THOMSON REUTERS EIKON ADFIN CREDIT CALCULATION GUIDE DOCUMENT NUMBER March 2011

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1 THOMSON REUTERS EIKON ADFIN CREDIT CALCULATION GUIDE DOCUMENT NUMBER March 2

2 Copyrght 25-2 Thomson Reuters. All Rghts Reserved. Republcaton or redstrbuton of Thomson Reuters content, ncludng by framng or smlar means, s prohbted wthout the pror wrtten consent of Thomson Reuters. "Thomson Reuters" and the Thomson Reuters logo are trademarks of Thomson Reuters and ts afflated companes. Acknowledgement s made to all other brand or product names referred to n the text that are regstered trademarks, trademarks, or trade names of ther respectve owners. Document Hstory DOCUMENT NUMBER DATE UPDATE March 2 Reformatted "Where" tables November 2 Renamed the calculaton gude to Thomson Reuters Ekon Adfn Credt Calculaton Gude

3 THOMSON REUTERS EIKON ADFIN CREDIT CALCULATION GUIDE TABLE OF CONTENTS THOMSON REUTERS EIKON ADFIN CREDIT CALCULATION GUIDE... 2 CREDIT DERIVATIVES OVERVIEW... 3 Credt Default Swaps (CDS)... 3 Credt Lnk Notes (CLN)... 4 RISK MODEL CALIBRATION... 5 Calbraton of the Credt Event Probablty Curve... 5 Calbraton Examples n Thomson Reuters Ekon Excel... 4 Calbraton wth Jarrow-Lando-Turnbull (JLT) Method... 8 CDS PRICING AND EVALUATION European CDS Prcng Amercan CDS Prcng CDS Bg Bang Impact on Prcng wth Adfn Credt Lnked Note Prcng Spread Calculaton... 5 COLLATERALIZED DEBT OBLIGATIONS... 5 CDO Prcng... 5 CDO Valuaton Impled Correlaton Example Usng Thomson Reuters Ekon Excel CDO References... 59

4 THOMSON REUTERS EIKON ADFIN CREDIT CALCULATION GUIDE The Adfn Credt Calculaton Gude explans the Thomson Reuters Ekon Adfn credt functons, and ther formulas to prce and analyze Credt Dervatves nstruments such as Credt Default Swaps (CDS). A credt dervatve s a dervatve securty for whch payoff depends on the occurrence of a credt event. The am of any model used n credt dervatves prcng s to express the credt rsk lnked to ths type of nstrument. For nstance, a rsky bond s a bond whose ssuer can default. The most commonly used reference for ndcatng the probablty of default s the ratng gven to the frm by a ratng agency, such as Standard & Poor s or Moody s. The lower the ratng, the greater the rsk of a bond ssued by ths frm defaultng before maturty. Ths rsk s referred to as the default probablty. The most common way of prcng a bond s dscountng all the cash flows usng a zero-coupon curve. In order to take nto account the default rsk of a rsky bond, the prncple wll be smlarly to dscount the cash flows usng a rsky zero-coupon curve. Snce ths rsky zero-coupon curve may not be avalable, we need to fnd a model for the default probablty, whch wll allow us to go from the non-rsky curve to the rsky one. The model chosen makes t possble to prce any nstrument and take nto account the rsk. Ths requres calculaton of the dscount factors of bonds, swaps and asset swaps. To avod repetton, ths book s organzed accordng to calbraton models, rather than credt dervatve types. The calbraton models descrbed are: Credt event probablty curve from a CDS spread curve or rsky zero-coupon curve Cox-Ingersoll-Ross (CIR) coeffcents from a CDS spread curve or rsky zero-coupon curve Jarrow-Lando-Turnbull (JLT) method to reflect transtons between ratngs and market nformaton From the calbraton model, the Net Present Value and CDS Spread calculatons are detaled usng practcal examples. Credt Dervatves Overvew Rsk Model Calbraton CDS Prcng and Evaluaton Collateralzed Debt Oblgatons 2 Thomson Reuters Ekon Adfn Credt Calculaton Gude

5 Credt Dervatves Overvew CREDIT DERIVATIVES OVERVIEW Termnology Credt Event Recovery rate CDS CLN A default event, whch can take dfferent forms: bankruptcy, ratngs downgrade, restructurng of debt, falure to meet a payment oblgaton... The percentage of a clam that s recoverable n the event of counterparty to a transacton defaultng. Notaton: R. Stands for Credt Default Swap. Stands for Credt Lnked Notes The type of contracts a buyer and seller can agree upon n case of a credt event are: Credt Default Swaps (CDS) Credt Lnk Notes (CLN) CREDIT DEFAULT SWAPS (CDS) Protecton Buyer Rsk Seller Premum Protecton Seller Rsk Buyer Nomnal Underlyng Asset Interest Contngent Payment In a credt default swap, the protecton seller agrees to pay the contngent payment f the default has happened. If there s no default before the maturty of the credt default swap, the protecton seller pays nothng. Three knds of contngent payment exst: Cash settlement: the notonal mnus the market value of the reference asset after the default. Bnary: a pre-agreed percentage of the notonal amount Physcal delvery: delvery of the reference asset. Extensons CDS callable: the rsk seller has the rght to cancel the CDS. The premum gven to the rsk buyer wll be hgher than for a straght CDS. CDS Quanto: the currency of the premum s dfferent from the one of the underlyng flows. 3

6 CREDIT LINK NOTES (CLN) A credt-lnked note s a combnaton of a default swap wth a bond ssued by the protecton buyer. Interest Protecton Buyer Rsk Seller Nomnal Protecton Seller Rsk Buyer Nomnal Interest Underlyng Asset Contngent Payment: Nomnal f no Credt Event Recovery Value f Credt Event 4 Thomson Reuters Ekon Adfn Credt Calculaton Gude

7 Rsk Model Calbraton RISK MODEL CALIBRATION Ths secton descrbes the calbraton of the dfferent rsk models, n order to set the default probablty term structure. The calbratons are: Calbraton of the Credt Event Probablty Curve Calbraton Examples n Thomson Reuters Ekon Excel Calbraton wth Jarrow-Lando-Turnbull (JLT) Method CALIBRATION OF THE CREDIT EVENT PROBABILITY CURVE Calbraton of the Credt Event Probablty Curve from a CDS Spread Curve The dea s to compute the default probabltes gven that an n-szed CDS spread curve, such as the net present value for each of these CDS, s equal to zero. The calbraton s done wth a classcal bootstrappng method: We assume that the default probablty at tme s. Gven the spread for a Y maturty CDS, we compute the default probablty at tme Y. Gven the spread for a ny maturty CDS, we compute the default probablty at tme ny. If there are gaps n the CDS curve, we use the nterpolaton method lnked to the Rsk Model n order to calculate the convenent default probabltes. Example n Thomson Reuters Ekon Excel CredtStructure RISKMODEL:CURVE ND:DIS NBDAYS:46 RECOVERY:.3 INSTTYPE:CDS RateStructure RM:YC ZCTYPE:DF IM LIN CDS Spread Curve 23-Sep-2 6M Sep-2 Y Sep-2 2Y Sep-2 3Y Sep-2 4Y Sep-2 5Y Sep-2 7Y Sep-2 Y 46.8 CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA 5

8 Rsk-Free Zero Coupon Curve 9-Sep-2.% 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep- 7.75% 9-Sep % 9-Sep % Syntax =AdCredtStructure(Rsk-Free Zero Coupon Curve, CDS Spread Curve, RISKMODEL:CURVE ND:DIS NBDAYS:46 RECOVERY:.3 INSTTYPE:CDS, RM:YC ZCTYPE:DF IM:LIN ) CredtStructure KEYWORD RISKMODEL:CURVE NBDAYS: RECOVERY:XX INSTTYPE:CDS SPECIFIES the credt event probablty curve the number of days per dscrmnaton nterval for prcng of Amercan CDS the recovery rate value n percentage the model calbraton by usng a credt default swap curve Result returned by the functon =AdCredtStructure() returns: MATURITY DEFAULT PROBA 9-Sep Mar % 23-Sep % 23-Sep % 23-Sep % 6 Thomson Reuters Ekon Adfn Credt Calculaton Gude

9 Rsk Model Calbraton MATURITY DEFAULT PROBA 25-Sep % 24-Sep % 23-Sep % 24-Sep % Manual Calculaton In ths part an explct example focused on the frst CDS s used to explan the calculaton. To retreve the whole default probablty curve by bootstrappng, the same method s used for all CDS. Default Probablty Curve 9-Sep Mar-23 P(t<T2) (unknown DP) =% Where P(t<T) s the probablty that the credt event occurs at tme t when T s the maturty date. We know that on the 9th September 22, the default value s. Then we compute the default probablty at perod 6M by solvng the followng problem: We assume that the unknown default probablty (P(t<T2)) on the 24th March 23 s %. Note that to use the Excel solver a value of P(t<T2) has to be gven as nput. The value of the P(t<T2) s the one whch enables to return a Net Present Value of the CDS equal to. We use the AdCdsNpv() functon for our calculatons. For more nformaton on how to calculate the Net Present Value of a Credt Default Swap refer to CDS Prcng and Evaluaton. Syntax =AdCdsNpv( 9SEP2, 23SEP2, 6M,38.7, Rsk-Free Zero Coupon Curve,Default proba curve, CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA, RISKMODEL:CURVE RECOVERY:.3 NBDAYS:46 ND:DIS, RM:YC ZCTYPE:DF CredtStructure KEYWORD RISKMODEL:CURVE NBDAYS: RECOVERY:XX CDSTYPE:AMERCD S SPECIFIES the credt event probablty curve the number of days per dscrmnaton nterval for prcng of Amercan CDS the recovery rate value n percentage an Amercan CDS 7

10 Result returned by the functon =AdCdsNpv() returns Npv To fnd P(t<T2) we wll solve the formula NPV= usng the Excel solver for nstance, we have: Default Probablty Curve DATE PROBABILITY 9-Sep Mar % To calculate the next default probablty, we use the part of the default probablty curve already bult. The date of the unknown default probablty corresponds to the maturty date of the CDS. Calbraton of the Credt Event Probablty Curve from a Rsky ZC Curve We use the drect formula lnkng a rsk-free zero-coupon prce B(,T), a rsky zero-coupon prce B (,T) and the recovery rate R: B(,T) B' (,T) DefaultP(T) ( R) B(,T) Example n Thomson Reuters Ekon Excel Rsk-Free Zero Coupon Curve DATE PROBABILITY 9-Sep-2.% 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep- 7.75% 9-Sep % 9-Sep % Rsky Zero Coupon Curve DATE PROBABILITY 8 9-Sep-2.% 9-Sep % Thomson Reuters Ekon Adfn Credt Calculaton Gude

11 Rsk Model Calbraton DATE PROBABILITY 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep- 5.5% 9-Sep % 9-Sep % Syntax =AdCredtStructure(Rsk-Free Zero Coupon Curve,Rsky Zero Coupon Curve,"RISKMODEL:CURVE ND:DIS RECOVERY:.3 INSTTYPE:DF, RM:YC ZCTYPE:DF IM:LIN" CredtStructure KEYWORD RISKMODEL:CURVE RECOVERY:XX INSTTYPE:DF SPECIFIES the credt event probablty curve the recovery rate value n percentage the model calbraton by usng a credt zero-coupon curve Result returned by the functon =AdCredtStructure() returns: MATURITY DEFAULT PROBABILITY 9-Sep-22 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9

12 MATURITY DEFAULT PROBABILITY 9-Sep % 9-Sep % 9-Sep % Manual calculaton Accordng to the formula, we have on the 9th September 23: B(, T) B' (, T) 96.73% 93.77% DefaultPT ( ) % ( R) B(, T) ( 3%) 96.73% To retreve the whole default probablty curve, the same method s used for all maturty dates. Calculaton of Default Probabltes from the Credt Event Probablty Curve Formula P( t 2 ) P( t ) P P( t ) WHERE P P(t) P(t2) DENOTES probablty to default between t and t2 nterpolaton from the nput default probablty curve at date t nterpolaton from the nput default probablty curve at date t2 Example n Thomson Reuters Ekon Excel The default probablty curve used n the followng example s the one calculated from the CDS spread curve n the prevous secton. Maturty Array: -Jan-3 -Jul-3 -Jan-4 -Jul-4 -Jan-5 -Jul-5 -Jan-6 -Jul-6 -Jan-7 Thomson Reuters Ekon Adfn Credt Calculaton Gude

13 Rsk Model Calbraton -Jul-7 -Jan-8 DEFAULT PROBA CURVE DFI * 9-Sep-2 24-Mar % 23-Sep % 23-Sep % 23-Sep % 25-Sep % 24-Sep % 23-Sep % 24-Sep % WHERE Perod Start Date: CredtStructure: AdMode: EQUALS JAN3 RISKMODEL:CURVE ND:DIS LAY:H Syntax =AdDefaultProba( JAN3,Maturty_Array,Default proba curve, RISKMODEL:CURVE ND:DIS, LAY:H CredtStructure KEYWORD RISKMODEL:CURVE SPECIFIES the credt event probablty curve Result returned by the functon =AdDefaultProba() returns: MATURITY DEFAULT PROBABILITIES -Jan-3.% -Jul % -Jan % -Jul %

14 MATURITY DEFAULT PROBABILITIES -Jan % -Jul-5.958% -Jan % -Jul % -Jan % -Jul % -Jan % Manual Calculaton Example on the frst default probablty (JUL3): t: JAN3 t2: JUL3 P(JAN3)=.264% (Ths can be calculated usng the AdInterp() functon) P(JUL3)= 3.45% Then usng the formula: P( t 2 ) P( t ) 3.45%.264% P 2.24% P( t ).264% To retreve the whole default probablty curve, use the same formula for all maturty dates. Calbraton of Cox Ingersoll Ross Coeffcents The dffuson concerns the default ntensty (notaton: ht), whch s defned as follows: Pr obablty ( default _ betw eent and _ t dt ) h The default ntensty dffuson process s gven by the CIR (Cox Ingersoll Ross) equaton: dh (h h ) dt t t ht dwt () t dt WHERE h Λ σ Wt DENOTES long term ntensty convergence speed volatlty coeffcent Brownan moton under the neutral rsk probablty The nterest of ths model s to ensure postve values for ht. If we consder a determnstc case, the equaton (2) becomes: dh (h 2 t h ) dt Thomson Reuters Ekon Adfn Credt Calculaton Gude t Usng equaton (), we can calculate the survval probablty at tme t: Q ( T) Eexp T h t s ds (3)

15 Rsk Model Calbraton Usng equaton (2), the two probabltes useful for credt dervatves prcng have the followng form: Survval probablty: t e Q( t) exp (h h ) h t (4) Default probablty: t e Q( t) exp (h h ) h t (5) WHERE h h t T DENOTES ntal ntensty long-term ntensty the current tme the maturty of the nstrumental, whch s unknown the tme of the default Parameters h, h and λ can be calbrated thanks to a CDS premum curve; the am beng the mnmzaton of the dstance between prces gven by the model and prces gven by the market. The credt dervatves market s not lqud enough to enable the calbraton of σ. Moreover ths parameter does not have much nfluence on credt dervatves prces; for ths reason, the non-stochastc verson of ths model s mplemented (sgma s consdered null). Levenberg-Marquardt Method Snce Cox-Ingersoll-Ross s a parametrc model and there s no smple formula to lnk the calbraton nputs and outputs, we need a method to approxmate the three coeffcents descrbng ths model. Adfn Analytcs allows you, va AdCdsSpread(), to calculate the three ntensty and convergence speed parameters of the Cox, Ingersoll, and Ross model, usng ether an algorthm that lnks the spread to the maturty or the followng approxmated formula (n order to mprove the performance): P I( R) e I h (h h ) T T WHERE I P R T DENOTES average default ntensty spread of the credt default swap recovery rate maturty of the credt default swap In order to do so, we chose the non-lnear square method of Levenberg-Marquardt. (Refer to the book Numercal Recpes, lsted n the Bblography on page 38). 3

16 CALIBRATION EXAMPLES IN THOMSON REUTERS EIKON EXCEL Calbraton of Cox-Ingersoll-Ross Coeffcents from a CDS Spread Curve CredtStructure RISKMODEL:CIR APPROX:YES NBDAYS:46 INSTTYPE:CDS RECOVERY:.3 RateStructure RM:YC ZCTYPE:DF IM:LIN CDS Spread Curve 23-Sep-2 6M Sep-2 Y Sep-2 2Y Sep-2 3Y Sep-2 4Y Sep-2 5Y Sep-2 7Y Sep-2 Y 46.8 CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA CLDR:EMU_FI DMC:M CDSTYPE:AMERCDS LFIXED FRQ:4 CCM:MMA Rsk-Free Zero Coupon Curve 9-Sep-2.% 9-Sep % 9-Sep % 9-Sep % 9-Sep % 4 Thomson Reuters Ekon Adfn Credt Calculaton Gude

17 Rsk Model Calbraton 9-Sep % 9-Sep % 9-Sep % 9-Sep- 7.75% 9-Sep % 9-Sep % Syntax =AdCredtStructure(Rsk-Free Zero Coupon Curve,Rsky Zero Coupon Curve, RISKMODEL:CIR APPROX:YES INSTTYPE:DF RECOVERY:.3, RM:YC ZCTYPE:DF IM:LIN, LAY:H ) CredtStructure KEYWORD RISKMODEL:CIR RECOVERY:XX INSTTYPE:DF SPECIFIES the Cox-Ingersoll-Ross model the recovery rate value n percentage the model calbraton by usng a credt zero-coupon curve Result returned by the functon =AdCredtStructure() returns: Default ntensty start value.324 Default ntensty long term value.434 Default ntensty convergence speed.83 Calculaton of Default Probabltes from Cox-Ingersoll-Ross Coeffcents Accordng to the prevous part, we know that: Default probablty equals to: t e Q( t) exp (h h ) h t WHERE h t DENOTES ntal ntensty the current tme 5

18 WHERE T DENOTES the maturty of the nstrumental, whch s unknown the tme of the default The default probablty curve used n the followng example s the one calculated from the CDS spread curve n the prevous secton. Maturty Array: -Jan-3 -Jul-3 -Jan-4 -Jul-4 -Jan-5 -Jul-5 -Jan-6 -Jul-6 -Jan-7 -Jul-7 -Jan-8 CIR Array Thomson Reuters Ekon Adfn Credt Calculaton Gude

19 Rsk Model Calbraton Perod Start Date CredtStructure AdMode JAN3 RISKMODEL:CIR ND:DIS LAY:H Syntax =AdDefaultProba( JAN3,Maturty_Array,CIR_Array, RISKMODEL:CIR ND:DIS, LAY:H ) CredtStructure KEYWORD RISKMODEL:CIR SPECIFIES the Cox-Ingersoll-Ross model Result returned by the functon =AdDefaultProba() returns: MATURITY DEFAULT PROBABILITY -Jan-3.% -Jul % -Jan % -Jul % -Jan % -Jul % -Jan % -Jul % -Jan % -Jul % -Jan % 7

20 Manual Calculaton Example on the frst default probablty (JUL3): So we have T.49589* h.489 h.7 λ.975 *Can be calculated usng the DfCountYears() functon. Then usng the formula: ( t ) ( ) e e Q( t) exp ( h h ) h t exp ( ) %.975 To retreve the whole default probablty curve, use the same formula for all maturty dates. CALIBRATION WITH JARROW-LANDO-TURNBULL (JLT) METHOD Markov Model The Jarrow, Lando and Turnbull model s based on the assumpton that the bankruptcy process follows a dscrete state space Markov chan n credt ratngs. The stablty of ths model has been enhanced by Kjma and Komorbayash. Markov Chan Let Q(K,K) be the transton matrx defnng the Markov process, where the (K )th lne corresponds to the lower credt ratng class, and the Kth lne represents the bankruptcy state: t s also called the absorbng state of ths process, snce the frm s ratng cannot change after bankruptcy. The element of the matrx Q, q,j, s the probablty for a bond whch ratng s to become rated j wthn a tme step. q,... q K, q, q q,k... K,K Approxmatons of the q,j can be found n Moody s or Standard & Poor s reports, avalable at Here s an example for bonds maturng n one year. The matrx lne stands for the ratng after transton, and the matrx column stands for the base ratng. Transton Matrx The am s now to compute the transton matrx for gong from step t to step t+. The matrx Q t,t+ s wrtten as follows : q' q' K,, (t,t )... (t,t ) q',2 (t,t ) q' q',k K,K (t,t )... (t,t ) 8 Thomson Reuters Ekon Adfn Credt Calculaton Gude

21 Rsk Model Calbraton Aa A Baa Ba B Caa D Aaa Aa A Baa Ba B Jarrow and al.(997) assume that the q,j(t,t+) can be calculated usng: q',j Where q' K,j j (t,t ) (t) q (t) (t,t ),j (6) s a determnstc functon of tme that ensures that: (t,t ),, j, q',j j Usng a matrx notaton ths can be summarzed by: Q' t,t I k (t) (QIk ) (7) Where (t) (t) s a dagonal matrx wth the on the dagonal and Ik s a null matrx wth n the last column (k). Evaluaton The am of ths part s to evaluate ( (t), K (t) for t, Let v (t,t) be the prce of a rsky zero-coupon of maturty T ssued by a frm n credt class I at tme t. We know that: v (t,t) p(t, T) R( R) Q' ( T) t Where p(t,t) s a default free zero coupon, R the recovery rate and before maturty. ) Q' t ( T) the probablty that the frm wll not default If we assume that we have for each credt class and for each maturty a default free zero coupon and a rsky zero coupon, (t) Kjma and al. gve formulas to fnd the values of the. For t=: () q j,k v (,) Rp(,) p(,) ( R) for =..K- (9) 9

22 Assumng that Q',t exsts, for.. t : K (t) q' j,j v (,t ) Rp(,t ) (,t) p(,t ) ( R) ( q ) Default Probablty The survval probablty can now be expressed as follows: Q' ( T) t K j q',j ' (t,t) q,k (t,t) Knowng ths probablty, the credt dervatves prces can be calculated. k Example n Thomson Reuters Ekon Excel We want to calbrate a downgrade probablty curve from a transton matrx usng Jarrow, Lando and Turnbull model for a sngle A bond that remans sngle A at the credt event date (RATING: to CREDITEVENT:). Transton matrx for Ratng: along Y-axs and Credt event:j along X-axs TRANSITION MATRIX A B C D A B..7.. C D Rsk Free Curve 7-Jul-2.% 7-Jul % 7-Jul-4 86.% RISKY CURVE A RISKY CURVE B RISKY CURVE C 7-Jul-2.%.%.% 7-Jul-3 89.% 84.5% 74.% 7-Jul-4 85.% 74.% 63.% 2 Thomson Reuters Ekon Adfn Credt Calculaton Gude

23 Rsk Model Calbraton Recovery rate: 5% RateStructure: CredtStructure: RM:YC ZCTYPE:DF RISKMODEL:CURVE INSTTYPE:DF RECOVERY:.5 RATING: CREDITEVENT: In ths example the return value of the AdJLTCredtStructure() functon s retreved and then the formula to recalculate the result s appled manually. Syntax =AdJLTCredtStructure(RskFree Curve,Rsky Curve, Transton Matrx, RISKMODEL:CURVE RECOVERY:.5 RATING: CREDITEVENT:, RM:YC ZCTYPE:DF ) CredtStructure KEYWORD RISKMODEL:CURVE RECOVERY:XX RATING: CREDITEVENT: SPECIFIES the credt event probablty curve the recovery rate value n percentage the ssung frm ratng expressed as the column number n the transton matrx the ratng whch corresponds to the credt event Result returned by the functon =AdJLTCredtStructure() returns: 7-Jul-2.% 7-Jul % 7-Jul-4 9.7% Manual Calculaton In ths part, an explct example focused on all transton matrx and probablty date s used to explan the calculaton. I k.95. Q WHERE Ik Q DENOTES A null matrx wth n the last column The transton matrx 2

24 22 Thomson Reuters Ekon Adfn Credt Calculaton Gude T=: Q (,) calculaton: Downgrade probablty matrx from RATING: to CREDITEVENT:j from the 7JUL2 to the 7JUL3 ) I (Q () I ' Q k k, Accordng to the equaton (9): %) ( 89.9% 89.9% 5% 89%. R) ( p(,) p(,) R (,) v q (), %) ( 84.5% 89.9% 5% 84.5%. R) ( p(,) p(,) R (,) v q () 2 2, %) ( 74% 89.9% 5% 74%. R) ( p(,) p(,) R (,) v q () 3 3,4 3 () 4 Wth the matrx L() and therefore () And I Q k Then we have: ) I ).(Q ( k I ) I ().(Q '(,) Q k k T=: Q (,2) calculaton: Downgrade probablty matrx from RATING: to CREDITEVENT:j from the 7JUL3 to the 7JUL4 ) I (Q () I ' Q k k,2

25 Rsk Model Calbraton 23 Accordng to the equaton (9): %) ( 86% 86% 5% 85% R) ( p(,2) p(,2) R (,2) v %) ( 86% 86% 5% 74% R) ( p(,2) p(,2) R (,2) v %) ( 86% 86% 5% 63% R) ( p(,2) p(,2) R (,2) v 3 Wth () L' Q '().. () L Q L and therefore, () And I Q k Then we have: ) I ).(Q ( k

26 Q'(,2) ().(Q Ik ) Ik Therefore the downgrade probablty matrx from RATING: to CREDITEVENT:j from the 7JUL2 to the 7JUL4 s: Q'(,2) Q'(,2).Q'(,) In our example, we need the downgrade probablty table (here gven to two decmal places) from RATING: to CREDITEVENT:. We can retreve these values from the matrces: Jul-2.%* 7-Jul %** 7-Jul-4 9.7%*** * On 7th July 22 the downgrade probablty from RATING: to CREDITEVENT: s equal to. ** T= *** T= 24 Thomson Reuters Ekon Adfn Credt Calculaton Gude

27 CDS Prcng and Evaluaton CDS PRICING AND EVALUATION CDS prcng and evaluaton s based on the default probablty term structure returned by rsk model calbraton. Refer to Rsk Model Calbraton for a descrpton of rsk models. NOTATION P( T) IS THE Probablty that the credt event occurs at tme t when T s the maturty date. P R CP DF(T) n t Settle Maturty NbDay Fx the CDS premum. recovery rate. Recovery = Contngent Payment. dscount factor at tme T. the number of cash flows of the CDS f no credt event. the tme of the default event. CDS settlement date. CDS maturty date. tme step. outrght f the CDS s cross-currency (fx= otherwse). The dfferent types of prcng are: European CDS Prcng Amercan CDS Prcng CDS Bg Bang Impact on Prcng wth Adfn Credt Lnked Note Prcng Spread Calculaton EUROPEAN CDS PRICING Analytcal Formulas For a European CDS, the contngent payment s pad at maturty: 25

28 -Recovery t Default tme T Tme P P P P P Fxed Leg Calculaton NPV n Fxed CF DF( T ) P( T ) fx WHERE Fx DENOTES Outrght f the CDS s cross-currency (fx= otherwse) P( T) The probablty to default after the tme T of the th cash flows payment DF(T) Dscount factor at tme T. Floatng Leg Calculaton NPV ( R) DF( T ) P( T) Float WHERE Fx DENOTES Outrght f the CDS s cross-currency (fx= otherwse) P( T) The probablty to default after the tme T of the th cash flows payment. DF(T) Dscount factor at tme T. CDS Net Present Value Calculaton PAID LEG NPV FORMULA Fxed NPV NPV Float NPV Fxed Floatng NPV NPV Fxed NPV Float 26 Thomson Reuters Ekon Adfn Credt Calculaton Gude

29 CDS Prcng and Evaluaton Example of a CDS Prcng from Cox-Ingersoll-Ross Coeffcents Example n Thomson Reuters Ekon Excel Consder a Credt Default Swap, whch expres on 5th September 24. The settlement date s on 23rd September 2, t starts on 5th September 22, the spread s 2 bass ponts, and the recovery rate s 3%. Settle: 23SEP, Start Date: 5SEP2, Maturty Date: 5SEP4, Spread=2, R=3%, CdsStructure CDSTYPE:EURCDS CLDR:NULL LFLOAT AOD:NO LFIXED FRQ:4 CredtStructure RISKMODEL:CIR RECOVERY:.3 ND:DIS RateStructure RM:YC ZCTYPE:DF CIR Array Sep-2.% 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep- 7.75% 9-Sep % 9-Sep % In ths example the return value of the AdCdsNpv() functon s retreved and then the formulas to recalculate the result s appled manually. 27

30 Syntax =AdCdsNpv( 23SEP, 5SEP2, 5SEP4,2,Rsk-Free DF curve,cir_array, CDSTYPE:EURCDS CLDR:NULL LFLOAT LFIXED FRQ:4, RISKMODEL:CIR RECOVERY:.3 ND:DIS, RM:YC ZCTYPE:DF ) CredtStructure KEYWORD CDSTYPE:EURCDS RISKMODEL:CIR RISKMODEL:CURVE RECOVERY:XX SPECIFIES a European CDS the Cox-Ingersoll-Ross model the credt event probablty curve the recovery rate value n percentage Result returned by the functon =AdCdsNpv() returns: NPV.2275 Manual Calculaton Fxed leg DATE I CF I P(T TI ) P(T T I )= - P(T T I ) DF(T I ) CFIXP(T T I )X DF(T I ) 5-Dec Mar Jun Sep Dec Mar Jun Sep In the table: TO CALCULATE USE EXAMPLE P( T) AdDefaultProba() =AdDefaultProba( 5SEP2,Date,CIR Array, LAY:H ) Dscount Factors AdRate() For example on the frst cash flow: DF2 = AdRate( 23SEP, 5DEC2, ZcDate:ZcRate, "RM:YC RATETYPE:CMP ZCTYPE:DF") = Thomson Reuters Ekon Adfn Credt Calculaton Gude

31 CDS Prcng and Evaluaton TO CALCULATE USE EXAMPLE AdInterp() Interpolaton on the Yeld Curve at the date (5DEC2) of the frst cash flow for example: AdInterp( 5DEC2 ;ZCDate;ZCRate;"IM:LIN") = Hence, NPV Fxed 8 CF DF( T ) P( T ).3678 Floatng leg DATE CONTINGENT PAYMENT DF P(T T) 5-Sep In the table TO CALCULATE USE EXAMPLE Dscount Factor at tme T AdRate() DF2=AdRate( 23SEP, 5SEP4, ZcDate:ZcRate, "RM:YC RATETYPE:CMP ZCTYPE:DF") = AdInterp() Interpolaton on the Yeld Curve at the maturty date (5SEP4): =AdInterp( 5SEP4 ;ZCDate;ZCRate;"IM:LIN") = Hence: NPV Float ( R) DF( T ) P( T) ( 3%) Therefore the Net Present Value of the CDS s: NPV NPV Float NPV Fxed Example of CDS Prcng from a Default Probablty Curve Example n Thomson Reuters Ekon Excel Consder a Credt Default Swap, whch expres on 5th September 24. The settlement date s on 23rd September 2, t starts on 5th September 22, the spread s 2 bass ponts, and the recovery rate s 3%. Settle: 23SEP, Start Date: 5SEP2, Maturty Date: 5SEP4, Spread=2, R=3%, CdsStructure CDSTYPE:EURCDS CLDR:NULL LFLOAT LFIXED FRQ:4 CredtStructure RISKMODEL:CURVE RECOVERY:.3 ND:DIS Ratestructure RM:YC ZCTYPE:DF 29

32 Rsk-Free DF Curve 9-Sep-2.% 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep- 7.75% 9-Sep % 9-Sep % Probalty Curve 9-Sep-2 % 24-Mar % 23-Sep % 23-Sep % 23-Sep-5 3.2% 25-Sep % 24-Sep % 23-Sep % 24-Sep % In ths example the return value of the AdCdsNpv() functon s retreved and then the formula to recalculate the result s appled manually. 3 Thomson Reuters Ekon Adfn Credt Calculaton Gude

33 CDS Prcng and Evaluaton Syntax =AdCdsNpv( 23SEP, 5SEP2, 5SEP4,2,Rsk-Free DF curve,proba Curve, CDSTYPE:EURCDS CLDR:NULL LFLOAT LFIXED FRQ:4, RISKMODEL:CURVE RECOVERY:.3 ND:DIS, RM:YC ZCTYPE:DF ) CredtStructure KEYWORD CDSTYPE:EURCDS RISKMODEL:CURVE RECOVERY:XX SPECIFIES a European CDS the credt event probablty curve the recovery rate value n percentage Result returned by the functon =AdCdsNpv() returns: NPV.2263 Manual Calculaton Fxed leg DATE I CF I P(T TI ) P(T T I )= - P(T TI) DF(T I ) CFIXP(T T I )X DF(T I ) 5-Dec Mar Jun Sep Dec Mar Jun Sep In the table TO CALCULATE USE EXAMPLE P( T) AdDefaultProba() =AdDefaultProba( 5SEP2,Date,CIR Array, LAY:H ) Dscount Factors AdRate() For example on the frst cash flow: DF2 = AdRate( 23SEP, 5DEC2, ZcDate:ZcRate, "RM:YC RATETYPE:CMP ZCTYPE:DF") =

34 TO CALCULATE USE EXAMPLE AdInterp() Interpolaton on the Yeld Curve at the date (5DEC2) of the frst cash flow for example: AdInterp( 5DEC2 ;ZCDate;ZCRate;"IM:LIN") = Hence, NPV Fxed 8 CF DF( T ) P( T ).3678 Floatng leg DATE I CF I P(T TI ) P(T T I )= - P(T TI) 5-Sep In the table: TO CALCULATE USE EXAMPLE Dscount Factor at tme T AdRate() AdInterp() DF2=AdRate( 23SEP, 5SEP4, ZcDate:ZcRate, "RM:YC RATETYPE:CMP ZCTYPE:DF") = Interpolaton on the Yeld Curve at the maturty date (5SEP4): =AdInterp( 5SEP4 ;ZCDate;ZCRate;"IM:LIN") = Hence, NPV Float ( R) DF( T ) P( T) ( 3%) Therefore the Net Present Value of the CDS s: NPV NPV Float NPV Fxed Example of CDS Prcng from a Default Probablty Curve Example n Thomson Reuters Ekon Excel Consder a Credt Default Swap, whch expres on 5th September 24. The settlement date s on 23rd September 2, t starts on 5th September 22, the spread s 2 bass ponts, and the recovery rate s 3%. Settle: 23SEP, Start Date: 5SEP2, Maturty Date: 5SEP4, Spread=2, R=3%, CdsStructure CDSTYPE:EURCDS CLDR:NULL LFLOAT LFIXED FRQ:4 CredtStructure RISKMODEL:CURVE RECOVERY:.3 ND:DIS Ratestructure RM:YC ZCTYPE:DF 32 Thomson Reuters Ekon Adfn Credt Calculaton Gude

35 CDS Prcng and Evaluaton Rsk-Free DF Curve 9-Sep-2.% 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep- 7.75% 9-Sep % 9-Sep % Probablty Curve 9-Sep-2 % 24-Mar % 23-Sep % 23-Sep % 23-Sep-5 3.2% 25-Sep % 24-Sep % 23-Sep % 24-Sep % In ths example the return value of the AdCdsNpv() functon s retreved and then the formula to recalculate the result s appled manually. Syntax =AdCdsNpv( 23SEP, 5SEP2, 5SEP4,2,Rsk-Free DF curve,proba Curve, CDSTYPE:EURCDS CLDR:NULL LFLOAT LFIXED FRQ:4, RISKMODEL:CURVE RECOVERY:.3 ND:DIS, RM:YC ZCTYPE:DF ) 33

36 CredtStructure KEYWORD CDSTYPE:EURCDS RISKMODEL:CURVE RECOVERY:XX SPECIFIES a European CDS the credt event probablty curve the recovery rate value n percentage Result returned by the functon =AdCdsNpv() returns: NPV.2263 Manual Calculaton Fxed leg DATE I CF I P(T TI ) P(T T I )= - P(T TI) DF(T I ) CFIXP(T T I )X DF(T I ) 5-Dec Mar Jun Sep Dec Mar Jun Sep In the table: TO CALCULATE USE EXAMPLE P( T) AdDefaultProba() =AdDefaultProba( 5SEP2,Date, CIR Array, LAY:H ) Dscount Factor at tme T AdRate() For example on the frst cash flow: DF2 = AdRate( 23SEP, 5DEC2, ZcDate:ZcRate, "RM:YC RATETYPE:CMP ZCTYPE:DF") = AdInterp() Interpolaton on the Yeld Curve at the date (5DEC2) of the frst cash flow for example: AdInterp( 5DEC2 ;ZCDate;ZCR ate;"im:lin") = Thomson Reuters Ekon Adfn Credt Calculaton Gude

37 CDS Prcng and Evaluaton Hence, NPV Fxed Floatng leg 8 CF DF( T ) P( T ).3678 DATE CONTINGENT PAYMENT DF P(T T) 5-Sep In the table: TO CALCULATE USE EXAMPLE Dscount Factor at tme T AdRate() AdInterp() DF2=AdRate( 23SEP, 5SEP4, ZcDate:ZcRate, "RM:YC RATETYPE:CMP ZCTYPE:DF") = Interpolaton on the Yeld Curve at the maturty date (5SEP4): Hence: =AdInterp( 5SEP4 ;ZCDate;ZCRate;"IM:LIN") = NPV Float ( R) DF( T ) P( T) ( 3%) Therefore the Net Present Value of the CDS s: NPV NPV Float NPV Fxed AMERICAN CDS PRICING Analytcal Formulas For an Amercan CDS, the contngent payment s pad at the tme of the default event: -Recovery t Default tme T Tme P P P P P Fxed leg calculaton NPV n Fxed CF DF( T ) P( T ) fx WHERE Fx DENOTES Outrght f the CDS s cross-currency (fx= otherwse) P( T ) whch s the probablty to default after the tme T of the th cash flows payment. DF(T )= dscount factor at tme T. 35

38 Floatng leg calculaton Maturty StartDate n nt( ) NbDays Maturty StartDate dt n NodeDate StartDate dt for =,,n NPV Float m ( R) DF( T ) P( T j j j ) P( Tj ) WHERE P( T ) j DENOTES The probablty to default after the tme Tj- of the (j-)th dscretzaton step. P( Tj ) The probablty to default after the tme Tj of the jth dscretzaton step. DF(T ) Dscount factor at tme T j. CDS Net Present Value calculaton PAID LEG NPV FORMULA Fxed Floatng NPV NPV Float NPV Fxed NPV NPV Fxed NPV Float Example of CDS prcng from Cox-Ingersoll-Ross Coeffcents Example n Thomson Reuters Ekon Excel Consder a Credt Default Swap, whch expres on 5th September 24. The settlement date s on 23rd September 2, t starts on 5th September 22, the spread s 2 bass ponts, and the recovery rate s 3%. So we have: Settle 23SEP Start Date 5SEP2 Maturty Date 5SEP4 Spread 2 R 3% NbDays Thomson Reuters Ekon Adfn Credt Calculaton Gude

39 CDS Prcng and Evaluaton CdsStructure CDSTYPE:AMERCDS CLDR:NULL LFLOAT AOD:NO LFIXED FRQ:4 CredtStructure RISKMODEL:CIR RECOVERY:.3 NBDAYS:46 ND:DIS Ratestructure RM:YC ZCTYPE:DF RISK-FREE DF CURVE 23-Sep-.% 23-Sep % 23-Sep % 23-Sep % 23-Sep % 23-Sep % 23-Sep % 23-Sep % 23-Sep % 23-Sep % 23-Sep- 63.9% CIR ARRAY In ths example the return value of the AdCdsNpv() functon s retreved and then the formula to recalculate the result s appled manually. Syntax =AdCdsNpv( 23SEP, 5SEP2, 5SEP4,2,Rsk-Free DF curve,cir_array, CDSTYPE:AMERCDS CLDR:NULL LFLOAT AOD:NO LFIXED FRQ:4, RISKMODEL:CIR RECOVERY:.3 NBDAYS:46 ND:DIS, RM:YC ZCTYPE:DF ) 37

40 CdsStructure KEYWORD CDSTYPE:AMERCDS AOD:YES SPECIFIES an Amercan CDS that the accrued s pad at the credt event date CredtStructure KEYWORD RISKMODEL:CIR RISKMODEL:CURVE RECOVERY:XX SPECIFIES the Cox-Ingersoll-Ross model the credt event probablty curve the recovery rate value n percentage Result returned by the functon =AdCdsNpv() returns: NPV.2263 Manual Calculaton Fxed leg DATE I CF I P(T TI ) P(T T I )= - P(T T I ) DF(T I ) CF I XP(T T I )X DF(T I ) 5-Dec Mar Jun Sep Dec Mar Jun Sep In the table: TO CALCULATE USE EXAMPLE P( T) AdDefaultProba() =AdDefaultProba( 5SEP2,Date,CIR Array, LAY:H ) Dscount Factors AdRate() For example on the frst cash flow: DF2 = AdRate( 23SEP, 5DEC2, ZcDate:ZcRate, "RM:YC RATETYPE:CMP ZCTYPE:DF") = Thomson Reuters Ekon Adfn Credt Calculaton Gude

41 CDS Prcng and Evaluaton TO CALCULATE USE EXAMPLE AdInterp() Interpolaton on the Yeld Curve at the date (5DEC2) of the frst cash flow for example: AdInterp( 5DEC2 ;ZCDate;ZCRate;"IM:LIN") = Hence, NPV Fxed 8 CF DF( T ) P( T ).3678 Floatng leg DATE CONTINGENT PAYMENT DF P(T T) 5-Sep In the table: TO CALCULATE USE EXAMPLE Dscount Factor at tme T AdRate() AdInterp() DF2=AdRate( 23SEP, 5SEP4, ZcDate:ZcRate, "RM:YC RATETYPE:CMP ZCTYPE:DF") = Interpolaton on the Yeld Curve at the maturty date (5SEP4): =AdInterp( 5SEP4 ;ZCDate;ZCRate;"IM:LIN") = Floatng leg Cash flows dates calculaton for the floatng leg are: Cash flows dates calculaton for the floatng leg are: Maturty StartDate 5SEP 4 5SEP 2 n nt( ) nt( ) 5 NbDays 46 5SEP 4 5SEP 2 dt 46 5 NodeDate StartDate dt 5SEP 2 46 for =,,5 Hence we have the table: DATEJ CONTINGENT PAYMENT DF(T J ) P(T T J ) P(T T J )= - P(T T J ) DF(T J )X[P(T T J -) - P(T T J )] 5-Sep Feb Jul Nov Apr Sep In the table: 39

42 TO CALCULATE USE EXAMPLE Dscount Factor at tme T AdRate() AdInterp() DF2=AdRate( 23SEP, 8FEB3, ZcDate:ZcRate, "RM:YC RATETYPE:CMP ZCTYPE:DF") = Interpolaton on the Yeld Curve at the at the date (8FEB3): =AdInterp( 8FEB3 ;ZCDate;ZCRate;"IM:LIN") = Hence, NPV Float ( R) ( 3%) m j DF( T ) j P( T ) P( T ) ( 3%) j Therefore the Net Present Value of the CDS s: NPV NPV Float NPV Fxed Example of a CDS Prcng from a Default Probablty Curve Example n Thomson Reuters Ekon Excel j Consder a Credt Default Swap, whch expres on 5th September 24. The settlement date s on 23rd September 2, t starts on 5th September 22, the spread s 2 bass ponts, and the recovery rate s 3%. So we have: Settle Start Date Maturty Date 23SEP 5SEP2 5SEP4 Spread 2 R 3% NbDays 46 CdsStructure CredtStructur e Ratestructure CDSTYPE:AMERCDS CLDR:NULL LFLOAT AOD:NO LFIXED FRQ:4 RISKMODEL:CURVE RECOVERY:.3 NBDAYS:46 ND:DIS RM:YC ZCTYPE:DF RISK-FREE DF CURVE 9-Sep-2.% 9-Sep % 9-Sep % 9-Sep % 4 Thomson Reuters Ekon Adfn Credt Calculaton Gude

43 CDS Prcng and Evaluaton RISK-FREE DF CURVE 9-Sep % 9-Sep % 9-Sep % 9-Sep % 9-Sep- 7.75% 9-Sep % 9-Sep % PROBA CURVE 9-Sep-2 % 24-Mar % 23-Sep % 23-Sep % 23-Sep-5 3.2% 25-Sep % 24-Sep % 23-Sep % 24-Sep % In ths example the return value of the AdCdsNpv() functon s retreved and then the formula to recalculate the result s appled manually. Syntax =AdCdsNpv( 23SEP, 5SEP2, 5SEP4,2,Rsk-Free DF curve,proba Curve, CDSTYPE:AMERCDS CLDR:NULL LFLOAT AOD:NO LFIXED FRQ:4, RISKMODEL:CURVE RECOVERY:.3 NBDAYS:46 ND:DIS, RM:YC ZCTYPE:DF ) CdsStructure KEYWORD CDSTYPE:AMERCDS AOD:YES SPECIFIES an Amercan CDS that the accrued s pad at the credt event date 4

44 CredtStructure KEYWORD RISKMODEL:CURVE RECOVERY:XX SPECIFIES the credt event probablty curve the recovery rate value n percentage Result returned by the functon =AdCdsNpv() returns: NPV.2263 Manual Calculaton Fxed leg DATE I DFI * P(T TI ) P(T T I )= - P(T T I ) DF(T I ) CF I XP(T T I )X DF(T I ) 5-Dec Mar Jun Sep Dec Mar Jun Jun In the table TO CALCULATE USE EXAMPLE P( T) AdDefaultProba() =AdDefaultProba( 5SEP2,Date,CIR Array, LAY:H ) Dscount Factor at tme T AdRate() For example on the frst cash flow: DF2 = AdRate( 23SEP, 5DEC2, ZcDate:ZcRate, "RM:YC RATETYPE:CMP ZCTYPE:DF") = AdInterp() Interpolaton on the Yeld Curve at the date (5DEC2) of the frst cash flow for example: AdInterp( 5DEC2 ;ZCDate;ZCRate;"IM:LI N") = Thomson Reuters Ekon Adfn Credt Calculaton Gude

45 CDS Prcng and Evaluaton Hence, NPV Fxed 8 CF DF( T ) P( T ) Floatng leg: Cash flows dates calculaton for the floatng leg are: Maturty StartDate 5SEP 4 5SEP 2 n nt( ) nt( ) 5 NbDays 46 5SEP 4 5SEP 2 dt 46 5 NodeDate StartDate dt 5SEP 2 46 for =,,5 Hence we have the followng table: DATE J CONTINGENT PAYMENT DF(T J )* P(T T J ) P(T T J )= - P(T T J ) DF(TJ)X[P(T T J- ) - P(T T J )] 5-Sep Feb Jul Nov Apr Sep In the table: TO CALCULATE USE EXAMPLE Dscount Factor at tme T AdRate() DF2=AdRate( 23SEP, 8FEB3, ZcDate:ZcRate, "RM:YC RATETYPE:CMP ZCTYPE:DF") = AdInterp() Interpolaton on the Yeld Curve at the at the date (8FEB3): =AdInterp( 8FEB3 ;ZCDate;ZCRate;"IM:LIN") = Hence, NPV Float ( R) ( 3%) m j DF( T ) j P( T ) P( T ) ( 3%) j Therefore the Net Present Value of the CDS s: NPV NPV Float NPV Fxed j 43

46 CDS BIG BANG IMPACT ON PRICING WITH ADFIN The Adfn prcng lbrary replcates the methodology used by the CDS prcer publshed on For example, note that the accrual on default of the protecton leg s calculated under assumptons that permt an exact ntegraton, whereas the old JPM method approxmated the ntegral. There are three mportant concepts to defne: the Par Spread, the Upfront Payment, and the Conventonal Spread. The Par spread s the spread at whch the CDS s traded on the market, meanng the CDS that gves an NPV equal to for the default probablty curve and for all tenors The upfront payment (n %) s the amount of cash to be pad to the protecton seller at the settlement of the deal for a CDS wth a fxed coupon ( or 5). It can be ether postve or negatve The conventonal spread s the spread that makes the NPV equal to the upfront wth a fxed coupon payment. Unlke the par spread, each tenor has ts own default probablty curve. Ths calculaton s standardzed usng a flat hazard rate and the ISDA Zero Coupon rate. How to match the ISDA/Markt calculators Buld your zero coupon curve usng AdTermStructure() Change keywords n CDSStructure to reflect new contract specfcatons Accrued has been added to AdCdsNpv() as a fourth output In the nputs, you have to enter the market data for dscount factors. For example, ths data can be obtaned from For example, on the trade date of 4-Mar-2, we obtan the followng nputs: M M,M.228% M 2M,M.2394% M 3M,M.259% M 6M,M.3832% M 9M,M.5976% M Y.8344% S 2Y.368% S 3Y.635% S 4Y 2.38% S 5Y % S 6Y 2.887% S 7Y 3.487% S 8Y % 44 Thomson Reuters Ekon Adfn Credt Calculaton Gude

47 CDS Prcng and Evaluaton S 9Y 3.525% S Y % S 2Y 3.96% S 5Y 4.45% S 2Y 4.35% S 25Y 4.395% S 3Y 4.425% Usng the AdTermStrucure() functon, we get the followng curve (please refer to the Adfn Term Structure Calculaton Gude for more nformaton): 8-Mar- 8-Apr May Jun Sep Dec Mar Mar Mar Mar Mar Mar Mar

48 8-Mar Mar Mar Mar Mar Mar Mar Mar Features of the CDS: Trade Date 4-Mar- Last Coupon Date 2-Dec-9 Maturty Date 2-Mar-5 Coupon (bps) 5 Recovery 4% Notonal,,. Converson Examples Convertng a conventonal spread of 5 to an upfront Buld the credt structure usng AdCredtStructure() 4-Mar- 2-Mar Thomson Reuters Ekon Adfn Credt Calculaton Gude

49 CDS Prcng and Evaluaton To obtan the upfront, use the AdCdsNpv() functon: Syntax =AdCdsNpv("4mar","2dec9","2mar5",5,IsdaCurve,CredtStructure, "FRCD:467 CLDR:WEEKEND CFADJ:Yes CFADJ:4282:NO DMC:F AOD:YES MDILD:YES AODMT:Exact STEPIN: CDSTYPE:AMERCDS CASHSETTLEDELAY:3WD LFIXED FRQ:4 CCM:MMAO ALIMIT:NO ACC:AO PX:C","RISKMODEL:CURVE IMPROBA:CFT NBDAYS: RECOVERY:.4","RM:YC ZCTYPE DF IM:LOG CLDRADJ:CLDR') Add or change keywords n the CDS Structure to reflect new contract specfcatons: KEYWORD AODMT:EXACT COMMENT Specfes the calculaton of accrual on default MDILD:YES Specfes that the end date s ncluded n the contract STEPIN: Specfes that T+ s used to determne accrual dates CASHSETTLEDELAY:3WD To dscount premum and accrued accordngly CFADJ:YES CFADJ:[MaturtyDate]-:NO Specfes that the maturty date must not be adjusted Fnally, we obtan the upfront: Upfront % For the sake of comparson, here s the correspondng Markt Calculator result: Upfront% = Clean Prce% = % = % 47

50 Convertng an upfront to a conventonal spread We buld the CDS structure usng AdCredtStructure() as follows: FRCD:467 CLDR:WEEKEND CFADJ:YES CFADJ:4282:NO DMC:F AOD:YES MDILD:YES AODMT:EXACT STEPIN: CDSTYPE:AMERCDS CASHSETTLEDELAY:3WD LFIXED FRQ:4 CCM:MMA ALIMIT:NO ACC:A PX:C UPFRONT:-.983 We obtan: 4-Mar- 2-Mar To calculate the conventonal spread, we use the AdCdsSpread() functon: Syntax =AdCdsSpread('4mar","2dec9","2mar5",,IsdaCurve,CredtStructure, "FRCD:467 CLDR:WEEKEND CFADJ:YES CFADJ:4282:NO DMC:F AOD:YES MDILD:YES AODMT:EXACT STEPIN: CDSTYPE:AMERCDS CASHSETTLEDELAY:3WD LFIXED FRQ:4 CCM:MMAO ALIMIT:NO ACC:AO PX:C","RISKMODEL:CURVE IMPROBA:CFI NBDAYS: RECOVERY:.4","RM:YC ZCTYPE:DF IM"LOG CLDRADJ:CLDR") Fnally, we obtan the conventonal spread: Conventonal Spread Here s the correspondng Markt calculator: Conventonal spread = Thomson Reuters Ekon Adfn Credt Calculaton Gude

51 CDS Prcng and Evaluaton Prcng a CDS The accrual on default s an ntegral. Its value can be approxmated numercally, or calculated exactly wth some assumptons. Adfn takes the latter approach. There are several ways of ntegratng the accrual on default of the fee leg and the contngent leg. Most fnancal software products approxmate all ntegrals n ths calculaton numercally. The ISDA xll lbrary (whch s based on a JPM model) uses the exact ntegral descrbed here. We have the followng formula for the fxed (also known as premum) leg: Premum Leg N N T N N T l T S DF s S DF( l) ds( l) T T WHERE DF DENOTES Dscount factor S N Fxed coupon/spread for the CDS of maturty TN Year fracton between T and T + S Probablty that the reference entty survves tme T, observed at tme The hazard rate s assumed to be a pecewse constant functon composed of ndvdual hazard rates k. The forward rate curve s assumed to be determnstc, and also pecewse constant composed of the rates f k. Under these assumptons, the ntegral for accrual on default can be wrtten as follows: N n () sk DFk sk DFk Accrual on Default SN Tk T Tk T T T k k fk k fk k fk k f k We have the followng formulas for the contngent leg: Amercan case: payment at default tme Protecton Leg T N LGD DF( l) ds( l), wth LGD=( recovery) European case: payment at maturty tme Protecton Leg LGD DFN( sn), wth LGD=( recovery) We agan assume that the hazard rate and forward rate are pecewse constant functons, as defned for the fxed leg. The Amercan ntegral can then be calculated exactly as follows: Protecton Leg e f 49 N ( f )( T T ) s DF

52 CREDIT LINKED NOTE PRICING Snce a CLN s a bond plus a CDS, ts evaluaton can be vewed as the sum of a classcal bond prcng and a CDS prcng detaled upwards. SPREAD CALCULATION The spread calculaton s a solver on the NPV calculaton. The frst solver used s Newton; when Newton fals, we use dchotomy. Example n Thomson Reuters Ekon Excel We use the same feature of the Amercan CDS descrbed upwards but wth a new NPV equals to.2. Syntax =AdCdsSpread( 23SEP, 5FEB2, 5SEP4,.2,Rsk-Free DF curve,cir_array, CDSTYPE:AMERCDS CLDR:NULL LFLOAT AOD:NO LFIXED FRQ:4, RISKMODEL:CIR RECOVERY:.3 NBDAYS:46 ND:DIS, RM:YC ZCTYPE:DF ) Result returned by the functon =AdCdsSpread() returns: Spread Manual Calculaton As explaned above, the spread s calculated by solvng the followng problem: We use the AdCdsNpv() functon for our calculaton. Syntax =AdCdsNpv( 23SEP, 5FEB2, 5SEP4,XX,Rsk-Free DF curve,cir_array, CDSTYPE:AMERCDS CLDR:NULL LFLOAT AOD:NO LFIXED FRQ:4, RISKMODEL:CIR RECOVERY:.3 NBDAYS:46 ND:DIS, RM:YC ZCTYPE:DF ) For the moment we assume that XX (unknown spread) s. In the example, the value of XX s the one, whch enables to return a NPV of the CDS equal to.2. By usng the Excel solver, we have: Spread Thomson Reuters Ekon Adfn Credt Calculaton Gude

53 Collateralzed Debt Oblgatons COLLATERALIZED DEBT OBLIGATIONS A collateralzed debt oblgaton (CDO) s smlar to a CDS; n exchange for a set of perodc payments, one buys protecton aganst losses due to default. The dfference between the two s that a CDO groups a large number of reference nstruments (bonds, loans, CDSs, etc.) nto a sngle debt nstrument. A CDO s sold n slces of dfferent rsks. Each slce s known as a tranche. Adfn Analytcs functons apply to several types of CDOs: cash CDOs, synthetc CDOs, and sngle-tranche CDOs such as ndexes (CDX, Traxx, etc.). The followng sectons dscuss CDOs n detal: CDO Prcng CDO Valuaton Impled Correlaton Example Usng Thomson Reuters Ekon Excel CDO References CDO PRICING The prcng model for a CDO s a multfactor copula model. Adfn Analytcs calculates the dstrbuton of the default loss condtonally on the factors of the model. The factors are random varables whch descrbe the behavor of the credt market. The basc formula that descrbes the model s 2 x a M a W () where x s a random varable. Ths basc formula defnes a correlaton structure between the x, whch here depend partally on a sngle common factor M. M and W are also random, wth ndependent, zero-mean, unt varance dstrbutons. The coeffcents a are constraned: a [,] Input factors Adfn Analytcs needs several nput factors, whch should be determned before calculaton: 5 The default probablty of each ssuer n the portfolo. The copula model decouples the ndvdual dstrbutons from the correlaton structure. Adfn Analytcs assumes that the margnal default probabltes are known (most lkely derved from the credt default swap market). The correlaton values (a ). Dscount factors or zero-bond prces for known maturtes. Recovery rates. These can be determned from data publshed by ratng agences. The type of copula you want to use: Gaussan or Student s t. The Gaussan copula s the model descrbed above, but more genercally Adfn Analytcs permts up to three common factors. The Student s t copula n Adfn Analytcs s an extenson of the Gaussan copula, agan for up to three common factors. The underlyng vector (Z, Z 2... Z N ) s assumed to follow a Student s t-dstrbuton wth Z degrees of freedom. Gx, where G follows an nverse /2 Gamma dstrbuton wth parameters.

54 Expected Loss: Both the spread and contngent payment legs of a CDO strongly depend on the expected global losses n the portfolo. The total loss s dvded nto dscrete tme steps (t, t 2... t K ), and Adfn Analytcs calculates the probablty at each step usng a recursve approach. Default correlaton model Let us assume that we have a debt portfolo of N loans or bonds. Defne the default tme of the th company or ssuer F (t) the probablty that default of the th company occurs before the date t. If Q s the cumulatve dstrbuton of x, then x can be mapped to usng a percentle-to-percentle transformaton. That s, Q ( x) F ( t) (2) Combnng ths relaton wth the basc model for x, we get P( t / M ) F W Q ( F ( t)) am 2 a (3) Ths equaton can be generalzed to many common factors: P( t / M, M 2,..., M n) F W Q ( F ( t)) 2 aj a j M j (4) Fnally, an ntegral over the common factors must be performed to compute the cumulatve dstrbuton of the default tme. For that we need the jont probablty densty of all common factors. Fortunately, we know that the common factors are ndependent. Ths fact makes the ntegraton easer. The user s choce of copula model s mportant. The standard Gaussan copula model s not very accurate n some crcumstances. In the next secton, we talk about some other copula models the end-user can choose to prce CDO tranches. Copula Models Gaussan copula Consder Equaton () for all ssuers. If we assume that M s a standard Gaussan varable, and (W, W 2 W N ) s a Gaussan vector, then the model s called a one-factor Gaussan copula. However, M may be extended to a Gaussan vector of common factors. Ths model s called a mult-factor Gaussan copula (Equaton 4). 52 Thomson Reuters Ekon Adfn Credt Calculaton Gude

55 Collateralzed Debt Oblgatons Student's t copula Ths model s an extenson of the Gaussan copula. Defne the vector (W, W 2 W N ) as follows: W Gx (5) where G follows an nverse Gamma dstrbuton wth /2 parameters. The vector (W, W 2 W N ) s assumed to follow a Student s t dstrbuton wth degrees of freedom. Furthermore, we only consder the case of symmetrc random varables W. Let W Gx (6) The vector (W, W 2 W N ) s assumed to follow a Student s t dstrbuton wth degrees of freedom. Furthermore, we only consder the case of symmetrc random varables W. Let 2 x a M a V (6) If S s the cumulatve dstrbuton functon of W, then the condtonal probablty that a default of the th ssuer occurs before date t gven (M, G) s /2 G S F() t am P( t M, G) FV 2 a (7) Ths probablty s ntegrated over M and G to determne the probablty of default. Ths s a two-factor model. Double t copula In ths model, the default tmes are modeled from a latent random vector (x, x 2 x N ) such that m 2 v 2 m v x a M a V (8) M and V are ndependent. They follow Student s t-dstrbutons wth probablty that the th company defaults before date t s m and v degrees of freedom respectvely. The v H F ( t) am ( m 2) / P( t M, G) S v v 2 a 2 (9) The probablty should be ntegrated over the factor(s) M and V, as wth the other copula models. Ths s the most commonly used type of copula for prcng CDO tranches. Other varants exst, such as the Clayton copula. In the future we may mplement other models to enrch the choce for the end-user. 53

56 CDO VALUATION The loss gven default In the "Copula Models" secton, we ntroduced varous formulas to compute the probablty that a gven company defaults before a fxed date t. Anyone who prces CDO tranches s manly nterested n the global loss rsk n the debt portfolo. A CDO tranche [A;B] absorbs any loss greater than A and less than B that occurs n the debt portfolo. The protecton buyer makes perodc nterest payments on a nomnal value to the protecton seller. If a default occurs, the protecton seller must pay the amount of the loss to the protecton buyer. The protecton seller's future nterest payments on the nomnal value are reduced by the amount he pad to compensate the loss. The fxed leg s pad by the seller of protecton. The floatng leg s pad by the protecton buyer. The ssue s to calculate the expected loss, as both the floatng and fxed legs strongly depend on ts value. Assume that the recovery rate of the th company s R and that ts nomnal value s M. If ths company defaults on date t, there s a loss equal to ( R )M. The loss functon at tme t can be wrtten as ( R ) M t The aggregate portfolo loss at tme t s L( t) ( R ) M t In tranche [A;B], expected loss absorbed at tme t s l t E L t B A AB, ( ) mn( ( ), ) () () Now, f there s a tenor structure {t, t, t k, t K }, at tme t k, the floatng leg cash flow s l ( tk ) l( tk ) and the fxed leg cashflow s kr B A l( tk ) l( tk ) 2 Here k s the accrual factor and r s the rate or the premum. The present value of the CDO s Recursve approach As the prevous formula shows, we must calculate l(t k ) for all k[ K] Adfn takes a recursve approach. The boundares of the loss nterval are (zero loss) and max (l ). To set up the recursve calculaton, we must frst choose an approprate dscretzaton step for ths nterval. Sdenus argues for a dscretzaton method wth lttle mpact on the computatonal effort of the ntegraton. Hs method also depends on a senstvty threshold, whch s arbtrarly chosen. Adfn adopts the same method, but nstead of a senstvty threshold sets a maxmum teraton number for the loop. Every company or oblgor of the pool has a possble loss equal to ( R )M. 54 Thomson Reuters Ekon Adfn Credt Calculaton Gude