Centre for Investment Research Discussion Paper Series

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1 Cenre for Invesmen Research Discussion Paper Series Discussion Paper # 07-01* Converible Arbirage: Risk and Reurn Mark Huchinson Universiy College Cork, Ireland Liam Gallagher Dublin Ciy Universiy, Ireland 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).

2 Converible Arbirage: Risk and Reurn Mark Huchinson* Universiy College Cork, Ireland Liam Gallagher** Dublin Ciy Universiy, Ireland Absrac This paper specifies a simulaed converible arbirage porfolio o characerise he risks in converible arbirage. For ou-of-sample comparison he risk profile of converible arbirage hedge fund indices is also examined. Resuls indicae ha converible arbirage is posiively relaed o defaul and erm srucure risk facors. These risk facors are augmened wih he simulaed converible arbirage porfolio, mimicking a passive invesmen in converible arbirage, o assess he risk and reurn of individual hedge funds. As he simulaed porfolio s excess reurn exhibis negaive skew and excess kurosis i helps accoun for he non-normaliy in individual fund reurns. Two facor models of converible arbirage fund performance are esimaed. The firs model specifies lagged and conemporaneous observaions of he risk facors, conrolling for illiquidiy in he securiies held by funds. In he second model a facor mimicking illiquidiy risk is also specified. We find weak evidence of abnormal risk adjused reurns in he individual fund daa and no evidence of ou-performance in he hedge fund indices. JEL Classificaion: G10, G19 Keywords: Arbirage, Converible bonds, Trading, Hedge funds, Facor models We are graeful o SunGard Trading and Risk Sysems for providing Monis Converibles XL converible bond analysis sofware and converible bond erms and condiions. *Address for Correspondence: Mark Huchinson, Deparmen of Accouning and Finance, Universiy College Cork, College Road, Cork, Ireland. Telephone: , m.huchinson@ucc.ie **Address for Correspondence: Liam Gallagher, DCU Business School, Dublin Ciy Universiy, Dublin 9, Ireland. Telephone: , liam.gallagher@dcu.ie 1

3 Converible Arbirage: Risk and Reurn Absrac: This paper specifies a simulaed converible arbirage porfolio o characerise he risks in converible arbirage. For ou-of-sample comparison he risk profile of converible arbirage hedge fund indices is also examined. Resuls indicae ha converible arbirage is posiively relaed o defaul and erm srucure risk facors. These risk facors are augmened wih he simulaed converible arbirage porfolio, mimicking a passive invesmen in converible arbirage, o assess he risk and reurn of individual hedge funds. As he simulaed porfolio s excess reurn exhibis negaive skew and excess kurosis i helps accoun for he non-normaliy in individual fund reurns. Two facor models of converible arbirage fund performance are esimaed. The firs model specifies lagged and conemporaneous observaions of he risk facors, conrolling for illiquidiy in he securiies held by funds. In he second model a facor mimicking illiquidiy risk is also specified. We find weak evidence of abnormal risk adjused reurns in he individual fund daa and no evidence of ou-performance in he hedge fund indices. 1. Inroducion Converible arbirageurs aemp o capure profi by combining long posiions in converible bonds wih shor posiions in he issuer s equiy. The posiions are designed o generae reurns from wo sources: (i) income from he converible bond coupon and shor ineres, and (ii) long volailiy exposure from he opion componen of he converible bond. In his paper, we provide esimaes of he abnormal reurns o converible arbirage hedge fund invesmens, and also describe he risks associaed wih hese reurns. Income from he converible bond comes from he coupon paid periodically by he issuer o he holder of he bond and ineres on he proceeds of he shor sock sale. As he coupon is generally fixed i leaves he holder of he converible bond exposed o erm srucure risk. As he converible bond remains a deb insrumen unil convered, he holder of he converible bond is also exposed o he risk of defaul by he issuer. The reurn from he long volailiy exposure comes from he equiy opion componen of he converible bond. To capure he long volailiy exposure, he arbirageur iniiaes a dynamic hedging sraegy. The hedge is rebalanced as he sock price and/or converible price move. Previous research has highlighed ha hedge fund reurns conain saisical feaures unusual in financial ime series. 1 Hedge fund reurns are generally non-normally disribued exhibiing negaive skewness and excess kurosis. Linear analysis of non-normal reurns using sandard normally disribued asse benchmarks yields inefficien resuls, leading o erroneous conclusions abou hedge fund performance. To address his issue previous research has specified 1 Ka and Lu (2001) and Brooks and Ka (2001) amongs ohers documen hese characerisics in hedge fund reurns. 2

4 risk facors ha have non-normal characerisics correcing for much of he non-normaliy in he reurn disribuion of he funds. Fung and Hsieh (2001) focus on he rend following sraegy specifying lookback sraddles as risk facors and Michell and Pulvino (2001) focus on he risk arbirage sraegy consrucing a risk arbirage porfolio which serves as a benchmark of risk arbirage performance. The ask of performance evaluaion is furher complicaed when looking a converible arbirage as funds ypically follow quie differen sraegies 2 and he reurns of converible arbirage hedge funds exhibi serial correlaion. Ka and Lu (2001) and Gemansky e al. (2004) hypohesise ha he observed auocorrelaion in hedge fund reurns is due o illiquidiy in he securiies held by hese funds. In he case where he securiies held by a fund are no acively raded, he reurns of he fund will appear smooher han rue reurns, be serially correlaed, resuling in a downward bias in esimaed reurn variance and a consequen upward bias in performance when he fund is evaluaed using mean-variance analysis. Overall, exising academic sudies find ha converible arbirage hedge funds generae significan abnormal reurns. 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 0.42% per monh. Fung and Hsieh (2002) esimae he converible arbirage hedge fund index generaes alpha of 0.74% per monh. These findings sugges ha financial markes exhibi significan inefficiency in he pricing of converible bonds. 3 However, here are wo alernaive non-compeing explanaions for he large abnormal reurns documened in previous sudies. The firs explanaion is ha converible arbirage funds are receiving a risk premium for bearing risks, which are unique o he sraegy and have no been fully adjused for in previous sudies. The second explanaion is ha he illiquidiy in he securiies held by individual hedge funds leads o underesimaion of risk facor coefficiens and upward biased esimaes of performance. In his paper we aemp o address hese issues. To assess converible arbirage hedge fund performance we specify a simulaed converible arbirage porfolio augmened wih defaul and erm srucure risk facors o capure he reurn generaing process common o converible arbirage hedge funds. By defining a se of risk facors ha mach an invesmen sraegy s aims and reurns, individual fund s exposures o 2 Ka and Lu (2001) provide evidence ha he cross correlaions beween hedge fund reurns wihin sraegies are low. 3 Ammann e al. (2004) and King (1986) documen evidence of converible bond under pricing on he French and US converible bond markes. Kang and Lee (1996) also find evidence of converible bond under pricing a issue. 3

5 variaions in he reurns of he risk facors can be idenified. Following he idenificaion of exposures, he effeciveness of he manager s aciviies can be compared wih ha of a passive invesmen in he risk facors. For ou-of-sample comparison we demonsrae empirically ha he simulaed converible arbirage porfolio reurns srongly resemble he reurns of converible arbirage hedge fund indices. As he simulaed porfolio is consruced as a passive 4 converible arbirage invesmen and also shares he characerisics of he hedge fund indices, bu conains none of he biases, i serves as a useful benchmark risk facor of individual fund performance. 5 Furhermore, as he simulaed porfolio exhibis negaive skewness and posiive excess kurosis is specificaion as a risk facor also helps accoun for he non-normaliy in he reurns of individual converible arbirage hedge funds. The second explanaion for he high abnormal reurns o converible arbirage repored in previous sudies is ha he illiquidiy in he securiies held by he funds leads o underesimaion of risk facor coefficiens and a corresponding overesimaion of performance. Alhough previous sudies have idenified he serial correlaion in hedge fund reurns and aribued his o illiquidiy, sudies of converible arbirage performance have made he implici assumpion ha conemporaneous risk facors fully capure he risk in converible arbirage invesmens despie he presence of auocorrelaion. Drawing on he non-synchronous rading lieraure on bea esimaion in he presence of hin rading we specify conemporaneous and lagged observaions of he risk facors in a converible arbirage performance evaluaion model. 6, 7 Furhermore, o correc for he serial correlaion in hedge fund reurns we specify a lag of he individual hedge fund reurn as an explanaory variable. This variable can be inerpreed as a facor mimicking illiquidiy risk. Esimaes of abnormal reurn o converible arbirage from his model are no significanly differen from zero for he hedge fund indices, and are 1.8% per annum, on average, for he individual hedge funds. The remainder of he paper is organised as follows. In he nex secion we describe he consrucion of he simulaed porfolio. Secion 3 provides a definiion of he risk facor models specified o es he ou of sample properies of he simulaed porfolio, and Secion 4 presens 4 No analysis is underaken on he relaive valuaions of he converible bonds. 5 The difficuly wih he use of hedge fund benchmark reurns o define he characerisics of a sraegy and measure he performance of individual funds is hedge fund daa conains hree main biases, insan hisory bias, selecion bias and survivorship bias as discussed in deail by Fung and Hsieh (2000). 6 Asness e al. (2001) demonsrae ha lagged S&P500 reurns are significan explanaory variables for several hedge fund indices. 7 Techniques for esimaing beas so as o conrol for hin rading bias have been proposed by Scholes and Williams (1977) and Dimson (1979) amongs ohers. 4

6 resuls from esimaion of risk facors on he simulaed converible arbirage porfolio and he converible arbirage hedge fund indices. Secion 5 describes he converible arbirage performance measuremen models and Secion 6 presens resuls from he esimaion of converible arbirage risk and performance. Secion 7 concludes he paper. 2. Consrucing he Converible Arbirage Benchmark Porfolio To provide a benchmark for he converible arbirage sraegy we consruc a simple converible arbirage porfolio, designed o capure income and volailiy. The porfolio combines long posiions in converible bonds wih dela neural hedged shor posiions in he issuer s equiy. These hedges are hen rebalanced daily, mainaining he dela neural hedge. The simulaed porfolio focuses exclusively on he radiional converible bond as his allows us o use a universal hedging sraegy across all insrumens in he porfolio. Due o daa consrains, we focus exclusively on converible bonds lised in he Unied Saes beween 1990 and To enable he forecasing of volailiy, issuers wih equiy lised for less han one year were excluded from he sample. 8 Any non-sandard converible bonds and converible bonds wih missing or unreliable daa were removed from he sample. The final sample consiss of 503 converible bonds, 380 of which were live a he end of 2002, wih 123 dead. The erms of each converible bond, daily closing prices and he closing prices and dividends of heir underlying socks were included. Converible bond erms and condiions daa were provided by Monis. Closing prices and dividend informaion came from DaaSream and ineres rae informaion came from he Unied Saes Federal Reserve Saisical Releases. The converible bond porfolio is an equally weighed porfolio of dela neural hedged long converible bonds and shor sock posiions. In order o iniiae a dela neural hedge for each converible bond he dela for each converible bond is esimaed on he rading day i eners he porfolio. 9 The dela esimae is hen muliplied by he converible bond s conversion raio o calculae i he number of shares o be sold shor in he underlying sock (he hedge raio) o iniiae he dela neural hedge. On he following day he new hedge raio, i + 1, is calculaed, and if i + 1> i hen i + 1- i shares are sold, or if i + 1< i, hen i i shares are 8 GARCH(1,1) is specified o esimae volailiy. There is a variey of volailiy forecasing models such as GARCH, EGARCH, IGARCH, A-GARCH, NA-GARCH, V-GARCH in he lieraure. Poon and Granger (2003) provide a comprehensive review of volailiy forecasing. None of he varians consisenly ouperforms he GARCH model of Bollerslev (1986). 9 Dela esimaes are generaed using Monis ConveriblesXL converible bond pricing sofware. 5

7 purchased mainaining he dela neural hedge. The dela of each converible bond is hen recalculaed daily and he hedge is readjused mainaining he dela neural hedge. Daily reurns were calculaed for each posiion on each rading day up o and including he day he posiion is closed ou. A posiion is closed ou on he day he converible bond is delised from he exchange. Converible bonds may be delised for several reasons: he company may be bankrup, he converible may have expired or he converible may have been fully called by he issuer. The daily reurns for a posiion i on day are calculaed as follows. R i CB Pi P + C P P + D + r 1S i, 1 = (1) CB U U i 1 i i 1 ( i i 1 i ) CB U Pi 1 + i 1Pi 1 where R i is he reurn on posiion i a ime, U Pi is he underlying equiy closing price a ime, CB Pi is he converible bond closing price a ime, Ci is he coupon payable a ime, Di is he dividend payable a ime, i 1 is he dela neural hedge raio for posiion i a ime 1 and r is he ineres on he shor proceeds from he sale of he shares. Daily reurns are hen 1S i, 1 compounded o produce a posiion value index for each hedged converible bond over he enire sample period. The value of he converible bond arbirage porfolio on a paricular dae is given by he formula. V i N = i= Wi PVi = 1 F (2) where V is he porfolio value on day, Wi is he weighing of posiion i on day, PVi is he value of posiion i on day, F is he divisor on day and N is he oal number of posiion on day. W i is se equal o one for each live hedged posiion. On he incepion dae of he porfolio, he value of he divisor is se so ha he porfolio value is equal o 100. Subsequenly he porfolio divisor is adjused o accoun for changes in he consiuens in he porfolio. Following a porfolio change he divisor is adjused such ha equaion (3) is saisfied. 6

8 i= N i= W ib b PV i= N i i= = 1 W ia a PV i 1 (3) F F where PVi is he value of posiion i on he day of he adjusmen, Wib is he weighing of posiion i before he adjusmen, W is he weighing of posiion i afer he adjusmen, F is he divisor ib b before he adjusmen and F a is he divisor afer he adjusmen. Thus he pos adjusmen index facor F a is hen calculaed as follows: F a = F x b i= 1 i= N i= 1 i= N W W ia ib PV PV i i (4) As he margins on he sraegy are small relaive o he nominal value of he posiions converible bond arbirageurs usually employ leverage. Calamos (2003) and Ineichen (2000) esimae ha for an individual converible arbirage hedge fund his leverage may vary from wo o en imes equiy. However, he level of leverage in an efficienly run porfolio is no saic and varies depending on he opporuniy se and risk climae. Khan (2002) esimaes ha in mid 2002 converible arbirage hedge funds were a an average leverage level of 2.5 o 3.5 imes, whereas Khan (2002) esimaes ha in lae 2001 average leverage levels were approximaely 5 o 7 imes. From a sraegy analysis perspecive i is herefore difficul o ascribe a se level of leverage o he porfolio. Changing he leverage applied o he porfolio has obvious effecs on reurns and risk as measured by sandard deviaion. We apply leverage of wo imes o he porfolio as his produces a porfolio wih a similar average reurn o indices of converible arbirage hedge fund reurns. Finally monhly reurns 10 were calculaed from he index of converible bond porfolio values. Inser Table 1 abou here Summary saisics for he monhly reurns on he simulaed converible arbirage porfolio in excess of he risk free rae of ineres, CBRF, are presened in Panel A of Table 1 wih 10 All monhly reurn calculaions are logarihmic. 7

9 summary saisics for he excess reurn on wo hedge fund indices; he HFRI Converible Arbirage Index, HFRIRF; and, he CSFB Tremon Converible Arbirage Index, CSFBRF. The CSFB Tremon Converible Arbirage Index is an asse-weighed index (rebalanced quarerly) of converible arbirage hedge funds beginning in 1994 whereas he HFRI Converible Arbirage Index is equally weighed wih a sar dae of January Alhough he CSFB Tremon indices conrols for survivor bias, according o Ackermann e al. (1999), HFR did no keep daa on dead funds before January This will bias upwards he performance of he HFRI index pre The average reurn on CBRF is 0.33% per monh wih a variance of The average reurn is lower and he variance higher han he wo converible arbirage hedge fund indices, CSFBRF and HFRIRF. CBRF is negaively skewed and has posiive kurosis as do he wo hedge fund indices. 3. Tesing he Robusness of he Converible Arbirage Benchmark Porfolio In his secion six asse pricing models are employed o es he ou of sample properies of he simulaed porfolio: he marke model derived from he Capial Asse Pricing Model (CAPM) described in Sharpe (1964) and Linner (1965), he Fama and French (1993) hree facor sock model, he Fama and French (1993) hree facor bond model, he Fama and French (1993) combined sock and bond model, he Carhar (1997) four facor model and Eckbo and Norli s (2005) liquidiy facor model. This secion briefly describes hese models, providing an explanaion of he expeced relaionship beween converible arbirage excess reurns and he individual facors. The marke model is a single index model, which assumes ha all of a sock s sysemaic risk can be capured by one marke facor. The inercep of he equaion, α, is commonly called Jensen s (1968) alpha and is usually inerpreed as a measure of ou- or under-performance. The equaion o esimae is he following: y = α + β RMRF + ε (5) RMRF where y = R R f, R is he reurn on he hedge fund index a ime, R f is he risk free rae a monh, RMRF is he excess reurn on he marke porfolio on monh, is he error erm α and 11 For deails on he consrucion of he CSFB Tremon Converible Arbirage Index see For deails on he consrucion of he HFRI Converible Arbirage Index see 8

10 β RMRF are he inercep and he slope of he regression, respecively. As converible arbirageurs are exposed o credi risk, which is ypically srongly relaed o equiy marke reurns, here should be a significanly posiive β MKT coefficien. The Fama and French (1993) hree facor sock model is esimaed from an expeced form of he CAPM model. This model exends he CAPM wih he inclusion of wo facors o accoun for size and marke o book raio of firms. I is esimaed from he following equaion: y = α + β RMRF + β SMB + β HML + ε (6) RMRF SMB HML where SMB is he facor mimicking porfolio for size (small minus big) a ime and HML is he facor mimicking porfolio for book o marke raio (high minus low) a ime. 12 Capocci and Hübner (2004) specify he HML and SMB facors in heir models of hedge fund performance. Moreover, Agarwal and Naik (2004) specify he SMB facor in a model of converible arbirage performance and find i has a posiive relaion wih converible arbirage reurns. As he opporuniies for arbirage are greaer in he smaller less liquid issues ex ane i would be expeced ha a posiive relaionship beween converible arbirage reurns and he size facor. There is no ex ane expecaion of he relaionship beween he facor mimicking for book o marke equiy and converible arbirage reurns hough Capocci and Hübner (2004) repor a posiive HML coefficien for converible arbirage. Fama and French (1993) also propose a hree facor model for he evaluaion of bond reurns. They draw on he seminal work of Chen e al. (1986) o exend he CAPM incorporaing wo addiional facors aking he shifs in economic condiions ha change he likelihood of defaul and unexpeced changes in ineres raes ino accoun. This model is esimaed from he following equaion y = α + β RMRF + β DEF + β TERM + ε (7) RMRF DEF TERM where DEF is he difference beween he overall reurn on a marke porfolio of long-erm corporae bonds 13 minus he long erm governmen bond reurn 14 a monh. TERM is he facor 12 For deails on he consrucion of SMB and HML see Fama and French (1992, 1993). 13 The reurn on he CGBI Index of high yield corporae bonds is used raher han he reurn on he composie porfolio from Ibboson and Associaes used by Fama and French (1993) due o is unavailabiliy. 9

11 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. 15 I is expeced ha converible arbirage reurns will be posiively relaed o boh of hese facors as he sraegy generally has erm srucure and credi risk exposure. The growh of he credi derivaive marke has provided he faciliy for arbirageur s o hedge credi risk. The magniude and significance of he DEF coefficien, (β DEF ) should indicae o wha degree hedge funds have availed of his faciliy. Fama and French (1993) also esimae a combined model when looking a he risk facors affecing sock and bond reurns. As a converible bond is a hybrid bond and equiy insrumen we also esimae his model using he following equaion: y = α + β RMRF + β SMB + β HML + β DEF + β TERM + ε (8) RMRF SMB HML DEF TERM As arbirageurs aemp o hedge equiy marke risk, i is expeced ha he bond marke facors will be he mos significan in explaining converible arbirage excess reurns in his model. Carhar s (1997) four facor model is an exension of Fama and French s (1993) sock model. I akes ino accoun size, book o marke and an addiional facor for he momenum effec. This momenum effec can be described as he buying of asses ha were pas winners and he selling of asses ha were pas losers. This model is esimaes using he following equaion: y = α + β RMRF + β SMB + β HML + β UMD + ε (9) RMRF SMB HML UMD where UMD is he facor mimicking porfolio for he momenum effec. UMD is consruced in a slighly differen manner o Carhar s (1997) momenum facor 16. Six porfolios are consruced by he inersecion of wo porfolios formed on marke value of equiy and hree porfolios formed on prior welve monh reurns. UMD is he average reurn on he wo high prior reurn porfolios and he wo low prior reurn porfolios. There is no ex ane expecaion for he relaionship 14 The reurn on he Lehman Index of long erm governmen bonds is used raher han he reurn on he monhly long erm governmen bond reurn from Ibboson and Associaes used by Fama and French (1993) due o is unavailabiliy. 15 The reurn on he Lehman Index of shor erm governmen bonds is used raher han he one monh reasury bill rae from he previous monh used by Fama and French (1993). 16 Carhar (1997) consrucs his facor as he equally weighed average of firms wih he highes hiry percen eleven-monh reurns lagged one period minus he equally weighed average of firms wih he lowes hiry percen eleven monh reurns lagged by one period. 10

12 beween converible arbirage reurns and he momenum facor. Capocci and Hübner (2004) repor a negaive coefficien for converible arbirage hedge funds. The final model employed is Eckbo and Norli s (2005) exension of he Carhar model incorporaing a liquidiy facor. Eckbo and Norli (2005) esimaed he following equaion: y = α + β RMRF + β SMB + β HML + β UMD + β TO + ε (10) RMRF SMB HML UMD TO where TO is he reurn on a porfolio of low-liquidiy socks minus he reurn on a porfolio of high-liquidiy socks. 17 Arbirageurs generally operae in less liquid issues so a negaive relaionship beween he liquidiy facor and converible arbirage reurns is expeced. Table 1, Panel B presens summary saisics of he explanaory facor reurns. 18 The average risk premium for he risk facors is simply he average values of he explanaory variables. UMD he momenum facor produces a large 1.14% average reurn bu his facor also has he larges variance and sandard error. The wo bond marke facors DEF and TERM have low sandard errors bu of he wo only DEF exhibis an average reurn (0.54%) significanly differen from zero a sandard levels. Oher han SMB and TO all of he explanaory variables reurns have significanly negaive skew and all have posiive kurosis oher han RMRF, TERM and TO. Inser Table 2 abou here Table 2, Panel A presens a correlaion marix of he explanaory variables. The firs hing ha should be noed is he poenial for mulicollineariy. There is a high absolue correlaion beween TO and several facors, RMRF, SMB and DEF. DEF is also significanly posiively correlaed wih RMRF, SMB and UMD he momenum facor is negaively correlaed wih HML. Table 2, Panel B presens he correlaions beween he hree dependen variables, CBRF, CSFBRF and HFRIRF and he explanaory variables. All of he variables are highly correlaed as eviden by cross correlaions ranging from 0.32 o 0.80, all significan a he 1% level. All are posiively relaed o DEF he defaul risk facor and SMB he facor proxy for firm size. CBRF 17 For deails on he consrucion of TO see Eckbo and Norli (2005). 18 Daa on SMB, RMRF, HML and UMD was provided by Kenneh French. Liquidiy facor daa was provided by Øyvind Norli. 11

13 and HFRIRF are posiively correlaed wih RMRF and all are negaively relaed o TO he liquidiy facor. 4. Resuls of Esimaing Risk Facor Models In his secion, he resuls of esimaing he risk facor models defined in he previous secion on he simulaed converible arbirage porfolio are presened. Ou-of-sample comparison resuls are also presened from esimaing he risk facor models on wo indices of converible arbirage hedge fund reurns. Inser Table 3 abou here Table 3 presens resuls of he OLS esimaion of he risk facor models discussed above on CBRF, he simulaed converible arbirage porfolio excess reurns, from January 1990 o December The error erm of he reurn regression is poenially heeroskedasic and auocorrelaed. Alhough he condiional heeroskedasiciy and auocorrelaion are no formally reaed in he OLS esimae of he parameer, he -sas in parenhesis below he parameer esimaes are heeroskedasiciy and auocorrelaion consisen due o Newey and Wes (1987). 19 Ljung and Box (1978) Q-Saisics, esing he join hypohesis ha he firs en lagged auocorrelaions of he residual are all equal o zero, are repored. The firs resul is from esimaing he marke model. The marke coefficien value of 0.20 is significanly posiive indicaing ha here is a posiive relaionship beween converible bond arbirage reurns and he marke porfolio. This is a finding consisen wih Capocci and Hübner (2004) who esimae a significanly posiive marke coefficien for converible arbirage hedge funds of However he low adjused R 2 indicaes ha his one facor model may no fully capure he risk in converible bond arbirage. The second resul is from esimaion of he Fama and French (1993) hree facor sock model. The facor loadings on all hree facors are significanly posiive, consisen wih Capocci and Hübner s (2004) findings for converible arbirage. I should be highlighed ha he SMB coefficien indicaes ha converible arbirageurs appear o favour issues from smaller companies perhaps due o he greaer arbirage opporuniies. The nex resul is from esimaing he Carhar (1997) four facor model. The momenum facor 19 For all he ime-series analysis in his chaper, adjusing he auocorrelaion beyond a lag of 3 periods does no yield any maerial differences. A -sa based on 3 lags is adoped for regressions. 12

14 adds lile explanaory value o he regression and he Ecko and Norli (2005) TO facor adds no explanaory power o he model. The penulimae resul is from esimaion of he Fama and French (1993) bond facor model. The coefficiens on boh facors, DEF and TERM, are highly significan, wih coefficien weighings greaer han 0.20 and he overall explanaory power of he regression improves wih an adjused R 2 of 37.1%. The resuls indicae ha converible arbirageurs have significan erm srucure and credi risk. Wih he improvemen in model fi he esimaed alpha coefficien has reduced o 0.07% per monh. The final resul is an esimaion of he combined Fama and French s (1993) bond and sock facor models. The coefficiens for RMRF, SMB and HML are all significanly differen from zero alhough he inclusion of hese facors adds lile o he explanaory power of he model. Consisen wih he evidence presened by Brooks and Ka (2001) of serial correlaion in converible arbirage reurns he Q-Sas are significan a he 1% level indicaing ha he residuals of he esimaed regressions presened in Table 3 exhibi serial correlaion. Inser Tables 4 and 5 abou here For ou-of-sample comparison, Table 4 and 5 repors resuls from he same series of regressions, only his ime on he HFRI Converible Arbirage Index from January 1990 o December 2002 and he CSFB Tremon Converible Arbirage Index from January 1994 o December Resuls are srikingly similar o he simulaed porfolio bu he explanaory power of he regressions is lower. Again he major risks faced by he arbirageur are defaul risk, erm srucure risk and he risk from invesing in he issues of small companies. The residuals of all esimaed regressions exhibi auocorrelaion and he Q-Sas are higher han hose repored for he simulaed porfolio residuals. The resuls reveal ha of he facors specified, defaul and erm srucure risk facors are he mos significan risk facors in converible arbirage reurns. This resul is robus for he simulaed converible arbirage porfolio and wo indices of converible arbirage hedge fund reurn, providing evidence o suppor he simulaed converible arbirage porfolio capuring he key risk characerisics of he converible arbirage sraegy. 5. Converible Arbirage Performance Measuremen Models 13

15 By specifying risk facors wih reurns which capure he daa generaing process of he converible arbirage sraegy, we are able o evaluae he performance of he hedge fund indices and individual converible arbirage hedge funds relaive o his porfolio. In his secion he converible arbirage performance models, which specify he excess reurns of he simulaed porfolio (CBRF) and defaul (DEF) and erm (TERM) srucure risk facors are defined. As CBRF does no include non-radiional converible bonds, DEF and TERM are specified o capure he risk from invesing in he converible securiies no included in CBRF. We consider wo risk facor models, a model incorporaing lags of he risk facors, and a model incorporaing lags of he risk facors augmened wih a one period lag of he hedge fund reurn. In he iniial model converible arbirage reurns are assumed o be linearly relaed o he reurns on a se of asse class facors described as: y = α + β 0 CBRF + β 1 DEF + β 2 TERM + ε (11) where y is he excess reurn on he hedge fund, DEF = (DEF, DEF -1, DEF -2 ), TERM = (TERM, TERM -1, TERM -2 ) and CBRF = (CBRF, CBRF -1 and CBRF -2 ). The β coefficien is he sum of he conemporaneous β and lagged β s. Lags of he risk facors are specified in (11) o increase efficiency in he esimaion of he risk facor coefficiens, given illiquidiy in he securiies held by converible arbirage hedge funds. This specificaion is inended o accoun for he poenial for mis-measuremen of marke risk when assessing porfolios conaining illiquid asses. Asness, e al (2001) and Gemansky e al (2004) show ha omiing lagged marke observaions can lead o downward biased esimaes of marke risk and upward biased esimaes of hedge fund performance. This model is hen augmened in (12) wih he one period lag of he hedge fund reurn o furher correc for serial correlaion in converible arbirage reurns. 20 Gemansky e al. (2004) argue ha i is illiquidiy (and possible reurn smoohing by hedge fund managers) ha causes he perceived serial correlaion. In he case where he securiies held by a fund are no acively raded, he reurns of he fund will appear smooher han rue reurns and be serially correlaed. Assuming serial correlaion is caused by illiquidiy, if hedge funds hold only liquid securiies hen he reurns a ime should be unrelaed o reurns a ime -1. A posiive coefficien on he one period lag of he hedge fund s excess reurn indicaes ha he fund is receiving a risk premium for 20 A similar resul would be achieved by esimaing he facor model using a saisical auocorrelaion correcion procedure such as he Corchane-Orcu (1949) procedure. However, a disadvanage of his saisical procedure is ha he resuls canno be inerpreed easily as funcions of risk. 14

16 bearing liquidiy risk. The coefficien on his erm should also capure illiquidiy premium received by invesors for lockups and oher share resricions imposed on invesor redempions. 21 The second model we esimae is: y = α + β 0 CBRF + β 1 DEF + β 2 TERM + β 3 y -1 + ε (12) where y is he excess reurn on he hedge fund, DEF = (DEF, DEF -1, DEF -2 ), TERM = (TERM, TERM -1, TERM -2 ), CBRF = (CBRF, CBRF -1 and CBRF -2 ) and y -1 is he one period lag of he hedge fund excess reurn. The β coefficien is he sum of he conemporaneous β and lagged β s. Resuls from esimaion of (11) and (12) for he HFRI and CSFB Tremon hedge fund indices and individual converible arbirage funds from he HFR daabase are presened in he following secion. 6. Esimaion of Converible Arbirage Fund Performance In his secion of he paper we presen resuls from esimaing he converible arbirage performance measuremen models (11) and (12). We iniially esimae he performance of he wo hedge fund indices before examining he performance of he individual funds. Inser Table 6 abou here Table 6 presens he resuls from OLS esimaion of he wo performance measuremen models for he HFRI (Panel A) and CSFB Tremon (Panel B) converible arbirage hedge fund indices. Panel A, row 1 displays he coefficiens from esimaing (11) for he HFRI index (wih corresponding P-Values from he -ess ha α = 0 and β i + β i-1 + β i-2 = 0 in row 2). 22 The coefficiens on CBRF, DEF and TERM are all significan from zero a, a leas, he 5% level. The inercep is significan from zero a he 1% level indicaing abnormal performance of 32 basis poins per monh. Panel A, row 3 conains he coefficiens from esimaing (12) for he HFRI index (wih corresponding P-Values from he -ess ha α = 0, β i + β i-1 + β i-2 = 0 for I = CBRF, DEF and TERM and β Y = 0 in row 2). Again all β coefficiens are significan from zero, wih he expeced sign, bu here he measure of abnormal performance, α, is no significanly differen from zero. 21 Aragon (2006) documens a negaive relaionship beween share resricions and he liquidiy of a fund s porfolio. 22 Tes saisics are auocorrelaion and heeroskedasiciy consisen due o Newey and Wes (1987). 15

17 The resuls for he CSFB Tremon index are displayed in Panel B. Resuls from esimaing (11) are presened in row 1 (wih corresponding P-Values in row 2) and (12) is presened in row 3 (wih corresponding P-Values in row 4). Again all β coefficiens are significan from zero wih he anicipaed sign, bu for boh models he α is no significan differen from zero a accepable saisical levels. Resuls from esimaing hese models, for boh he HFRI and CSFB Tremon index, find, a bes, weak evidence of abnormal performance by converible arbirageurs. The explanaory power of all models is higher han he risk facor specificaions esimaed in Table 4 and 5, demonsraing he increase in efficiency of hese performance models. Nex we esimae he risk and performance of individual converible arbirage hedge funds. The individual fund daa was sourced from he HFR daabase. The original daabase consised of 105 funds. However, many funds have more han one series in he daabase. Ofen his appears o be due o a dual domicile. (E.g. Fund X Ld and Fund X LLC wih almos idenical reurns.) To ensure ha no fund was included wice, he cross correlaions beween he individual funds reurns were esimaed. If wo funds had high correlaion coefficiens hen he deails of he funds were examined in deail. 23 Finally, in order o have adequae daa o run he facor model ess, any fund ha does no have 24 consecuive monhly reurns beween 1990 and 2002 is excluded. The final sample consised of fory-six hedge funds. Of hese fory-six funds, weny were sill alive a he end of December 2002 and weny-six were dead. Inser Table 7 abou here Descripive saisics on each hedge fund are repored in Table 7. The mean number of observaions is fify-eigh monhs up o a maximum of sixy-nine. The mean monhly reurn is 0.95% and he minimum monhly reurn by a fund over he sample period was -34%. The maximum monhly reurn was +23%. The mean skewness is and he mean kurosis is The Ljung and Box (1978) Q-Saisic ess he join hypohesis ha he auocorrelaions of up o an order of en are all equal o zero. The resuls rejec his hypohesis for weny of he hedge funds. Inser Table 8 abou here 23 These correlaions are no repored bu are available on reques from he auhors. In wo cases high correlaion coefficiens were repored due o a fund reporing wice, in USD and in EUR. In his siuaion he EUR series was deleed. 16

18 Table 8 presens resuls from esimaing he risk facor model (11) for individual converible arbirage hedge funds. 24 The mean explanaory power of he model is 27% (adjused R 2 ). 25 The coefficiens on DEF, TERM and CBRF are significanly differen from zero for weny-wo, weny-one and weny-five hedge funds, respecively. The mean coefficien on DEF is 0.21, compared o a range of 0.17 o 0.25 for he converible arbirage porfolio and indices. The mean coefficien of TERM is 0.16 compared o a range of 0.19 o 0.30 for he converible arbirage porfolio and indices and he mean coefficien on CBRF is The alphas are significanly posiive for weny-four hedge funds and significanly negaive for one hedge fund. Furhermore, he mean alpha, for he fory-six hedge funds, is a saisically significan 0.28% per monh. 26 Inser Table 9 abou here Table 9 presens he resuls of repeaing his analysis wih he inclusion of he ime -1 hedge fund excess reurn as an explanaory variable. The DEF coefficiens are significan for nineeen hedge funds (mean coefficien of 0.23 compared o 0.17 for he model omiing y -1 ), he coefficiens on TERM (mean coefficien 0.21 compared o 0.14 for he model omiing y -1 ), CBRF (mean coefficien of 0.43 compared o 0.48 for he model omiing y -1 ) and he y -1 coefficiens (mean coefficien 0.22) are significan for approximaely half of hedge funds. The mean adjused R 2 of he model is 33%. Wih he inclusion of he facor mimicking for illiquidiy in he securiies held by hedge funds he alphas generaed by he converible bond hedge funds are significanly posiive for weny hedge funds and significanly negaive for four hedge funds. However, he mean alpha, for he fory-six hedge funds, is 0.15% per monh a he 10% saisical significance level, or on an annualised basis of 1.8%, compared o a significanly posiive alpha of 0.28% per monh for he lagged model omiing he lag of y. All oher coefficiens are significan a he 1% level. The resuls repored here for boh hedge fund indices and he individual funds are similar demonsraing he robusness of our performance measuremen models. The coefficiens on CBRF, DEF and TERM are all saisically significan, posiive and of similar magniude. When 24 As he resuls are noisy a he individual fund level we concenrae our discussion of Tables 8 and 9 on he mean coefficiens repored in row 1 of boh ables. 25 Wih several lags of he risk facors specified he model is likely o be over-parameerized for some funds leading o lower adjused R 2 s. 26 All of he coefficiens are significan a he 1% level wih he excepion of β TERM which is significan a he 5% level. 17

19 he model (11) is esimaed wihou specifying he lag of he hedge fund reurn we find some evidence of converible arbirage abnormal performance. The HFRI index and he HFR funds exhibi abnormal performance of approximaely 30 basis poins per monh. When his model is specified for he CSFB Tremon index we find no evidence of abnormal risk adjused performance. However, when he lag of he hedge fund index is also specified we find no evidence of abnormal performance for eiher of he hedge fund indices. In he individual fund daa we find evidence o sugges weak abnormal performance of 15 basis poins per monh or approximaely 1.8% per annum. 6. Conclusion In his paper we generaed a simple converible arbirage porfolio o idenify sources of converible arbirage risk. This porfolio shares he risk characerisics of converible arbirage benchmark indices bu conains none of he biases. Evidence from esimaing risk facor models on his porfolio and he hedge fund indices finds suppor for he simulaed porfolio capuring he key characerisics in he reurn generaing process of converible arbirage. Since he simulaed porfolio shares he risk profile of converible arbirage, i serves as a useful benchmark of hedge fund performance. The reurns on he simulaed porfolio also exhibi negaive skewness and excess kurosis, helping o accoun for he non-normaliy in converible arbirage hedge fund reurns. Evidence from examining he HFRI and CSFB Tremon hedge fund indices and individual hedge funds from he HFR daabase finds suppor for he defaul risk facor, erm srucure risk facor and he excess reurn on he simulaed porfolio being significan in individual converible arbirage hedge fund reurns, paricularly if boh lagged and conemporaneous observaions of he risk facors are specified. This is a resul which suppors Asness e al. s (2001) findings, ha o efficienly esimae he risks faced by hedge funds a model which includes lags of he explanaory variables should be specified. When a non-synchronous model of hedge fund performance is esimaed resuls indicae ha converible arbirage hedge funds generae a saisically significan alpha of 0.28% per monh, or 3.4% per annum. However, residuals from he esimaed regressions exhibi auocorrelaion. The one period lag of he hedge fund s reurn is hen included, correcing for serial correlaion in hedge fund reurns. When his model is specified for he hedge fund indices hey we find no evidence of abnormal performance. For he individual funds he mean esimae of abnormal performance from his model is lower (1.8% per monh) han ha repored for he model excluding he serial correlaion 18

20 correcion hough remains saisically significan from zero a he 10% level. Considering he previously documened survivorship bias in he HFR daabase 27, his suggess ha converible arbirage hedge funds generaed, a bes, only modes abnormal (risk-adjused) reurns over he sample period. 27 Liang (2000) examines wo large daabases (HFR and TASS) and finds an upward bias of 2% per annum. 19

21 References Ackermann, C., R. McEnally and D. Ravenscraf (1999), The Performance of Hedge Funds: Risk, Reurn and Incenives, Journal of Finance, Vol.54, No. 3 (June), pp Agarwal, V. and N.Y. Naik (2004), Risks and Porfolio Decisions Involving Hedge Funds, Review of Financial Sudies, Vol.17, No.1 (Spring), pp Ammann, M., A. Kind and C. Wilde (2004), Are Converible Bonds Underpriced? An Analysis of he French Marke, Journal of Banking and Finance, Vol.27, No.4 (April), pp Aragon, G.O. (2006), Share Resricions and Asse Pricing: Evidence from he Hedge Fund Indusry, Journal of Financial Economics, Forhcoming. Asness, C., R. Krail and J. Liew (2001), Do Hedge Funds Hedge?, Journal of Porfolio Managemen, Vol.28, No.1 (Fall), pp Bauer, R., K. Koedijk and R. Oen (2005), Inernaional Evidence on Ehical Muual Fund Performance and Invesmen Syle, Journal of Banking and Finance, Vol.29, No.7 (July), pp Bollerslev, T. (1986), Generalised Auoregressive Heeroskedasiciy, Journal of Economerics Vol.31, No.3 (April), pp Brennan, M.J. and A. Subrahmanyam (1996), Marke Microsrucure and Asse Pricing: On he Compensaion for Illiquidiy in Sock Reurns, Journal of Financial Economics, Vol.41, No.3 (July), pp Brennan, M.J., T. Chordia and A. Subrahmanyam (1998), Alernaive Facor Specificaions, Securiy Characerisics and he Cross-Secion of Expeced Reurns, Journal of Financial Economics, Vol.49, No.3 (Sepember), pp Brooks, C. and H.M. Ka (2001), The Saisical Properies of Hedge Fund Index Reurns and heir Implicaions for Invesors, Working paper (CASS Business School). 20

22 Brown, S.J., W. Goezmann and R.G. Ibboson (1999), Offshore Hedge Funds: Survival and Performance , Journal of Business, Vol.72, No.1 (January), pp Calamos, N. (2003), Converible Arbirage: Insighs and Techniques for Successful Hedging, (New Jersey: John Wiley and Sons). Capocci, D. and G. Hübner (2004), Analysis of Hedge Fund Performance Journal of Empirical Finance Vol.11, No.1 (January), pp Carhar, M.M. (1997), On Persisence in Muual Fund Performance, Journal of Finance, Vol.52, No.1 (March), pp Chen, N-F., R. Roll and S. Ross (1986), Economic Forces and he Sock Marke, Journal of Business, Vol.59, No.3 (July), pp Cochrane, D. and G.H. Orcu (1949), Applicaion of Leas Squares Regression o Relaionships Conaining Auocorrelaed Error Terms, Journal of American Saisical Associaion, Vol.44, No.245 (March), pp Davies, J.L. (2001), Muual Fund Performance and Managemen Syle, Financial Analyss Journal, Vol.57, No.1 (January/February), pp Dimson, E. (1979), Risk Measuremen when Shares are Subjec o Infrequen Trading, Journal of Financial Economics, Vol.7, No.2 (June), pp Eckbo, B. and Ø. Norli (2005), Liquidiy Risk, Leverage and Long-Run IPO reurns, Journal of Corporae Finance, Vol.11, No.1-2 (March), pp Fama E.F. and K.R. French. (1992), The Cross-Secion of Expeced Sock Reurns, Journal of Finance, Vol.47, No.2 (June), pp (1993), Common Risk Facors in he Reurns on Socks and Bonds, Journal of Financial Economics, Vol.33, No.1 (February), pp

23 Fung, W. and D.A. Hsieh (1997), Empirical Characerisics of Dynamic Trading Sraegies: he Case of Hedge Funds, Review of Financial Sudies, Vol.10, No.2 (Summer), pp (2000), Performance Characerisics of Hedge Funds: Naural vs. Spurious Biases, Journal of Financial and Quaniaive Analysis, Vol.35, No.3 (Sepember), pp (2001), The Risk in Hedge Fund Trading Sraegies: Theory and Evidence from Trend Followers, Review of Financial Sudies, Vol.14, No.2 (Summer), pp (2002), Hedge Fund Benchmarks: Informaion Conen and Biases, Financial Analyss Journal, Vol.58, No.1 (January/February), pp Gemansky M., A.W. Lo and I. Makarov (2004), An Economeric Model of Serial Correlaion and Illiquidiy in Hedge Fund Reurns, Journal of Financial Economics, Vol.74, No.3 (December), pp Ineichen, A. (2000) In Search of Alpha (Unied Kingdom: UBS Warburg Research Publicaion). Kang, J.K. and Y.W Lee (1996), The Pricing of Converible Deb Offerings, Journal of Financial Economics, Vol.41, No.2 (June), pp Ka, H.M. and S. Lu (2002), An Excursion ino he Saisical Properies of Hedge Funds, Working Paper (CASS Business School). Khan, S.A. (2002), A Perspecive on Converible Arbirage, Journal of Wealh Managemen, Vol.5, No.2 (Fall), pp King, R. (1986), Converible Bond Valuaion: an Empirical Tes, Journal of Financial Research Vol.9, No.1 (March), pp Liang B. (2000), Hedge Funds: he Living and he Dead, Journal of Financial and Quaniaive Analysis, Vol.35, No.3 (Sepember), pp

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25 Table 1: Summary Saisics RMRF is he excess reurn on Fama and French s (1993) marke proxy, SMB and HML are Fama and French s facor-mimicking porfolios of size and marke o book equiy. UMD is he Carhar (1997) facor mimicking porfolio for one-year momenum. TERM and DEF are Fama and French s proxies for he deviaion of long-erm bond reurns from expeced reurns due o shifs in ineres raes and shifs in economic condiions ha change he likelihood of defaul. TO is he facor mimicking porfolio for liquidiy. CSFBRF is he excess reurn on he CSFB Tremon Converible Arbirage index, HFRIRF is he excess reurn on he HFRI Converible Arbirage index and CBRF is he excess reurn on he simulaed converible arbirage porfolio. All of he variables are monhly from January 1990 o December 2002 excep he CSFB Tremon Converible Arbirage Index which is from January 1994 o December Mean T-Sa Variance Sd Error Panel A: Dependen Variables Skew Kurosis Jarque- Bera CSFBRF 0.440*** *** 4.61*** *** HFRIRF 0.538*** *** 3.28*** *** CBRF 0.325** *** 9.00*** *** Panel B: Explanaory Reurns RMRF *** *** SMB ** 1.72*** 24.49*** HML *** 5.58*** *** UMD 1.144*** *** 5.46*** *** DEF 0.540*** * 2.59*** 47.2*** TERM * TO *** ***, ** and * indicae significance a he 1%, 5% and 10% level respecively. Saisics are generaed using RATS