Loss Given Default Implied by Cross-sectional No Arbitrage

Transcription

1 Loss Given Defaul Implied by Cross-secional No Arbirage Graduae School of Business Columbia Universiy Jeong Song November 15, 2007 Absrac I develop various frameworks for he separaion of loss given defaul and defaul inensiy presen in securiies wih credi risk. They include spo and forward credi defaul swaps, digial defaul swaps and bonds. Cross-secional no-arbirage resricion beween differen securiies exracs he pure measure of defaul inensiy and loss given defaul no conaminaed by he oher. Using spo and forward CDS premium daa of 10 emerging marke sovereigns, I find ha 75% level of loss given defaul prevails in he sovereign CDS markes across counries over ime. Posiive correlaion beween loss given defaul and defaul inensiy is only found in Brazil and Venezuela during he period of poliical urmoil in each counry. This resul is puzzling considering diverse fundamenals across counries and ime variaion of he marginal rae of subsiuion. Loss given defaul below (above) he 75% generaes negaive (posiive) pricing errors in forward CDS and he magniude of hem is economically significan. This persisen negaive (posiive) pricing errors wih mis-specified loss given defaul higher (lower) han he rue one are consisen wih he model developed. The auhor would like o hank Joyce Chang, Global Head of Emerging Markes Research, Foreign Exchange, and Commodiies a J.P. Morgan Securiies, for affording him access o he JPMorgan Daabase. 1

2 1 Inroducion For he recen several years, credi derivaives markes have grown explosively. A recen repor by he Briish Bankers Associaion (2006) esimaes ha by he end of 2006, he size of marke would be USD 20 rillion, which is far beyond is own predicion of USD 8.2 rillions made in I expecs ha a he end of 2008 he global credi derivaives marke will have expanded o USD 33 rillion and coninue o grow. 1 In addiion o he fas growh of he credi derivaives markes, he composiion of obligors also is shifing. AAA-BBB raing classes represen 59% in 2006 falling from 65% in 2004 and i is expeced o coninue o fall o 52% by end of In conras, under invesmen grade classes have expanded and expeced o reach nearly half of he marke. 2 The developmen of credi derivaives in Emerging Markes parallels ha of he global credi derivaives marke. 3 Two main componens of defaul risk are (risk neural) defaul inensiy (λ) and (risk neural) loss given defaul(l). A number of sudies have focused on modeling defaul inensiy, bu research on loss given defaul are rare. 4 The difficulies in disenangling defaul inensiy and loss given defaul have been well known since Duffie and Singleon (1999). In order o idenify defaul inensiy, boh academics and praciioners ofen assume loss given defaul as a consan. Longsaff e al. (2005) pre-specify loss given defaul as 50% in heir sudy on credi defaul swaps and bonds for invesmen grade corporaes. For sovereigns, Adler and Song (2007) se loss given defaul as 75% in heir sudy on he dynamics of emerging markes sovereign CDSs and bonds. Zhang (2004) and Pan and Singleon (2006) also ake loss given defaul as a consan in heir work on Sovereign CDS, bu heir sudies are differen from 1 Single name credi defaul swap (CDS) represens a subsanial porion of he marke. I represens 51% of he oal marke in year 2004 and 33% in year 2006 respecively. Index rades, he second larges produc, represen 30% in year 2006 growing from 9% in year Synheic CDOs (collaeralized deb obligaion) come in he hird place a 16% of he marke for boh year. 2 In conras, he BB-B classes have expanded from 13% o 23% and are expeced o grow o 27% by end 2008 according o he repor. Under B classes represen he remaining porion of he marke. 3 CDSs are he mos basic produc for Emerging Markes as well. They are based on sandard ISDA conrac documenaion, and enjoy an acive broker marke wih dealers quoing wo-way pricing for sandard conrac sizes (see Dages e al. (2005) for more deails). More ineresingly and imporanly, here are quoes for CDS premiums from 1-10 years for he EM sovereigns, whereas he quoes are heavily concenraed on he 5 year conrac for he corporae boh in he U.S. and in he Emerging Markes. For more deails, see Packer and Suhiphongchai (2003) and Pan and Singleon (2006). 4 Alman and Kishore (1996) and Acharya, Bharah, and Srinivasan (2004), for example, provide analysis on acual recoveries of defauled securiies. Noe ha his sudy is on he acual loss given defaul, no he risk neural one. For a survey paper on he recovery risk, see Das (2005). 2

3 he previous ones in ha hey esimae i raher han pre-specify i. However, he separaion of defaul inensiy and loss given defaul relying only on CDS may be difficul as shown in Duffie (1999). Despie of difficulies of separaion of defaul inensiy and loss given defaul, i is crucial in many pricing circumsances. Even he basic CDS requires he separaion when raders do mark-o-marke of premium paymen leg of a conrac. We need o calculae PV01, he presen value of a one-basis-poin annuiy wih he mauriy of he credi defaul swap ha erminaes following a credi even, for he mark-o-marke purpose. And PV01 is a funcion of defaul inensiy, no a funcion of loss given defaul. CDS swapion and digial defaul swap (DDS) also require a separae measure of defaul inensiy in heir pricing. Furhermore, as he marke for credi risk coninues o develop, here will be more rading of coningen credi securiies ha depend on defaul inensiy and loss given defaul separaely, raher han in combinaion. In his paper, I sugges frameworks for he separaion by imposing cross-secional noarbirage resricions on differen securiies. I develop, firs, pricing models for various credi insrumens including spo and forward credi defaul swap (CDS), digial defaul swap (DDS) and defaulable bonds. Pricing funcions of hose securiies are derived in erm of defaul inensiy and loss given defaul of underlying reference eniy. Then, I impose cross-secional no-arbirage resricions beween hem. Due o cross defaul provisions and absolue prioriy rules, each credi insrumen wih he same level of senioriy is exposed o he same level of risk ou of defaul inensiy and loss given defaul of a cerain reference eniy. I makes possible for defaul inensiy or loss given defaul in a pricing model for a cerain ype of securiy o be replaced wih (observable) prices of oher securiies. Once eiher defaul inensiy or loss given defaul is idenified, he remaining componen can be sequenially esimaed. Forward CDS premiums, implied by no-arbirage, are derived in erms of defaul inensiy and oher CDS premiums. Loss given defaul is canceled ou wih a cross-secional noarbirage resricion beween spo and forward CDSs, and only defaul inensiy remains. One of he meris of his framework is ha he separaion does no require any assumpion on he process specificaion of loss given defaul. I can be a consan or ime varying. 3

4 Furhermore, i can be correlaed or uncorrelaed o oher model parameers such as defaul inensiy. The fac ha spo and forward CDS conracs are wih consan mauriies is also convenien since we do no need o do mauriy mach of hem. Sand alone digial defaul swap (DDS) premiums reveals a pure measure of defaul inensiy, since loss given defaul in DDS is a conracually fixed number. Proecion buyer pays premiums unil he mauriy or he ime of defaul. In exchange, he proecion seller pays a pre-specified loss amoun o he proecion buyer in he even of defaul. Differences beween DDS and CDS are, firs, he amoun of paymen from he proecion seller is prespecified. In addiion o i, DDS conrac is usually cash seled, while CDS conrac, in many cases, requires physically delivery of reference obligaion when defaul occurs. Therefore, DDS ransfer of defaul even risk while usual CDS ransfer defaul loss risk among counerparies. A cross-secional no-arbirage resricion beween CDS and DDS, ineresingly, leads o a measure of expeced loss given defaul. Bond price can also be used in exracing a pure measure of defaul inensiy wih crosssecional resricions wih CDS. When bond is floaing rae noe and a par, i does no add addiional informaion over CDS. However, when i is no a par, he pariy relaion beween bond yield spread and CDS premiums does no hold as poined ou in Adler and Song (2007). I find ha loss given defauls around 75% prevails in he sovereign CDS markes based on he cross-secional resricion beween spo and forward CDSs. 75% level of loss given defaul persisenly generaes he smalles pricing error in he pricing of forward CDS premiums for all sovereigns in he sample; 10 emerging marke sovereigns are in he sample and sample periods are from 1999 o This finding is in conras wih he resul found in Pan and Singleon (2006). Esimaes in Pan and Singleon (2006) are 24%, 23% and 83% for Turkey, Mexico, and Korea. Loss given defaul around 25% causes he pricing error of abou bp in forward CDS premiums for Turkey while loss given defaul of 75% leads o only abou 10bp pricing error. 6 For Mexico, loss given defaul around 25% causes he pricing error of abou 20bp in forward CDS premiums for Turkey while loss given defaul of 75% leads o 5 Counries include Bulgaria, Brazil, Colombia, Korea, Mexico, Malaysia, Philippines, Poland, Turkey, and Venezuela. 6 Bid-ask spreads for 5 year CDS for corresponding periods are abou 25-50bp. 4

5 only abou 2bp pricing error. 7 The difference in pricing error depending on loss given defaul is less han bid ask spread for Korea. The remainder of his paper is organized as follows. Secion 2 presens pricing models for sand-alone spo credi defaul swaps, forward credi defaul swaps, digial defauls swaps, and defaulable bond. Secion 3 invesigaes he loss given defaul in he acual and risk neural probabiliy space in a simple hree sae economy. This secion also illusraes how cross secional resricions work using CDS and bond. Secion 4 provides he framework for he separaion of defaul inensiy and loss given defaul by imposing cross-secional no-arbirage resricions. No-arbirage resricions are imposed beween spo CDS and forward CDS, resuling in he pure measure of defaul inensiy. The pure measure of loss given defaul is obained by imposing he resricion beween spo CDS and digial defaul swaps. No arbirage resricion beween CDSs and bonds also lead o he separaion of defaul inensiy. Secion 5 documens he empirical findings on loss given defaul in he CDS marke. Secion 6 summarizes he resuls and offers concluding remarks. 2 Pricing Models In his secion, I derive pricing formulae for credi defaul swaps, forward credi defaul swaps, digial defaul swaps, and defaulable bonds. The sand-alone pricing formula for each securiy is derived in erms of defaul inensiy and loss given defaul. Digial defaul swap is an excepion in ha i provides defaul inensiy no conaminaed by he loss given defauls. I is because loss given defaul in digial defaul swaps is conracually pre-fixed. Oher han ha, he separae esimaion of defaul inensiy and loss given defaul is no feasible and hey should be joinly esimaed. However he accuracy of he join esimaion is quesionable (see Longsaff e al. (2005), Duffie (1999) and Pan and Singleon (2006) for more deails). Though he difficulies of idenificaion of defaul inensiy and loss given defaul in sand-alone pricing model are well known, sand-alone models form bases for he separaion when hey are used in combinaions via cross-secional no-arbirage resricions. I, firs, se-up he common noaions o be used in his paper. A probabiliy space 7 Bid-ask spreads for 5 year CDS for corresponding periods are abou 10bp. 5

6 (Ω, F, Q) is well defined, where he filraion F = {F 0 T } saisfies F T = F and i is complee, increasing and righ coninuous where Q is he equivalen maringale measure. Suppose also a locally risk-free shor rae process r. Le χ(τ) = 1 ξ τ be a defaul indicaor funcion of a reference eniy, where ξ is he sopping ime ha characerizes he ime of defaul by he reference eniy. An risk neural defaul inensiy process λ(τ) for a sopping ime ξ is characerized by he propery ha he following is he maringale, τ ( ) χ(τ) 1 χ(µ) λ(µ)dµ 0 L denoe he risk-neural fracional loss of face value on a reference obligaion in he even of a defaul. 2.1 Spo Credi Defaul Swap In his secion, I derive a pricing model for CDSs. 8 Suppose ha wo paries make a spo CDS conrac a ime wih mauriy of τ c. A buyer of proecion periodically pays premiums, s,τc, o a seller. The paymen is made imes per uni ime unil any one of he following evens happens: he underlying reference eniy defauls on is reference obligaion or he mauriy of he CDS conrac comes. The paymen begins a + τc. The seller of proecion receives he premium paymen and is presen value a is s,τc (τ c ) E Q ( )) B() B( + j 1 χ ( + j F ) where B(τ) = e τ r(s)ds. Then he value of he premium leg is s,τc (τ c ) E Q e + j r(s)+λ(s)ds The buyer of proecion will receive a uni face value of he reference obligaion in exchange 8 My model is differen from he previous model by Longsaff e al. (2005) and Zhang (2004) in ha i explicily reflec he discree premium paymen in acual conracs. Their models assume a coninuous paymen of premiums. My model is also differen from he model proposed by Pan and Singleon (2006) in ha i does no assume loss given defaul as a consan in he risk neural space. 6

7 of he physical delivery of he obligaion when a credi even happens. The payoff process, D(), follows dd() = (1 χ())λ()l()d + dm D () where M D () s a maringale wih respec o Q. Then he presen value of he proecion paymen is τc E Q L(µ)λ(µ)e µ r(s)+λ(s)ds dµ Since he ne presen value of a spo CDS a is iniiaion be zero, he spo CDS premium can be obained by equaing he value of he wo legs E s Q τc L(µ)λ(µ)e µ,τc = (τ c ) E Q e + r(s)+λ(s)ds dµ j r(s)+λ(s)ds (1) 2.2 Forward Credi Defaul Swap In his secion, I develop a pricing model for forward CDSs. 9 A forward CDS conrac is an obligaion o buy or sell a CDS on a specified reference eniy for a specified spread a a specified fuure ime. Suppose ha wo paries make a forward CDS conrac a ime wih mauriy of τ c. A buyer of proecion will begin o pay premiums, s τf,τ c, o a seller a a cerain pre-se fuure ime (expiry of a forward conrac), which is denoed as τ f. The paymen is made imes per uni ime unil any one of he following evens happens: he underlying reference eniy defauls on is reference obligaion or he mauriy of he forward CDS conrac comes. 9 The previous model developed by Hull and Whie (2003) assumes a consan loss given defaul and independence beween he shor rae and defaul probabiliy. I exend he model wih more flexible srucure. The new model does no require eiher a consan loss given defaul or he independence beween he shor rae and defaul probabiliy. The flexible feaures of he model allow he models o be valid regardless of various model resricions made in he previous lieraure. 7

8 The seller of proecion receives he premium paymen and is presen value a is s τf,τc (τ c τ f ) E Q ( )) B() B(τ f + j 1 χ (τ f + j F ) where B(τ) = e τ r(s)ds. Then he value of he premium leg is s τf,τc (τ c τ f ) E Q e τ f + j r(s)+λ(s)ds The buyer of proecion will receive a uni face value of he reference obligaion in exchange of he physical delivery of he obligaion when a credi even happens. The payoff process, D(), follows dd() = (1 χ())λ()l()d + dm D () where M D () s a maringale wih respec o Q. Then he presen value of he proecion paymen is E Q τc τ f L(µ)λ(µ)e µ r(s)+λ(s)ds dµ Since he ne presen value of a forward CDS a is iniiaion is zero, he forward CDS premium can be obained by equaing he values of he wo legs s τf,τc = E Q τc τ f L(µ)λ(µ)e µ (τ c τ f ) E Q r(s)+λ(s)ds dµ e τ f + j r(s)+λ(s)ds (2) Pricing formula for a forward CDS is quie similar o he one for a spo CDS; They are virually idenical excep he beginning poin of he premium paymen. Wih τ f =, he equaion (2) is idenical o he equaion (1). 8

9 2.3 Digial Defaul Swap In his secion, I develop a pricing model for digial defaul swaps. 10 Digial defaul swaps conrac is an obligaion ha a proecion seller pays a pre-specified dollar amoun o a proecion buyer in he even of defaul. Spo and forward credi defaul swaps and digial defaul swaps are designed o proec agains differen ypes of risk. The credi defaul swaps ransfer he risk of loss of he obligaion holder in he ime of defaul. Therefore pricing funcion conains boh defaul inensiy and loss given defaul. However, digial defaul swaps ransfer only he risk of a defaul even. Regardless of he realizaion of he loss, pre-specified amoun is paid, and as a resul he pricing funcion conains only defaul inensiy. Suppose ha wo paries make a digial defaul swaps conrac a ime wih he mauriy of τ c. A buyer of proecion will begin o pay premiums, s D,τ c, o a seller a ime + τc. The paymen is made imes per uni ime unil any one of he following evens happens: he underlying reference eniy defauls on is reference obligaion or he mauriy of he CDS conrac comes. In he even of defaul, he seller of a proecion pay he pre-specified L o he buyer. The seller of proecion receives he premium paymen and is presen value a is s D,τc (τ c ) E Q ( )) B() B( + j 1 χ ( + j F ) where B(τ) = e τ r(s)ds. Then he value of he premium leg is s D,τc (τ c ) E Q e + j r(s)+λ(s)ds The buyer of proecion will receive he pre-specified dollar amoun L when a credi even 10 Berd and Kapoor (2003) also derived he pricing formula for digial defaul swaps. Their model is differen from mine wih wo respecs. Their model is derived under he acual probabiliy space while mine is under he risk neural probabiliy space. For he consisency wih oher pricing model in his paper and fuure use in he nex secion, a model under risk neural measure is necessary. More imporanly, my model is a model for he absolue pricing. Their model is derived using he hedge raio in he relaive pricing se up. Their pricing funcion is expressed wih hedging insrumen. 9

10 happens. The payoff process, D(), follows dd() = L (1 χ())λ()d + dm D () where M D () s a maringale wih respec o Q. Then he presen value of he proecion paymen is τc L E Q λ(µ)e µ r(s)+λ(s)ds dµ Since he ne presen value of a digial defaul swaps a is iniiaion is zero, he digial defaul swap premium can be obained by equaing he value of he wo legs s D,τc = L E Q τc λ(µ)e µ (τ c ) E Q r(s)+λ(s)ds dµ e + j r(s)+λ(s)ds (3) Noe ha pricing formula DDS premiums is similar o ha of spo CDS. The only difference is ha loss given defaul in he numeraor of he equaion (1) is pulled ou as a consan L in he equaion (3). 2.4 Defaulable Bond Le P denoe a bond price a ime wih he mauriy of τ b. A bond holder receives a periodic coupon paymen unil mauriy condiional on no defaul a each coupon paymen ime. Coupon is paid M b imes a year and C denoes an annualized coupon raes for he bond. If here is no defaul unil is mauriy, he invesor receives he principal (normalized as one). In he even of defaul, he bond holder only receive a fracional recovery of face value, 1 L a he ime of defaul. The value of coupon paymen unil defaul is C M b M b (τ b ) E Q e j + M b r(s)+λ(s)ds 10

11 The value of principal condiional on no defaul unil mauriy is E Q e τ b r(s)+λ(s)ds Finally, he value of recovery a defaul is τb E Q ( ) 1 L(µ) λ(µ)e µ r(s)+λ(s)ds dµ Then, he price of he bond is derived by summing up he value of coupon, principals and recovery. P = C M b (τ b ) M b E Q e j + M b r(s)+λ(s)ds τb +E Q ( ) 1 L(µ) λ(µ)e µ + E Q r(s)+λ(s)ds dµ e τ b r(s)+λ(s)ds (4) 3 Loss Given Defaul Alhough loss given defaul is criical o he pricing of credi-relaed securiies as shown in he previous secion, convenion wihin boh academic analysis and indusry pracice is o rea i as a consan (e.g., Zhang (2004) and Pan and Singleon (2006)) and i is ofen pre-specified based on a hisorical average (e.g., Longsaff e al. (2005) and Adler and Song (2007)). I is se o lie in he 50 60% range for U.S. corporaes, and abou 75% for sovereigns (see Das and Hanouna (2006) and Pan and Singleon (2006) for more deails). However, hisorical averages of loss given defaul are measured in he acual probabiliy space. Loss given defaul in he acual probabiliy space canno proxy loss given defaul in he risk neural space, unless here is no risk premiums on recovery risk. In his secion, I conras loss given defaul in he acual and risk neural probabiliy space and show ha expeced loss given defaul in he risk neural space is generally bigger han ha in he acual probabiliy space. I also discuss how he hisorical average could have been used as a proxy for he risk neural one. 11

12 3.1 LGD in acual and risk neural probabiliy space In his secion, I assume a hree-sae economy for he exposiion of he relaion beween loss given defaul in he acual and risk neural probabiliy space. By consrucion, he payoff space is complee and he unique sae price is defined. However, he resul holds when he asse span is no complee, which allowing many combinaion of sae prices, as long as here is no-arbirage. Suppose a hree-sae economy. Sae one represens no defaul of an obligor. Sae wo represens he case of defaul of he obligor and a big loss (L B ). Finally he las sae is for he case of defaul of he obligor wih a small loss (L S ). The marginal rae of subsiuion (he pricing kernel) is respecively denoed as m 1, m 2, and m 3 for each sae wih probabiliy of P P 1, PP 2, and PP 3. Le PP D denoe he probabiliy of defaul in he acual probabiliy space. Le P P D B denoe he join probabiliy of defaul wih a big loss and PP D S denoe he join probabiliy of defaul wih a small loss in acual probabiliy space. For sae prices denoed q 1, q 2, and q 3, following equaions hold. q 1 = P P 1 m 1 = (1 P P D ) m 1 q 2 = P P 2 m 2 = P P D B m 2 q 3 = P P 3 m 3 = P P D S m 3 Risk neural probabiliy for each sae is defined as P Q 1 = P Q 2 = P Q 3 = q 1 q 1 + q 2 + q 3 = (1 P Q D ) q 2 = P Q D B q 1 + q 2 + q 3 q 3 = P Q D S q 1 + q 2 + q 3 where P Q, and PQ (PQ ) respecively denoes he probabiliy of defaul and he join prob- D D B D S 12

13 abiliy of defaul wih a big (small) loss in he risk neural probabiliy space. Then P Q = q 2 + q 3 D q 1 + q 2 + q 3 ( P P D PP m B D 2 + P P 1 P P D B D = (1 P P) m D 1 + P P D PP m B D 2 + P P D ) m 3 ( 1 P P B D ) m 3 P Q = D B = P Q = D S = q 2 q 1 + q 2 + q 3 P P D PP B D m 2 (1 P P D ) m 1 + P P D PP B D m 2 + P P D q 3 q 1 + q 2 + q 3 ( ) P P 1 P P D B D m 3 (1 P P D ) m 1 + P P D PP B D m 2 + P P D ( ) 1 P P m B D 3 ( ) 1 P P m B D 3 The condiional probabiliy of big loss given defaul in he risk neural probabiliy space is P Q B D = PQ D B P Q D = P P D PP B D m 2 P P D PP B D m 2 + P P D ( 1 P P B D ) m 3 = PP D S P P D B m3 m 2 = PP S D P P B D m3 m 2 P Q B D depends no only on he condiional probabiliy of loss given defaul in he acual probabiliy space, bu also on he raio of marginal rae of subsiuion of small and big loss saes. For he expecaion of loss given defaul L in he risk neural probabiliy space, E Q L = L B P Q + L B D S PQ S D P P D = (L B L S ) PP m B D 2 ( ) + L P P D PP m B D 2 + P P 1 P P S (5) m D B D 3 For he expecaion of loss given defaul L in he acual probabiliy space, E P L = L B P P B D + L S PP S D = (L B L S ) PP B D + L S (6) 13

14 Then he difference beween he expecaion of loss given defaul in he risk neural and acual probabiliy space is P P E Q L E P D L = (L B L S ) PP m B D 2 ( ) (L P P D PP m B D 2 + P P 1 P P B L S ) P P B D m D B D 3 ( ) = (L B L S ) (m 2 m 3 ) 1 P P P P B D B D ( ) (7) P P m B D P P m B D 3 I is noable from equaion (7), E Q L = E P L if and only if L B = L S or m 2 = m 3. Significan cross-secional variaions in loss given defaul in he acual probabiliy space found in Alman e al. (2005) implies ha L B L S. 11 In addiion, loss given defaul on recen sovereign defauls also exhibis significan variaions according o Moody s (2006). 12 Marginal rae of subsiuion (MRS) for he big loss sae and small loss sae differs unless here is no loss a he aggregae level. Higher MRS for he big loss sae implies ha E Q L > E P L. 3.2 Idenificaion of Loss Given Defaul In spie of he shorfall of he hisorical average as a proxy, i has been used mainly because of he economerical infeasibiliy of he separaion of wo componens; when conracs are priced under he fracional recovery of marke value convenion (RMV) inroduced by Duffie and Singleon (1999), he produc of defaul inensiy and loss given defaul deermines prices. Arbirary choice of loss given defaul is compensaed by he corresponding adjusmen of defaul inensiy. In his case, random choice of a fixed number for loss given defaul does 11 Alman e al. (2005) find defaul probabiliy and loss given defaul are posiively correlaed a he aggregae level. They aribue he correlaed relaion in U.S. corporaes o a business-cycle. 12 Loss given defaul is as following wih defauled year in parenhesis: Dominican Republic (2005) 8%; Ukraine (2000) 31%; Moldova (2001) 34%; Uruguay (2003) 34%; Grenada (2004) 35%; Pakisan (1998) 35%; Ecuador (1999) 56%; Argenina (2001) 67%; Ivory Coas (2000) 82%, Russia (1998) 82%. I is he average, issuer weighed, rading price on a sovereign s bonds hiry days afer is iniial missed ineres paymen, or in cases in which he iniial defaul even was he disressed exchange iself, i repors he average price shorly before he disressed exchange. Ineresingly, a he announcemen of exchange offers, which ofen occurred monhs afer he firs defaul even, he loss were subsanially lower excep Ukraine and Argenina: Dominican Republic 5%; Ukraine 40%; Moldova N/A; Uruguay 15%; Grenada N/A; Pakisan 35%; Ecuador (1999) 40%; Argenina 70%; Ivory Coas N/A, Russia 50%. For valuing CDS conracs, i is he loss in value on he underlying bonds wihin a monh when an acual physical delivery occurs beween he insurer and he insured. Pan and Singleon (2006) quoes raders ha recovery depends on he size of he counry (and he size and disribuion of is exernal deb). 14

15 no affec he saisical fi of he model being esed. CDS is priced wih he framework of fracional recovery of face value (RFV). And defaul inensiy and loss given defaul can be idenified in principle. However, a pracical level, several ses of loss given defaul and defaul inensiy provide equally good fis for observed CDS premiums. 13 Pan and Singleon (2006) joinly esimaes loss given defaul and defaul inensiy using CDS daa for sovereigns. They ake loss given defaul as a consan and esimae i using he (quasi) maximum likelihood esimaion mehod. Esimaes for loss given defaul in heir sudy are 23%, 24% and 83% for Mexico, Turkey and Korea. However, i should be noed he maximum likelihood wih unresriced loss given defaul is abou he same wih he case where hey impose he 75% resricion. Likelihood are (32.126), (27.700), and (36.626) for Mexico, Turkey and Korea in resriced (unresriced) model. I illusraes he difficulies of idenificaion of loss given defaul and defaul inensiy by solely using CDS daa. However, a cerain se of {L,λ} providing good fis for CDS may no work well for oher securiies, e.g. classic bulle bonds. Figure 1 illusraes his. For he simpliciy, he shor rae is se as a consan, 5%. Bond pays semi-annual coupon wih coupon rae, 8%. Defaul inensiy, λ, is se wih a range of 0.01,0.2 and loss given defaul, L is se wih 0.1,0.9. Wih his se-up, he range of CDS premiums generaed is wihin 0.0,0.2 (figure 1(a)). I should be noe ha many combinaion of {L,λ} fi a given CDS premium. The inersecion beween he plane, premium = f(l,λ), where f is he graph such ha f : (L,λ) premiums and he oher plane premium = a consan is generally a line, no a poin. All poins on he line are combinaions of {L,λ} ha perfecly fi a given CDS premium. As wih CDS, many combinaion of {L, λ} fi a given bond price. The inersecion beween he plane, Price = g(l,λ), where g is he graph such ha g : (L,λ) Price and he oher plane Price = a consan is generally a line, no a poin. All poins on he line are combinaions of {L,λ} ha perfecly fi a given bond price. Wih he same range of defaul inensiy and loss given defaul, bond price is in 0.5,1.2 (figure 1(b)). 13 see Duffie (1999) for deails. Inser Figure 1 here 15

16 However, combinaion ha maches boh CDS premiums and bond price are no as many as hose which mach only one of hem. In his example, cross-secional resricion beween CDS and bond no only explores he addiional price informaion in boh securiies, bu also significanly improves he idenificaion of loss given defaul and defaul inensiy. This fac is illusraed in figure 2; Given a pair of observed bond price and CDS premium, he L is uniquely idenified. I is also noable ha when CDS premiums are low, bond prices does no vary a lo as L varies. However, as CDS premiums ge high, bond prices significanly varies. Inser Figure 2 here From above, I show ha cross-secional resricions among differen securiies improve he idenificaion. Usually, for a cerain reference eniy, here are several ypes of securiies raded wih exposure o he credi risk of ha eniy. Wih cross-secion of hese securiies, we can improve he idenificaion of loss given defaul and defaul inensiy. 4 Separaion of Defaul Inensiy and Loss Given Defaul In his secion, I develop frameworks for he separae idenificaion of defaul inensiy and loss given defaul. The separae idenificaion comes from he cross-secional no-arbirage resricion beween securiies wih exposure o he credi risk of he common reference eniy. Spo CDS, forward CDS, digial defaul swaps (DDS) and defaulable bonds are among he mos common single name securiies wih credi exposure. Imposing he resricions beween sand-alone pricing funcions of each securiy, which are derived in erms of defaul inensiy and loss given defaul in he previous secion, I derive new pricing formulae for forward CDS, DDS and defaulable bonds in erms of spo CDS premiums and one of he wo componens, eiher defaul inensiy or loss given defaul. These pricing mehods are differen from hose in he previous secion. While he pricing funcions in he previous secion are absolue pricing in he sense ha hey are derived in erms of defaul inensiy and loss given defaul of he reference eniy, he pricing funcions in his secion are relaive pricing in ha hey are expressed wih oher securiies price Prior sudies such as Madan and Unal (1998), Unal, Madan, and Gunay (2001) and Bakshi e al. (2001) use muliple deb securiies for he separaion. Madan and Unal (1998) requires he exisence of wo deb 16

17 4.1 Implied Forward CDS Premiums: No Arbirage Resricion beween Spo CDS and Forward CDS In his secion, I impose a cross-secional no-arbirage resricion beween spo CDS and forward CDS and develop a new pricing framework for forward CDS. I call he new pricing equaion for he forward CDS premiums implied forward CDS premiums, since he price is implied by he no-arbirage resricion. The new pricing formula has wo disincive feaures. Firs, i allows he separaion beween defaul inensiy and loss given defaul. The difficulies of he separaion are well documened in he previous lieraure (e.g. Duffie and Singleon (1999), Longsaff e al. (2005), Pan and Singleon (2006) and Le (2007)). In equaion (2), he forward CDS premium is derived in erms of he shor ineres rae, defaul inensiy, and loss given defaul. However, he implied forward CDS premiums are derived in erms of oher observable spo CDS premiums and defaul inensiy. In he derivaion of he implied CDS premiums, loss given defaul is canceled ou leading o he separae idenificaion of defaul inensiy. In addiion, he cross-secional no-arbirage resricion beween spo and forward CDS does no require any assumpion on he process specificaion of defaul inensiy and loss given defaul. Previous sudies(e.g. Duffie (1999), Zhang (2004), Longsaff e al. (2005), Pan and Singleon (2006)) assume he independence among he shor rae, defaul inensiy and loss given defaul. Furhermore, hey assume ha loss given defaul is consan, no ime varying. Suppose a forward CDS conrac, wih expiry τ f, o buy and sell a CDS wih ime o mauriy (τ c τ f ). Suppose also wo spo CDS conracs, a ime, wih ime o mauriy (τ f ) and (τ c ). From equaion (1), wo spo CDS premiums wih ime o mauriy (τ f ) and (τ c ) are respecively priced as below. s,τf s,τc (τ f ) (τ c ) E Q E Q e + j r(s)+λ(s)ds e + j r(s)+λ(s)ds τf = E Q L(µ)λ(µ)e µ τc = E Q L(µ)λ(µ)e µ r(s)+λ(s)ds dµ r(s)+λ(s)ds dµ (8) (9) securiies wih differen senioriies. Bakshi e al. (2001) needs large cross-secion of bonds in heir esimaion. 17

18 From equaion (2), he following equaion holds for he forward CDS. s τf,τc (τ c τ f ) E Q e τ f + j r(s)+λ(s)ds τc = E Q τ f L(µ)λ(µ)e µ r(s)+λ(s)ds dµ (10) When I add equaion (8) o equaion (10), i leads o he equaion(9). I is noable ha he equaliy should always hold regardless of he process specificaion of defaul inensiy and loss given defaul, for he righ hand side of summaion. τf E Q L(µ)λ(µ)e µ τc = E Q L(µ)λ(µ)e µ r(s)+λ(s)ds dµ r(s)+λ(s)ds dµ + E Q τc τ f L(µ)λ(µ)e µ r(s)+λ(s)ds dµ For he lef hand side, s,τf = s,τc (τ f ) (τ c ) E Q E Q e + j M r(s)+λ(s)ds e + j r(s)+λ(s)ds + s τ f,τc (τ c ) E Q e τ f + j M r(s)+λ(s)ds Then he forward CDS premiums, s f,τc, is s τf,τc = s,τc (τ c ) E Q e + j r(s)+λ(s)ds (τ c τ f ) E Q s,τf (τ f ) E Q e τ f + j r(s)+λ(s)ds e + j r(s)+λ(s)ds (11) I should be noed ha, in equaion (11) for he forward CDS premium, s τf, loss given,τc defaul is no presen. Wih he implied forward premiums, defaul inensiy, no conaminaed by loss given defaul, is derived in erms of oher observable spo and forward CDS premiums. More imporanly, he equaion (11) holds regardless of he process specificaion of he defaul arrival inensiies and loss given defaul. Unlike he separaion hrough he 18

19 raio of CDS premiums wih differen mauriy, proposed by Pan and Singleon (2006), he assumpion ha L(µ) is a consan, is no necessary any more. 4.2 Implied DDS Premiums: No Arbirage Resricion beween CDS and DDS In his secion, I impose a cross-secional no-arbirage resricion beween CDS and digial defaul swaps (DDS) and develop a new pricing framework for DDS. I call he new pricing equaion for DDS premiums implied DDS premiums, since he price is implied by he noarbirage resricion. The implied DDS premiums have a remarkable feaure ha hey resul in he separaion where loss given defaul remains wih he absence of defaul inensiy. We can direcly ge a measure of expeced loss given defaul by comparing he CDS and DDS premiums. I is noable ha in he equaion (3), DDS premiums are derived in erms of defaul inensiy, no loss given defaul. However, implied DDS premiums derived in his secion are expressed in erm of loss given defaul, no defaul inensiy. Suppose ha wo paries make a digial defaul swaps conrac wih mauriy τ c a ime, wih premiums s D,τ c. The paymen is made imes per uni ime unil any one of he following evens happens: he underlying reference eniy defauls on is reference obligaion or he mauriy of he CDS conrac comes. In he even of defaul, he seller of a proecion pay he pre-specified amoun, L, o he proecion buyer. Then from he equaion (3), s D,τc = L E Q τc λ(µ)e µ (τ c ) E Q r(s)+λ(s)ds dµ e + j r(s)+λ(s)ds (12) From he equaion (1), a spo CDS conrac wih he same mauriy wih premiums, s,τc, is prices as below E s Q τc L(µ)λ(µ)e µ,τc = (τ c ) E Q e + r(s)+λ(s)ds dµ j r(s)+λ(s)ds (13) 19

20 From equaions (12) and (13), s D,τc = L s,τc EQ τc λ(µ)e µ E Q τc L(µ)λ(µ)e µ r(s)+λ(s)ds dµ r(s)+λ(s)ds dµ (14) When we assume ha L(µ) is no correlaed wih λ(µ)e µ r(s)+λ(s)ds, s D,τc = L s,τc (15) E L(µ) Q Noe ha he implied DDS premiums are derived in erms of risk neural expecaion of loss given defaul. Exension o he forward DDS premium pricing is sraighforward; replace subscrip in equaion (15) wih τ f, he expiry of he forward conrac. 4.3 Implied Bond Price: No Arbirage Resricion beween CDS and Bond In his secion, I impose a cross-secional no-arbirage resricion beween CDS and bond, and develop a new pricing framework for bond. I call he new pricing equaion for bond price implied bond price, since he price is implied by he no-arbirage resricion. The implied bond price is derived in erm of defaul inensiy, no loss given defaul. Suppose a bond wih an annualized coupon C, number of paymen M b and mauriy τ b. From he equaion (4), bond price P is as below P = C M b M b (τ b ) E Q e j + M b r(s)+λ(s)ds τb +E Q ( ) 1 L(µ) λ(µ)e µ + E Q r(s)+λ(s)ds dµ e τ b r(s)+λ(s)ds (16) Suppose a spo CDS conrac wih premiums s,τc, number of paymen and mauriy τ c. 20

21 From he equaion (1), E s Q τc L(µ)λ(µ)e µ,τc = (τ c ) E Q e + r(s)+λ(s)ds dµ j r(s)+λ(s)ds (17) When M b = and τ b = τ c, P = C s,τc M b M b (τ b ) E Q τb +E Q λ(µ)e µ e j + M b r(s)+λ(s)ds r(s)+λ(s)ds dµ + E Q e τ b r(s)+λ(s)ds (18) Wih cross secional no-arbirage resricion beween CDS and bond, defaulable bond price is derived in erm of defaul inensiy ha is no conaminaed by loss given defaul. The equaion (18) holds regardless of he process specificaion of he defaul inensiy and loss given defaul. When we assume ha L(µ) is no correlaed wih λ(µ)e µ r(s)+λ(s)ds, same mauriies of CDS and bond are no necessary. In his case bond price is as below P = C M b (τ b ) M b s,τc (τ c ) E Q e j + M b r(s)+λ(s)ds E Q e + j r(s)+λ(s)ds + E Q 5 Esimaion of Loss Given Defaul e τ b r(s)+λ(s)ds E Q τb λ(µ)e µ E Q τc λ(µ)e µ τb + E Q λ(µ)e µ r(s)+λ(s)ds dµ r(s)+λ(s)ds dµ 5.1 Loss Given Defaul in Spo and Forward CDS of Sovereigns r(s)+λ(s)ds dµ (19) In his secion, I provide he empirical esimae of loss given defaul prevailing in he CDS marke. Unlike he previous sudies, I impose a cross-secional resricion beween spo and forward CDS premiums. The cross secional resricion beween spo CDSs and forward CDSs is crucial, since i cancel ou loss given defaul in he pricing model. For forward CDS conrac, he premiums are funcion of he shor rae, defaul inensiy and loss given defaul 21

22 as in he equaion (20). s τf,τc = E Q τc τ f L(µ)λ(µ)e µ (τ c τ f ) E Q r(s)+λ(s)ds dµ e τ f + j r(s)+λ(s)ds (20) However, wih cross-secional resricion, forward CDS premiums are derived in erms of observable spo CDS premiums and defaul inensiy, as in he equaion (21). s τf,τc = s,τc (τ c ) E Q e + j r(s)+λ(s)ds (τ c τ f ) E Q s,τf (τ f ) E Q e τ f + j r(s)+λ(s)ds e + j r(s)+λ(s)ds (21) Daa Daily daa from 1999 o 2005 for spo and forward CDS premiums are supplied by J.P. Morgan Securiies, one of he leading players in he CDS marke. Counries in he sample are Bulgaria, Brazil, Colombia, Korea, Mexico, Malaysia, Philippines, Poland, Turkey, and Venezuela. These CDS conracs are sandard ISDA conracs for physical selemen for Emerging Marke (EM) Sovereigns. The noional value of conrac (lo size) is beween five o en million USD for a large marke like Brazil, while i is ypically beween wo o five million for small markes. The prices hold a close of business. For riskless raes, I collec daa for he consan mauriy rae for six-monh, one-year, wo-year, hree-year, five-year, seven-year, and en-year raes from he Federal Reserve Summary Saisics Table 1 provides he basic saisic for spo CDS premiums wih one, hree, five, seven and en year mauriies. Noiceable paern in spo CDS premiums is ha mean, median, minimum of CDS premiums increase in mauriy for all counries. However, maximum of CDS premiums wih shor mauriies are ofen higher han ones wih long mauriies, which resuls 22

23 from he invered CDS premiums curve during he high credi risk period. Pan and Singleon (2006) documen he posiive slope of spread curve as a prominen feaure of he CDS daa. Spread curves for Mexico and Korea never show inversion in heir sudy even hough hey find inversion for Turkey (Figure 3). In my sample, curves for Korea and Mexico are also invered during he Long Term Capial Managemen and Russian Defaul crisis (Figure 4 and 5). Inser Table 1 here Inser Figure 3 here Inser Figure 4 here Inser Figure 5 here Table 2 provides he basic saisic for forward CDS premiums wih one, hree, five, and seven year expiry. They all have he same mauriy of en years. Similar paerns o spo CDS premiums appear in forward CDS premiums; mean, median, minimum of CDS premiums increase in mauriy for all counries. Bu he inversions occurs for maximum of CDS premiums, paricularly wih high level of CDS premiums. One ineresing movemen occurs for Brazil during Spo CDS premiums reached he highes levels in he sample; one year premiums are 4,645bp and en year premiums are 3,315bp. Around he highes premium period, he level of premiums of one year expiry and en year mauriy forward CDS also peaks. However, oher forward premiums wih longer expiry decreased raher han increased (Figure 6). This paern implies ha marke believe he near erm defaul is very likely, bu condiional on no defaul in near erm, he defaul likelihood is no high for longer erm and i would even ge lower. Inser Table 2 here Inser Figure 6 here The bid-ask spreads for spo CDS wih 5 year mauriy range beween 10 and 110 for Brazil, 10 and 60 for Korea, 10 and 90 for Mexico, 10 and 15 for Malaysia, 6 and 60 for 23

24 Panama, 30 and 60 for Turkey, and 2 and 120 for Venezuela. The bid-ask spreads for conracs wih oher mauriies are comparable in magniude o hose of he five-year conracs. Inser Table 3 here Esimaion of Loss Given Defaul CDS premiums wih mauriy τ c is s,τc = = L E Q τc L(µ)λ(µ)e µ (τ c ) E Q E Q τc λ(µ)e µ (τ c ) E Q r(s)+λ(s)ds dµ e + j r(s)+λ(s)ds r(s)+λ(s)ds dµ e + j r(s)+λ(s)ds (22) when L(µ) is assumed no o be correlaed wih λ(µ)e µ r(s)+λ(s)ds. L denoes E Q L. Using 10 spo CDS conrac wih one o en year mauriy, I boosrap E Q e + j r(s)+λ(s)ds, j = {1,2,3,..., (τ c )} on each day. L (0,1) and i is varied by 0.01 (equivalen o 1%). I define he pricing error in forward CDS premiums wih expiry τ f and mauriy τ c given loss given defaul L as ǫ(, τ f, τ c ; L) = s τf,τc s,τc (τ c ) E Q e + j r(s)+λ(s)ds, L (τ c τ f ) E Q e τ f + (τ s f ),τf E Q j r(s)+λ(s)ds, L e + j r(s)+λ(s)ds, L (23) Pricing error, ǫ(,τ f,τ c ; L), is obained on he daily basis. As shown in he Figure 7, loss given defaul of 75% prevails in CDS markes for Mexico. Similar paerns are observed for oher counries. Inser Figure 7 here 24

25 E Q L is a funcion of loss given defaul in he acual probabiliy space and he marginal rae of subsiuion. 15 E Q L increases eiher when E P L increases or when marginal rae of subsiuion a big loss sae ges bigger han he one a small loss sae. Defaul probabiliy and loss given defaul are posiively correlaed a he aggregae level in acual probabiliy space as found in Alman e al. (2005). This posiive correlaion can be aribued he correlaed relaion in U.S. corporaes o a business-cycle. 16 Recen severe credi deerioraions of emerging marke sovereigns, however, come from idiosyncraic poliical circumsances raher han a business cycle. Spike in CDS premiums in Brazil corresponds o he period when Luiz Inacio Lula da Silva, known as Lula, won presidenial elecions and began o lead he firs lef-wing governmen in 40 years. Venezuelan crisis also corresponds o he period of poliical urmoil in he counry. 17 Poliical insabiliy may lead o Sovereign defaul. Furhermore i may adversely affec he negoiaion process afer he defaul, leading o he high loss. I find he high loss given defaul accompanying high defaul inensiy in Brazil and Venezuela. Oher han hese wo counries, I do no find significan ime variaion in loss given defaul during he sample periods. Inser Figure 8 here Based on he finding ha loss given defaul are no ime-varying in mos cases, I calculae RMSE(L), defined as below, o find ou he prevailing loss given defaul and pricing error 15 In previous secion, i is shown ha P P E Q L E P D L = (L B L S ) PP B D ( m2 ) (L P P D PP B D m2 + PP B L S ) P P 1 P P B D m D B D 3 ( ) = (L B L S ) (m2 m3) 1 P P P P B D B D ) (24) P (1 P B D m2 + P P m B D 3 in a hree sae economy. 16 In recession, defauls are correlaed aggregae level of defaul is high. Clusered defauls lead o disress asse sales wih high loss given defaul. Chichilnisky and Wu (2006) show how individual risk even can be propagaed and magnified ino a major widespread defaul. They show ha in an open se of economies, individual defaul leads o a widespread defaul no maer how large he economy is. The propagaion of defaul may cause he devaluaion of he asses, leading o he posiive correlaion beween loss given defaul and defaul inensiy. 17 Armed forces head announced Chavez has resigned and Chavez was aken ino miliary cusody in April, A few days laer, Chavez reurned o office. Opposiion pary demanded ha Chavez resign. 25

26 associaed wih i. 1 RMSE(L) = 1 ǫ 2 (,τ f,τ c ; L) (25) Table 4 repors he RM SE for various L wih various mauriies. Some noiceable paerns are observed. Firs of all, loss given defaul around 75% provides he smalles pricing error for mos cases. Furhermore, pricing errors exhibis V shaped paern: hey iniially decrease in loss given defaul, reach he boom wih L around 75%, and increase hereafer. Pricing errors ge larger wih lower level of loss given defaul. 25% level of loss given defaul generaes he pricing error amouning o several muliples of bid-ask spreads of corresponding spo CDS. Considering diverse economic fundamenals across counries in he sample, he uniform 75% resul is surprising. Inser Table 4 here Ineresingly, 25% of loss given defaul persisenly generaes he negaive pricing error as shown in figure I is noiceable ha 50% of loss given defaul also leads o he negaive pricing error wih smaller magniude. Pricing errors urn ino he posiive numbers as he loss given defaul increases. 75% loss given defaul persisenly generaes he small pricing error over ime. The reason ha loss given defaul below 75% generaes he negaive pricing errors is ha he corresponding defaul probabiliy fiing spo CDS premiums is higher han he ones markes believe. From equaion (23), ǫ(, τ f, τ c ; L) = s τf,τc s,τc (τ c ) E Q = s τf,τc s,τc ( s,τc s,τf ) e + j r(s)+λ(s)ds, L (τ c τ f ) E Q (τ f ) E Q (τ c τ f ) E Q (τ s f ),τf E Q e + j r(s)+λ(s)ds, L j r(s)+λ(s)ds, L r(s)+λ(s)ds, L (26) r(s)+λ(s)ds, L e τ f + e + j e τ f + j In he equaion (26), (s s τf ) > 0 wih upward sloping CDS erm srucure. So,τc,τc does (s,τc s,τf ) > 0 as well. Negaive pricing errors are induced by he large value of 18 Same paern are observed for all oher counries in he sample. 26

27 (τ f ) E Q e + j r(s)+λ(s)ds / (τ, L c τ f ) E Q e τ f + j ( r(s)+λ(s)ds, L o be ) denoed as A(,τ f,τ c,r,λ; F, L) hereafer during he normal ime wih upward sloping CDS erm srucure. Lower level of loss given defaul is associaed wih higher level of defaul inensiy when i is fied o he CDS premiums, which leads o he higher value of A(,τ f,τ c,r,λ; F, L). Therefore, negaive pricing errors come as a resul wih lower level of loss given defaul han one prevailing in he CDS marke. Same reasoning applies o he posiive pricing errors associaed wih loss given defaul higher han 75%. Anoher feaure of he resul is ha differen level of loss given defaul does no induce he pricing error beyond is ransacion cos for counries wih good credi qualiy. In my sample, Korea, Malaysia and Panama exhibi relaively fla and small pricing errors comparing o he oher counries. These hree counries are hose wih bes credi qualiy in he sample. 19 Wih loss given defaul varying from 25% o 75%, he differences in he pricing errors are only abou a few basis poins, which could be negligible considering he ransacion cos. As already shown in he equaion (26), A(,τ f,τ c,r,λ; F, L) is he main deerminan of he magniude of pricing errors in forward CDSs. Difference beween he values of A(,τ f,τ c,r,λ; F, L) wih rue λ prevailing in he marke and mis-esimaed λ is small wih low level of defaul inensiy, which leads o small pricing errors. 6 Conclusion Two major componens of credi risk, defaul inensiy and loss given defaul, can be separaely idenified wih cross-secional no-arbirage resricions. I develop various frameworks for he separaion by imposing cross secional no-arbirage resricions beween credi insrumens, including spo and forward credi defaul swaps, digial defaul swaps and bonds. These frameworks allow he pure measure of defaul inensiy no conaminaed by he loss given defaul. Paricularly, he resricion beween spo and forward CDS provides he separaion of defaul inensiy independen of process specificaion of loss given defaul; i allows ime 19 They have he smalles CDS premiums during he sample period where boh spo and forward CDS premiums are available. 27

28 varying loss given defaul and various correlaion srucures beween loss given defaul and defaul inensiy. Using spo and forward CDS premiums of 10 emerging marke sovereigns, I find ha 75% level of loss given defaul prevails in he sovereign CDS markes across counries over ime. Posiive correlaion beween loss given defaul and defaul inensiy is found in Brazil and Venezuela during he period of poliical urmoils in each counry. Loss given defaul below 75% generaes negaive pricing errors in forward CDS and he magniude of hem are economically significan. These persisen negaive pricing errors wih mis-specified loss given defaul higher han he rue one are consisen wih he model developed. Assessing he loss given defaul wih oher securiies such as bonds remains for furher research. 28

29 References Michael Adler and Jeong Song. The behavior of emerging marke sovereigns credi defaul swap premiums and bond yield spreads. Columbia Universiy, Edward Alman, Brooks Brady, Andea Resi, and Andrea Sironi. The link beween defaul and recovery raes: Theory, empirical evidence and implicaions. Journal of Business, 78 (6): , G. Bakshi, D. Madan, and F. Zhang. Invesigaing he sources of defaul risk: Lessons from empirically evaluaing credi risk models. Working Paper, Universiy of Maryland and Federal Reserve Board, M. Arhur Berd and Vivek Kapoor. Degial premium. The Journal of Derivaives, 10(3): 66 76, May Briish Bankers Associaion. Credi derivaives repor Graciela Chichilnisky and Ho-Mou Wu. General equilibrium wih endogenous uncerainy and defaul. Journal of Mahemaical Economics, 42: , Gerard Dages, Damon Palmer, and Shad Turney. An overview of he emerging marke credi derivaives marke. Federal Reserve Bank of New York, May Sanjiv R. Das. Recovery risk. Journal of Invesmen Mangemen, 3(1): , Sanjiv R. Das and Paul Hanouna. Implied recovery. Sana Clara Universiy and Villanova Universiy, Darrel Duffie. Credi swap valuaion. Financial Analyss Journal, 55(3):73 87, Jan/Feb Darrell Duffie and Kenneh J. Singleon. Modeling erm srucures of defaulable bonds. Review of Financial Sudies, 12: , John C. Hull and Alan Whie. The valuaion of credi defaul swap opions. Universiy of Torono,

30 Anh Le. Separaing he componens of defaul risk: A derivaive-based approach. New York Universiy, Francis A. Longsaff, Sanjay Mihal, and Eric Neis. Corporae yield spreads: Defaul risk or liquidiy? new evidence from he credi defaul swap marke. Journal of Finance, LX(5): , Oc D. Madan and H. Unal. Pricing he risks of defaul. Review of Derivaives Research, 2: , Moody s. Defaul and recovery raes of sovereign bond issuers, Moody s Invesors Service, April Frank Packer and Chamaree Suhiphongchai. Sovereign credi defaul swaps. BIS Quarerly Review, pages 79 88, Dec Jun Pan and Kenneh Singleon. Defaul and recovery implici in he erm srucure of sovereign cds spreads Frank X. Zhang. The relaionship beween credi defaul swap spreads, bond yields, and credi raing announcemens. Federal Reserve Board,

31 Figure 1: CDS Premiums and Bond Prices (a) CDS Premiums (b) Bond Prices 31

32 Figure 2: Idenificaion of L Q 32