lexander L. Baranovski, Carsen von Lieres and ndré Wilch 8. May 2009/ Defaul inensiy model Pricing equaion for CDS conracs Defaul inensiy as soluion of a Volerra equaion of 2nd kind Comparison o common pracise: The boosrapping approach revisied Deriving defaul inensiy from CDS spread observaions. Special case of Nelson Siegel: Exac analyical soluion and numerical recipe Simple approximaion formula for defaul probabiliies Fiing of sochasic models of defaul inensiy: CIR Model Page
Credi Defaul Swap (CDS) Reference firm Exposed o credi risk (which may include bankrupcy, failure o make principal or ineres paymens, resrucering, ) Proecion buyer Periodically pays a premium (CDS-spread) unil mauriy or a credi even Proecion seller Provides proecion from credi losses by paying a compensaion in case of a credi even before mauriy Conrac deails: Noional, Mauriy, Paymen frequency (e.g. quarerly), Credi even ype, Selemen (physical / cash) Premium conains informaion on marke s esimae of credi risk. Our goal is o exrac his informaion from CDS spread observaions. Page 2
Defaul Inensiy Model Defaul probabiliy (disribuion of defaul ime ) pdf of P( τ = exp λ( u) du 0 ( ) p dp = d ( τ defaul inensiy (defaul hazard rae) l is he condiional defaul arrival rae given no defaul up o ime : P( < τ + p( P( τ + τ > = λ( ( > ) > ( τ ) P τ 0 P Page 3
Valuaion of risky cash flows Le he risk free rae (shor rae) r( be given. Today s price of a riskless zero coupon bond wih mauriy is hen (under he assumpion of an arbirage free world) D( = exp r( u) du 0 More generally, suppose a riskless invesmen provides cash flows C a imes. Then he presen value of his invesmen is,..., C n,..., n PV riskless ( C,..., C Now suppose ha cash flows are coningen on he survival of an obligor wih defaul ime. (Their iming could also depend on.) Then he presen value is D( where denoes he indicaor funcion of he se (even τ >. I > ) = n i i n i= C n PV risky ( C,..., C ) = n C D( i i ) I τ > i i= τ { } ) Page 4
Valuaion of he Premium leg of a CDS The presen value of premium paymens PV n Pr emium = s i D( i ) Iτ > i= where s denoes he CDS spread and i = i i he ime inerval beween premium paymens and, as above, he survival indicaor. I τ >i i i By definiion of he discoun facors D( i ) and since E( I τ > ) = P( τ > i i ) = exp λ( u) du 0 he expecaion of he presen value of he premium paymens is E( PV λ n i Pr emium ) = s i exp r( u) + ( u) i= 0 du Page 5
Coninuous premium paymens Wih he noaion f ( = exp r( u) + λ( u) du 0 we ge E( PV n Pr emium) = s i f ( i ) i= For 0 his may be approximaed by i E( PVPr emium ) s f ( d Using his approximaion we implicily assume coninuous, accrued premium paymens T 0 Page 6
Valuaion of he defaul leg of a CDS The presen value of he defaul paymen is given by PV Defaul = δ D( τ ) Iτ T where d denoes he loss given defaul (fixed), T he mauriy of he conrac and he defaul indicaor. I T τ The expecaion of he presen value of defaul paymens is given by T E( PVDefaul ) = δ E( D( τ ) Iτ T ) = δ λ D( λ( exp ( u) du 0 0 Wih f defined as above (previous slide) his may be rewrien as E( PV ) = δ λ( f ( d Defaul T 0 d Page 7
Pricing equaion Under he above consideraions he pricing equaion E ( PVPr emium ) = E( PVDefaul ) for he CDS conrac becomes (IEQ) T s( T ) f ( d = δ λ( f ( d T 0 0 which is an inegral equaion for he funcion f. Wih he noaion s = s(t) we emphazize he dependency of he CDS spread on he mauriy T. Page 8
Piecewise consan defaul inensiy In case of a consan deaful inensiy he inegral equaion (IEQ) simplifies o λ = s( T ) [ 0, T ] δ Now suppose he CDS spread observaions s(t ) for differen mauriies T < T 2 < < T m are given (e.g. m = 5, T =, T 2 = 3, T 3 = 5, T 4 = 7 and T 5 = 0 years) and λ m ( = k I k k= λ, ( ( T T ] where T 0 = 0. Then from (IEQ) applied o each spread observaion we obain a se of equaions s k ( ) ( ) ( ) = k ( ) T ( ) k J λ exp λi + λi Ti δ λ J λ exp λi + λi Ti = i= k = i= where J l,,l m. T ( λ ) = D( exp( λ T d (k =,2,,m) which may be solved ieraively for Page 9
Piecewise consan defaul inensiy (con.) k = : derive λ from equaion s( T = δ λ ) λ2 k = 2: derive from and equaion s λ 2 ( ) ( ) ( ) = 2 ( ) T2 J λ exp λi+ λi Ti δ λ J λ exp ( λi+ λi ) T i = proceed like his unil k = m. i= Disadvanage of his procedure (which is ofen referred o as boosrap ): Defaul inensiy disconinuous (as a sep funcion) No smoohing of noisy daa = i= Page 0
Solving he inegral equaion Our approach relies on firs smoohing CDS spread observaions (e.g. by fiing a Nelson-Siegel ype curve o he observaions). Then we solve he inegral equaion (IEQ) for f( Once we have done ha we may use he relaion (log f() = (l( + r() o obain l( = f ( / f( r( dvanage of his approach: Defaul inensiy is a coninuous funcion of ime Smoohing of noisy observaions is done in a preceding sep Page
Volerra equaion Using he relaion l( = f ( / f( r( equaion (IEQ) may by pu ino he form s( f ( = + r( x) f ( x) dx δ which is a Volerra equaion of he 2 nd kind. 0 Insead of solving he inegral equaion direcly we sugges o ransform he problem o he problem of solving an ordinary linear differeial equaion of 2 nd order which is numerically easier o handle Page 2
Differenial equaion Tranformaion ino an ordinary linear differenial equaion of 2 nd order: (ODE) f '' + f ' g + f h = 0 wih ime dependend coefficiens where s( g ( = r( + + δ s''( s'( and h( = r'( + 2s'( δ s''( r( + s'( s( δ s( : CDS spread curve r( : reference curve (risk free rae) wih iniial condiions: f ( 0) = and f '(0) = λ(0) r(0) where λ(0) = s(0) δ Page 3
Special case of Nelson Siegel: Exac analyical soluion and numerical recipe NS curves are fied o he CDS quoes as well as he shor raes 5 4.5 4 3.5 EURIBOR_09 shorrae_09 EURIBOR_08 800.00 700.00 600.00 BBB BB B BBB rae [% ] 3 2.5 2 shorrae_08 C D S spread [bps ] 500.00 400.00 300.00.5 200.00 0.5 00.00 0 5 9 3 7 2 25 29 33 37 4 45 49 53 57 6 65 69 73 77 8 85 89 93 97 0 05 09 3 7 2 enor [monhs] 0.00 5 9 3 7 2 25 29 33 37 4 45 49 53 57 6 65 69 73 77 8 85 89 93 97 0 05 09 3 7 enor [monhs] he fied curves in a form shor rae on 20.03.08: r( = 0.045 (0.00063 + 0.0059 exp(-0.67 shor rae on 20.03.09: r( = 0.037 - (0.026+ 0.07 exp(-.032 CDS spread s( = a (b + c exp(-d Page 4
Implied defaul inensiy erm srucure he fied curves s( and r( as well as LGD δ =0.6 as an inpu of he ODE f '' + f ' g + f h = 0 defaul inensiy λ( = f f r( is an oupu of he ODE CDS implied defaul rae on 20.03.08 BBB BB B CDS implied defaul rae on he 20.03.2009 BBB BB B 9.00% 8.00% 20.03.2008 20.03.2009 40 35 r a e [ % ] 7.00% 6.00% 5.00% 4.00% 3.00% 2.00%.00% r a e [% ] 30 25 20 5 0 5 0.00% 5 9 3 7 2 25 29 33 37 4 45 49 53 57 6 65 69 73 77 8 85 89 93 97 0 05 09 3 7 ime [monhs] 0 5 9 3 7 2 25 29 33 37 4 45 49 53 57 6 65 69 73 77 8 85 89 93 97 0 05 09 3 7 ime [monhs] Page 5
Comparison o boosrapping approach defaul raes 0.04 smooh defaul inensiy curve implied by he proposed procedure 0.03 0.02 0.0 piece-wise consan defaul inensiy by boosrapping 0 2 4 6 8 0 years Defaul inensiy for European -raed corporaes as of as of 20.03.2008 for mauriies below 5 years: piece-wise consan defaul inensiy displays an economically uninuiive behaviour (large ump sizes) and is poenially vulnerable o anomalies in he daa (zig-zag behaviour). These disadvanages are avoided in he Nelson Siegel fiing approach. Page 6
Implied PD erm srucure PD PD( = P( τ = exp λ( u) du 0 PD erm srucure PD erm srucure on he 20.03.2009 BBB BB B BBB BB B 60.00% 50.00% 20.03.2008 00 90 80 20.03.2009 40.00% 70 60 P D [ % ] 30.00% 20.00% P D [ % ] 50 40 30 0.00% 20 0 0.00% 5 9 3 7 2 25 29 33 37 4 45 49 53 57 6 65 69 73 77 8 85 89 93 97 0 05 09 3 7 ime [monhs] 0 5 9 3 7 2 25 29 33 37 4 45 49 53 57 6 65 69 73 77 8 85 89 93 97 0 05 09 3 7 ime[monhs] defaul rae and PD can be done in a closed form PD,20.03.2008) = e ( r γ ) 0.025 ν ( ) ν ( ( ) = exp(, r = 4.34%; F ( a. b; z) γ = 0.369; ν γ 0.995 0.66 + + ( ) 0.97 e ν 0.52 0.025 ν ( ( ) F ( 2,.85, 0.025ν ( ) is he Kummer confluen hypergeomeric funcion Page 7
Implied erm-srucure of cumulaive defaul probabiliies for - raed corporaes Raing defaul probabiliis 0.35 PD 0 y 0.30 0.25 PD 7 y 0.20 PD 5 y 0.5 PD 3 y 0.0 0.05 PD y 0 50 00 50 200 days 20.03.08 < τ (days) < 20.03.09 Page 8
-raed CDS erm-srucure Raing 300 250 CDS cds 0 y cds 7 y cds 5 y cds 3 y cds y 200 50 00 50 00 50 200 20.03.08 < τ (days) < 20.03.09 days Page 9
Simple approximaion formula for defaul probabiliies We consruc a raio µ raing ( enor, dae) = CDS PD raing raing ( enor, dae) ( enor, dae) sable over ime for a large range of raing classes!!! 0.40 0.35 0.30 0.25 0.20 µ raing dae = raing ( enor, dae) : 0.5y { 20.03.08,..., 20.03.09}, = {,,BBB} enor 0y, 0.5 4 6 8 0 enor PD R 0.039 0.222 (, x) = ( 0.065 + 0.468e + 0.835e ) CDS (, x) x, R = BBB, 0.5 [ years] 0 R, Page 20
Fiing of sochasic models of defaul inensiy: CIR Model-I We consider a case of a sochasic defaul inensiy ( θ λ) d σ λdw dλ = α + before: f ( = exp r( u) + λ ( u) du 0 now: f ( = E exp r( u) + λ( u) du 0 f ( = D( E exp λ( u) du λ( 0) D( exp( ( λ( 0) C( ) CIR + 0 ffine erm srucure ( ) = 2 β ( β + α ) + 2β ( e ) C ( 2αθ ln σ ( β + α ) / 2 2β e β ( β + α ) ( e ) + 2β = 2 2 β = α + 2 2σ Page 2
Fiing of sochasic models of defaul inensiy: CIR Model-II ( = D( exp( ( λ( 0) C( ) f CIR + solves f '' + f ' g + f h = 0 wih he inpu parameers s( (CDS), r( (shor rae), δ (LGD) i f i i PD( i ) ( ) i i min { α, θ, σ } ( ) D ( ) exp ( ) s(0) δ + C( ) 2 survival probabiliy.00 0.95 0.90 α = 2.74, σ = 450.73 and θ = -2.38 on 20.03.2008 0.85 0.80 Raing 0.75 0 2 4 6 8 0 years Page 22
Conclusion New mehod for deriving defaul inensiy from CDS spread observaions The mehod has wo main advanages: The defaul inensiy naurally becomes a coninuous funcion of and no economically uninuiive disconinuiies arise. The procedure is sable w.r.. ouliers and noisy daa (e.g. due o erroneous CDS-quoes) because is relies on a preceding smoohing procedure. The new esimaion procedure also serves as a sable basis for fiing sochasic defaul inensiy models like CIR. Page 23