Distance-to-Default (According to KMV model)

1 Distance-to-Default (AccordingtoKMVmo del)TeterevaAnastasijaNumericalIntro ductoryCourseScho olofBusinessandEconomicsHumb oldt Universit :CalculationofDistance-to-Defaultaccordi ngtoKMVmo del(KealhoferMertonVasicekmo del)Theaim:Europ eancompanies,b othnon-/defaulted,b othnon-/ nancialContents: TheoreticalbackgroundIKMVmo del ApplicationusingrealdataIComputationofDi stance-to-DefaultIComputationofprobabili tyofdefaultDistance-to-DefaultTheoretica lbackground2-1 Whatisdefault?Defaulthapp rm' "measure" rm' "measure" delIdea:Firm'sequitycanb ondholdersreceivetheirdebts, : observablevalueandvolatilityofequity(VEa nd E), unobservablevalueandvolatilityof rm'sasset(VAand A). BasedonBlack-Scholesoptionpricingtheory. Equityisacalloptiononthevalueofassetsoft hecompany(VAconsideredasC,VEasS). Debt(D)istakenasastrikeprice(Dconsidered asK). distance -to-DefaultTheoreticalbackg round2-4 KMVmo delAssumptions: Debt:homogeneouswithtimeofmaturityT Capitalstructure:VA(t) =D(t) +VE(t) Marketp erfection:ignorecoup onsanddividends,nop enaltytoshortsales.

2 Dynamicoftheasset:assetsaretradedandfoll owgeometricBrownianmotiondVA= AVAdt+ , Aitsvolatility, AdriftanddWisaWienerpro delDuetoBlack-Scholesoptionpricingtheory analogicallytoC(t) =S(t) (d1) e r(T t) K (d2)valueofequatycanb epricedasVE(t) =VA(t) (d1) e r(T t) D (d2)(1)UsingIto'sformulaonecanshow E=VAVE VE VA A(2)VA(VE)-valueoftheasset(equity)d1=log (VA(t)D)+(r 12 2A)(T t) A T t A( E)-volatilityofasset(equity)d2=d1 A T tr-risk-freerateT-timeofdebt'smaturityDi stance-to-DefaultTheoreticalbackground2- 6 KMVmo del-nonlinearsystemofequationsThus,to ndunobservablevalueandvolatilityoftheass etoneshouldsolvethenonlinearsystemofequa tions:{f1(VE, E) =VA(t) (d1) e r(T t) D (d2) VE(t) =0f2(VE, E) =VAVE (d1) A E=0 Thesolutionisuniqueas f1 VA= (d1)(analogicallyto inBlack- scholes )f1isincreasingfunctionof VA f1(VA) ,f2( E) del- distance -to-DefaultDefaulthapp enswhenthevalueofcompany'sassetfallsb elow"defaultp oint"(valueofthedebt).}

3 Distance-to-Default -distanceb etweentheexp ectedvalueoftheassetandthedefaultp (t) =log(VAD) + (r 12 2A)(T t) A T tAndprobabilityofdefault:PD(t) =P[VA D] = = ( DD) distance -to-DefaultTheoreticalbackgro und2-8 KMVmo del- distance -to-CapitalDistance-to-Capit al-pro cessedfromDistance-to- default -reason:DDd o esnotincludecomplexitiesrelatedto nancial rms-accordingto[Larsen&Mange,2008]comput edas:DC(t) =log(VA D) + (r 12 2A)(T t) A T twhere =11 PCAR-PCAR capitalrequirement(AccordingtoBaselCapit alAccordIsetto8%)-forDDwetake = ,VA(t) oint, ,TSource:[Crosbie&Bohn,2004] distance -to- DefaultApplicationusingrealdata-DATADESC RIPTION3-1 erofsto (almost)defaulted nancial:Commerzbank non nancial:ArcandorInondefaulted nancial:Credit-Suisse non (short-andlong-termdebts) distance -to-Def aultApplicationusingrealdata-DATADESCRIP TION3-2 ( ) nancialassistance-6thJun2009announcedina bilitytopayrentsforstores-9thJun2009bank ruptcyDistance-to-DefaultApplicationusin grealdata-DATADESCRIPTION3-3 Realdata:Credit-Suisse-swissinternationa l nancialcompany-2009 BankoftheYearbytheInternationalFinancing ReviewVolvo-Swedishpro ducerofcars,trucks.

4 -rapidgrowthinlastyears,2007b oughtNissanDistance-to-DefaultApplicatio nusingrealdata-CALCULATION4-1 Calculation: (1year) cks sto +onehalfoflong- (inR) : (1year) cks sto +onehalfoflong- (inR) osingthemetho dExample:VE=4740291, E= ,D=33404048,r= ,T t=1,VA-?, A-? Startingvalue(VA=4740291, A= )INewton's(8023027, ),3iterationsIBroyden(8023027, ),3iterationsIIterations(8023027, ),6iterationsIOne-dimensional(8023027, ),7iterations Startingvalue(VA=0, A=0)INewton's(4740291, ),JacobianissingularIBroyden(4740291, ),JacobianissingularIIterations(8023027, ),6iterationsIOne-dimensional(8023027, ),7iterationsDistance-to-DefaultApplicat ionusingrealdata-Rco des5-1 Newton's,Broyden12> f n e w t o n < - f u n c t i o n ( x ) {3+ y < - n u m e r i c ( 2 )4+ d 1 = ( l o g ( x [ 1 ] / Z ) + ( r + x [ 2 ] ^ 2 / 2 ) * T )}

5 / x [ 2 ] / s q r t ( T )5+ d 2 = d 1 - x [ 2 ] * s q r t ( T )6+ y [ 1 ] < - S 0 - ( x [ 1 ] * p n o r m ( d 1 ) - e x p ( - R * T ) * D * p n o r m (d 2 ) )7+ y [ 2 ] < - s i g m a S * S 0 - p n o r m ( d 1 ) * x [ 2 ] * x [ 1 ]8+ y9+ }10> n l e q s l v ( c ( V E , S E ) , f n e w t o n , c o n t r o l = l i s t ( b t o l =.))

6 0 1 ) ,11m e t h o d = " B r o y d e n " ) \ $ x12[ 1 ] 8 . 0 2 3 0 2 7 e + 0 6 1 . 4 1 6 1 8 5 e - 0 213> n l e q s l v ( c ( V E , S E ) , f n e w t o n , c o n t r o l = l i s t ( b t o l = . 0 1 ) ,14m e t h o d = " N e w t o n " ) \ $ x15[ 1 ] 8 . 0 2 3 0 2 7 e + 0 6 1 . 4 1 6 1 8 5 e - 0 2 distance -to-DefaultApplicationusingreald ata-Rco des5-2 Iteration1> D 1 < - f u n c t i o n ( V 0 , Z , r , s i g m a V , T )2+ { ( l o g ( V 0 / Z ) + ( r + s i g m a V ^ 2 / 2 ) * T ) / s i g m a V / s q r t ( T ) }3> D 2 < - f u n c t i o n ( d 1 , s i g m a V , T ) { d 1 - s i g m a V * s q r t ( T ) }4> f 1 < - f u n c t i o n ( V a )5+ { V a * p n o r m ( D 1 ( V a , D , R , S A , 1 ) ) - e x p ( - R )}

7 * D * p n o r m ( D 2 ( D 1 (V a , D , R , S A , 1 ) ,6+ S E , 1 ) ) - V E }7> f 2 < - f u n c t i o n ( S a ) { V A / V E * p n o r m ( D 1 ( V A , D , R , S a , 1 ) ) * S a -S E }8> I T 1 < - V E; I T 2<- S E ; c o u n t e r<- 09> w h i l e ( s q r t ( ( S A - I T 1 ) ^ 2 + ( V A - I T 2 ) ^ 2 ) > 0 . 1 * ( 1 + s q r t (I T 1 ^ 2 + I T 2 ^ 2 ) )10+ a n d c o u n t e r < 1 0 0 0 )11> { S A < - I T 2; I T 1<- u n i r o o t ( f 1 , c ( 0 , V E * 1 0 0 ) ) \ $ r o o t12+ V A < - I T 1.

8 I T 2<- u n i r o o t ( f 2 , c ( 0 , S E * 1 0 0 ) ) \ $ r o o t13+ c o u n t e r < - c o u n t e r + 1 } distance -to-DefaultApplicationusingrealdata-Rco des5-3 Reductiontoone-dimensionalcase1> f < - f u n c t i o n ( x ) {2+ V A = x [ 1 ]3+ S A = x [ 2 ]4+ d 1 = ( l o g ( V E / D ) + ( R + S A ^ 2 / 2 ) * T ) / S A / s q r t ( T )5+ d 2 = d 1 - S A * s q r t ( T )6+ e 1 = V E - ( V A * p n o r m ( d 1 ) - e x p ( - R * T ) * D * p n o r m ( d 2 ) )7+ e 2 = S E * V E - p n o r m ( d 1 ) * S A * V A8+ r e t u r n ( e 1 ^ 2 + e 2 ^ 2 )9+ }10> n l m i n b ( c ( V E , S E ))

9 , f , l o w e r = c ( 0 , 0 ) , u p p e r = c ( 1 E 1 0 , 1E 3 ) , ) \ $ p a r11[ 1 ] 8 . 0 2 3 0 2 7 e + 0 6 1 . 4 1 6 1 8 5 e - 0 2 distance -to-DefaultApplicationusingreald ata-RESULTS6-105010015020025002040608010 0 COMMERZBANKTimeProbability of DefaultFigure1:DefaultprobabilityusingDi stance-to-DefaultDistance-to-DefaultAppl icationusingrealdata-RESULTS6-2050100150 200250020406080100 COMMERZBANKTimeProbability of DefaultFigure2:DefaultprobabilityusingDi stance-to-CapitalDistance-to-DefaultAppl icationusingrealdata-RESULTS6-3050100150 200250020406080100 ARCANDORTimeProbability of DefaultFigure3:DefaultprobabilityusingDi stance-to-DefaultDistance-to-DefaultAppl icationusingrealdata-RESULTS6-4050100150 200250020406080100 ARCANDORTimeProbability of DefaultFigure4 of DefaultFigure5 of DefaultFigure6:DefaultprobabilityusingDi stance-to-CapitalDistance-to-DefaultAppl icationusingrealdata-RESULTS6-7050100150 200250020406080100 VOLVOTimeProbability of DefaultFigure7.

10 DefaultprobabilityusingDistance-to-Defau ltDistance-to-DefaultApplicationusingrea ldata-RESULTS6-8050100150200250020406080 100 VOLVOTimeProbability of DefaultFigure8:DefaultprobabilityusingDi stance-to-CapitalDistance-to-DefaultAppl icationusingrealdata-CONCLUSIONS7-1 Conclusions: Some nancialproblemscanb efore. For nancialcompaniesDistance-to-Capitalismor eappropriateforcalculatingtheprobability ofdefault. Mo delismoreusefulforratingthanpredicting. Mostof : Estimationofvolatilityindi erentways. Estimationofinterestrateindi erentways. Di erentfrequenciesofdata. ComparisonofUSandEurop eancompanies. Di erenttimehorizon. Impactofcrisisondi erentindustries. ImpactofBaselI :priceandnumb erofsto cksdatabase:DatastreamEurib ordatabase:DatastreamDistance-to- default Sources9-2 Referencesandarticles: CrosbieP.,BohnJ.(2004):Mo dellingdefaultrisk,PublishedbyMo o dy'sKMVC ompany ,ShumwayT.(2004):ForecastingDefaultwitht heKMV-MertonMo del,UniversityofMichigan LuY.