Centre for Investment Research Discussion Paper Series

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1 Cenre for Invesmen Research Discussion Paper Series Discussion Paper # 07-04* Regime Change and Converible Arbirage Risk Mark C. Huchinson Universiy College Cork Liam A. Gallagher Dublin Ciy Universiy Cenre for Invesmen Research O'Rahilly Building, Room 3.02 Universiy College Cork College Road Cork Ireland T +353 (0) /2765 F +353 (0) /3920 E cir@ucc.ie W *These Discussion Papers ofen represen preliminary or incomplee work, circulaed o encourage discussion and commens. Ciaion and use of such a paper should ake accoun of is provisional characer. A revised version may be available direcly from he auhor(s). 1

2 Regime Change and Converible Arbirage Risk Mark C. Huchinson * Accouning and Finance, Universiy College Cork, College Road, Cork, Ireland. Telephone: , m.huchinson@ucc.ie Liam A. Gallagher DCU Business School, Dublin Ciy Universiy, Dublin 9, Ireland. Telephone: , liam.gallagher@dcu.ie Absrac This paper analyses he daa generaing process of he converible arbirage hedge fund sraegy. Wihin a nonlinear framework, we allow for alernae regimes of converible arbirage risk using smooh ransiion auoregressive (STAR) models. In one regime he converible arbirage sraegy exhibis relaively large exposure o defaul and erm srucure risk facors and negaive alpha. In he alernae regime he sraegy exhibis relaively low exposure o marke risk facors and posiive alpha. Significanly, over he sample period he sraegy generally exiss in he low risk/high alpha regime. We sugges ha evidence repored in his paper accouns for abnormal reurns repored for he sraegy in previous sudies. Keywords: smooh ransiion, hedge fund, converible arbirage JEL classificaion: G11, G12, C32 1. Inroducion Academic lieraure on hedge fund performance has generally focused on linearly modelling he relaionship beween he reurns of hedge funds and he asse markes and coningen claims on hose asses in which hedge funds operae. Recenly, several sudies model he reurns of hese funds using echniques which are no resriced by assumpions of normaliy. * Corresponding auhor. We would like o hank Niall O Sullivan, Lucio Sarno and paricipans a he 2005 and 2006 European Financial Managemen Associaion Annual Meeings for helpful commens. We are also graeful o he Irish Research Council for he Humaniies and Social Sciences (IRCHSS) for research funding and SunGard Trading and Risk Sysems for providing converible bond daa and pricing sofware. 2

3 In his paper we are ineresed in invesigaing wheher an alernaive non-linear model specificaion increases efficiency in he modelling of converible arbirage hedge fund reurns. In paricular we focus on he smooh ransiion auoregressive (STAR) family of models which have he advanage, over alernaive non-linear regime swiching specificaions when modelling financial daa, of allowing a smooh ransiion beween regimes. 1 Many sudies of hedge funds have documened non-lineariy in heir reurns (See for example, Liang (1999), Agarwal and Naik (2000), Brooks and Ka (2001), Ka and Lu (2002) and Fung and Hsieh (1997, 2000)). One avenue of research has modelled his non-lineariy in a linear asse pricing framework using non-gaussian risk facors. Fung and Hsieh (2001, 2002) presen evidence of hedge fund sraegy payoffs sharing characerisics wih lookback sraddles, and Michell and Pulvino (2001) documen he reurns from a merger arbirage porfolio exhibiing similar characerisics o a shor posiion in a sock index pu opion. Using opion payoffs as risk facors, Agarwal and Naik (2004) demonsrae he non-linear relaionship beween hedge fund reurns and risk facors. Modelling he reurns of converible arbirage hedge funds Huchinson and Gallagher (2007) and Agarwal, Fung, Loon and Naik (2007) consruc facor porfolios mimicking converible arbirage invesmens. In addiion o he linear facor model lieraure several sudies uilize models whose funcional specificaion, raher han facor specificaion, capures he non-normal characerisics of hedge funds. Raher han specifying facors wih non-normal disribuions, hese sudies relax he assumpion of a linear relaionship beween he risk facor and hedge fund reurns. Ka and Miffre (2005) employ a condiional model of hedge fund reurns which allows he risk coefficiens and alpha o vary. Kazemi and Schneeweis (2003) also aemp o explicily address he dynamics in hedge fund rading sraegies by specifying condiional models of hedge fund performance. Kazemi and Schneeweis (2003) employ he sochasic discoun facor model which 1 In financial markes wih many paricipans operaing independenly and a differen ime horizons, movemens in asse prices are likely o be smooh. 3

4 has previously been employed in he muual fund lieraure. Alernaely, Amin and Ka (2003), imposing zero resricions on he disribuion of he funds reurns, evaluae hedge funds from a coningen claims perspecive. STAR models were developed by Teräsvirsa and Anderson (1992) for modelling non-lineariies in he business cycle and offer several advanages over a Hamilon (1989) Markov swiching model. STAR models incorporae a leas wo alernae risk regimes, allowing for a smooh ransiion from one risk regime o anoher. When esimaing he STAR model no ex ane knowledge of he hreshold variable is required. These models have been specified exensively o model economic ime series (see for example Saranis (1999), Skalin and Terasvira (1999), Ocal and Osborn (2000) and Holmes and Maghrebi (2004)) and sock reurns (see for example McMillan (2001), Bradley and Jansen (2004) and Bredin and Hyde (2007)). Overall, exising academic sudies find ha converible arbirage hedge funds generae significan abnormal reurns. In sudies of general hedge fund performance, Capocci and Hübner (2004) and Fung and Hsieh (2002) provide some evidence of converible arbirage performance. Capocci and Hübner (2004) specify a linear facor model o model he reurns of several hedge fund sraegies and esimae ha converible arbirage hedge funds earn an abnormal reurn of 0.4% per monh. Fung and Hsieh (2002) esimae he converible arbirage hedge fund index generaes alpha of 0.7% per monh. Focusing exclusively on converible arbirage hedge funds Huchinson and Gallagher (2007) find evidence of individual fund abnormal performance bu no abnormal reurns in he hedge fund indices. Agarwal, Fung, Loon and Naik (2007) documen posiive abnormal reurns which hey accoun for wih new issue converible bond under pricing daa. These mixed findings sugges ha eiher financial markes exhibi significan inefficiency in he pricing of converible bonds, or prior sudies have failed o specify a funcional model which correcly explains converible arbirage risk. Financial heory suggess ha he relaionship beween converible arbirage reurns and risk facors is non-linear. Being long a converible bond and shor an underlying sock, funds are hedged agains equiy marke risk bu are lef 4

5 exposed o a degree of downside defaul and erm srucure risk. When he converible bond is above a cerain hreshold i acs more like equiy han bond. However, when he converible bond falls in value i acs more like bond han equiy. Effecively, he converible arbirageur is shor a credi pu opion. 2 Highlighing his non-lineariy Agarwal and Naik (2004) provide evidence ha converible arbirage hedge fund indices reurns are posiively relaed o he payoff from a shor equiy index opion. In his paper evidence is presened of a non-linear relaionship beween converible arbirage hedge fund reurns and defaul and erm srucure risk facors. This non-linear relaionship is modelled using logisic smooh ransiion auoregressive (LSTAR) models. We also provide evidence ha he specificaion of hese models provides increases in efficiency over an alernae linear specificaion. Nine converible arbirage hedge fund series are modelled, including five hedge fund indices and four porfolios made up of individual converible arbirage hedge funds. To ensure he robusness of hese resuls he model is also specified for a simulaed converible arbirage porfolio and again evidence is presened supporing he hypohesis of non-lineariy in he relaionship beween he reurns of converible arbirage and risk facors. The remainder of his paper is organised as follows. The nex secion conains deails of he daa. Secion 3 provides a review of he smooh ransiion auoregressive models. Secion 4 provides deails of he esimaion resuls. Secion 5 concludes. 2. Daa In his secion of he paper we presen deails of he converible arbirage series and explanaory risk facors. To model he converible arbirage hedge fund sraegy we specify five indices of converible arbirage hedge funds, four porfolios made up of converible arbirage hedge funds 2 Some converible arbirage funds hold credi defaul swaps o hedge credi risk. However, Huchinson and Gallagher (2007) documen significan exposure amongs converible arbirage hedge funds o defaul and erm srucure risk facors. 5

6 from he HFR daabase and a simulaed converible arbirage porfolio. 3 All of he series have differen sar daes. The sample period runs from he sar of each series o December The indices specified are he CSFB Tremon Converible Arbirage Index, he HFRI Converible Arbirage Index, he Van Hedge Converible Arbirage Index, he Barclay Group Converible Arbirage Index and he CISDM Converible Arbirage Index. The CSFB Tremon Converible Arbirage Index is an asse weighed index (rebalanced quarerly) of converible arbirage hedge funds beginning in 1994, he CISDM Converible Arbirage Index represens he median fund performance, whereas he HFRI, Van Hedge and Barclay Group Converible Arbirage Indices are all equally weighed indices of fund performance. The four porfolios are: HFR EQL, an equally weighed porfolio of converible arbirage hedge funds; HFR LRG, an equally weighed porfolio made up of he larges funds, ranked by monh - 1 asses under managemen; HFR MID, an equally weighed porfolio made up of he mid ranking funds, ranked by monh -1 asses under managemen; and, HFR SML, an equally weighed porfolio made up of he smalles funds, ranked by monh -1 asses under managemen. Finally CBARB, he simulaed porfolio is an equally weighed porfolio consruced of long posiions in converible bonds combined wih dela neural hedged shor posiions in he underlying socks. For deails on he consrucion and saisical characerisics of his porfolio see Huchinson and Gallagher (2006, 2007). Descripive saisics and cross correlaions for he en converible arbirage excess reurn series are presened in Table 1 and Table 2. 5 Of he indices he Van Hedge index has he larges mean reurn, 0.83, and he CSFB Tremon index has he larges variance, The porfolios formed from he HFR daabase have similar mean reurns bu HFR SML, made up of smaller hedge funds has he larges variance, more han wice he magniude of he oher size porfolios. Finally all of he series exhibi posiive kurosis and, wih he excepion of he Barclay Group index, which 3 For a review of he differen hedge fund indices see Golz, Marellini and Vaissié (2007) 4 The number of observaions for each series varies and is repored in Table 1. 5 The one monh T-bill rae is specified for he risk free reurn. 6

7 does no cover Ocober 1998, negaive skewness. The cumulaive reurns of he en series are repored in Figure 1. The explanaory variables specified in his sudy are RMRF, SMB, HML, DEF and TERM. Huchinson and Gallagher (2006) provide evidence ha hese five equiy and bond marke facors drive he converible arbirage daa generaing process. RMRF, SMB and HML are Fama and French (1992, 1993) marke, size and book-o-marke facors, respecively. 6 DEF and TERM represen defaul and erm srucure risk facors (Chen, Roll and Ross (1986)). DEF is he difference beween he overall reurn on a marke porfolio of long-erm corporae bonds minus he long erm governmen bond reurn a monh. TERM is he facor proxy for unexpeced changes in ineres raes. I is consruced as he difference beween monhly long erm governmen bond reurn and he shor erm governmen bond reurn. Descripive saisics and cross correlaions of he risk facors are presened in Table 3. All of he mean reurns are posiive. Jacque and Bera (1987) saisics indicae ha four of he five risk facors exhibi nonnormaliy. DEF and TERM exhibi auocorrelaion. In he nex secion of he paper we discuss he STAR mehodology specified in his sudy o model converible arbirage reurns. 3. Mehodology This secion of he paper provides a review of hreshold model mehodology focusing on he smooh ransiion auoregressive (STAR) model firs proposed by Chan and Tong (1986) and exended by Teräsvira and Anderson (1992) for modelling non-lineariy in he business cycle. STAR models are specified in his sudy for hree principle reasons. (1) They incorporae wo alernae regimes, corresponding wih he heoreical relaionship beween converible arbirage reurns and risk facors. One regime where he porfolio is more exposed o defaul and erm srucure risk and a second regime where he porfolio is less exposed o defaul and erm 6 The daa on RMRF, SMB and HML are downloaded from Kenneh French s websie. 7

8 srucure risk and more exposed o he converible arbirage risk facor. (2) They incorporae a smooh ransiion from one risk regime o anoher. In financial markes wih many paricipans operaing independenly and a differen ime horizons, movemens in asse prices and risk weighings are likely o be smooh raher han sharp. (3) When esimaing he STAR model no ex ane knowledge of he hreshold variable level, c, is required. This hreshold level is esimaed simulaneously wih he coefficiens of he model. The only ex ane expecaion of he level of he hreshold is ha i lies beween he minimum and maximum of he hreshold variable. In his sudy we specify he one period lag of he converible arbirage series reurn as he hreshold variable. The converible arbirage series proxy aggregae hedged converible bonds held by arbirageurs. If he series generaes negaive reurns hen aggregae hedged converible bonds held by arbirageurs have fallen in value. This fall in value is caused eiher by a decrease in he value of he shor sock posiion in excess of he increase in he value of he long corporae bond posiion or, more likely, a decrease in he value of he long converible bond posiion in excess of he increase in he value of he shor sock posiion. When he one period lag of he converible arbirage benchmark reurn is below he hreshold level, converible bond prices and delas have decreased. As converible bond prices fall, we expec he arbirageur s porfolio o be more exposed o defaul and erm srucure risk. When he one period lag of he converible arbirage benchmark reurn is above he hreshold level, converible bond prices and delas increase and we expec he porfolio o exhibi less fixed income characerisics, wih relaively smaller coefficiens on he defaul and erm srucure risk facors. Consider he following NLAR model. y = α ' x + β ' x f ( z ) + e (1) 8

9 Where α = (α 0,, α m ), β = (β 0,, β m ), x = (y,, y -p ; x 1,, x k ) and he variable z is he ransiion variable. If f ( ) is a smooh coninuous funcion he auoregressive coefficien (α 1 + β 1 ) will change smoohly along wih he value of y -1. This ype of model is known as a smooh ransiion auoregressive (STAR) model. The wo paricularly useful forms of he STAR model ha allow for a varying degree of auoregressive decay are he LSTAR (Logisic-STAR) and ESTAR (Exponenial-STAR) models. Choosing f( z ) [1 exp( ( z c))] 1 = + γ yields he logisic STAR (LSTAR) model where γ is he smoohness parameer (i.e. he slope of he ransiion funcion) and c is he hreshold. In he limi as γ approaches zero or infiniy, he LSTAR model becomes a linear model since he value of f(z ) is consan. For inermediae values of γ, he degree of decay depends upon he value of z. As z approaches -, θ approaches 0 and he behaviour of y is given by y = α ' x + e. As z approaches +, θ approaches 1 and he behaviour of y is given by ( α ' + β ')x + e. Choosing f z = z c yields he exponenial STAR (ESTAR) model. For he 2 ( ) 1 exp( γ ( ) ) ESTAR model, as γ approaches infiniy or zero he model becomes a linear model as f(z ) becomes consan. Oherwise he model displays non-linear behaviour. I is imporan o noe ha he coefficiens for he ESTAR model are symmeric around z = c. As z approaches c, f(z ) approaches 0 and he behaviour of y is given by y = α ' x + e. As z moves furher from c, θ approaches 1 and he behaviour of y is given by ( α ' + β ')x + e. The esimaion of STAR models consiss of hree sages (Granger and Teräsvira (1993)): (a) Specificaion of a linear model. The iniial sep requires he specificaion of he linear model (4). y = α + β x + ε (4) 9

10 Where y is he excess reurn on he hedge fund index, and x is a marix of converible arbirage risk facors. (b) Tesing lineariy The second sep involves esing lineariy agains STAR models using he linear model specified in (a) as he null. To carry ou his es he auxiliary regression is esimaed: u = β ' x + β ' xz + β ' xz + β ' xz (5) Where he values of u are he residuals of he linear model specified in he firs sep and z is he ransiion variable. The null hypohesis of lineariy is H 0 : β 1 = β 2 = β 3 = 0. 7 (c) Choosing beween LSTAR and ESTAR If lineariy is rejeced he selecion beween LSTAR and ESTAR models is based on he following series of nesed F ess. H3: β 3 = 0 (6) H2: β 2 = 0 β 3 = 0 (7) H1: β 1 = 0 β 2 = β 3 = 0 (8) Acceping (6) and rejecing (7) implies selecing an ESTAR model. Acceping boh (6) and (7) and rejecing (8) leads o an LSTAR model as well as a rejecion of (6). Granger and Teräsvira (1993) argue ha sric applicaion of his sequence of ess may lead o incorrec conclusions and sugges he compuaion of he P-values of he F-ess of (6) o (8) and make he choice of he STAR model on he basis of he lowes P-value. 7 Equaion (5) can also be used o selec he ransiion variable z. We conduced his es for each candidae for he ransiion variable drawing from he marix of converible arbirage risk facors. As i leads o he smalles P-value for each of he series, we fail o rejec he lag of he hedge fund series reurn as he choice of z. These resuls are available from he auhors on reques. 10

11 We esimae he STAR models using non-linear leas squares in he RATS programme. RATS specifies he Marquard variaion of he Gauss-Newon o solve he non-linear leas squares regression. In he nex secion of he paper we discuss he empirical resuls from our applicaion of he STAR mehodology o he converible arbirage series. 4. Empirical resuls In his secion of he paper we presen he empirical resuls from esimaing he STAR models for he en converible arbirage series. The remainder of his secion is divided ino hree subsecions. Subsecion 4.1 presens resuls from esimaion of he linear model; Subsecion 4.2 presens he lineariy es resuls; and finally, Subsecion 4.3 presens resuls from esimaing he STAR models Linear model resuls We begin wih he resuls from esimaing (9) for each of he converible arbirage hedge fund series. y = δ 0 + δ 1 YLAG + δ 2 RMRF + δ 3 SMB + δ 4 HML + δ 5 DEF + δ 6 TERM + ε (9) Where YLAG is he one period lag of he hedge fund series reurn a ime. Resuls from esimaing his model are presened in Table 4. Looking firs a he equiy marke facors, RMRF, he excess reurn on he marke porfolio is significanly posiively relaed o five of he en converible arbirage series. SMB, he size relaed facor is significanly posiive for all of he funds series. This suggess ha he sraegy generaes reurns from he smaller issuers. HML, he book-o-marke equiy facor is significan for only wo series, HFR EQL and CBARB. The wo bond marke facors, DEF and TERM, are significanly posiive for nine of he en hedge fund series. HFR SML is he only series wih no exposure o hese facors. Finally, YLAG is 11

12 significan for eigh of he en converible arbirage series. Seven of he hedge fund series exhibi significanly posiive alphas, ranging from 15 o 41 basis poins per monh. This finding of abnormal performance for he converible arbirage sraegy from a linear specificaion is consisen wih prior sudies. The explanaory power of he linear model (adjused R 2 ) ranges from 8% for HFR SML o 59% for he CISDM series. Jacque and Bera saisics indicae seven of he en series residuals are significanly non-normal, and we fail o rejec ARCH effecs for all of he esimaed regression residuals. 4.2 Lineariy ess The lineariy ess for each of he series are displayed in Table 5. To ensure ha he mos appropriae lag of he converible arbirage series is specified as he ransiion variable we begin by seing z = y -d where d is he delay parameer. We hen conduc lineariy ess for values of he delay parameer over he range 1 d 8. P-values for he lineariy es are calculaed and displayed in row one of each panel in Table 5. The delay parameer d is chosen by he lowes P- value. The ess for he choice beween ESTAR and LSTAR for each series are shown in rows 2 o 4 of each Panel in Table 5. Lineariy ess of he HFRI index are repored in Panel A of Table 5. Lineariy is rejeced a levels of d = 1, 2, 3 and 8 bu he lowes P-value is for d = 1 so, consisen wih expecaions, y -1, he one period lag of he hedge fund series reurn, is chosen as he ransiion variable z. A d = 1, he only significan P-value is for H 1 indicaing an LSTAR model. For each of he oher converible arbirage series we find evidence o rejec lineariy a muliple lags. Consisen wih our finding for HFRI, he lowes P-value for each series is for d = 1. Choosing beween ESTAR and LSTAR models for each series is no as sraighforward. CSFB, VANHEDGE, BRCLYGRP, HFL LRG and HFR MID are all LSTAR. For CISDM and HFR EQL a d = 1, he lowes P-value is for H 1, again indicaing an LSTAR model. Finally, as he 12

13 resul for CBARB is less conclusive we assume i follows an LSTAR specificaion consisen wih he oher nine converible arbirage series. In he nex subsecion we presen resuls from esimaion of he LSTAR models. 4.3 Smooh ransiion auoregressive model The LSTAR model parameer esimaes ogeher wih he diagnosic saisics are repored in Table 6. Figure 2 displays he ransiion funcions ploed agains ime and he ransiion variable. We idenify wo regimes which we erm he negaive alpha regime and he posiive alpha regime. The ransiion beween he wo regimes is relaively smooh (3.92 < γ < 9.14). The level of he hreshold lies beween for he HFRI series and for he CBARB series. Looking a Fig. 2 here are several disinc periods when he converible arbirage series move ino he negaive alpha regime, 1990 o 1992; 1994; 1998; and, 2001 o These coincide wih severe financial evens which are likely o have negaively affeced credi spreads o 1992 coincides wih he collapse of he Exchange Rae Mechanism (ERM) in he Eurobloc; 1994 coincides wih he Mexican Peso crisis; 1998 coincides wih he Asian and Long Term Capial Managemen crises; and finally, 2001 o 2002 coincides wih he ending of he docom bubble and he Argenina financial crisis. Examining he coefficien esimaes in Table 6, wih he excepion of HFR SML which has no exposure o hese risk facors, coefficiens on DEF and TERM decrease markedly as he series swiches from he negaive alpha regime o he posiive alpha regime. There is no clear paern for he remaining risk facors. The equiy marke facors are no significan in eiher regime for four of he series (HFRI, VanHedge, CISDM and HFR EQL) and only one coefficien is significan in each sae for hree series (BRCLYGRP, HFR LRG and CBARB). HFR MID and HFR SML have large exposure o he hree equiy marke risk facors, RMRF, SMB and HML in he negaive alpha regime and his exposure decreases markedly in he posiive alpha regime. 13

14 Finally, for he CSFB series he equiy marke facors show no clear paern, wih RMRF and HML coefficiens increasing and he SMB coefficien decreasing in he posiive alpha regime. The specificaion of he LSTAR models improves efficiency over he linear specificaion for all of he converible arbirage series wih adjused R 2 range from 45% o 71%. There is also a reducion in AIC and SBC for all of he series. ρ* he raio of residual sandard deviaions of he LSTAR and linear models demonsraes he efficiency gain from he LSTAR specificaion. 8 This raio ranges from 0.54 for he HFR SML series o 0.88 for he CBARB and VANHEDGE series. 9 For six of he series we can now rejec he presence of ARCH effecs in he residuals, alhough here remains evidence of non-normaliy in seven of he series. Consisen wih heoreical expecaions, he resuls for all of he converible arbirage series provide evidence o suppor he exisence of a non-linear relaionship beween converible arbirage reurns and explanaory risk facors. We idenify wo alernae risk regimes for he sraegy; a negaive alpha regime; and, a posiive alpha regime. In he negaive alpha regime, wih z < c (i.e. prior monh converible arbirage reurns are below he hreshold level) he converible arbirage series have increased risk coefficiens and negaive alpha. In his regime he porfolio generally exhibis increased exposure o fixed income risks. This regime also appears o coincide wih incidences of marke sress, wih a corresponding decrease in liquidiy, such as he 1994 Peso crisis and he 1998 Asian currency crisis. In he posiive alpha regime, wih z > c (i.e. prior monh converible arbirage reurns are above he hreshold level) he defaul and erm srucure risk coefficiens generally decrease and he sraegy exhibis posiive alpha. In his regime he porfolio exhibis less fixed income risk characerisics and is characerised by relaively benign financial markes. 8 ρ* he raio of he residual sandard deviaions is calculaed as σ nl /σ lin, where σ nl is he LSTAR esimaed residual sandard deviaion and σ lin is he residual sandard deviaion from he esimaed linear model. The smaller he ρ* he greaer he efficiency gain. 9 The smaller he ρ* he greaer he efficiency gain. 14

15 The presence of hese wo risk regimes has imporan implicaions for invesors in converible arbirage hedge funds. Though hese funds have hisorically offered high reurns wih relaively low sandard deviaion and exposure o marke risk facors, his appears due o he favourable marke condiions since The evidence presened in his paper indicaes ha in fuure periods of marke sress he sraegy will become significanly exposed o fixed income risk facors, and, more imporanly, under-perform a passive invesmen in hese facors. 5. Conclusions The ess conduced in his paper have rejeced lineariy for he converible arbirage hedge fund series. These hedge fund series are classified as logisic smooh ransiion auoregressive (LSTAR) models. The esimaed LSTAR models provide a saisfacory descripion of he nonlineariy found in converible arbirage hedge fund reurns and have superior explanaory power relaive o linear models. For all of he hedge fund series he esimaed LSTAR model improves efficiency relaive o he linear alernaive. The esimaes of he ransiion parameer indicae ha he speed of ransiion is relaively slow from one regime o anoher bu he facor loadings become relaively large, and alphas become negaive, as previous monh s hedge fund reurns move below he hreshold level. Hisorically he swich ino he negaive alpha regime coincides wih several severe financial crises. We make wo key conribuions o he undersanding of converible arbirage and hedge fund risk and reurns in his paper. We idenify wo risk regimes and we also idenify marke condiions where arbirageurs under-perform. Previous research has idenified only one risk regime for converible arbirage. The evidence presened here suppors he exisence of wo alernae risk regimes, a negaive alpha regime, wih higher defaul and erm srucure risk when monh -1 reurns are below a hreshold level, and a posiive alpha regime, wih lower defaul and erm srucure risk when monh -1 reurns are above a hreshold level. 15

16 Prior research has also documened he sraegy generaing eiher significanly posiive alpha or alpha insignifican from zero. Our finding of negaive alpha in he higher risk regime is imporan for invesors in converible arbirage hedge funds. While converible arbirageurs ouperform a passive invesmen in risk facors in relaively benign financial markes, when arbirageurs are more exposed o defaul and erm srucure risk, hey can under-perform relaive o a passive invesmen in he risk facors. 16

17 References Agarwal, V. and Naik, N.Y., Muli-period performance persisence analysis of hedge funds, Journal of Financial and Quaniaive Analysis, Vol. 35, 2000, pp Agarwal, V. and Naik, N.Y., Risks and porfolio decisions involving hedge funds, Review of Financial Sudies, Vol. 17, 2004, pp Agarwal, V., Fung, W., Loon, Y-C and Naik, N. Y., Liquidiy provision in he converible bond marke: Analysis of converible arbirage hedge funds, Working Paper (SSRN, 2007). Amin, G. S. and Ka, H. M., Hedge fund performance : Do he money machines really add value?, Journal of Financial and Quaniaive Analysis, Vol. 38, 2004, pp Bradley, M. D. and Jansen, D. W., Forecasing wih a nonlinear dynamic model of sock reurns and indusrial producion, Inernaional Journal of Forecasing, Vol. 20, 2004, pp Bredin, D. and Hyde, S., Regime change and he role of inernaional markes on he sock reurns of small open economies, European Financial Managemen, 2007, forhcoming. Brooks, C., and Ka, H. M., The saisical properies of hedge fund index reurns and heir implicaions for invesors, Working Paper (CASS Business School, 2001). Capocci, D. and Hübner, G., Analysis of hedge fund performance, Journal of Empirical Finance, Vol. 11, 2004, pp Chan, K. S and Tong, H., On esimaing hresholds in auoregressive models, Journal of Time Series Analysis, Vol. 7, 1986, pp Chen, N-F, Roll, R. and Ross, S., Economic forces and he sock marke, Journal of Business, Vol. 59, 1986, pp Choi, D., Gemansky, M. and Tookes, H., Converible bond arbirage, liquidiy exernaliies and sock prices, Working Paper (Yale ICF, 2007) 17

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19 Holmes, M. J. and Maghrebi N., Asian real ineres raes, nonlinear dynamics, and inernaional pariy, Inernaional Review of Economics & Finance, Vol. 13, 2004, pp Huchinson M. C. and Gallagher L., Simulaing converible bond arbirage porfolios, Applied Financial Economics, 2006, forhcoming. Huchinson M. C. and Gallagher L., Converible arbirage: risk and reurn, Working Paper (Cenre for Invesmen Research, Universiy College Cork, 2007). Jacque, C. M. and Bera, A. K., A es for normaliy of observaions and regression residuals, Inernaional Saisical Review, Vol. 55, 1987, pp Ka, H. M. and Lu, S., An excursion ino he saisical properies of hedge funds, Working Paper (CASS Business School, 2002) Ka, H. M. and Miffre, J., Hedge fund performance: he role of non-normaliy risks and condiional asse allocaion, Working Paper (CASS Business School, 2005) Kazemi, H. and Schneeweis, T., Condiional Performance of Hedge Funds, Working Paper (CISDM, Isenberg School of Managemen, Universiy of Massachuses, 2003) Liang, B., On he performance of hedge funds, Financial Analyss Journal, Vol. 55, 1999, pp McMillan, D. G., Non-linear predicabiliy of sock marke reurns: evidence from nonparameric and hreshold models, Inernaional Review of Economics and Finance, Vol. 10, 2001, pp Michell, M. and Pulvino, T., Characerisics of risk and reurn in risk arbirage, Journal of Finance, Vol. 56, 2001, pp Ocal, N. and Osborn, D., Business cycle non-lineariies in UK consumpion and producion, Journal of Applied Economerics, Vol. 15, 2000, pp

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21 Table 1 Converible arbirage series summary saisics This able repors summary saisics for he monhly converible arbirage excess reurn series specified in his analysis. HFRI is he HFR Converible Arbirage Index of hedge funds, CSFB is he CSFB Tremon Converible Arbirage Index of hedge funds, VANHEDGE is he VanHedge Converible Arbirage Index of hedge funds, BRCLY GRP is he Barclay Group Converible Arbirage Index of hedge funds and CISDM is he CISDM Converible Arbirage Index of hedge funds. HFR EQL is an equally weighed porfolio of converible arbirage hedge funds from he HFR daabase, HFR LRG, HFR MID & HFR SML are equal weighed porfolios of large, medium and small size (asses under managemen) converible arbirage hedge funds from he HFR daabase. CBARB is a simulaed converible arbirage porfolio. N is he number of observaions. JB Sa is he Jacque-Bera normaliy es saisic. Q-Sa is he Ljung-Box auocorrelaion es Q Saisic for welve lags of each series. ***, ** and * indicae significance a he 1%, 5% and 10% levels respecively. N Mean Variance Skewness Kurosis JB Q Sa Sa HFRI *** 99.99*** CSFB *** 59.84*** VANHEDGE *** 30.25*** BRCLY GRP * CISDM *** 54.11*** HFR EQL *** 76.87*** HFR LRG *** 46.96*** HFR MID *** 40.75*** HFR SML *** 30.42*** CBARB *** 62.37*** 21

22 Table 2 Converible arbirage series correlaion marix This able repors linear correlaion coefficiens for he monhly converible arbirage excess reurn series in his analysis. HFRI is he HFR Converible Arbirage Index of hedge funds, CSFB is he CSFB Tremon Converible Arbirage Index of hedge funds, VANHEDGE is he VanHedge Converible Arbirage Index of hedge funds, BRCLY GRP is he Barclay Group Converible Arbirage Index of hedge funds and CISDM is he CISDM Converible Arbirage Index of hedge funds. HFR EQL is an equally weighed porfolio of converible arbirage hedge funds from he HFR daabase, HFR LRG, HFR MID & HFR SML are equal weighed porfolios of large, medium and small size (asses under managemen) converible arbirage hedge funds from he HFR daabase. CBARB is a simulaed converible arbirage porfolio. HFRI CSFB VAN BRCLY CISDM HFR HFR HFR HFR CBARB HEDGE GRP EQL LRG MID SML HFRI CSFB VAN HEDGE BRCLY GRP CISDM HFR EQL HFR LRG HFR MID HFR SML CBARB

23 Table 3 Risk facor summary saisics and correlaion marix This able repors summary saisics and linear correlaion coefficiens for monhly financial variables. Panel A repors he summary saisics, while Panel B repors linear correlaions. RMRF, SMB and HML are facors represening marke, size and book-o-marke risk premia (Fama and French, 1992). DEF and TERM are risk facors for defaul and erm srucure risk (Chen, Roll and Ross, 1986). JB Sa is he Jacque-Bera normaliy es saisic. Q-Sa is he Ljung-Box auocorrelaion es Q Saisic for welve lags of each series. ***, ** and * indicae significance a he 1%, 5% and 10% levels respecively. Panel A: Summary saisics Mean Variance Skewness Kurosis JB Sa Q Sa RMRF *** SMB *** HML *** DEF *** 53.32*** TERM ** Panel B: Correlaion marix RMRF SMB HML DEF TERM RMRF SMB HML DEF TERM

24 Table 4 Linear AR(1) Model This able repors he OLS esimaion of he linear firs order auoregressive model. YLAG is he one period lag of he dependen variable. ***, ** and * indicae coefficien significance a he 1%, 5% and 10% levels respecively. σ is he residual sandard deviaion, Adj. R 2 is he adjused R 2. JB is he Jacque-Bera es for normaliy and ARCH(4) is he LM es up o lag 4. JB and ARCH es resuls are P-Values. AIC and SBC are he Akraike Informaion Crierion and he Schwarz Bayesian Crierion respecively. Variable HFRI CSFB VANHEDGE BRCLY GRP CISDM HFR EQL HFR LRG HFR MID HFR SML CBARB δ ** *** 0.41*** 0.23*** 0.29*** *** 0.46** 0.15 δ YLAG 0.52*** 0.61*** 0.51*** 0.42*** 0.51*** 0.35*** 0.52*** 0.32*** δ RMRF 0.02* * 0.04*** 0.06*** *** δ SMB 0.04** 0.05* 0.05** 0.06* 0.04*** 0.07*** 0.04* 0.06*** 0.07** 0.08*** δ HML * *** δ DEF 0.16*** 0.26** 0.16*** 0.10*** 0.08*** 0.08** 0.22*** 0.10* *** δ TERM 0.20*** 0.28*** 0.19*** 0.15*** 0.12*** 0.16*** 0.23*** 0.18*** *** Diagnosics σ Adj. R JB ARCH(4) AIC SBC

25 Table 5 Lineariy and STR ess This able presens resuls from a sequence of F-ess carried ou for each of he converible arbirage series afer esimaion of he following auxiliary regression, u = β 0 z + β 1 z x + β 2 z x β 3 z x Where he values of u are he residuals from he linear AR(d) model y = α 0 + θ y - d + λ' x + u. The null hypohesis of lineariy is H 0 : β 1 = β 2 = β 3 = 0. The selecion beween L-STAR and E- STAR models is based on he following series of nesed F-ess. H 3 : β 3 = 0 H 2 : β 2 = 0 β 3 = 0 H 1 β 1 = 0 β 2 = β 3 = 0 ***, ** and * indicae coefficien significance a he 1%, 5% and 10% levels respecively. Panel A: HFRI D H ** 0.03** 0.08* * H H * * 0.11 H *** 0.01*** Panel B: CSFB TREMONT D H *** 0.00*** H * 0.77 H H *** 0.00*** 0.02** Panel C: VANHEDGE D H *** 0.00*** 0.00*** *** * H *** *** H ** H *** 0.00*** 0.07* 0.06* *** Panel D: BRCLYGRP D H * H H * *** H *** 0.00*** Panel E: CISDM D H *** 0.02** *** * H * 0.10* H ** 0.09* H *** 0.00*** ** ** Panel F: HFR EQL D H *** ** H H * ** H *** 0.01*** *

26 Table 5. Coninued. Panel G: HFR LRG D H *** 0.00*** 0.07* H H ** H *** 0.00*** Panel H: HFR MID D H ** * H * H ** 0.05* H *** * Panel I: HFR SML D H *** 0.00*** 0.00*** 0.01** 0.00*** 0.04** 0.00*** 0.18 H ** 0.00*** ** 0.01*** *** 0.17 H *** *** * 0.01*** H *** ** 0.04** ** 0.33 Panel J: CBARB D H *** *** ** 0.01*** 0.00*** H ** *** ** H * *** *** 0.03** H * ** 26

27 Table 6 Smooh ransiion auoregressive regression model This able repors he non-linear leas squares (NLLS) esimaion of he smooh ransiion auoregressive model, y = α x + F(z )β x + e ; F(z ) = {1 + exp[ γ(z c)]} -1 ; where γ > 0 ***, ** and * indicae coefficien significance a he 1%, 5% and 10% levels respecively. σ is he residual sandard deviaion, Adj. R 2 is he adjused R 2. JB is he Jacque-Bera es for normaliy and ARCH(4) is he LM es up o lag 4. JB and ARCH es resuls are P-Values. ρ* demonsraes he efficiency gain. I is compued as σ NL /σ L where σ NL and σ L is he residual sandard deviaion from he non-linear and linear models respecively. AIC and SBC are he Akraike Informaion Crierion and he Schwarz Bayesian Crierion respecively. Variable HFRI CSFB VANHEDGE BRCLYGRP CISDM HFR EQL HFR LRG HFR MID HFR SML CBARB α ** ** α YLAG 0.49*** 0.44** *** 0.35* 0.39** 0.32* α RMRF ** *** 0.28* 0.95** 0.28* α SMB *** ** *** 0.83*** α HML ** *** 2.60*** α DEF 0.29*** 0.57*** 0.75*** 0.43** 0.29*** 0.30*** 0.68*** 0.69*** α TERM 0.36*** 0.63*** 0.44** 0.84*** 0.28*** 0.44*** 0.64*** 0.86*** ** β ** *** ** 1.70** ** β YLAG *** * 0.08 β RMRF * *** -0.25* -0.93** β SMB * ** *** -0.77** 0.17 β HML ** *** -2.62*** 0.26* β DEF -0.26** -0.50*** -0.66*** *** -0.32*** -0.64*** -0.66*** β TERM -0.27** -0.52*** *** -0.22** -0.42*** -0.57*** -0.75*** c 0.33*** ** * ** -0.78*** -1.40*** -1.64*** γ 7.33** 3.92** 6.10*** 6.37*** 8.17*** 5.72** 6.82*** 9.14* 6.51*** 3.96*** Diagnosics σ R JB ARCH(4) ρ* AIC SBC

28 Fig. 1. Cumulaive Reurns of he converible arbirage series This figure plos he cumulaive reurns for each of he converible arbirage series over he sample period. 28

29 Fig. 1. Coninued 29

30 F(z ) A: HFRI F(z ) Time B: CSFB z F(z ) Time C: VANHEDGE z Time z Fig 2. Transiion funcion for he smooh ransiion auoregressive (STAR) models Lef hand panel plos he ransiion funcion f(z ) agains ime. Righ hand panel plos f(z ) agains he ransiion variable z for each of he converible arbirage series. 30

31 F(z ) D: BRCLYGRP F(z ) Time E: CISDM z F(z ) Time F: HFR EQL z Time z Fig 2. Coninued. 31

32 F(z ) G: HFR LRG F(z ) Time H: HFR MID z F(z ) Time I: HFR SML z Time z Fig 2. Coninued. 32

33 F(z ) J: CBARB Time z Fig 2. Coninued. 33