Copulas and bivariate risk measures : an application to hedge funds

Transcription

1 Copulas ad bivariate risk measures : a applicatio to hedge fuds Rihab BEDOUI Makram BEN DBABIS Jauary 2009 Abstract With hedge fuds, maagers develop risk maagemet models that maily aim to play o the effect of decorrelatio. I order to achieve this goal, compaies use the correlatio coefficiet as a idicator for measurig depedecies existig betwee (i) the various hedge fuds strategies ad share idex returs ad (ii) hedge fuds strategies agaist each other. Otherwise, copulas are a statistic tool to model the depedece i a realistic ad less restrictive way, takig better accout of the stylized facts i fiace. This paper is a practical implemetatio of the copulas theory to model depedece betwee differet hedge fud strategies ad share idex returs ad betwee these strategies i relatio to each other o a "ormal" period ad a period durig which the market tred is dowward. Our approach based o copulas allows us to determie the bivariate VaR level curves ad to study extremal depedece betwee hedge fuds strategies ad share idex returs through the use of some tail depedece measures which ca be made ito useful portfolio maagemet tools. JEL classificatio : C13; C14; C15; G23. Keywords : Hedge fud strategies, share idex, depedece, copula, tail depedece, bivariate Value at Risk. EcoomiX, Uiversity of Paris Ouest Naterre La Défese, 200 aveue de la République, Naterre Cedex Frace. [email protected]. Tel: , Fax: Special thaks to the Professor Valérie Migo for support ad helpful commets. SDFI, Uiversity of Paris Dauphie ad CNP Isurace, Place du Maréchal de Lattre de Tassigy, PARIS CEDEX 16 Frace. [email protected]. Tel:

2 1 Itroductio Academic research bega to focus o sources of returs ad risk of hedge fuds i 1997 with the pioeerig work of Fug ad Hsieh (1997). Brow, Goetzma ad Ibbotso (1999) studied a sample of offshore hedge fuds betwee 1989 ad 1995 ad foud a positive risk adjusted. Their results did ot highlight either the effect of maagers talet, or the persistece of the performace of some others. Other studies have focused o the performace of hedge fuds without takig ito accout the differet factors, styles ad characteristics associated with abormal returs. Capocci (2004) argued that may hedge fud strategies are desiged to be weakly correlated with share idex returs ad bod idex returs ad he also studied the correlatio betwee various hedge fud strategies. He showed that the correlatio betwee hedge fud strategies ad share ad bod idexes varies greatly from oe strategy to aother ad betwee the various strategies. The high yield idex is more strogly correlated with most idexes. To ivestigate the role of the period uder study, Capocci (2004) studied the correlatios o the sub-period Jauary December 2002 (bear market) ad he oticed that correlatios geerally ted to decrease comparig to the "ormal" period eve if this is ot true for all idexes ( for istace short sellig or emergig market, see Capocci ad Hüber (2003) ad Capocci ad Mahieu (2003)). Capocci (2004) foud that the short sellig idex is egatively correlated with all other strategies whe it was ot with the share idex. Moreover, the correlatios betwee idividual strategies remai relatively the same for the two periods. All strategies (without cosiderig the short sellig strategy) are positively correlated with the share idex but the mortgage backed idexes, equity ad multi-market strategy are oly very weakly correlated as well as these strategies have little or o correlatio with the bod idex ad the raw material idex. However, this depedece coefficiet which is ofte used by practitioers has several disadvatages. Ideed, the correlatio coefficiet is ot defied if the secod order momets of radom variables are ot fiite. Besides, a correlatio coefficiet of zero does ot ecessarily imply idepedece betwee the variables studied. I additio, we ca measure oly liear correlatio usig this coefficiet. Furthermore, the liear correlatio coefficiet is ot ivariat by a cotiuous ad icreasig fuctio ( such as the logarithm fuctio, Embrechts et al (1999)) ad it does ot take ito accout the depedece of extremal values. The study of comovemets betwee hedge fud strategies o the oe had ad comovemets betwee them ad the share idex o the other had, has bee made usig the correlatio coefficiet. This paper proposes to use a more appropriate methodology based o copulas theory. Our aim is firstly to model the structure of depedece betwee the returs of differet hedge fud strategies ad secodly betwee returs of each strategy ad the market o two periods: a "ormal" period ad a period represetig the occurrece of a rare ad extreme evet (whe the market tred is dowward) as well as modellig tail depedece betwee these variables through the use of extremal depedece coefficiets. Moreover, we are iterested i determiig the level curves of the bivariate Value at Risk (VaR) betwee (i) differet hedge fuds strategies ad the margial rate of substitutio (MRS) betwee the VaR of two hedge fuds strategies ad (ii) the VaR of a particular hedge fuds strategy ad the share idex for a give risk level by usig copulas theory. Hece, our methodology eables us specifically to take ito accout the extreme values ad to study the impact of depedece o various measures of risk without makig restrictive assumptios about the liearity ad mootoocity of these series. Copulas fuctios are a statistical tool which has may advatages. First, copulas make it possible to determie the ature of depedece of the series, be it liear or ot, mootoe or ot. I additio to the fact that they offer a great flexibility i the implemetatio of the multivariate 2

3 aalysis, copulas authorize a wider selectio of the margial distributios of the fiacial series. Secod, they allow a less baal represetatio of the statistical depedece i fiace based o the traditioal correlatio measure ( see Embrechts et al. (1999)). Third, they authorize less restrictive uivariate probability distributios which make it possible to better accoutig for the stylized facts i fiace (leptokurticity, asymmetry, tail depedece). Fourth, they cosider very geeral multivariate distributios, idepedetly of the laws of the margial oes which ca have differet laws ad be uspecified. Furthermore, the copulas approach eables us to ease the implemetatio of multivariate models. Ideed, this approach allows the decompositio of the multidimesioal law ito its uivariate margial fuctios ad a depedece fuctio that would make possible extesios of some results obtaied i the uivariate case to the multivariate case. Hece, copula is a exhaustive statistic of the depedece. The structure of the paper is as follows. Sectio 2 describes the cocept of copulas ad their basic properties. Sectio 3 presets the empirical aspect of our work. We preset the hedge fuds data ad highlight a certai umber of depedecies betwee the share idex ad the various hedge fuds strategies, as well as depedeces existig betwee the various strategies. We idetify the copulas that model these depedeces over two differet periods: oe "ormal" period ad a period durig which the market tred is dowward. Moreover, this modelig of the depedeces will eable us to determie the level curves of the bivariate VaR betwee the hedge fuds strategies. Lastly, we evaluate tail depedeces betwee the market idex, the hedge fuds idex ad the strategy dedicated shorts through the use of various extremal depedece coefficiets. Sectio 4 is devoted to the coclusio. 2 Depedet Models : a copula approach A copula is a multidimesioal uiform distributio. It is a relatively old statistical tool itroduced by Sklar (1959), brought up to date by Geest et Mackay (1986). As defied by Nelse (1998), "Copulas are fuctios that joi or couple multivariate fuctios to their oe dimesioal margis. Copulas are distributio fuctios whose oe dimesioal margis are uiform". I what follows ad with referece to the work of Nelse (1998), Geest & MacKay (1986), Deuit & Charpetier (2004),we attempt to give a more precise defiitio to copulas ad to preset some mai properties. 2.1 Defiitio of a bivariate copula A bivariate copula C is 2 icreasig fuctio o the uit 2-cube I 2 = [0, 1] [0, 1]. Formally, it is a fuctio C : I 2 I with the followig properties: : For all u, v I, C (u, 0) = 0 = C (0, v) ad C (u, 1) = u, C (1, v) = v For all u 1, u 2, v 1 ad v 2 I such as u 1 u 2 ad v 1 v 2 : C (u 2, v 2 ) C (u 1, v 2 ) C (u 2, v 1 ) + C (u 1, v 1 ) 0 Thus: u 2 u 1 v 2 v 1 C ( x, y ) 0 3

4 2.2 Basic properties of Copula fuctio Theorem of Sklar Let H be a bivariate cumulative desity fuctio with uivariate margial cumulative desity fuctios F ad G. There exists a copula C such that for all x, y R : H ( x, y ) = C ( F (x), G ( y )) If F are G cotiuous, the C is uique. This theorem shows that ay bivariate cumulative desity fuctio H ca be writte i the form of a copula fuctio. Therefore, it is possible to costruct a wide rage of multivariate distributios by choosig the margial distributios ad a apropriate copula Property of Ivariace Let X 1 ad X 2 be two cotiuous radom variables with margis F 1 et F 2, liked by a copula C. Let h 1 ad h 2 two strictly icreasig fuctios,the : Theorem 1 (Fréchet bouds) C (F 1 (h 1 (x 1 )), F 2 (h 2 (x 2 ))) = C (F 1 (x 1 ), F 2 (x 2 )) Let C be a bivariate copula, the for all (u, v) Dom C : max (u + v 1, 0) C (u, v) mi (u, v) Fréchet (1951) shows that there exists upper ad lower bouds for a copula fuctio. I two dimesios, both of the Fréchet bouds are copulas themselves, but as soo as the dimesio icreases, the Fréchet lower boud is o loger -icreasig fuctio Theorem 2 Let C be a copula, for all u 1, u 2, v 1, v 2 Dom C : C (u 2, v 2 ) C (u 1, v 1 ) u 1 v 2 + u 1 v 1 for all a I : t C (t, a) is o -decreasig: Horizotal sectio of C i a t C (a, t) is o-decreasig : Vertical sectio of C i a 3 hedge fuds ad study of depedeces This sectio is devoted to the empirical part of our work. After presetig the data, we seek to highlight a umber of depedecies betwee the share idex ad the hedge fud idex o the oe had, ad the differet hedge fuds strategies o the other had. We select suitable copulas represetig these depedecies over two differet periods: a "ormal" period ad a period durig which the market tred is dowward. Besides, the copulas approach eables us ot oly to determie the level curves of the bivariate VaR betwee the hedge fuds strategies but also to evaluate tail depedeces betwee the market idex, the hedge fuds idex ad the strategy dedicated short via extremal depedece coeficiets. 4

5 3.1 Data The period uder study exteds from Jauary 1994 to December Regardig the data of alterative assets, we use the historical returs of the 13 mothly idexes divided ito 2 categories: a overall idex of hedge fuds for all strategies, ad a set of idexes represetig the 12 strategies of hedge fuds that make up the database CSFB / TREMONT. Accordig to Fug ad Hsieh (2000), these idexes are less affected by the survivor bias tha the idividual fuds data. Ulike other idexes, CSFB / Tremot idexes take ito accout the retur of the committee weighted by the size of fuds i the basket of fuds. To characterize the share market, we use the iteratioal idex share MSCI World whose returs are take from the TASS database. 3.2 Modelig depedece over the etire period We carry out the study of the depedece structure betwee the hedge fud idexes ad share idex. We cosider two periods, a "ormal" period ad a period durig which the market tred is dowward, i order to aalyse the deformatio of the structure of depedece. A goodess of fit test allows us to validate the choice of copula selected. The search of depedecies focuses o both hedge fuds strategies ad share idex as well as o hedge fud strategies i relatio to each other. Through the applicatio of various tests of adjustmet, it appears that the ormal distributio provides the best adjustmet for the cosidered variables. Figure 1 shows the adjustmet of the distributio of share ad hedge fuds idexes to a Gaussia distributio usig two graphic approaches : respectively hitograms goodess of fit ad Quatile-Quatile Plot (QQPlot). We Figure 1: Adjustemet of the share idex distributio ad the hedge fuds distributio to a Gaussia distributio remark more "picked" cetral values ad heavier left tails of share ad hedge fuds idexes 5

6 distributios tha the Gaussia desity fuctio. I fact, the empirical distributio presets rare observatios with a decrease slower tha the expoetial decrease of the ormal distributio. However the use of the Jarque-Bera test of the ull hypothesis that the sample i the empirical vector comes from a ormal distributio with ukow mea ad variace, agaist the alterative that it does ot come from a ormal distributio, leads us to assume a ormal distributio for the share ad hedge fuds idexes Graphical aalysis The choice of the suitable copula is the first difficulty i the implemetatio of modelig depedece. Figures 2, 3 ad 4 make it possible to apprehed the form of the depedeces which exist o the oe had betwee the share idex, the hedge fuds idex ad the hedge fuds strategies ad the structure of depedece betwee the differet strategies hedge fuds o the other had. I additio to the sig ad the itesity of the depedeces, these graphs provide us a first idicatio o the tail depedece which we will treat thereafter. Figure 2: Scatter plot of the hedge fud idex returs ad the share idex returs The set of poits is very close to the first bisector, illustratig a positive depedece betwee the share idex ad the hedge fuds idex. Moreover, the set of poits is very cocetrated o the 2 dials of the first bisector. Positive values for oe variable coicide with positive values for the other variable. This result seems cotradictory to what is expected sice the fuds maagers urge the lack of correlatio betwee the share market ad hedge fuds returs. 6

7 Figure 3: Scatter plot of the Dedicated Short strategy ad the share idex The set of poits is very close to the secod bisector ad the upper tail of the distributio is highly cocetrated. The largest values i the upper tail for oe variable coicide with large values of the same sig for the other variable. We ote that the depedece betwee the share idex ad the dedicated short idex is egative. This is ot surprisig sice this strategy aims to maitai a short positio o assets. Figure 4: Scatter plot of the share idex, the Dedicated Short strategy ad other hedge fuds strategies 7

8 Figure 4 shows that there exists positive depedeces betwee share idex ad the Equity Market strategy, the Multi-strategy ad the Emergig Market strategy as well as a egative depedece betwee the Dedicated Short Bias strategy ad the Equity Market Strategy. Relyig o this graphical approach (Figures 2, 3 ad 4) as a first step, the ormal copula seems to be the most suitable copula that models the form of depedece betwee the share idex ad hedge fud idex ad betwee hedge fud strategies i relatio to each other. We will use differet goodess-of-fit tests to choose the adequate copula to our variables Estimatio of copulas parameters The Normal copula is defied as follows. Let Φ 1 be the iverse foctio of a stadard ormal distributio ad H α the distributio fuctio of the bivariate ormal distributio with correlatio coefficiet α. The ormal copula C N α is : C N α (u, v) = H ( α Φ 1 (u), Φ 1 (v) ) f or u, v [0, 1], α [ 1, 1]. There is o explicit form for the distributio fuctio of the bivariate ormal law. To estimate the parameters of copulas which model the depedecies betwee these variables, we use the parametric method Iferece Fuctios for Margis (IFM). This method was itroduced by Shih & Louis (1995). It ca reduce the estimatio problem i two steps : - The estimatio of the parameters θ 1,..., θ of the margis. - The estimatio of the parameter θ C of the copula. Let us deote : θ = (θ 1,..., θ, θ C ) We begi by determiig the maximum likelihood estimators of margis: θ i = argmax θi k=1 ( f i x k i, θ ) i We the itroduce these estimators i the copula part of the log-likelihood fuctio, which leads to: θ C = argmax θc l ( c ( ( F 1 x k 1, θ ) ( 1,..., F1 x k, θ ) )), θc k=1 Other procedures ca be applied, such as o-parametric estimatio of margis followed by maximum likelihood estimatio for the parameter of the copula. We refer the readers to Geest et al. (1995) ad Shih ad Louis (1995) procedures. There are other o-parametric estimatio methods such as empirical copula (see Deheuvels (1979)) ad parametric estimatio methods like the momets method, the maximum likelihood method ad the CML Caoical Maximum Likelihood (see Bouye ad al. (2000)). The IFM method we use has the advatage of beig based o calculatios less cumbersome tha maximum likelihood procedure. Nevertheless, the determiatio of iformatio matrix of Godambe ca be very complicated because it creates multiple derived calculatios. For the same reasos as for the maximum likelihood method, a possible error i margis estimatio i this method may lead to erroeous estimator of the copula parameter. 8

9 Tables 1 to 3 iclude the values of the estimated parameters of copulas. Table 1 : Estimated copulas parameters of the share idex ad hedge fuds strategies The share idex hedge fuds idex Dedicated short strategy Equity market strategy Muti Strategy emergig markets strategy Table 2 :Estimated copulas parameters of the dedicated short strategy ad the other hedge fuds strategies Dedicated short strategy Equity market strategy Muti Strategy emergig markets strategy Table 3 : Estimated copulas parameters of hedge fuds strategies Muti Strategy Dedicated short strategy Equity market strategy We ote a positive depedece betwee the share idex ad the hedge fuds idex o the oe had ad the share idex with the equity market strategy ad multi-strategy o the other had. However, the stregth of this depedece differs from oe strategy to aother. There is a egative depedece betwee the share idex ad dedicated short strategy. The upper tail of the distributio is very cocetrated. Our results also show a egative depedece betwee dedicated short strategy ad the other hedge fuds strategies ad a positive depedece betwee the equity market strategy ad multi-strategy. This meas that returs of share idex coicide with the returs of the same sig for the hedge fuds idex, the equity market strategy ad multi-strategy. Therefore losses ad gais i these variables occur simultaeously. By cotrast, returs of the dedicated short strategy coicide with the returs of the opposite sig for the share idex, hedge fuds idex, the equity market strategy ad multi-strategy. Thus, losses of the dedicated short strategy occur simultaeously with gais of the share idex, hedge fuds idex, the equity market strategy ad multi-strategy. These observatios offer useful portfolio maagemet tools Goodess of fit tests Choosig the suitable copulas that model the depedecies betwee our radom variables is of paramout importace. The questio is the followig : what is the best structure of depedece that ca be adapted to the pheomeo studied? Goodess of fit tests for copulas are relatively recet. There are few papers o this issue, but the field is i costat developmet. Let cosider a sample of radom vectors iid X i R d as X i = ( X 1,i,..., X d,i ) X iid. Let H be the distributio fuctio of X ad C the copula of X, thus : 9

10 H (x 1,..., x d ) = C (F 1 (x 1 ),..., F d (x d )) Geerally, a statistic test distiguishes betwee two assumptios : Null hypothesis H = H 0 or C = C 0 Alterative hypothesis H H 0 or C C 0 A first solutio is to compare the empirical copula defied as : Ĉ ( k1,..., k ) p = 1 card { } i R 1,i k 1,..., R p,i k p to the estimated parametric copula C, where R j,i, j = 1,..., p & i = 1,..., deotes the rak of X j,i amog the observatios X j,1,..., X j,. Depedecy accepted is oe that esures that C is as close as possible to Ĉ. Let { ( ) x k, y k } be a sample of size of pairs of radom variables. The empirical copula is the k=1 fuctio C defied as : 1 C ( i, j ) = Number of pairs ( x, y ) i the sample as x x i, y y j The empirical desity fuctio of the copula C is give by: ( i c, j ) = 1 if ( x i, y j ) is a elemet of the sample. 0 otherwise The lik betwee C ad c is defied as follows : 2 : c ( i, j ) = C ( i, j ) C ( i, j ) = i p=1 q=1 j ( p c, q ) ( i 1 C, j ) ( i C, j 1 ) ( i 1 + C, j 1 ) The empirical copulas are useful to provide o-parametric estimators of measures of depedece such as the ρ of Spearma, ad the τ of Kedall. Ideed, these two measures are determied empirically as follows: ρ = i=1 ( ( i C, j ) i j ) 1 The empirical copulas were origially itroduced by (Deheuvels (1979)). 2 To demostrate this propositio, see Nelse (1998). j=1 10

11 τ = 2 i 1 j 1 ( ( i c 1, j ) ( p c, q p=1 q=1 ) ( i c, q ) ( p c, j )) Figure 5 presets the empirical copula that models the depedece betwee the hedge fud idex (IHF). It also reports the parametric copula allowig the adjustmet of the empirical copula to the ormal copula of parameter alpha = estimated by the IFM method. We ote a good fit betwee the empirical copula ad the ormal oe. This validates the choice of the ormal copula to model depedece betwee share ad hedge fud idexes. Figure 5: Empirical copula versus parametric copula for the hedge fuds idex (IHF) ad the share idex Figure 6 shows the desity of the copula that models depedece betwee the hedge fuds idex ad the share idex. There is a strog cocetratio i the upper tail of the distributio, ad to a lesser extet i the lower tail distributio which idicates that positive largest values from the share idex ad hedge fuds idex occur together. By cotrast, the probability of the simultaous occurece of egative largest values is smaller tha the probability of the positive oes. I additio to these graphical methods, we use the goodess of fit test of Geest, Quessy ad Remillard (2006) to validate the choice of copula selected. Geest, Quessy & Rémillard (2006) have expaded the work of Wag ad Wells (2000) providig alterative statistics give by : ad S = 1 0 K (t) 2 k (θ, t) dt where k (θ, t) is thedesity f uctio associated to K (θ, t) T = sup 0 t 1 K (t) S is based o the distace of Cramèr Vo Mises, while the statistic T is based o that of Kolmogorov Smirov. It is importat to ote that these statistics have several advatages: - Some simple formulas are available for S ad T i terms of the rak of obsevatios, which 11

12 Figure 6: Desity fuctio of the copula hedge fuds idex (IHF)- share idex is ot the case with the origial statistics of Wag ad Wells (2000). - The selectio procedures are ot iflueced by a exteral costat whose selectio ad ifluece o the limit distributio of the test statistic were ot cosidered by Wag ad Wells (2000). - S ad T distributios ca be determied ot oly for archimedia bivariate copulas but also for copulas with dimesio greater tha 2 ad for copulas satisfyig the geeral coditio of regularity. - The parametric bootstrap method is valid ad ca be used to approximate the P-values associated with fuctioal K ad i particular with S ad T. Performig a parametric bootstrap for statistics S ad T, it is possible to obtai approximate thresholds associated with these statistical assumptios of type C (C θ ), ie : the copula C associated to F belogs to the family of copulas (C θ ). I the paper of Geest ad Rémillard (2006), oe cofirms the validity of this bootstrap approach for both types of the most commo goodess of fit tests : (i) tests where we compare the distace betwee a multivariate empirical distributio ad parametric estimatio uder the ull hypothesis or (ii) those where we compare the distace betwee the empirical estimators ad parametric uivariate pseudoobservatios such as W i, obtaied through the itegral trasformatio of probability. Geest et al. (2006) showed that : S = j=1 i=1 K 2 ( ) ( ( j K θ, j + 1 ) ( K θ, j )) 1 ( ) ( j K (K 2 θ, j + 1 ) ( K 2 θ, j )) 12

13 ad T = ( ( ) ( j max i=0,1 & 0 j 1 K K θ, j + 1 ) ) The test cocludes to the rejectio of the ull hypothesis H 0 : C = C 0 whe the observed values S ad T are above the quatile of order 1 α of their distributios uder the assumptio H 0. After applyig this test, the ormal copula is the suitable copula modelig the depedece betwee the share idex ad the differet hedge fuds strategies ad betwee these strategies i relatio to each other. Note that there exists other goodess of fit tests : the parametric bootstrap method i the approach of Wag ad Wells (2000), the parametric bootstrap method based o C θ ( see Geest ad Rémillard (2005)), the goodess of fit test of copulas based o the trasformatio of Rosemblatt (old fashio) (see Breyma ad al. (2003)) ad the goodess of fit test of copulas based o the trasformatio of Rosemblatt (ew fashio) (see Klugma ad Parsa (1999), Ghoudi ad Rémillard (2004), Geest ad al. (2008). 3.3 Modelig depedece o the subperiod Jauary 2000-December 2002 I order to try to aswer the questio of the depedece structure variatio whe the market tred is dowward, a first possibility is to estimate the parameters of bivariate copulas existig betwee the share idex, hedge fuds idex ad hedge fuds strategies via the Iferece Fuctios for Margis method. As previously, we fid that the ormal copula is the suitable copula that models the depedece structure betwee the share idex, hedge fuds strategies ad hedge fud idex. Moreover, this modelig of depedece provides extra iformatio o the evolutio of the degree of depedece i relatio with the market treds. Tables 4, 5 ad 6 show the differet values of the estimated copulas parameters i a crisis period (Jauary 2000-December 2002). Table 4 : Estimated copulas parameters of the share idex ad hedge fuds strategies i a crisis period Share idex Hedge fud idex Dedicated short strategy Equity market strategy Muti Strategy emergig markets strategy Table 5 : Estimated copulas parameter of the dedicated short strategy ad the emergig market strategy i a crisis period Dedicated short strategy emergig markets strategy

14 Table 6 : Estimated copulas parameter of the equity market strategy ad the multi strategy i a crisis period Equity market strategy Multi Strategy These results show that the depedece parameter has icreased for the ormal copulas that model (i) depedece betwee the share idex ad the strategies dedicated short, multi strategy ad emergig market ad (ii) the depedecies betwee the hedge fuds strategies. I other words, more the copula parameter is high, the greater the depedecy is. This meas that the share idex, the dedicated short strategy ad emergig market strategy have become more depedet whe the market tred is dowward. However, durig the "ormal" period, the depedece betwee the share idex ad the equity market strategy was positive but the sig of depedece has becomig egative, whe a extreme evet had occurred ( September 11, 2001). It is the same for the sig of depedece betwee the equity market strategy ad the multi-strategy, which idicates that losses ad, respectively, gais of the share idex have a higher probability to coicide with gais ad, respectively, losses of the dedicated short strategy ad emereget market strategy whe the market tred is dowward tha i the "ormal" period. While i the "ormal" period, the returs of the equity market strategy coicide with returs of the same sig for the share idex ad the multi-strategy, whe the market tred is dowward the returs of the equity market strategy occur simultaously with returs of the opposite sig for the share idex ad the multi-strategy. 3.4 Copulas ad bivariate VaR Value at Risk (VaR) has become the stadard measure that fiacial aalysts use to asses market risk. The VaR is defied as the maximum potetial loss due to adverse market movemets for a give probability. The use of copulas allows us to determie the level curves of the bivariate VaR ad examie for a give threshold level, the margial rate of substitutio betwee the VaR of two uivariate risks. Ideed, sice we have margial distributios of returs of differet hedge fud strategies, ad the share idex, it is possible to trace the level curves correspodig to the miimum copula (atimootoicity), maximum copula (comootoicity) ad the idepedece copula. Let r A ad r B be the returs of the series A ad B. Let F A ad F B be the uivariate distributio fuctios of returs of respectively r A ad r B. We have, for ay threshold α [0, 1] : { (ra, r B ) ; max (F A (r A ) + F B (r B ) 1, 0) = α}, ati mootoicity; {(r A, r B ) ; F A (r A ).F B (r B ) = α}, idepedece; {(r A, r B ) ; mi (F A (r A ), F B (r B )) = α}, comootoicity. The level curves from the empirical copula are give by : {(r A, r B ) ; C (F A (r A ), F B (r B )) = α} The level curves are used to determie the margial rate of substitutio betwee the two uivariate VaR. The more the empirical curve is high approachig the case ati-mootoicity, the more is the depedece betwee the returs A et B ad the more is the compesatio effect. However, 14

15 the more the curves are close to their lower limit, correspodig to the case of comootoicity or positive depedece, the more the returs ted to move i the same directio, the depedece betwee losses (correlatio) is therefore very high. Regardig the curves of multiplicatio, they correspod to diversificatio (see Cherubii ad Luciao (2001)). We determie the 95% level curves of the bivariate VaR betwee the share idex ad the hedge fuds idex ( Figure 7) ad the level curves of bivariate VaR of differet hedge fuds strategies ( Figures 8 ad 9). We ote from the Figure 7 that the 95% level curve of the empirical copula is closer to Figure 7: Bivariate VaR of share idex ad hedge fuds idex that correspodig to the case co-mootoy (the lower limit) or positive depedece. It follows that the returs of the share idex ad the hedge fuds idex operate i the same directio, the correlatio betwee the losses is therefore high. As a cosequece, it is preferably ot to put these two elemets ito a sigle portfolio. We remark accordig to the Figure 8 that the 95% level curve of the empirical copula is closer to the level curve of multiplicatio or idepedece which is the case of diversificatio: the losses of the Dedicated short strategy ad the Equity Market Neutral strategy are ot correlated. I order to guaratee diversificatio, it is best to put these two strategies i the same portfolio. Fially, we ote from Figure 9 that the 95% level curve of the empirical copula is high ad closer to that of the ati-mootoicity case. Cosequetly, the depedece betwee the returs of the Dedicated Short strategy ad the Log Short Equity strategy is egative ad the effect of adjustmet ad compesatio will take place. To esure a better allocatio i its portfolio, it is advisable to combie these two strategies i the same portfolio of fuds. We ote that the maximum potetial loss, calculated through the risk measure VaR is higher i the depedece case compared to the case of idepedece. This meas that assumig the o-correlatio betwee hedge fud strategies ad share idex uderestimate the portfolio risk measured by VaR. 15

16 Figure 8: Bivariate VaR of Dedicated short strategy ad Equity Market Neutral strategy Figure 9: Bivariate VaR of Dedicated Short strategy ad the Log Short Equity strategy 16

17 3.5 Tail depedece : extremal coefficiets χ ad χ The study of the tail depedece allows us to describe the depedece i the tails of distributio ad to examie the simultaeous occurrece of extreme values. We use two coefficiets of tail depedece eablig us to measure the asymptotic depedece betwee hedge fuds strategies ad the share idex. The depedece measures χ ad χ were itroduced by Coles et al. (1999). After trasformatio of (X,Y)which are the two series to study ito (U, V) havig uiform margial distributios, we get : 1 Pr (U < u, V < u) logc (u, u) Pr(V > u U > u) = 2 2 f or 0 u 1. 1 Pr (U < u) log (u) The depedece measure χ (u) is defied as χ (u) = 2 logc (u, u) log (u) f or 0 u 1. The fuctio χ (u) is thus a quatile-depedet measure of depedece. The sig of χ (u) determies whether the variables are positively or egatively associated to the quatile level u. - χ (u) is bouded as follows : 2 log (2u 1) log (u) χ (u) 1 The lower boud is iterpreted as for u 1/2, ad 0 for u = 1. A sigle parameter measure of extremal depedece is give by χ = lim u 1 χ (u) Loosely stated, χ is the probability of oe variable beig extreme give that the other is extreme. - I the case χ = 0, the variables are asymptotically idepedet. Thus, we require a complemetary depedece measure to assess extremal depedece withi the class of asymptotically idepedet variables. By aalogy with the defiitio of χ (u), comparaiso of joit ad margial survivor fuctios of (U, V) leads to : χ (u) = 2logPr (U > u) 2log (1 u) 1 = 1 f or 0 u 1, logpr (U > u, V > u) logc (u, u) where 1 < χ (u) < 1 for all 0 u 1. To focus o extremal characteristics, we also defie : χ = lim u 1 χ (u) The measures χ ad χ are related to the Ledford & Taw (1996, 1998) characterisatio of the joit tail behavior, through η, the tail depedece coefficiet ad L (t), the relative stregth of limitig depedece : χ = 2η 1 17

18 { c i f χ=1 ad L(t) c>0 as t, χ = 0 i f χ=1 ad L(t) 0 as t, ad i f χ<1. We have the followig classificatio : - χ [0, 1]: the set (0, 1] correspods to asymptotic depedece; - χ [ 1, 1] ; the set [ 1; 1) correspods to asymptotic idepedece. Thus the complete pair (χ, χ) is required as a summary of extremal depedece: - (χ > 0; χ = 1) implies asymptotic depedece, i which case the values of χ ad χ determie the stregth of depedece withi the class; - (χ = 0; χ < 1) implies asymptotic idepedece. I practice, we first assess χ : - χ < 1 asymptotic idepedece ; - χ = 1 asymptotic depedece, here we must also estimate χ. We calculate the extremal depedece coefficiet χ i order to examie the simultaeous occurrece of extreme values ad to measure the asymptotic depedece of the miimum ad the maximum betwee the share idex ad the hedge fuds idex as well as the asymptotic depedece betwee the share idex ad the strategy Dedicated Short. Table 7 : Measure χ Share idex hedge fuds idex dedicated short Strategy The results which are grouped i Table 7, lead us to coclude that the value of χ is lower tha 1 for the share idex ad the hedge fuds idex ad for the share idex ad strategy Dedicated Short. Therefore, there is a asymptotic idepedece betwee the share idex ad the hedge fuds idex o the oe had ad betwee the share idex ad strategy Dedicated Short o the other had. Loosely speakig, kowig extreme loss for the share idex, there is zero probability that loss with a comparable itesity to take place cocurretly for the hedge fud idex ad the Dedicated Short strategy. 4 Coclusio There is a regular expasio of the hedge fuds debate towards the fiacial istitutios. Faced with the ever-chagig eviromet, maagers of hedge fuds are expected to adapt by modelig the structure of depedece betwee differet hedge fud strategies i relatio to each other ad i relatio to the stock market. This paper provides elemets ad techiques o this debate by relyig o the theory of copulas. We have proposed a study of depedece ad tail depedece of the differet hedge fuds strategies ad the share idex o oe had ad of the differet hedge fuds strategies i relatio to each other o the other had. We also assess the risk of differet hedge fuds strategies i various cases of depedece through the use of bivariate Value at Risk. Cotrary to what the fuds maagers precoize, our results highlighted a certai umber of 18

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