Cov ( Ri , Rm2 ) , while co-kurtosis





3 is defined as momimmm = E  ( Ri − Ri )( Rm − Rm )  = Cov ( Ri , Rm3 ) .





In Eq. (1), covariances must be scaled to arrive at relative measures of risk. The wellknown definition of the beta of asset i is: betai =

Cov ( Rm , Ri )

(2)

Var ( Rm )

which is the scaling of the first Cov(.) term on the RHS of Eq.(1). The two other covariance expressions in this equation give rise to (scaled) co-skewness (systematic skewness) and cokurtosis (systematic kurtosis), respectively (Fang and Lai, 1997): co − skewnessi =

co − kurtosisi =

Cov ( Rm2 , Ri )

(3)

Var ( Rm ) 

1.5

Cov ( Rm3 , Ri ) Var ( Rm ) 

(4)

2

To make our three measures of risk time-varying, we rely on the multivariate GARCH procedure (MGARCH, Bollerslev et al., 1988). In this regard, to compute the conditional covariance measure in Eq. (2), we resort to the following simple system: Rit = γ 1 + ε t Rmt = γ 2 + ξ t

(5)

where γ 1 and γ 2 are constants and ε t and ξ t are innovations. We apply a MGARCH to system (5) to compute Cov ( Rm , Ri ) . To compute Cov ( Rm2 , Ri ) in Eq.(3) and Cov ( Rm3 , Ri ) in Eq.(4), we 9

3 substitute Rmt2 and Rmt to Rmt in system (5), respectively. We use the BEKK algorithm (Engle

and Kroner, 1995) to estimate system (5) since it is a parsimonious approach in terms of the number of parameters to estimate. We then scale these estimated conditional covariances with the conditional variance of Rmt , obtained with a simple GARCH model, according to Eq. (2), (3) and (4). Note that we do not impose any structure to the covariances in system (5) since we aim at examining how they respond to macroeconomic and financial shocks. Our risk measures being generated variables, they may lead to biases in the estimation process (Pagan, 1984, 1986; Pagan and Ullah, 1988). However, in our VAR models, all variables are considered endogenous since they are regressed on predetermined or lagged variables—i.e., on instruments—and therefore, the possible interactions between the explanatory variables and the innovations are accounted for. 3.2 The theoretical model Since we adopt a vector autoregressive (VAR) approach14, our model must be parsimonious with respect to the number of explanatory variables. Indeed, given that each variable is lagged over several periods, this quickly reduces the number of degree of freedom available when adding more variables. The main objective of our paper is to study the response of hedge fund strategies’ moments—i.e., beta, co-skewness and co-kurtosis—to macroeconomic and financial shocks. To conduct our experiments, we thus borrow from the empirical APT model of Chen et al. (1986), and from the multifactorial model proposed by Gregoriou (2004) to study the market timing of hedge funds: yt = λ0 + λ1output _ gap + λ2 credit _ spread + λ3term _ spread + λ4VIX + ς t

(6)

where yt are our measures of multi-moment risk—i.e., beta, co-skewness and co-kurtosis; output_gap is computed using U.S. quarterly GDP15; credit_spread is the spread between BBB and AAA corporate bond yields; term_spread is the spread between the ten-year interest rate and the three-month Treasury bills rate, VIX measures the volatility of the U.S. stock market, and ς is the innovation. In our model, we have substituted the VIX for the inflation rate which is a component of the original Chen et al.’s (1986) model16. Note that the choice of the term spread, the 14

For more detail on the computation of the linear VAR with hedge fund applications, see Racicot and Théoret (2018). To compute the quarterly output gap, we first take the log of real GDP. We then detrend this transformed series with the Hodrick-Prescott filter using a smoothing coefficient (λ) equal to 1600—the trend of the series being a measure of potential output. The resulting residuals are the output gap measure. 16 The reasons for this substitution are the following. First, because over our sample period, inflation is not a matter of concern as it was when Chen et al. (1986) conducted their study. Second, given its high level of volatility, the significance of the inflation rate was usually quite weak in the Chen et al.’s (1986) estimations or at least lower than the other included explanatory variables. Third, since we run our model with VARs, its formulation ought to be parsimonious. Fourth, the VIX is one of the most important variables explaining hedge fund returns since these funds are much involved in structured products. 15

10

credit spread and the industrial production growth rate—here replaced by the output gap, which is more stable than the industrial production growth—is supported by many researchers in the field of hedge funds (Kat and Miffre, 2002; Amenc et al., 2003; Brealy and Kaplanis, 2010; Bali et al., 2014; Lambert and Platania, 2016). Furthermore, Agarwal et al. (2017a) and Lambert and Platania (2016) find that the implied volatility (VIX) of the U.S. stock market is an important driver of hedge fund returns and hedge fund exposures to the stock market and Fama and French factors. Finally, according to Gregoriou (2004), the model we use is justified by the fact that hedge funds trade options (VIX), and have significant exposure to credit risk (credit_spread) and term spread risk (term_spread). In addition to these variables, we include the output gap in order to study the procyclicality of the risk borne by hedge fund strategies. It seems reasonable to assume that hedge funds decrease their exposure to risk in bad states—i.e., recession or crisis—and increase it in good states—i.e., expansion (Stafylas et al., 2018). Hedge fund portfolio managers should aim at decreasing their beta and co-kurtosis as well as increasing their co-skewness in bad states and reversing the signs of these exposures in good states. In our canonical model given by Eq. (6), bad states are obviously associated with a decrease in the output gap and a rise in the credit spread, which results in an increase in investors’ risk aversion and a rise in firms’ bankruptcies. Moreover, an increase in the term spread should also entail a decrease in hedge fund risk taking, since we may relate a rise in this spread to an increase in the liquidity premium (Lambert and Plantania, 2016). The slope of the yield curve is particularly steep in recession. Indeed, the liquidity constraint of the economic agents is then binding and the liquidity premium increases for long-term investments (Gollier, 2001). Finally, an increase in VIX also co-moves with bad states owing to the Black (1976) leverage effect, whereby an increase in VIX is associated with a drop in the stock market. Note that short-selling transactions and the use of structured products may help adjust risk exposures to the stance of the business cycle. For instance, performing short sales when markets are falling may result in a rise in co-skewness—i.e., a decrease in tail risk. However, this operation requires good forecasting skills, to say the least. Moreover, according to Agarwal et al. (2017b) and Bali et al. (2017), the use of puts during crises may reduce tail risk. There are two other ways opened to hedge funds to manage their risk during weak market states which are substitutes to hedging (Infante et al., 2018). Indeed, hedge funds can increase their liquidity ratio and deleverage by selling risky (illiquid) assets and decrease their balance sheet leverage ratio17 (Shleifer and Vishny, 2010; Brunnermeier and Sannikov, 2014). These two last measures

17 For instance, it is well-known that a decrease in a firm’s leverage results in a reduction of its stock beta. See Copeland and Weston (1988), chap. 13, pp. 455-462.

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contribute to increase systemic risk in the hedge fund industry when they are performed simultaneously by most strategies, which is usually the case during crises. 3.3 The nonlinear local projection method18 In the classical linear VAR model, the coefficients of the IRF function are computed recursively over the horizon {1,..., h} , the recursive equation to compute the IRF coefficients being (Kilian and Kim, 2011): h

θh = Φh P -1 = ∑ Φh-s Bs P -1

(7)

s =1

with Φ0 = I k and where the coefficient matrix Bs is obtained via the following reduced form of the VAR y t = B1 y t-1 + B 2 y t-2 + .... + Bp y t-p + µ t , with y t = ( y1t ,..., ykt ) ' and P is the matrix which orthogonalizes shocks. For instance, θ1 = B1 P -1 , θ 2 = ( B12 + B 2 ) P -1 , θ 3 = ( B13 + B1 B 2 + B 2 B1 + B 3 ) P -1 , and so on19. If the VAR includes only one lag for each endogenous variable—i.e., if p = 1—we then have:

θ1 = B1 P -1 , θ 2 = ( B12 ) P -1 , θ3 = ( B13 ) P -1 , and so on. In this case, the coefficients of the IRF can be viewed as multipliers of the individual structural shocks20.

In addition to the standard linear VAR model which is used as the benchmark in our study, we rely on a nonlinear local projection (LP) algorithm to compute the impulse response functions of hedge fund return moments. In the linear LP method, the coefficients are not computed recursively, as in the standard linear VAR, but they are rather based on the following expression of an optimal forecast (Jordà, 2004, 2005; Misina and Tessier, 2008; Tessier, 2015) which, incidentally, corresponds to the general definition of an impulse response (Hamilton, 1994; Koop et al., 1996): IR ( t , s, d i ) = E ( y t +s v i = d i ; Xt ) − E ( y t +s v i = 0 ; Xt ) s = 0,1, 2,...

(8)

where IR is the impulse response at time t+s; E(.) denotes the best mean-squared error predictor; yt is the k x 1 vector of endogenous variables; X t = ( yt −1 , yt − 2 ,...) ' ; 0 is of dimension k x 1; vi is the k x 1 vector of reduced-form disturbances, and di is the vector of the experimental shocks. An impulse response is thus a difference between two forecasts: (i) the forecast of yt+s when the instantaneous structural shocks are equal to di; (ii) the forecast of yt+s when the structural shocks are equal to 0. 18 To implement the nonlinear local procedure, we rely on a MATLAB code produced by one of our research assistants, Vincent Beauséjour. This code is available upon request. All the nonlinear impulse response figures associated with the local projection method are plotted using MATLAB. 19 See Judge et al. (1988) for more detail on the computation of these multipliers. These authors refer to this recursive computation as innovation accounting. 20 Actually, in the classical linear VAR, we often view the coefficients of the IRF as multipliers even if p>1.

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In order to compute the IRFs associated with the LP method, we must retain the last s+1 observations of the sample as dependent variables of the local projection—i.e., from yt to yt+s. We regress sequentially these vectors on the same set of lagged variables—i.e.,

(y

t-1

, y t-2 , ..., y t-p ) . We thus obtain the following sequence of regressions (Jordà, 2004, 2005;

Kilian and Kim, 2011)21: y t+h = F1h y t-1 + F2h y t-2 + ... + Fph y t-p + µ t +h

h = 0,..., H

(9)

The corresponding structural impulse responses are22: Ωh = φh P -1 = F1h P -1

h = 0,..., H

(10)

where P-1 is the Cholesky triangular matrix used to retrieve the structural shocks (Sims, 1980). Note that the impulse response coefficients involve the terms yt-1 only. Hence, they are not computed recursively, as in the classical linear VAR model. According to Jordà (2004, 2005), they are more robust to misspecifications of the data generating process, thereby avoiding an increase in the misspecification error through the nonlinear calculation of the standard VAR technique as the forecast horizon increases. The non-linear LP method we use is based on the following transformation of the LP model (Jordà, 2004, 2005): s+1 s +1 2 s y t+s = B1s +1 y t-1 + B s+1 2 y t-2 + ... + B p y t-p + Q1 y t-1 + ξ t+s

s = 0,1, 2,..., h

(11)

where the observations from t to t+h of our time series are used to compute the IRFs. We thus introduce a quadratic term on the first lag of yt-1 to account for the possible nonlinearities linking the endogenous variables23. The IRFs associated with Eq. (11) are easily computed as follows: ) ˆ s ( 2y d + d 2 ) s = 0,1, 2,..., h IR ( t , s, d i ) = B1s d i + Q 1 t-1 i i

(12)

where IR are the impulse response values. The impulse responses thus depend on the history of the time series as measured by yt-1. As usual, the reduced-form shocks have to be orthogonalized before plotting the IRFs of the endogenous variables24. The non-linear LP model presents many advantages over the classical linear VAR procedure—i.e, asymmetry, non-proportionality and state dependence (Jordà, 2004, 2005; Misina and Tessier, 2008). None of the restrictions imposed to shocks by the linear VAR are desirable in the context of our study. Indeed, it is well-known that the behavior of economic (financial) We neglect the constant term in this equation. As mentioned by Kilian and Kim (2011), Jordà (2005) does not make an explicit distinction between structural and reduced-form impulse responses. 23 Note that we can add quadratic terms on lagged values of higher order and also introduce cubic terms in Eq. (11). However, the formulation given by Eq. (11) performs the best in the context of our study. Moreover, as noted by Jordà (2004, 2005), since the impulse response coefficients involve only the terms yt-1 in the local projection technique, it seems parsimonious to restrict nonlinearities to the first lag alone. 24 See Kilian and Kim (2011) for the computation of the IRF confidence intervals. 21 22

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time series is asymmetric—i.e., these series react more to negative than to positive shocks (Neftci, 1984; Sichel, 1987; Hamilton, 1989, 2005; Ferrara and Guérin, 2018). A negative shock will also produce a different effect on hedge fund risk depending on whether the economy is in expansion or recession. In our linear VAR model, we regress each moment of a hedge fund strategy—i.e., beta, co-skewness and co-kurtosis—on the variables of our canonical asset pricing model (Eq. (6)). In our nonlinear LP approach, this equation becomes: yt = ϕ0 + ϕ1output _ gap + ϕ 2 output _ gap 2 + ϕ3 credit _ spread + ϕ4 credit _ spread 2 + ... +ϕ5term _ spread + ϕ6 term _ spread 2 + ϕ7VIX + ϕ8VIX 2 + ξt

(13)

In Eq. (13), the non-squared variables are associated with risk and the squared or quadratic variables are associated with uncertainty (noise) (Racicot and Théoret, 2018, 2019). Insert Table 1 here

4. Data and stylized facts 4.1 Data Data on quarterly hedge fund returns are drawn from the database managed by Greenwich Alternative Investment (GAI). GAI manages one of the oldest hedge fund databases, containing more than 13,500 records of individual hedge funds. Returns provided by the database are net of fees. Our database runs from the first quarter of 1988 to the second quarter of 2016, for a total of 116 observations—a reasonable number on which to perform our VAR analyses. We rely on quarterly data for at least two reasons. First, the GAI quarterly database spans a longer period than its monthly counterpart. Second, since our empirical methodology is based on VARs, using quarterly data leads to less noise or instability in the computation of the impulse response functions than resorting to monthly data. Indeed, VARs are usually estimated with quarterly time series in the economic literature. We can therefore more easily capture fundamental trends with quarterly data than with monthly ones. In addition to the weighted composite index (general index), our database includes 11 strategies25. The description of these strategies appears in Table 1. Data for U.S. macroeconomic and financial variables are taken from the FRED database which is managed by the Federal Reserve Bank of St.-Louis. There are many biases which must be addressed when using hedge fund data, the major one being the survivorship bias—i.e., a bias which is created when a database only reports information on operating funds (Capocci and Hübner, 2004; Fung and Hsieh, 2004; Patton, 25 We retain the strategy classification of GAI to perform our experiments. This classification spans the spectrum of the main hedge fund strategies. Discarded strategies—as mergers—have a too short data sample compared to the others, which make the comparison unfeasible.

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Ramadorai and Streatfield, 2015; Racicot and Théoret, 2018). This bias is accounted for in the GAI database as index returns for periods since 1994 include defunct funds26. Other biases which are tackled for in the GAI database are the self-selection bias and the early reporting bias (Capocci and Hübner, 2004; Fung and Hsieh, 2004). Other problems related to hedge fund returns are due to illiquidity and the practice of return smoothing (Asness et al., 2001; Amihud, 2002; Pástor and Stambaugh, 2003; Getmansky et al., 2004; Brown et al., 2012). The problems may lead to an underestimation of risk in the hedge fund industry. However, according to Asness et al. (2001), some biases like the one due to the practice of return smoothing should be less serious for quarterly data—which is the frequency used in this paper—than for monthly data— the most popular frequency in hedge fund studies. This is another reason to resort to quarterly data in the framework of our study27. Insert Table 2 here 4.2 Descriptive statistics Table 2 provides the descriptive statistics of hedge fund strategies included in our database. During the entire sample period, the annual return of the hedge fund weighted index, henceforth general index, is higher than the one of the S&P500—i.e., 8.79% vs 6.68%28. The average returns displayed by the strategies are quite comparable. However, two strategies succeed in delivering an annual return higher than 10%: growth and value index. The standard deviation of their returns is much higher than the average. The returns of the futures and shortsellers strategies have also a standard deviation which exceeds the average one. Incidentally, these strategies reported substantial positive returns during the subprime crisis—i.e., 8.61% and 8.86%, respectively. This performance is associated in large part with short sales. Note however that the average annual return of short-sellers, at -2.63%, is negative over the whole sample period. Indeed, their return is usually negative when the stock market trends upward. Given the “hedging operations” of hedge funds, their beta is usually low, the average conditional beta of the general index being equal to 0.37. The growth, value index and event driven strategies have the highest (positive) betas, being 0.83, 0.58 and 0.43, respectively. Many strategies have average betas below the 0.20 mark—i.e., equity market neutral, fixed income, long-short credit and macro. Moreover, two strategies display a negative average beta: futures (-0.02), and especially short-sellers (-0.99).

Source: Greenwich Alternative Investment website (2016). Another bias is the backfill one or the instant history bias. An increase in the incubation period—i.e., the time between inception and the listing in a database—leads to a higher backfill bias. 28 However, since 2010, the S&P500 has shown a better performance than the hedge fund general index, the average annual returns being 9.92% and 3.84%, respectively. 26 27

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Table 2 also provides the first-order autocorrelation coefficient of the returns of hedge fund strategies. According to Getmansky et al. (2004), the degree of liquidity of a portfolio is negatively related to this coefficient. We may conjecture that the most “liquid” hedge fund strategies according to this measure—i.e., futures, macro, short-sellers, value index and growth—are the most likely to succeed in rebalancing their portfolios following macroeconomic and financial shocks. Note that the returns of these strategies have also the highest standard deviation. Hence, a liquid portfolio may foster a higher return standard deviation and reduce the practice of return smoothing which can mitigate the responses of hedge fund sources of risk to macroeconomic and financial shocks. Turning to higher moments, we note that the general index displays less tail risk—as measured by skewness and kurtosis—than the market index over our sample period. Indeed, the skewness and kurtosis of the S&P500 are equal to -1.88 and 9.65, respectively, while the corresponding statistics for the hedge fund general index are 0.26 and 4.47. The strategies embedded with the highest (positive) skewness are: macro, growth and futures. Due to the subprime crisis, the fixed income and convertibles strategies have the lowest level of skewness and the highest level of kurtosis. In other respects, the two strategies most involved in short sales—i.e., shortsellers and futures—display low tail risk from 1988 to 2016. As explained earlier, we do not focus on the standard skewness and kurtosis statistics to analyse the dynamic behavior of the higher moments of hedge fund strategies, but rather on coskewness and co-kurtosis which are corrected for idiosyncratic risk. As noted previously, a higher level of co-skewness and a lower level of co-kurtosis are associated with lower systematic tail risk. In this instance, the hedge fund general index embeds less tail risk than the S&P500 (Table 2). Note that the level of a strategy’s co-kurtosis tends to co-move with its beta. In this respect, the growth, value index and event driven strategies have simultaneously the highest betas and co-kurtosis among the strategies listed in Table 2. Two strategies present a negative co-kurtosis: short-sellers and futures. These strategies are thus well-positioned to capture positive payoffs during crises (Jurczenko et al., 2006; Jurczenko and Maillet, 2006). Co-skewness is not correlated with skewness at the strategy level. For instance, the growth strategy has a high positive skewness (0.71) but it shows the highest negative coskewness in absolute value (-1.39). We note the same relationship for the macro and shortsellers strategies. However, in line with their skewness, the futures and equity market neutral strategies display a positive co-skewness. Hence, co-skewness and co-kurtosis measures do not classify strategies in the same order as skewness and kurtosis with respect to tail risk. Insert Figure 1 here

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4.3 Stylized facts Figure 1 provides the plots of co-kurtosis for key hedge fund strategies. We note that most strategies’ co-kurtosis tends to display the same pattern over the business cycle. However, the amplitude of co-kurtosis differs by strategy. For strategies with the highest betas—e.g., growth and value index—co-kurtosis has a greater amplitude than for strategies with low betas—e.g., equity market neutral29. The co-kurtosis of two strategies—i.e., short-sellers and futures—moves in the opposite direction of the others. Moreover, similarly to their average betas, these two strategies have a negative co-kurtosis while the others have a positive one. More importantly, note that strategies’ co-kurtosis tends to return to zero during recessions, which are shaded in Figure 1. This corresponds to less risk-taking by strategies’ managers. Their behavior thus becomes more homogenous during crises, an obvious source of systemic risk in the hedge fund industry. Also evidenced in Figure 1, we note that the co-kurtosis of the hedge fund general index is generally higher than the market one but that it was under better control during the subprime crisis. Insert Figure 2 here Co-kurtosis thus acts as a leverage on the performance of strategies. It increases during expansion and decreases in recession. Some strategies—i.e., growth and value index—rely more on this source of leverage in expansion but, in line with the others, they deleverage in recession. In this respect, Figure 2 plots the relationship between beta and co-kurtosis over the sample period for the strategies reported in Figure 1. We find that a strategy’s beta tends to be proportional to its co-kurtosis when this higher moment is positive. During crises—i.e., in periods of rising uncertainty—the leverage effect as measured by co-kurtosis tends to decrease for most strategies, and this move is associated with a decline in beta. In periods of expansion or of reduced uncertainty, leverage increases, leading to a rise in beta. This relationship also holds for the equity market neutral strategy, which is usually considered as non-directional (Agarwal et al., 2017a). Figure 2 shows that its beta and co-kurtosis are clearly procyclical (Duarte et al., 2007; Patton, 2009). However, since this strategy aims at minimizing its market exposure in the long run, the fluctuations of its return moments are subdued. Two strategies are quite different from the others in terms of the fluctuations of their beta and co-kurtosis: short-sellers and futures. Similarly to the other strategies, their beta and co-kurtosis tend to decrease in absolute value during recessions (Figure 2). However, during

29 In order to avoid overloading the figure, we have not included all the strategies which are listed in Table 1. The amplitude of the co-kurtosis of the distressed and long-short credit strategies is close to the general index one (gi) while the co-kurtosis of the macro and fixed income strategies co-move very closely with the equity market neutral one.

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normal times, the co-movements between their beta and co-kurtosis are less tight than the ones observed for the other strategies. Insert Figure 3 here Figure 3 displays the co-skewness plots of selected strategies. We note much less regularity in the behavior of co-skewness from one strategy to the next than it was the case for beta and co-kurtosis. The co-skewness of the hedge fund general index is generally stable and close to zero. The co-skewness of the S&P500 is much more unstable and tends to deteriorate sometimes substantially—especially during crises. Turning to strategies, the value index and futures have quite favourable patterns for their co-skewness which remains positive most of the time. In contrast, the co-skewness of growth and short-sellers are usually negative and high in absolute value. While the co-kurtosis of the equity market neutral strategy is quite stable, its coskewness fluctuates much more and may drop considerably. This result is in line with Duarte et al. (2007) who, contrary to expectations, find that the equity market neutral strategy may be risky during crises. More importantly, co-skewness appears less manageable than co-kurtosis and beta during crises. In this respect, during the subprime crisis, short-sellers and growth strategies succeed in increasing their co-skewness, whereas, similarly to the market index, the co-skewness of the value index and the equity market neutral strategies have deteriorated. Hedge funds must thus make an arbitrage between the moments of their return distributions. Insert Figure 4 here It may be interesting to examine how the standard measures of kurtosis and skewness— which include idiosyncratic risk—behaved during the subprime crisis in comparison with the corresponding measures of co-kurtosis and co-skewness. Figure 4 shows that kurtosis increased markedly during the subprime crisis for the general index and three selected strategies—i.e., growth, value index and equity market neutral. However, except for the latter, kurtosis decreased sharply after its peak. Kurtosis thus seems to be less tractable than co-kurtosis which started to decrease at the beginning of the subprime crisis. The representative hedge fund appears unable to smooth the shock related to idiosyncratic risk embedded in kurtosis. However, kurtosis decreased for the futures strategy during the subprime crisis and was quite under control for the short-sellers strategy, in line with our previous observations on these two strategies. This fact is consistent with Asness et al. (2001) who find, over their shorter sample period, that the futures strategy display good market-timing skills30. Insert Figure 5 here

30

i.e., the futures strategy loads positively on the squared market return.

18

The behavior of hedge fund risk associated with skewness is similar to the one related to kurtosis during the subprime crisis (Figure 5). For the general index, the growth and value index strategies, skewness deteriorated markedly during the first part of the subprime crisis and recovered substantially thereafter: shocks were thus quickly absorbed. In line with its kurtosis, the skewness of the equity market neutral strategy continued to deteriorate after the subprime crisis, which suggests that this strategy may have difficulties to smooth its skewness following macroeconomic and financial shocks (Duarte et al., 2007; Patton, 2009). However, the skewness of the futures and short-sellers strategies evolved favourably during the crisis, suggesting once more that these two strategies may help hedging a portfolio during a crisis by endogenizing31 the impact of negative shocks.

5. Impulse response functions computed with the benchmark linear VAR and the non-linear local projection method In this section, we study the IRFs of hedge fund risk measures—beta, co-kurtosis and co-skewness—computed with the nonlinear local projection method (LP) first over an expansion period—i.e., from 2013 to 2016—and then over the subprime crisis—i.e., from 2007 to 2010. Since the presence of squared lagged variables introduces noise in the estimation process, we select a 90% confidence interval to test the significance of the IRFs. Moreover, to analyze the impact of nonlinearities, we also report the IRFs built with the benchmark linear VAR procedure. After many experiments on different orderings involving the classical information criteria to select a VAR model—i.e., the AIC, AICc32 and SIC—we order the variables of our setting (equation (13)) in the following way in order to identify the structural shocks, from the most exogenous to the most endogenous: output gap → term spread → credit spread → hedge fund risk measure → VIX

The output gap is thus considered as the most exogenous variable while the VIX, which is very sensitive to economic news, is viewed as the most endogenous. At the impact, the risk measure reacts to the output gap, the term spread and the credit spread but not to the VIX. Note that this ordering is empirical since there does not exist a comprehensive theoretical framework on the co-movements between the variables in our VAR model. Insert Figure 6 here

31

These strategies may endogenize external shocks by an opportune use of derivatives or by synchronized short sales. The AICc is a corrected version of the AIC criterion proposed by Hurvich and Tsay (1993) and specifically designed for VARs. See also Jordà (2005).

32

19

5.1 IRFs over an expansion period (2013-2016) 5.1.1

IRFs of strategies’ betas For our simulations over the expansion period, on the basis of the usual information

criteria—i.e., the AIC, AICc and SIC—we compute the impulse response functions of our models with three lagged values for our vector of explanatory variables. Figure 6 provides the results of our experiments for the betas of the hedge fund general index and strategies. The most significant IRF for the general index beta is obtained with the output gap. The beta is quite procyclical and this relationship does not seem to depend on nonlinearities in expansion since the IRF computed with the standard linear VAR is very similar to the one generated by the nonlinear LP method. As expected, the representative hedge fund takes more systematic risk in expansion— i.e., its beta increases. According to Figure 6, the beta of most strategies is also procyclical. This procyclicality is particularly strong for the growth, macro, event driven and value index strategies. Surprisingly, the beta of the equity market neutral strategy is also quite procyclical even if this strategy aims at hedging market risk in the long run (Duarte et al., 2007; Patton, 2009). However, two groups of strategies differ: those more involved in fixed income securities—i.e., fixed income and convertibles—and those which focus more on short sales—i.e., futures and short-sellers. The first category seems to have a countercyclical beta, probably because the increase in interest rate, which is usually associated with a rise in the output gap, is detrimental to the business lines of these strategies. In other respects, the beta of the futures strategy seems quite immunized against the business cycle. This is not the case for short-sellers, its beta increasing in absolute value following an output gap shock. The beta associated with the general index decreases after a VIX shock, a term spread shock and a credit spread shock. The representative hedge fund thus appears to control the risk associated with these shocks. Once more, the strategies which are the most procyclical show a similar reaction: value index, growth, macro and event driven, the portfolios of these strategies being relatively liquid. However, for some strategies, the response of their beta to these shocks may be quite different. Actually, strategies involved in short sales—i.e., short-sellers and futures— increase their beta after a VIX shock. According to the Black (1976) leverage effect, an increase in the VIX is associated with falling stock markets, which obviously benefits to shortsellers. Not surprisingly, a rising VIX has a weak impact on the betas of the equity market neutral strategies and of strategies focusing on bond markets—especially fixed income and convertibles. Moreover, a positive term spread shock induces strategies more involved in bond markets—i.e., convertibles, distressed, fixed income and long-short credit—to increase their beta. 20

They seek to capture the illiquidity premium associated with the term spread in times of expansion, while economic uncertainty is low (Lambert and Plantania, 2016). Similarly, the beta of three strategies increases after a credit spread shock: distressed, fixed income and futures. In times of low uncertainty, these strategies—which embed a great deal of credit risk—are induced to capture the credit risk premium. This behavior is consistent with Agarwal et al. (2017b) who argue that the payoffs of many hedge fund strategies behave as short puts in good times. For instance, strategies may capture the credit risk premium by selling credit default swaps (CDS). Hence, they act as insurance sellers in economic expansion. According to Bali et al. (2017), the kurtosis of a short put computed in the risk-neutral universe decreases in good times while its skewness also decreases. There is thus a trade-off between the higher moments of strategies that behave as short puts in good times. Summarizing, in expansion, hedge funds tend to adjust their beta to shocks either to manage their risk or to increase their payoffs depending on the nature of their business lines. During good times, the output gap shock has the most important impact, the representative hedge fund increasing its beta to capture the gains associated with increased economic growth. Although the beta of the representative hedge fund tends to decrease following a VIX shock, a term spread shock and a credit spread shock, some specialized strategies adopt an opposite behavior to benefit from these shocks. For instance, an increase in the VIX is profitable for strategies involved in short sales—i.e., futures and short-sellers. Strategies focusing more on bond markets—i.e., convertibles, distressed, fixed income, and long-short credit—tend to increase their beta in expansion in order to capture the premia associated with term spread and credit spread shocks33, the risk associated with this move being lower in expansion. Insert Figure 7 here

5.1.2 IRFs of higher moments Since a strategy beta tends to co-move tightly with its co-kurtosis, Figure 7 only provides the IRFs of the general index and of some key strategies. Not surprisingly, the pattern of the IRFs of the general index co-kurtosis is very similar to its beta, albeit less significant since the behavior of co-kurtosis is less smooth than beta. Co-kurtosis is procyclical—i.e., the representative hedge fund takes more fat-tail risk in expansion. In line with beta, the hedge fund gen33

i.e., the illiquidity risk and credit risk premia, respectively.

21

eral index co-kurtosis responds negatively to VIX and credit spread shocks. Turning to strategies, we note that co-kurtosis responds more to shocks for strategies with a higher beta—e.g., growth—than for strategies with a lower one—e.g., equity market neutral and futures. Therefore, as discussed previously, strategies with a higher beta also tend to leverage risk in good times. Insert Figure 8 here It is well-known that hedge funds must trade-off higher moments when constructing optimal portfolios. They cannot simultaneously increase their return and co-skewness while decreasing their beta and co-kurtosis. The standard efficient frontier obtained when trading higher moments is thus lower than the one computed with only the first two moments of the return distribution34. In our setting, this necessary arbitrage seems to be performed at the expense of risk associated with co-skewness. In Figure 8, we note that the co-skewness of the general index improves markedly after a positive output gap shock, which corresponds to a decrease in the level of tail risk. However, it deteriorates after VIX and credit spread shocks. There is therefore an arbitrage or trade-off between beta and co-kurtosis, on the one hand, and coskewness, on the other hand. Following these shocks, the representative hedge fund succeeds in reducing its risk associated with its beta and co-kurtosis but this decrease in risk is at the expense of increased risk related to co-skewness. The co-skewness of the value index responds in a similar way to shocks as the one of the general index. However, the growth strategy behaves quite differently. Its co-skewness responds negatively to an output gap shock, suggesting that it is countercyclical35. Also in contrast to the general index, it responds positively to VIX and credit spread shocks. This pattern is consistent with the findings of Campbell et al. (2010) who argue that growth stocks display hedging properties with respect to market reversals. The high liquidity of the portfolio of this strategy may facilitate these hedging operations. The IRFs of the macro strategy—another “liquid” strategy—seem to share analogous properties. Similarly to growth and macro, the co-skewness of strategies for which it is countercyclical tends to respond positively to adverse VIX and credit spread shocks, which is associated with a decrease in tail risk. It is especially the case for strategies focusing on the bond market: distressed, event-driven and fixed income. This result is not surprizing since the performance of the bond market is countercyclical36. Finally, the co-skewness of strategies which benefit from a term spread shock—especially distressed, fixed income and long-short-credit—decreases followi.e., hedge funds must trade-off higher return against lower tail risk. This reaction may be peculiar to the composition of the portfolio held by this strategy. Indeed, following an output gap shock, the managers of this strategy may buy growth stocks whose returns are negatively skewed. 36 Note that the managers of the distressed securities strategy can benefit from credit and VIX shocks which tend to increase the yield of speculative securities like high-yield bonds. 34 35

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ing such a shock, suggesting that capturing the illiquidity risk premium does not go without some risk. Once more, this result is in line with Agarwal et al. (2017b) who assert that many hedge fund strategies behave as insurance sellers during good times by deliberately increasing their tail risk. 5.2 IRFs over the subprime crisis (2007-2010) To better grasp the effects of nonlinearities on the behavior of hedge fund multimoment risk, we also perform a nonlinear local projection of our model during the subprime crisis, which stretches from the second quarter of 2007 to the fourth quarter of 201037. The information criteria indicate that the optimal lag of the endogenous variables increases during the subprime crisis compared to the expansion period. Based on such criteria, we select a lag equal to six quarters for the crisis period, which suggests that the impact of shocks lasts longer during crises than during normal times. We also use a lag of six quarters for the explanatory variables of our benchmark linear VAR model. Insert Figure 9 here

5.2.1

IRFs of strategies’ betas During the subprime crisis, the IRFs associated with the beta of the general index

evolve in the same direction as in expansion following macroeconomic and financial shocks (Figure 9). However, the responses of the beta are quicker, more important, and last longer. The confidence intervals of the IRFs are also tighter. Hence, beta definitively decreases after a negative output gap shock. Similarly, VIX and credit spread shocks result in a drastic reduction in the beta of the representative hedge fund in the short-run. This response is almost exclusively driven by nonlinearities since the magnitude of the IRFs associated with the benchmark VAR is very low compared to the nonlinear LP method. However, according to the forecast provided by this method, the beta reverses its downward movement after four quarters, suggesting that hedge funds may experience difficulties in timing their systematic risk during a crisis. Strategies’ beta tends to modulate its behavior on the one of the general index after an output gap shock. In other respects, the difficulty to track the impact of VIX and credit spread shocks is particularly high for some strategies which are confronted to a quick and significant upward reversal of their beta after these shocks—i.e., event driven, equity market neutral, fu37 In line with Bekaert and Hodrick (2012), we also consider the year 2010 as part of the subprime crisis even if the U.S. recession officially ended in 2009 according to the National Bureau of Economic Research (NBER). Indeed, the subprime crisis had protracted effects and the year 2010 is not satisfying the norms of a strong recovery.

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tures, value index, and growth. In expansion, risk management was easier to implement. Interestingly, the effect of the term spread shock is usually higher and more significant in the subprime crisis than during the expansion period. This result may be related to the liquidity constraint which is more at play during crises. For instance, the beta of some strategies—i.e., distressed, event driven, equity market neutral, fixed income, growth, short-sellers, and value index—jump after a term spread shock, which suggests that the deterioration of their liquidity position increases the systematic risk they bear38. However, portfolio managers of a majority of these strategies seem to regain the control of their beta after six quarters. In contrast, the macro strategy succeeds quite well in mastering risk related to a term spread (liquidity) shock. 5.2.2 IRFs of higher moments Similarly to the expansion period, we only reproduce the IRFs of the co-kurtosis of some key strategies. In line with the beta, a negative output gap39 shock leads to a decrease in the co-kurtosis of the general index during the subprime crisis. The VIX and credit spread shocks have a much more important and significant negative effect on the general index cokurtosis during the crisis than in expansion, and in line with the beta, this effect is mainly driven by nonlinearities. Once more, we note that hedge funds may have difficulties in keeping their control on co-kurtosis, while tracking risk is easier during expansion. In this respect, a liquidity shock—as measured by the term spread—entails a rise in hedge fund co-kurtosis risk at the impact. However, this effect is reversed after four quarters whereas co-kurtosis decreases very significantly, suggesting that hedge funds regain the control over their fat-tail risk after a liquidity shock. Insert Figure 10 here Regarding strategies, the same comments apply to the plots of their co-kurtosis than to the ones of their betas. Strategies having a positive co-kurtosis in Figure 10—i.e., equity market neutral, growth and macro—reduce significantly their fat-tail risk following a negative output gap shock40. In the short-run, VIX and credit shocks give way to a decrease in these strategies’ co-kurtosis which is more important and significant than during the expansion period. However, co-kurtosis resumes its upward trend after four quarters. Similarly to the general index, the co-

38 Actually, another interpretation is possible. A positive loading for the term spread in the beta IRF of a strategy may also suggest that this strategy seeks to capture the illiquidity risk premium, which is high during crises. However, we prefer our alternative interpretation related to the binding liquidity constraint during crises, which is more in line with the rest of our analysis and with theoretical macroeconomic models including a financial sector (e.g., Brunnermeier and Sannikov, 2014). 39 In contrast to the expansion period, we consider only negative output gap shocks during the subprime crisis. 40 Similarly to the expansion period, the magnitude of this reduction is proportional to the level of the beta of the corresponding strategy.

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kurtosis of these strategies jumps temporarily after a term spread shock. They thus have difficulties in controlling liquidity risk in the short-run during a crisis. Insert Figure 11 here In line with co-kurtosis, the responses of co-skewness to shocks are stronger, quicker and more significant during the subprime crisis (Figure 11). Even if hedge funds succeed in reducing their beta and co-kurtosis during the crisis after an output gap shock—at least temporarily—they must arbitrage these moves with co-skewness risk. Subsequently to VIX and credit shocks, the co-skewness of the general index decreases, which corresponds to a rise in the level of tail risk. Nonlinearities clearly dominate these responses. However, after four quarters, hedge funds regain the control over their co-skewness since it becomes significantly positive. Interestingly, while a term spread shock does not impact significantly co-skewness in expansion, this kind of shock influences positively and significantly the general index co-skewness in the shortrun. A similar reaction is shared by the value index, futures and short-sellers strategies. Exploiting the mispricing of stocks or focusing on short sales may thus benefit from a reduction in market liquidity. Strategies also react more strongly to shocks during the local projection performed over the subprime crisis than during the expansion period. Similarly to the expansion period, most strategies whose co-skewness is countercyclical—i.e., distressed, event-driven, equity market neutral, growth and macro—present very interesting dimensions for a risk averse investor during a crisis. Following a negative output gap shock, the co-skewness of these strategies increases, suggesting that they can act as good hedges during recessions or crises. Moreover, in the short-run, their co-skewness responds positively to VIX and credit spread shocks, which makes them even more attractive for risk-averse investors. This favourable behavior of the coskewness of these strategies may be explained by the important weight of bond transactions in their business lines and the business opportunities associated with turmoil—e.g., distressed, event driven—, by the relative skills of their managers—i.e., macro—, or by their hedging capacity—i.e., equity market neutral and growth. Note that, even if the fixed income strategy tends to display a countercyclical co-skewness, it particularly suffered from the subprime crisis, being too invested in mortgage-backed securities. Our forecast shows a deterioration of its coskewness after a VIX shock and especially after a credit shock. In our experiments performed over an expansion period, the term spread has poor explanatory power in the co-skewness equations of the strategies. Indeed, in expansion, the liquidity constraint is usually not binding. It is not the case during the subprime crisis. In the shortrun, a term spread shock leads to a significant decrease in the co-skewness of strategies which were insensitive to this kind of shock in expansion—i.e., event driven, equity market neutral, 25

growth and macro. A decrease in the liquidity of the financial markets is thus detrimental to these strategies from the point of view of co-skewness risk. As explained earlier, illiquidity drives endogenous risk during crises (Brunnermeier and Sannikov, 2014). Let us summarize the main results obtained from our experiments during the subprime crisis. First, the impact of macroeconomic and financial shocks on hedge fund multi-moment risk is much longer during the crisis than during the expansion period. Second, nonlinearities are much more at play during the crisis—especially for VIX and credit spread shocks— regardless of the risk measure used. Third, the IRFs associated with shocks are generally much more significant and of greater amplitude during the subprime crisis. Fourth, monitoring multimoment risk seems more difficult during the crisis. In many occasions, hedge fund may lose temporarily the control over their measures of risk. Overall, during crises or recessions, hedge funds seek to reduce their tail risk by becoming insurance buyers or “hedgers” but this alchemy requires skill, to say the least (Agarwal et al., 2017b). Our results show that some strategies— i.e., futures, short-sellers, macro and growth—succeed better in transforming the profile of their payoffs from short to long puts when transiting from expansion to crisis, or, in other terms, to transform their business lines from insurance sellers in order to capture risk premia to insurance buyers in order to hedge their operations. According to Bali et al. (2017), the kurtosis of (long) puts decreases in bad times but their skewness also decreases. During crises, this trade-off is desirable because investors mostly fear a large negative payout if an extreme (tail) event occurs. The decrease in the kurtosis of puts on the S&P500 was particularly high during the subprime crisis and especially during the Lehman Brothers bankruptcy in September 2008, an obvious extreme (rare) event (Bali et al., 2017).

6. Systemic risk related to the moments of the hedge fund strategies’ return distributions If financial institutions behave more homogenously as regards to risk when macroeconomic and financial uncertainty increases, systemic risk—i.e., risk related to contagion among financial institutions—rises in the financial system. For instance, during crises, financial institutions may deleverage simultaneously, which results in fire sales of assets (Shleifer and Vishny, 2010; Brunnermeier and Sannikov, 2014). This contributes to decrease these institutions’ net worth and increase the probability of bankruptcies, with detrimental spillover effects on the economy. Using a methodology popularized by Beaudry et al. (2001), by Baum et al. (2002, 2004, 2009) and by Caglayan and Xu (2016a, 2016b), Racicot and Théoret (2016) find that the cross26

sectional dispersion of hedge fund strategies’ betas tends to decrease in times of rising macroeconomic and financial uncertainty—an obvious source of systemic risk in the hedge fund industry. In this study, we go a step further and analyze the factors which may explain the more homogenous (or heterogeneous) behavior not only of strategies’ betas, but also of their higher moments—i.e., co-skewness and co-kurtosis. We still rely on Eq. (13) to implement our experiments. On the LHS of this equation, yt becomes the cross-sectional dispersions of the strategies’ betas, co-skewness and co-kurtosis. These dispersions are simply the standard deviations of these measures of risk computed using our eleven strategies over each quarter of our sample. As previously, we rely on our benchmark linear VAR model and nonlinear LP method to run our regressions. Insert Figure 12 here Figure 12 provides the plots of the cross-sectional dispersions of our strategies’ risk measures from the first quarter of 1988 to the second quarter of 2016. Analogously to Racicot and Théoret (2016), the cross-sectional dispersion of strategies’ betas tends to decrease sharply during a recession and it even anticipated the tech-bubble and subprime crises. More importantly, the cross-sectional dispersion of strategies’ co-skewness, and especially strategies’ cokurtosis, display the same pattern during crises. The cross-sectional dispersion of co-kurtosis even decreased towards zero during the tech-bubble and subprime crises. It thus appears that hedge fund strategies become more homogenous with respect to risk during recessions or crises, regardless of the risk measure used. Insert Figure 13 here Figure 13 transposes our VAR methodology to the strategies’ cross-sectional dispersions of our three risk measures. We note that the output gap shock is quite important for explaining the dynamics of the beta dispersion. In line with Figure 12, this shock contributes to increase the beta cross-sectional dispersion in expansion but to reduce it during the subprime crisis. A VIX shock has an even stronger impact on the cross-sectional dispersion of beta in crises and we note that nonlinearities are more at play during turmoil. While this shock tends to increase the cross-sectional dispersion of the beta in expansion, it decreases it temporarily in crises. However, the dispersion rebounds thereafter. A VIX shock thus makes the behavior of hedge funds more homogenous in terms of the beta at the start of a crisis but this move reverses quickly. Albeit less important, a credit spread shock has a similar effect on the beta dispersion in expansion and during crises. Finally, a term spread shock makes strategies more heterogeneous in terms of the beta, both in expansion and during crises. As discussed previously, the response of the beta to a term spread shock may differ substantially by strategy. 27

Turning to co-kurtosis, we note that the output gap is the only shock having a substantial impact on its cross-sectional dispersion in expansion. Similarly to beta, this dispersion is procyclical. However, during the subprime crisis, the responses of the co-kurtosis crosssectional dispersion to shocks are much stronger and significant than for the beta dispersion. A VIX shock has a longer negative impact on the co-kurtosis dispersion than on the beta dispersion, and nonlinearities dominate the plot. Moreover, this dispersion responds negatively to a credit spread shock and, once more, this reaction lasts longer than in the case of the beta dispersion. Therefore, during the subprime crisis, shocks tend to make the behavior of strategies more homogenous in terms of co-kurtosis, which thus represents a major source of systemic risk. Even if a term spread shock increases the co-kurtosis cross-sectional dispersion at the impact, this behavior reverses quickly and significantly. Finally, the cross-sectional dispersion of strategies co-skewness displays responses similar to co-kurtosis, both in expansion and crises. Summing up, the cross-sectional dispersions of our three risk measures are clearly procyclical—i.e., they respond positively to an output gap shock. This macroeconomic shock thus contributes to make the behavior of strategies more heterogeneous in expansion but more homogenous during recessions or crises. Apart from the output gap shock, a VIX shock is another important factor driving cross-sectional dispersions. While this kind of shock tends to increase the cross-sectional dispersion of beta in crises, it tends to decrease the dispersions of the two other measures of risk, at least in the short-run. We note the same pattern when considering a credit spread shock. Therefore, tail risk seems to be a major source of systemic risk in the hedge fund industry. Focusing only on the behavior of the beta cross-sectional dispersion may lead to a severe understatement of systemic risk in this industry during crises.

7. Robustness check As a robustness check, we apply our nonlinear local projection approach to monthly data41. Indeed, the responses of hedge fund risk measures to shocks may change with data frequency since aggregation may increase autocorrelation in time series. According to Chen and Engel (2005), the resulting bias in the parameters estimation may be positive or negative. Moreover, the speed of adjustment may be altered when varying data frequency. The monthly database on GAI hedge fund strategies starts in January 1995. It is thus shorter than the corresponding quarterly database we use. This is one of the factors explaining our previous choice of the quarterly frequency due to the spanning argument advanced in Shiller 41

We thank an anonymous referee for suggesting to also perform our experiments on monthly data.

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and Perron (1985). In addition, according to Asness et al. (2001), some biases, like the one associated with the practice of return smoothing, should be less serious for quarterly data—i.e., the frequency used in this paper—than for monthly data—the most popular frequency in hedge fund studies. The output gap measure we rely on to estimate Eq. (13) at the monthly level is computed using the US real industrial production index. The other series are the monthly equivalents of the quarterly ones. Insert Figure 14 here Figures 14 and 15 provide the monthly estimation of our nonlinear local projection model using the hedge fund general index and three strategies—i.e., growth, equity market neutral and short-sellers—over the expansion period and the subprime crisis, respectively. The choice of these strategies is based on the value of their beta, the growth strategy having a high positive beta, the equity market neutral, a beta close to zero, and short-sellers displaying a high negative beta, thus covering the whole spectrum of beta values. As expected, the IRFs obtained with monthly data are much more volatile (cyclical) than those associated with quarterly data. We observe that in many cases the quarterly IRF represents the underlying trend of the monthly IRF. For instance, in the expansion period, the quarterly response of the general index to the output gap is hump-shaped (Figure 8) while the corresponding monthly plot (Figure 14) displays a cyclical movement around an upward trend. Once more, the output gap and the VIX are the main drivers of hedge funds’ moments in our monthly model. However, the VIX weights more in the monthly than in the quarterly model, which suggests that financial variables have more explanatory power at shorter frequencies and real sector variables are more at play at longer ones. Insert Figure 15 here As in the quarterly experiment, the responses of hedge fund moments to shocks are less significant and persistent in the expansion period than during the subprime crisis (Figure 15). In this respect, the confidence intervals of the IRFs are much tighter during the crisis. This result is consistent with the asymmetries observed previously when comparing expansion and crisis periods. Regarding strategies, we note that our results are similar to the quarterly experiments—at least with respect to VIX and output gap shocks—but that the monthly IRFs are more cyclical, which may be associated with the difficulties of hedge fund strategies’ managers to meet their target, an issue that is less apparent at the quarterly frequency. For instance, similarly to the quarterly projections, short-sellers and growth strategies manage quite well their risk in the monthly projections but we observe more volatility. In this regard, their co-skewness tends to react positively to unfavourable VIX and output gap shocks during the crisis, but the cyclicality of the IRFs suggests that tracking a positive co-skewness may be quite demanding. 29

In other respects, the equity market neutral strategy seems to lose the control of its coskewness during the crisis since the IRFs related to the VIX and output gap shocks are quite cyclical and may plunge temporarily. This behavior is more noticeable in the monthly than in the quarterly projections. Insert Figure 16 here Figure 16 provides the monthly projections of the moment cross-sectional dispersions. The IRFs associated with the VIX and output gap shocks—i.e., our two main shocks—are consistent with the quarterly ones. For instance, a negative output gap shock results in a decrease in the cross-sectional dispersions of our three risk measures—which represents an increase in systemic risk—and this impact is more important during the crisis. However, a VIX shock tends to rise the beta cross-sectional dispersion and this effect is once more stronger during the crisis. Analogously to our quarterly findings, this kind of shock leads to a decrease in the crosssectional dispersion of our two other measures of risk, and this impact is again more discernible during the crisis. In this regard, the cross-sectional dispersion of co-skewness registers a big drop at the impact during the crisis, signalling a substantial increase in systemic risk in the hedge fund industry originating from this moment.

8. Conclusion Our paper does not resort to a cross-sectional framework like the works concerned with the micro-dynamics dimensions of the timing of tail risk by hedge fund managers (Jiang and Kelly, 2012; Bali et al., 2014; Namvar et al., 2016; Agarwal et al., 2017b). It rather relies on a holistic (time series) nonlinear approach designed to analyze how macroeconomic and financial shocks influence three measures of hedge fund risk—i.e., beta, co-skewness and co-kurtosis. To the best of our knowledge, our paper is the first that really focuses on the macroeconomic dimensions of tail risk in the hedge industry using (i) new empirical measures of tail risk based on a MGARCH procedure; (ii) a nonlinear local projection approach to forecast the responses of their risk measures to shocks. Our study shows that managers of hedge fund strategies tend to behave as insurance sellers in economic expansions but become hedgers or insurance buyers in recessions or crises (Bali et al, 2017; Agarwal et al., 2017b). For instance, many strategies tend to increase their exposure to term spread and credit risks in expansion but reduce it in recession. Hence, regarding credit risk, they behave as if they were sellers of credit default swaps (CDS) in expansion and buyers of CDS in recession. However, our projection over the subprime crisis reveals that some strategies may have difficulties to control their tail risk during a crisis—especially risk 30

associated with co-skewness. In this respect, the equity market neutral strategy—which seeks to neutralize all systematic risk by derivatives or short sales in the long run—may be risky during crises and is thus not so neutral after all. Minimizing risk exposure by means of hedging does not always lead to the desired results (Instefjord, 2005; Duarte et al., 2007; Patton, 2009; Brunnermeier and Sannikov, 2014) and may underestimate portfolio losses. Moreover, the degree of underestimation might be substantial for portfolios with low volatility similar to the ones held by the equity market neutral strategy. Even if hedge funds may be considered as variance reducers as argued by George Hall representing the Managed Fund Association before the U.S. House Financial Services Committee in 2007, their difficulties to control tail risk in periods of crises may offset the lower variance of their returns for investors concerned with the impact of adverse macroeconomic and financial shocks on the level of their downside risk. Finally, we analyze systemic risk in the hedge fund industry using an underpinning proposed by Beaudry et al. (2001). We find that the cross-sectional dispersions of our three hedge fund risk measures tend to decrease during crises, which suggests that the behavior of managers of hedge fund strategies become more homogenous in times of turmoil. Thus, they tend to deleverage as a group by relying on fire sales of assets, among others, an obvious source of systemic risk (Shleifer and Vishny, 2010; Brunnermeier and Sannikov, 2014). The VIX seems to be the most important factor contributing to systemic risk in the hedge fund industry. Importantly, the cross-sectional dispersions of our three risk measures are procyclical. An increase in economic growth thus contributes to reduce systemic risk but, conversely, a decrease tends to rise it. Focusing only on the beta to track hedge fund systemic risk over the business cycle may thus result in a serious underestimation of this kind of risk and of the potential of failures in the hedge fund industry. Hence, investors and regulators should improve their framework for analyzing hedge fund risk, as proposed in this paper. In this regard, our analysis could be compared to another field of analysis of systemic risk which also relies on VAR, as in this article, but which is more focused on the variance decomposition of the VAR model. Giudici and AbuHashish (2019) and Giudici and Pagnottoni (2019) developed a methodology to tackle systemic risk through an analysis of connectedness applied to bitcoin exchange prices that could be transposed to the returns of hedge fund strategies. This is a fruitful avenue of future research.

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35

Tables Table 1 Description of hedge fund strategies Strategy convertibles (CV) Distressed securities (DS) Diversified event driven (ED) Equity market neutral (EMN) Fixed income (FI) Futures (FUT) Growth Long-short credit (LSC) Macro Short sellers (SS) Value index (VI)

Description They take a long position in convertibles and short simultaneously the stock of companies having issued these convertibles in order to hedge a portion of the equity risk. The managers buy equity and debt at deep discounts issued by firms facing bankruptcy. The manager tries to benefit from mispricings related to corporate events (e.g., spin-offs, mergers and acquisitions, bankruptcy reorganisations and share buybacks). The managers aim at obtaining returns with low or no correlation with equity and bond markets. They exploit the pricing inefficiencies between related equity securities. The managers follow a variety of fixed income strategies like exploiting relative mispricing between related sets of fixed income securities. They invest in MBS, CDO, CLO and other structured products. The manager utilizes futures contracts to implement directional positions in global equity, interest rate, currency and commodity markets. He resorts to leveraged positions to increase his return. The managers invest in companies experiencing strong growth in earnings per share. They take long and short positions in credit in spite of the unavailability of bonds. They invest in high-yield bonds, CDS and CDO, among others. These funds have a particular interest in macroeconomic variables. They take positions according to their forecasts of these variables. Managers rely on quantitative models to implement their strategies. Managers take advantage of declining stocks. Short-selling consists in selling a borrowed stock in the hope of buying it at a lower price in the short-run. Managers’ positions may be highly leveraged. Managers invest in securities which are perceived undervalued with respect to their “fundamentals”.

Sources: Greenwich Global Hedge Fund Index Construction Methodology, Greenwich Alternative Investment (2015); Saunders et al (2014).

36

Table 2 Descriptive statistics, Greenwich Alternative Investment hedge fund strategy returns and moments, 1988Q1–2016Q2

Mean return Annual Median Maximum Minimum Std. Dev. Skewness Kurtosis

ρ

CV 0.0211 8.44% 0.0241 0.1500 -0.2079 0.0449 -1.86 13.00 0.57***

DS 0.0238 9.54% 0.0334 0.0946 -0.1491 0.0410 -1.25 5.56 0.47***

ED 0.0249 9.96% 0.0291 0.1192 -0.1039 0.0443 -0.54 3.72 0.29***

EMN 0.0191 7.66% 0.0164 0.1111 -0.0579 0.0245 0.35 5.49 0.27***

FI 0.0216 8.66% 0.0230 0.0815 -0.0857 0.0239 -1.09 7.45 0.41***

FUT 0.0223 8.92% 0.0087 0.1583 -0.1034 0.0507 0.69 3.35 0.01

GROWTH 0.0272 10.87% 0.0277 0.3698 -0.1372 0.0816 0.71 5.77 0.19**

LSC 0.0177 7.10% 0.0210 0.0853 -0.0795 0.0249 -1.11 6.34 0.37***

MACRO 0.0133 5.33% 0.0108 0.1921 -0.1076 0.0411 0.76 7.04 0.04

SS -0.0066 -2.63% -0.0142 return -0.2984 0.0958 0.16 3.94 0.11*

VI 0.0270 10.80% 0.0275 0.1918 -0.1128 0.0579 0.00 3.59 0.18***

Mean 0.0192 7.69% 0.0189 0.1554 -0.1312 0.0482 -0.29 5.93 0.26***

GI 0.0220 8.79% 0.0230 0.1706 -0.0908 0.0418 0.26 4.47 0.25***

MKT 0.0167 6.68% 0.0228 0.1205 -0.3169 0.0656 -1.88 9.65 0.11

mean beta Std. Dev

0.253 0.1177

0.363 0.1138

0.429 0.1165

0.141 0.0489

0.157 0.1138

-0.018 0.0488

0.828 0.1996

0.147 0.0774

0.106 0.0351

-0.999 0.3446

0.578 0.1407

0.180 0.1233

0.370 0.0847

1.000

mean co-skewness Std. Dev

0.3330 0.3077

-0.5901 0.5348

-0.2444 0.1400

0.1249 0.5582

-0.0736 0.2987

0.2007 0.2209

-1.3895 0.6370

0.1807 0.1257

-0.2919 0.1499

-0.3645 0.2042

0.3069 0.1483

-0.1644 0.3023

0.0594 0.0423

-0.0897 0.2948

mean co-kurtosis Std. Dev

1.8156 1.3115

5.4175 3.1998

7.8545 4.6592

1.2149 0.7218

0.3742 0.2274

-4.6259 2.7447

7.8446 4.6460

5.0790 3.0094

1.7692 1.0537

-6.9348 4.0997

9.8434 5.8341

2.6956 2.8643

4.1171 2.4406

2.7528 1.5530

Notes: The acronyms of the strategies are reported in Table 1. GI is the GAI general index. MKT is the market return as measured by the S&P500. The conditional beta, co-skewness and co-kurtosis are computed using the multivariate GARCH procedure, as explained in Section 3.1. Std. Dev. stands for standard deviation. The ρ coefficient associated with a strategy is equal to the coefficient of autocorrelation of the first order degree of its return. *, ** and ***, significant at the 10%, 5% and 1% levels, respectively.

Figures Figure 1 Co-kurtosis of the hedge fund general index and of some key strategies, 1988-2016

37

30

20 VI GROWTH

10 GI MKT

0 EMN FUT

-10 SS

-20 88

90

92

94

96

98

00

02

04

06

08

10

12

14

16

Notes: The acronyms of the strategies are listed in Table 1. GI is the hedge fund GAI general index. MKT is the market return as measured by the S&P500. The conditional co-kurtosis is computed using the multivariate GARCH procedure, as explained in Section 3.1. The periods associated with recessions are shaded.

38

Figure 2 Co-kurtosis and beta of the hedge fund general index and of key strategies

12

.55

10

.50 .45

6

.40

30

.9

25

.8

20

.7

15

.6

10

.5

beta

8

co-kurtosis

Value index

beta

co-kurtosis

General index

4

.35

2

.30

5

.4

0

.25

0

.3

.20

-5

-2 88

90

92

94

96

98

00

02

04

06

08

10

12

14

.2 88

16

90

92

94

96

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Equity market neutral

04

06

08

10

12

14

16

Growth 24

1.2

2.5

.24

20

1.1

2.0

.20

16

1.0

12

0.9

8

0.8

4

0.7

0

0.6

-4

0.5

1.5

.16

1.0

.12

0.5

co-kurtosis

.28

beta

3.0

beta

co-kurtosis

02

co-kurtosis value index beta value index

co-kurtosis general index beta general index

.08

0.0

.04

-0.5

.00 88

90

92

94

96

98

00

02

04

06

08

10

12

14

-8

16

0.4 88

90

92

94

co-kurtosis equity market neutral beta equity market neutral

-2

.2

-4

.1

-6

.0

-8

-.1

-10

-.2

-12

co-kurtosis

.3

94

96

98

00

02

04

06

08

10

12

14

16

beta growth

02

co-kurtosis futures

04

06

08

10

12

14

0

0.0

-4

-0.4

-8

-0.8

-12

-1.2

-16

-1.6

-20

-.3 92

00

beta

0

beta

.4

90

98

Short-sellers

2

88

96

co-kurtosis growth

Futures

co-kurtosis

00

-2.0 88

90

92

94

96

98

00

02

04

06

08

10

12

14

16

16 co-kurtosis short-sellers beta short-sellers

beta futures

Notes: The conditional beta and co-kurtosis are computed using the multivariate GARCH procedure, as explained in Section 3.1. The periods associated with recessions are shaded.

Figure 3 Co-skewness of the hedge fund general index and of some key strategies, 1988-2016 Market return, General index, Equity market neutral, Growth

39

2 EMN

1 GI

0 MKT

-1

-2 GROWT H

-3 88

90

92

94

96

98

00

02

04

06

08

10

12

14

16

General index, Value index, Futures, Short-sellers 1.2 FUT

0.8

0.4

VI GI

0.0 SS

-0.4

-0.8

-1.2 88

90

92

94

96

98

00

02

04

06

08

10

12

14

16

Notes: The acronyms of the strategies are listed in Table 1. The conditional co-skewness is computed using the multivariate GARCH procedure, as explained in Section 3.1. The periods associated with recessions are shaded.

Figure 4 Strategies’ kurtosis General index

Value index

7

6

6 5

5

mkt

vi

4 gi

4 3

3

gi 2

2

1

1

88

90

92

94

96

98

00

02

04

06

08

10

12

14

16

88

40

90

92

94

96

98

00

02

04

06

08

10

12

14

16

Equity market neutral

Growth

7

4.8 4.4

EMN

6

gi

4.0

5

3.6 growth 3.2

4 2.8

3

2.4

gi

2.0

2 1.6

1

1.2

88

90

92

94

96

98

00

02

04

06

08

10

12

14

16

88

Futures

90

92

94

96

98

00

02

04

06

08

10

12

14

16

Short-sellers

7

5.0 4.5

fut

6

4.0

5

gi

3.5

4

gi

ss

3.0 2.5

3

2.0

2 1.5

1

1.0

88

90

92

94

96

98

00

02

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06

08

10

12

14

16

88

90

92

94

96

98

00

02

04

06

08

10

12

14

16

Notes: Kurtosis is computed using a rolling window of 12 quarters. The periods associated with recessions are shaded.

Figure 5 Strategies’ skewness General index

Value index

1.5

1.5

gi

1.0

1.0

gi

mkt 0.5

0.5

0.0

0.0

-0.5

-0.5

-1.0

-1.0

-1.5

-1.5

-2.0

vi

-2.0

88

90

92

94

96

98

00

02

04

06

08

10

12

14

16

88

Equity market neutral

90

92

94

96

98

Growth

41

00

02

04

06

08

10

12

14

16

1.5

1.5

1.0 1.0 growth

0.5 0.5

0.0 -0.5

0.0

gi -1.0 -0.5

gi

-1.5 -1.0

EMN

-2.0

-1.5

-2.5 88

90

92

94

96

98

00

02

04

06

08

10

12

14

88

16

Futures

90

92

94

96

98

00

02

04

06

08

10

12

14

16

Short-sellers

2.5

1.6

2.0

1.2

1.5

0.8

ss fut

0.4

1.0 0.5

0.0

0.0

-0.4 -0.8

-0.5 gi

-1.0

gi

-1.2 -1.6

-1.5 88

90

92

94

96

98

00

02

04

06

08

10

12

14

88

16

90

92

94

96

98

00

02

04

06

08

10

12

14

16

Notes: The acronyms of the strategies are listed in Table 1. Skewness is computed using a rolling window of 12 quarters. The periods associated with recessions are shaded.

42

Figure 6 Quarterly non-linear local projection IRFs of strategies’ betas over an expansion period (2013-2016) v/s standard linear VAR General index Own shock

VIX shock

term-spread shock

credit shock

Convertibles

Distressed

Event driven

Equity market neutral

Fixed income

Futures

Growth

Long-short credit

43

output gap shock

Macro

Short-sellers

Value index

Notes : These plots represent the impulse responses of the strategies’ betas to their own shocks and to shocks of the explanatory variables. The outer dotted lines (red lines) embed the 90% confidence interval associated with local projection. The inner dotted line (blue line) is the impulse response function related to the nonlinear local projection while the solid line (pink line) is the impulse response function related to the conventional linear VAR model. The horizon h of the forecast is fixed at 12 quarters.

Figure 7 Quarterly nonlinear local projection IRFs of strategies’ co-kurtosis over an expansion period (2013-2016) v/s standard linear VAR 44

General index Own shock

VIX shock

term-spread shock

credit shock

output gap shock

Equity Market Neutral

Futures

Growth index

Macro

Short-sellers

Notes : These plots represent the impulse responses of the strategies’ co-kurtosis to their own shocks and to shocks of the explanatory variables. The outer dotted lines (red lines) embed the 90% confidence interval associated with the nonlinear local projection. The inner dotted line (blue line) is the impulse response function related to the nonlinear local projection while the solid line (pink line) is the impulse response function related to the conventional linear VAR model. The horizon h of the forecast is fixed at 12 quarters.

Figure 8 Quarterly nonlinear local projection IRFs of strategies’ co-skewness over an expansion period (2013-2016) v/s standard linear VAR General index Own shock

VIX shock

term-spread shock

45

credit shock

output gap shock

Convertibles

Distressed securities

Event driven

Equity market neutral

Fixed income

Futures

Growth

Long-short credit

Macro 46

Short-sellers

Value index

Notes : These plots represent the impulse responses of the strategies’ co-skewness to their own shocks and to shocks of the explanatory variables. The outer dotted lines (red lines) embed the 90% confidence interval associated with the nonlinear local projection. The inner dotted line (blue line) is the impulse response function related to the nonlinear local projection while the solid line (pink line) is the impulse response function related to the conventional linear VAR model. The horizon h of the forecast is fixed at 12 quarters.

Figure 9 Quarterly nonlinear local projection IRFs of strategies’ betas over the subprime crisis v/s standard linear VAR General index Own shock

VIX shock

term-spread shock

Convertibles

47

credit shock

output gap shock

Distressed securities

Event driven

Equity market neutral

Fixed income

Futures

Growth

Long-short credit

Macro

Short-sellers

Value index 48

Notes : These plots represent the impulse responses of the strategies’ betas to their own shocks and to shocks of the explanatory variables. The outer dotted lines (red lines) embed the 90% confidence interval associated with the nonlinear local projection. The inner dotted line (blue line) is the impulse response function related to the nonlinear local projection while the solid line (pink line) is the impulse response function related to the conventional linear VAR model. The horizon h of the forecast is fixed at 12 quarters.

Figure 10 Quarterly nonlinear local projection IRFs of strategies’ co-kurtosis over the subprime crisis v/s standard linear VAR General index Own shock

VIX shock

term-spread shock

credit shock

Equity market neutral

Futures

Growth

Macro 49

output gap shock

Short-sellers

Notes : These plots represent the impulse responses of the strategies’ co-kurtosis to their own shocks and to shocks of the explanatory variables. The outer dotted lines (red lines) embed the 90% confidence interval associated with the nonlinear local projection. The inner dotted line (blue line) is the impulse response function related to the nonlinear local projection while the solid line (pink line) is the impulse response function related to the conventional linear VAR model. The horizon h of the forecast is fixed at 12 quarters.

Figure 11 Quarterly nonlinear local projection IRFs of strategies’ co-skewness over the subprime crisis v/s standard linear VAR General index Own shock

VIX shock

term-spread shock

credit shock

Convertibles

Distressed

Event driven

Equity market neutral

50

output gap shock

Fixed income

Futures

Growth

Long-short credit

Macro

Short-sellers

Value index

Notes : These plots represent the impulse responses of the strategies’ co-skewness to their own shocks and to shocks of the explanatory variables. The outer dotted lines (red lines) embed the 90% confidence interval associated with the nonlinear local projection. The inner dotted line (blue line) is the impulse response function related to the nonlinear local projection while the solid line (pink line) is the impulse response function related to the conventional linear VAR model. The horizon h of the forecast is fixed at 12 quarters.

51

Figure 12 Quarterly cross-sectional dispersions of strategies’ betas, co-kurtosis and coskewness Beta Strategies' betas cross-sectional dispersions .8

.7

.6

.5

.4

.3

.2 88

90

92

94

96

98

00

02

04

06

08

10

12

14

16

Co-kurtosis Strategies' co-kurtosis cross-sectional dispersion 12

10

8

6

4

2

0 88

90

92

94

96

98

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02

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Co-skewness Strategies' co-skewness cross-sectional dispersion 1.2

1.0

0.8

0.6

0.4

0.2

0.0 88

90

92

94

96

98

00

02

52

04

06

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16

Notes: These cross-sectional dispersions are explained in Section 5. The periods associated with recessions are shaded. For more detail on the computation of cross-sectional dispersions, see Racicot and Théoret (2016).

Figure 13 Quarterly nonlinear local projection IRFs of strategies’ moments cross-sectional dispersions over an expansion period (2013-2016) and over the subprime crisis v/s standard linear VAR Beta cross-sectional dispersion in expansion Own shock

VIX shock

term-spread shock

credit shock

output gap shock

Beta cross-sectional dispersion during the subprime crisis

Co-kurtosis cross-sectional dispersion in expansion

Co-kurtosis cross-sectional dispersion during the subprime crisis

Co-skewness cross-sectional dispersion in expansion

Co-skewness cross-sectional dispersion during the subprime crisis

Notes : These plots represent the impulse responses of the strategies’ cross sectional dispersions to their own shocks and to shocks of the explanatory variables. The outer dotted lines (red lines) embed the 90% confidence interval associated with the nonlinear local projection. The inner dotted line (blue line) is the impulse response function related to the nonlinear local projection while the solid line (pink line) is the impulse response function related to the conventional linear VAR model. The horizon h of the forecast is fixed at 12 quarters. For more detail on the computation of cross-sectional dispersions, see Racicot and Théoret (2016).

53

Figure 14 Monthly non-linear local projection IRFs of strategies’ moments over an expansion period (2013-2016) v/s standard linear VAR

General index beta

co-kurtosis

co-skewness

Growth beta

co-kurtosis

co-skewness

Equity market neutral beta

54

co-kurtosis

co-skewness

Short-sellers beta

co-kurtosis

co-skewness

Notes: These plots represent the impulse responses of the selected strategies’ betas, co-skewness and co-kurtosis to their own shocks and to shocks of the explanatory variables. The outer dotted lines (red lines) embed the 90% confidence interval associated with the nonlinear local projection. The inner dotted line (blue line) is the impulse response function related to the nonlinear local projection while the solid line (pink line) is the impulse response function related to the conventional linear VAR model.

55

Figure 15 Monthly non-linear local projection IRFs of strategies’ moments over the subprime crisis v/s standard linear VAR General index beta

co-kurtosis

co-skewness

Growth beta

co-kurtosis

co-skewness

Equity market neutral beta

56

co-kurtosis

co-skewness

Short-sellers beta

co-kurtosis

co-skewness

Notes: These plots represent the impulse responses of the selected strategies’ betas, co-skewness and co-kurtosis to their own shocks and to shocks of the explanatory variables. The outer dotted lines (red lines) embed the 90% confidence interval associated with the nonlinear local projection. The inner dotted line (blue line) is the impulse response function related to the nonlinear local projection while the solid line (pink line) is the impulse response function associated with the conventional linear VAR model.

57

Figure 16 Monthly non-linear local projection IRFs of strategies’ moments cross-sectional dispersions over an expansion period (2013-2016) and over the subprime crisis v/s standard linear VAR Beta cross-sectional dispersion in expansion

Beta cross-sectional dispersion during the subprime crisis

Co-kurtosis cross-sectional dispersion in expansion

Co-kurtosis-sectional dispersion during the subprime crisis

Co-skewness cross-sectional dispersion in expansion

Co-skewness cross-sectional dispersion during the subprime crisis

Notes : These plots represent the impulse responses of the strategies’ cross sectional dispersions to their own shocks and to shocks of the explanatory variables. The outer dotted lines (red lines) embed the 90% confidence interval associated with the nonlinear local projection. The inner dotted line (blue line) is the impulse response function related to the nonlinear local projection while the solid line (pink line) is the impulse response function related to the conventional linear VAR model. For more detail on the computation of cross-sectional dispersions, see Racicot and Théoret (2016).

58

Mean return Annual Median Maximum Minimum Std. Dev. Skewness Kurtosis

CV 0.0211 8.44% 0.0241 0.1500 -0.2079 0.0449 -1.86 13.00 0.57***

DS 0.0238 9.54% 0.0334 0.0946 -0.1491 0.0410 -1.25 5.56 0.47***

ED 0.0249 9.96% 0.0291 0.1192 -0.1039 0.0443 -0.54 3.72 0.29***

EMN 0.0191 7.66% 0.0164 0.1111 -0.0579 0.0245 0.35 5.49 0.27***

FI 0.0216 8.66% 0.0230 0.0815 -0.0857 0.0239 -1.09 7.45 0.41***

FUT 0.0223 8.92% 0.0087 0.1583 -0.1034 0.0507 0.69 3.35 0.01

GROWTH 0.0272 10.87% 0.0277 0.3698 -0.1372 0.0816 0.71 5.77 0.19**

LSC 0.0177 7.10% 0.0210 0.0853 -0.0795 0.0249 -1.11 6.34 0.37***

MACRO 0.0133 5.33% 0.0108 0.1921 -0.1076 0.0411 0.76 7.04 0.04

SS -0.0066 -2.63% -0.0142 return -0.2984 0.0958 0.16 3.94 0.11*

VI 0.0270 10.80% 0.0275 0.1918 -0.1128 0.0579 0.00 3.59 0.18***

Mean 0.0192 7.69% 0.0189 0.1554 -0.1312 0.0482 -0.29 5.93 0.26***

GI 0.0220 8.79% 0.0230 0.1706 -0.0908 0.0418 0.26 4.47 0.25***

MKT 0.0167 6.68% 0.0228 0.1205 -0.3169 0.0656 -1.88 9.65 0.11

mean beta Std. Dev

0.253 0.1177

0.363 0.1138

0.429 0.1165

0.141 0.0489

0.157 0.1138

-0.018 0.0488

0.828 0.1996

0.147 0.0774

0.106 0.0351

-0.999 0.3446

0.578 0.1407

0.180 0.1233

0.370 0.0847

1.000

mean co-skewness Std. Dev

0.3330 0.3077

-0.5901 0.5348

-0.2444 0.1400

0.1249 0.5582

-0.0736 0.2987

0.2007 0.2209

-1.3895 0.6370

0.1807 0.1257

-0.2919 0.1499

-0.3645 0.2042

0.3069 0.1483

-0.1644 0.3023

0.0594 0.0423

-0.0897 0.2948

mean co-kurtosis Std. Dev

1.8156 1.3115

5.4175 3.1998

7.8545 4.6592

1.2149 0.7218

0.3742 0.2274

-4.6259 2.7447

7.8446 4.6460

5.0790 3.0094

1.7692 1.0537

-6.9348 4.0997

9.8434 5.8341

2.6956 2.8643

4.1171 2.4406

2.7528 1.5530

ρ

Highlights



We analyze the response of hedge fund portfolio co-moments to macroeconomic and financial shocks.



We rely on the nonlinear projection model à la Jordà and Kilian and Kim to monitor the asymmetries in the exposures of strategies’ return higher co-moments to risk factors.



The VAR nonlinear models show that a deleveraging process is taking place during the low regime for most strategies.



Investing in the short-selling and futures strategies may be beneficial in terms of portfolio diversification during crises.