Estimating Default Probabilities Using Corporate. Bond Price Data

Transcription

1 HITOTSUBASHI UNIVERSITY Ceter for Fiacial Egieerig Educatio (CFEE) CFEE Discussio Paper Series No. 2- Estimatig Default Probabilities Usig Corporate Bod Price Data Reiko Tobe Ceter for Fiacial Egieerig Educatio (CFEE) Departmet of Ecoomics, Hitotsubashi Uiversity 2- Naka, Kuitachi Tokyo, Japa

2 Estimatig Default Probabilities Usig Corporate Bod Price Data Reiko Tobe March 3, 22 Abstract This paper discusses estimatig method of implied default probability from the iterest spread. Istead of usig Duffie & Sigleto(999) model directly, we use statistical estimatig procedure of Takahashi(2). We shall provide a complete proof of the cosistecy of the estimator i the Takahashi(2) model employig the method of Wald(949) which gives the proof of cosistecy of maximum likelihood estimate. We also verify regularity coditios for asymptotic properties of the estimator i the Weibull survival fuctio case. Furthermore, we provide a empirical aalysis of implied survival probability estimatig. I the empirical aalysis, we discuss asymptotic properties usig bootstrap method. A iterestig fidig is that the umber of the data ad the existece of the outlier data have a major effect o the bootstrap estimatig results. Whe we aalyze the impact of the bakruptcy of Lehma Brothers, we observe that the iflueces differ with the 5 idividual compaies. Keywords: Implied default probability; Statistical model; Parametric model; Pseudo maximum likelihood estimator; Cosistecy; Asymptotic ormality; Bootstrap; Itroductio I this paper we study estimatig method of default probabilities of idividual compaies ad the represetative default probabilities of ratig classes. May ratig agecies publish bod issuer s ratig usig their ow methodology i measurig creditworthiess ad a specific ratig scale. Typically, ratigs are expressed as letter grades that rage, for example, from AAA to D to commuicate the agecy s opiio of relative level of credit risk. For example, a corporate bod that is rated AA is viewed by some ratig agecy s as havig a higher credit quality tha a corporate bod with a BBB ratig. But the AA ratig is t a guaratee that it will ot default, oly that, i their opiio, it is less likely to default tha the BBB bod. Some ratig agecies also aouce aual default probabilities of each ratig class. For modelig credit risk, two classes of models exists; structural model ad reducedform model. Reduced-form model was origially itroduced by Jarrow & Turbull (992), ad subsequetly studied by Jarrow ad Turbull (995), Duffie ad Sigleto (999) amog others. By usig Duffie ad Sigleto(999) model, we may calculate implied default probability by calibratio. Implied default probability meas default Graduate School of Ecoomics, Hitotsubashi Uiversity

3 probability which is icorporated by the bod prices ad it gives us view ad expectatios i the market. Takahashi(2) itroduced statistical model for estimatig implied default probability. They assumed observed corporate bod price cotais some error. As a result of cotaiig error term, we ca discuss asymptotic properties of the estimator. I Takahashi(2), they gave a sketchy proof for cosistecy ad a complete proof for asymptotic ormality. They also recommeded usig bootstrap method for examiig asymptotic properties whe we apply their method to real data. The first objective of this paper is to give the complete proof of the cosistecy of the pseudo maximum likelihood estimator which is proposed by Takahashi(2). We employ the method of Wald(949) which gives the proof of cosistecy of maximum likelihood estimate. His proof assumes the compact property ad uses the strog law of large umbers. Similarly, we prove cosistecy of our estimator uder some regularly coditios usig the strog law of large umbers. Furthermore we shall verify regularity coditios for cosistecy ad asymptotic ormality of our estimator. Whe we estimate the statistical model of Takahashi(2), we may use parametric model for estimatig survival fuctio. The parametric models we cosider are expoetial model ad Weibull model. Needless to say, expoetial distributio is a special case of Weibull distributio, we eed oly to verify the coditios i the Weibull model. That is the secod objective of this paper. The last objective is to apply our model to real data to estimate default probabilities(survival probabilities). We preset several umerical results i this paper. First, i order to set kow costats of our model, we examie comparative study varyig the recovery rate ad the coefficiet of error term. Secod, we compare the results o two parametric survival fuctios; expoetial survival fuctio ad Weibull survival fuctio. The sum of squared residuals from the expoetial survival fuctio tured out much larger tha that of the Weibull survival fuctio. Third, we compare differet recovery models; RM recovery model proposed by Duffie & Sigleto(999) ad RT recovery model proposed by Jarrow & Turbull(995). The examiatio showed that there is o great differece betwee the two. Fourth, we show estimatig results ad bootstrap results usig 8 idividual compaies data ad 5 ratig classes data o April 6, 24. There is a close relatio betwee the distributio of bootstrap estimates ad the umber of data. For example, the umber of Toyota s data was oly 3 ad the estimated bootstrap distributio of Toyota was quite differet from ormal distributio. It seems the larger amout of data(bods) beig used, the more like the bootstrap distributio is ormal distributio. The existece of outlier data also ifluece upo the distributio of bootstrap estimates. For example, whe we excluded 3 outlier data from the origial AA class data, the warps are reduced ad the distributios of bootstrap parameter estimates look more like ormal distributio. Kolmogorov-Smirov test also supported the ifluece of outlier data o the estimatig results. Fially we apply our statistical estimatig method to time series data. Sice we are iterested i the impact of the bakruptcy of Lehma Brothers(September 5, 28), we use the data from Jauary 4, 28 to March 3, 29 ad select 5 compaies, which might be affected by the bakruptcy. I fact, all 5 compaies survival probabilities were decliig after the bakruptcy, but the iflueces differ with the idividual compay i some poits. This paper is orgaized as follows. Sectio 2 provides a review of statistical estimatig method of implied survival probabilities proposed by Takahashi(2). The proof of asymptotic property of our estimator is discussed i Sectio 3. Some empirical results are preseted i Sectio 4, ad the last Sectio is devoted to the summary ad cocludig remarks. 2

4 2 Review of the model ad some developmet I this sectio, we provide a brief explaatio of Duffie ad Sigleto(999), which derive the relatioship betwee the default probabilities ad the iterest yield. We review Takahashi(2) model which is the mai model of this paper. We also show the applicatio of the model to the differet recovery type case. 2. Setup ad results o implied default probabilities Throughout this chapter, W = {W(t), t } deotes stadard Browia motio, o a probability space (Π,F,P r ). The filtratio {F t } t is geerated by W. We suppose there are o arbitrage opportuities i the market ad the market is assumed to be complete. Let P(,t) be time price of the defaultable zero coupo bod (corporate bod) maturig at t ad P (,t) be time price of the correspodig default free bod (govermet bod). The spread (differece of the yield to maturity betwee corporate bod ad govermet bod) ϒ is thus defied by ϒ() = t log [ P(,t) P (,t) Let Q be the uique equivalet Martigale measure, the, [ t ]} P (,t) = E {exp Q r(v)dv where r(v) is a istataeous risk free spot rate ad E Q deotes the expectatio uder the measure Q (Harriso ad Pliska[98]). Let τ be the time of default, the the survival fuctio G(t) is give by G(t) = Q{τ > t} { = exp t ] } λ(u)du where λ(t) is the hazard fuctio. We also let δ t be the recovery rate i the evet of default at t. The formula betwee the survival fuctio ad the iterest spread varies with the defiitio of recovery model. We cosider the followig two recovery models which are i commo use. RM Type Recovery (cf. Duffie & Sigleto(999)) By assumig that δ t P(τ,t) is recovered at the time of default, we have [ T } P(t,T ) = E {exp Q (r(v) + ( δ v )λ(v))dv ] F t t where P(τ,t) is the price of corporate bod just before default. If λ is assumed to be o radom fuctio ad δ to be costat, we have [ t ]} P(,t) = E {exp Q (r(v) + ( δ)λ(v))dv 3

5 [ t ] [ t ]} = E {exp Q r(v)dv exp ( δ) λ(v)dv = P (,t)q{τ > t} δ = P (,t)g(t) δ Uder RM type recovery, spread becomes ϒ(t) = { } P(,t) t log P (,t) ad implied survival fuctio is = t log {G(t) δ }, G(t) = = (e tϒ) δ ( ) P(,t) δ P (,t) RT Type Recovery (cf. Jarrow & Turbull(995)) recovery rate δ t is formulated as Uder RT type recovery, δ t = ( L t )P (t,t ) for a give fractioal recovery process ( Lt). We assume default hazard rate process λ be idepedet (uder Q) of short rate r ad recovery process be costat. With costat δ ad o-radom λ, the payoff at maturity i the evet of default is δ. The we have { [ s ] [ s ] } P(u,s) = Eu Q exp r(v)dv δ I {τ s} + exp r(v)dv I {τ>s} τ > u u u { [ s ]} = Eu Q exp r(v)dv E Q { u δ I{τ s} + I {τ>s} τ > u } u = P (u,s) ( δ + ( δ)eu Q { I{τ>s} τ > u }) = δp (u,s) + ( δ)p (u,s)q u {τ > s} By settig u =, s = t, it follows that P(,t) = δp (,t) + ( δ)p (,t)q u {τ > s} = δp (,t) + ( δ)p (,t)g(t) Uder RT type recovery, spread becomes ϒ (t) = { } P(,t) t log P (,t) = log{δ + ( δ)g(t)}, t I this paper, implied survival fuctio meas survival fuctio which is icorporated by the bod price. 4

6 ad implied survival fuctio is G (t) = = ( ) e tϒ δ δ ( ) P(,t) δ P (,t) where we defie ϒ (t) is the spread uder RT type recovery ad G is survival fuctio uder RT type recovery. I both type of recovery model, the formula is derived which presets the relatioship of the default probabilities ad the iterest spread. Takahashi(2) proposed the statistical model of estimatig procedure of implied default probabilities uder this relatioship. We review the results i the ext subsectio. 2.2 Review of the Statistical Model by Takahashi(2) We have a exact estimate of the true survival fuctio or each t ad the distributio fuctio of default time τ uder the equivalet martigale measure Q. τ G(t : θ) θ Θ R d d Sice there may be may sources of errors such as a lack of our iformatio ad a oise i the market, we suppose observed P(,t i ) cotais some error. ˆP(,t i ) = P(,t i ) + σh ti ε ti i =,2,..., where σ is ukow costat ad h t is a kow fuctio with h = ad h t is a icreasig i t. We cosider h t = [ P (,t)] α for some α. O the other had, P (,t i ) is supposed to be observed without ay error. After havig observed ˆP(,t i ), i =,...,, we calculate observed spread ˆϒ(t i ) = { } t log p(,ti ) p (,t i ) + σ i ε ti h where i is t p (,t i ). Usig the observed spread ˆϒ(t i ), implied survial fuctio S(t) i stead of G(t) becomes S(t) = = = (e t ˆϒ ) δ (e t t log { p(,ti ) p (,t i ) +σ i ε ti }) δ (e log { p(,ti ) p (,t i ) +σ i ε ti }) δ It is coveiet to rewrite S(t) as S δ (t) = Gt δ + σ ε t 5

7 The we assume the followig parametric model. Let Θ be a subset of R d, where d deote the dimesio of the parameter space. We also let G(t : θ) = Q{τ > t : θ} be a survival fuctio parameterized by d-dimesioal vector θ = (θ,...,θ d ) Θ. After havig observed {st i,i =,...,}, the pseudo maximum likelihood estimate ˆϑ of θ is defied by Let ad ˆϑ = arg θ Θ the, ˆϑ may be rewritte as mi { } G(ti,θ) δ st δ 2 i. i Ξ(t,θ) = G(t,θ) δ X(t) = S δ t ad x(t) = s δ t, ˆϑ = arg θ Θ [mi {Ξ(t i,θ) x ti } 2 ]. ad X(t) as X(t) = Ξ(t,θ) + σε t. 2.3 Statistical Model uder RT Type Recovery We shall ext derive statistical model uder RT type recovery. We set the same assumptios to P(,t i ), P (,t i ) ad ˆϒ(t i ) as those uder RM type recovery. Usig the observed spread ˆϒ(t i ), implied survival fuctio S (t) uder RT type recovery i stead of G (t) becomes ( ) S (t) = = = δ δ δ e t ˆϒ δ { } (e t t log p(,ti ) ) p (,t i ) +σ i ε ti δ ( ) p(,t) p (,t) δ + δ σ i ε ti = G (t) + δ σ i ε ti After havig observed {st i,i =,...,}, the pseudo maximum likelihood estimate ˆϑ of θ is defied by ) ( δ) (G (t i,θ) s 2 t i ˆϑ = arg θ Θ mi. i 6

8 Let ad the, ˆϑ may be rewritte as Ξ (t,θ) = ( δ)g (t,θ) X (t) = ( δ)s t ad x (t) = ( δ)s t, ˆϑ = arg θ Θ [mi { Ξ (t i,θ) x t i } 2 ]. ad X (t) as X (t) = Ξ (t,θ) + σε t. 2.4 RM Type Recovery ad RT Type Recovery The pseudo maximum likelihood estimate uder RM type recovery ˆϑ RM of θ is defied by ˆϑ RM = arg θ Θ mi = arg θ Θ mi = arg θ Θ mi { } G(ti,θ) δ st δ 2 i i ( ( ) G(t i,θ) δ e t i ˆϒ(t i ) δ i { } G(t i,θ) δ e t i ˆϒ(t 2 i ) The pseudo maximum likelihood estimate ˆϑ RT of θ is defied by ) ( δ) (G (t i,θ) s 2 t i ˆϑ RT = arg θ Θ mi = arg θ Θ mi = arg θ Θ mi = arg θ Θ mi i i i ) δ 2 ( ( )) ( δ) G (t i,θ) δ e t i ˆϒ(t i ) 2 δ ( ) ( δ)g (t i,θ) e t i ˆϒ(t i ) 2 δ i { } ( δ)g (t i,θ) + δ e t i ˆϒ(t 2 i ) i 7

9 Statistical RM Statistical RT RM-RT G δ ( δ)g + δ G δ G (δ =.3) (δ =.3) ( δ)g + δ Table : Simulatio of RM model ad RT model Differece betwee ˆϑ RM ad ˆϑ RT is the term RM : G(t i,θ) δ RT : ( δ)g (t i,θ) + δ We cosider Weibull distributio for the survival fuctio G(t i,θ) ad G (t i,θ). Table is a simulatio of RM model ad RT model for G =,.99,...,.9. From Table we ca see the differece G δ {( δ)g + δ} is egative i all cases. The smaller the survival probability G gets, the larger the differece G δ {( δ)g + δ} becomes. 2.5 Calibratio We may also suppose a model i which error term is ot cotaied to the price of defaultable bod P(,t) (ad of course P(,t) ). It meas we assume that there is o oise i the market ad P(,t) is determied with o error i the market. Uder RM type recovery ad the assumptio of o error term, we ca estimate the survival fuctio by the followig equatio; G(t) = G(t,θ) θ Θ R d d. Observe ϒ(t i ) i =,..., ad the estimate θ from [ { ˆϑ = mi G(t i,θ) (e ) } 2 ] t iϒ(t i ) δ. θ Θ We ca estimate the survival fuctio uder RT type recovery i the similar way; G(t) = G(t,θ) θ Θ R d d. Observe ϒ(t i ) i =,..., ad the estimate θ from [ { ˆϑ = mi G(t i,θ) ( e t iϒ(t i ) δ) } ] 2. θ Θ δ 8

10 3 Asymptotic Properties The aim of this sectio is to justify the estimatig method described i the previous sectio. We shall first give a complete proof of the cosistecy of the pseudo maximum likelihood estimator. We employ the method of Wald(949) which gives the proof of cosistecy of maximum likelihood estimator. Furthermore, we give a verificatio of the coditios for the cosistecy ad the asymptotic ormality of our estimator i the case of Weibull survival fuctio. 3. Proof of the Cosistecy I this subsectio, we prove that uder certai regularity coditios, our estimator ˆϑ is strogly cosistet. Let, Q(t,X t,θ) = {Ξ(t,θ) X t } 2. For ay ρ >, let ad, for ay r >, let, Q(t,X t,θ,ρ) = if Q(t,X t,θ ), θ θ <ρ ϕ(t,x t,r) = if θ >r Q(t,X t,θ). We shall ow impose some regularity coditios for the cosistecy: [Regularity Coditio C2] (coditios for the cosistecy). If lim i θ i = θ, lim i Q(t,X t,θ i ) = Q(t,X t,θ) for all (t,x t ) except o a set of measure zero uder the true parameter vector θ. The ull set may deped o θ, but ot o the sequece {θ i,i =,2,...}. 2. For ay pairs θ θ, Ξ(t,θ) Ξ(t,θ ), at least oet 3. For sufficietly small ρ, the expected values where θ deotes the true parameter. Q(t,X t,θ,ρ)df(x t,θ ) <, 4. For the true parameter vector θ, we have Q(t,X t,θ )df(x t,θ ) <. 5. If lim i θ i =, the lim i Q(t,X t,θ i ) = for ay (t,x t ) except perhaps o a fixed set of measure zero accordig to the true parameter vector θ. 9

11 ad, Let, θ be the true parameter vector, the it follows that, E θ {Q(t,X t,θ )} = E θ { (Ξ(t,θ ) X t ) 2} = σ 2 () E θ {Q(t,X t,θ)} = E θ {(Ξ(t,θ) X t ) 2} = { (Ξ(t,θ E θ ) X t + Ξ(t,θ) Ξ(t,θ ) ) } 2 = { (Ξ(t,θ E ) 2 ( θ ) X t + Ξ(t,θ) Ξ(t,θ ) ) } 2 = σ 2 + ( Ξ(t,θ) Ξ(t,θ ) ) 2 (2) We cosider the case where Θ is compact. Let Θ be a compact subset of Θ, which does ot cotai the true parameter vector θ. [Lemma ] For ay θ Θ, E θ {Q(t,X t,θ)} > E θ {Q(t,X t,θ )}. (3) Proof. It follows from [C2]-3 ad [C2]-4 that the expectated values i (3) exsist. Lemma follows easily from (2) ad [C2]-2. [Lemma 2] lim E θ ρ {Q(t,X t,θ,ρ)} = E θ {Q(t,X t,θ)} (4) Proof. It follows from [C2]- that, lim Q(t,X t,θ,ρ) = Q(t,X t,θ). (5) ρ except o a set whose probability measure is zero. Sice Q(t,X t,θ,ρ) is a decreasig fuctio of ρ, ad Q(t,X t,θ,ρ) >, there exists a itegrable fuctio ψ such that Q(t,X t,θ,ρ) ψ(t,x t,θ) (6) for all ρ. It follows from (5) ad Lebesgue s Covergece Thorem 2, that lim ρ Q(t,X t,θ,ρ)df(x t,θ ) = Q(t,X t,θ)df(x t,θ ). (7) 2 To show lim ρ Q(t,X t,θ,ρ)df(x t,θ ) = Q(t,X t,θ)df(x t,θ ) is equivalet to show, for ay sequece {ρ ;ρ }, lim Q(t,X t,θ,ρ )df(x t,θ ) = Q(t,X t,θ)df(x t,θ ). Sice Q(t,X t,θ,ρ ) is itegrable from [C2]-3, the,(6) meas for ay sequese {ρ ;ρ }, Q(t,X t,θ,ρ ) Q(t,X t,θ,ρ ).

12 Thus, Lemma2 is proved. [Lemma 3] lim E r θ {ϕ(t,x t,r)} = (8) Proof. Accordig to [C2]-5, lim ϕ(t,x t,r) = lim if Q(t,X r r t,θ) = (9) θ >r for ay (t,x t ) except o a set of measure zero uder the true parameter vector θ. ϕ(t,x t,r) is a icreasig fuctio of r. If there exists r > such that the measure F of a set {(t,x t ) X ;ϕ(t,x t,r) = } is positive, the, lim r ϕ(t,x t,r) = o a set {(t,x t ) X ;ϕ(t,x t,r) = }. So, Lemma3 lim r ϕ(t,x t,r)df(t,x t,θ ) = obviously holds. Thus, we shall merely cosider the case whe ϕ(t,x t,r) < o {(t,x t ) X. Sice ϕ(t,x t,r) is a icreasig fuctio of r, we defie, for ay sequece r i ;r < r 2 <... < r i <... ϕ (t,x t,r ) = ϕ(t,x t,r ) ϕ (t,x t,r i ) = ϕ(t,x t,r i ) ϕ(t,x t,r i ) i = 2,3,... Sice ad ϕ(t,x t,r i ) = i j= ϕ (t,x t,r j ), ϕ (t,x t,r j ) lim r i ϕ(t,x t,r i )df(t,x t,θ ) = lim ri = lim = = = i ri j= j= i j= ϕ (t,x t,r j )df(t,x t,θ ) () ϕ (t,x t,r j )df(t,x t,θ ) ϕ (t,x t,r j )df(t,x t,θ ) ϕ (t,x t,r j )df(t,x t,θ ) j= = lim ϕ(t,x t,r i )df(t,x t,θ ) r i

13 The secod ad fifth equality are from the chage the order of ifiite summatio ad itegratio ad the 6th equal is from (9). Thus, Lemma3 is proved. [Theorem ] (The cosistecy theorem) { P θ lim ˆϑ = θ } =. () Proof. Let r be a positive umber chose such that E θ {ϕ(t,x t,r )} > E θ {Q(t,X t,θ )}. (2) The existece of such a positive umber follows from Lemma3. Let Θ be the subset of Θ cosistig of all parameter vector θ of Θ for which θ r. With each parameter vector θ Θ, we associate a positive value ρ θ such that E θ {Q(t,X t,θ,ρ θ )} > E θ { Q(t,Xt,θ ) }. (3) The existece of such ρ θ follows from Lemma ad Lemma2. To prove that ˆθ is strogly cosistet we have to show that { ifθ Θ P Q(t } i,x ti,θ) θ Q(t i,x ti,θ > = ε (4) ) for all sufficietly large. This meas that for all sufficietly large the values of θ which miimizes Q(t i,x ti,θ ), ˆθ, belogs to eighborhood of θ with probability oe. Sice Θ is compact, there exists a fiite ope coverig of Θ {O(θ (l),ρ (l) ),l =,...,L}, θ where, O(θ,ρ θ ) is a sphere with ceter θ ad radius ρ θ ; Θ O(θ (),ρ () θ ) O(θ (L),ρ (L) θ ). Clearly, if θ Θ Hece { ifθ Θ Q(t i,x ti,θ) P θ Q(t i,x ti,θ ) Q(t i,x ti,θ) mi l=,...,l Thus from (5) we deduce { ifθ Θ P Q(t i,x ti,θ) θ Q(t i,x ti,θ ) P θ = P θ Q(t i,x ti,θ (l),ρ (l) θ ). } { mil=,...,l > P Q(t i,x ti,θ (l),ρ (l) θ ) } θ Q(t i,x ti,θ > ) (5) } > { mil=,...,l Q(t i,x ti,θ (l),ρ (l) θ ) Q(t i,x ti,θ ) { mil=,...,l Q(t i,x ti,θ (l),ρ (l) θ ) Q(t i,x ti,θ ) > } } (6) 2

14 By the strog law of large umbers, (2) ad (3), the right-had side of (6) is smaller tha ε for all sufficietly large. This completes the proof of Theorem. 3.2 Verificatio of the Regularity Coditios for cosistecy We shall ow verify the regularity coditios for cosistecy ad the asymptotic ormality of our estimator whe we apply Weibull distributio to the survival fuctio. Whe we estimate the statistical model of Takahashi(2), we cosider expoetial model ad Weibull model for the parametric survival fuctio. Needless to say, expoetial distributio is the special case of Weibull distributio, we eed oly to verify the regularity coditios i the Weibull model. Weibull survival fuctio is defied i the form of G Wei (θ,θ 2,t) = exp { (tθ ) θ 2 }, t. We ow verify the coditios of [C2] which is proved i the previous subsectio. Verificatio of the coditios of [C2]. We shall verify the coditios for cosistecy [C2] i the Weibull distributio case. Coditio [C2]- holds clearly. To show [C2]-2, let (θ,θ 2 ) (θ,θ 2 ) ad defie θ 2 = θ 2 θ 2. The, (tθ ) θ 2 (tθ ) θ 2 = (tθ ) θ 2+ θ 2 (tθ ) θ 2 = tθ 2(t θ 2 θ θ 2 + θ 2 θ θ 2 ) at least oe t, so that (tθ ) θ 2 (tθ )θ 2 at least oe t. Sice G(t,θ) = e (tθ)θ2 is strictly mootoic, (tθ ) θ 2 (tθ )θ 2 implies G(t,θ) G(t,θ ), ad hece that Ξ(t,θ) Ξ(t,θ ) at least oe t. We ext show the existece of E[Q(t,X t,θ )] ([C2]-4). The itegrad of the expectatio E[Q(t,X t,θ )] = = Q(t,x t,θ )df(x t,θ ) {Ξ(t,θ ) x t } 2 df(x t,θ ). is writte as follows. {Ξ(t,θ ) x t } 2 = = = = { G(t,θ ) δ { (e (tθ ) θ2 ) δ x t } 2 x t } 2 { (e θ θ 2 ) δ (e tθ2 ) δ }2 s δ t { (e θ θ 2 ) δ (e tθ2 ) δ P (,t) h t s δ t P }2 (,t) h t 3

15 Note that (e θ θ 2 ) δ is positive costat. From t, θ 2 > ad < δ <, (e tθ2 ) δ. It follows that st δ ad < P (,t). Therefore {Ξ(t,θ ) x t } 2 as h t ; h t implies E[Q(t,x t,θ )] <. So [C2]-4 holds. [C2]-3 ca be verified i the same way. For [C2]-5, if lim i θ i =, Ξ(t,θ i ). [C2]-5 holds clearly. 3.3 Verificatio of the Regularity Coditios for Asymptotic Normality The asymptotic ormality is proved completely by Takahashi(2), so we describe the coditios for the asymptotic ormality without proof. [Regularity Coditio C3] (coditios for the asymptotic ormality). The pseudo data will be sampled i a way that the time sequece {t i,i =,2,...} are selected so that there is a positive defiite matrix Σ(θ ) ad B(θ ) for which lim Var{ψ(θ,t i,x ti )} = lim σ 2 Σ(θ,) = σ 2 Σ(θ ) (7) where (l,m) th elemet of Σ(θ,) = σ 2 Var{ψ(θ,t i,x ti )} is defied by Ξ(t i,θ ) Ξ(t i,θ ) θ l θ m ad, [ ] 2 lim θ 2 ψ(l) (θ,t i,x ti ) = lim B(θ,) = B(θ ) (8) [ ] where (k,m) th elemet of 2 ψ (l) (θ,t θ 2 i,x ti ) is give by [ 3 ] {Ξ(ti Ξ(t i,θ ),θ } ) X t θ k θ m θ l + [ 2 ][ ] Ξ(t i,θ ) Ξ(t i,θ ) θ m θ l θ k + [ 2 ][ ] Ξ(t i,θ ) Ξ(t i,θ ) θ l θ k θ m + [ 2 ][ ] Ξ(t i,θ ) Ξ(t i,θ ) θ k θ m θ l ad the (k,m) th elemet of B(θ,) is defied by [ 2 ][ ] Ξ(t i,θ ) Ξ(t i,θ ) θ m θ l θ k + [ 2 ][ ] Ξ(t i,θ ) Ξ(t i,θ ) θ l θ k θ m + [ 2 ][ ] Ξ(t i,θ ) Ξ(t i,θ ). θ k θ m θ l 4

16 2. For every θ Θ, the time sequece {t i,i =,2,...} are selected so that [ 2 ] 2 Ξ(t i,θ) coverges as, f or all l ad m (9) θ l θ m [ 3 ] 2 Ξ(t i,θ) coverges as, f or all k, l ad m. (2) θ k θ l θ m 3. The sequece { [ ] } ψ(θ,t i,x ti ) = ψ () (θ,t i,x ti ),...,ψ (d) (θ,t i,x ti ),i =,2,... obeys the multivariate cetral limit theorem; [ Var { ψ(θ,t i,x ti ) }] 2 ψ(θ,t i,x ti ) N(,I d ) as (2) where = (,...,) is the d dimesioal zero vector ad I d is the d d idetity matrix. Next we shall verify the coditios of [C3] i the case of Weibull survival fuctio. Verificatio of the coditios [C3]. We ext show the coditios of the asymptotic ormality i the Weibull distributio case. To show coditio[c3]-, cosider α = (α,α2). ( ) θ Ξ (α,α2) = (α,α2) = α = θ Ξ 2 ( ) δ e ( tθ )θ 2 ( δ)( tθ ) θ 2 θ 2 θ ρ ( ) t δ e ( tθ )θ 2 ( δ)( tθ ) θ 2 l[ tθ ] ( ) e ( tθ ) θ δ 2 ( δ)( tθ ) θ 2 θ 2 θ + α2 ( e ( tθ ) θ 2 ) δ ( δ)( tθ ) θ 2 ( e ( tθ ) θ 2 ) δ ( δ)( tθ ) θ 2 l[ tθ ] ( α θ ) 2 + α2l[ tθ ] θ ( ) Sice e ( tθ ) θ δ 2 >, δ >, ( tθ ) θ 2 <, θ 2 /θ >, l[ tθ ] < ad > ( ) θ Ξ by defiitio, (α,α2) is ozero o a iterval. Therefore, θ Ξ 2 { ( )} 2 θ Ξ (α,α2) = α Σ(θ )α > θ Ξ 2 5

17 Sice this coclusio is true for ay α, Σ must be positive defiite matrix. The existece of a positive matrix B(θ ) ca be showed i the similar way. ( Coditio [C3]-2 holds by ispectio of 2 2 ( θ l θ m Ξ(t i,θ)) ad 2, 3 θ k θ l θ m Ξ(t i,θ)) (l,m,k =,2). I the case of k =, l = ad m =, it follows that ( 2 = ) 2 Ξ(t i,θ) θ θ ( ( ) e ( tθ ) θ δ 2 t 2 i ( δ)( t i θ ) 2+θ 2 ( + θ 2 )θ 2 + ( e ( tθ ) θ 2 ) δ t 2 i ( δ) 2 ( t i θ ) 2+2θ 2 θ 2 2 ) 2, ad ( = 3 θ θ θ Ξ(t i,θ) + + ( ) 2 ( e ( tθ ) θ 2 ) δ t 3 i ( δ)( t i θ ) 3+θ 2 ( 2 + θ 2 )( + θ 2 )θ 2 ( e ( tθ ) θ 2 ) δ t 3 i ( δ) 3 ( t i θ ) 3+2θ 2 ( + θ 2 )θ 2 2 ( e ( tθ ) θ 2 ) δ t 3 i ( δ) 3 ( t i θ ) 3+3θ 2 θ 3 2 ( ) e ( tθ ) θ δ 2 t 3 i ( δ) 2 ( t i θ ) 3+2θ 2 θ2 2( 2 + 2θ 2) ) 2 ( ) 2 2 θ θ Ξ(t i,θ) Sice < t i < T <, i = ( P (t i )) α P (t i ), < P (t i ) < ad α >, ( ad 3 2 θ θ θ Ξ(t i,θ)) coverges as. The rest case ca be verified i the similar way. By the equatio (7), coditio [C3]-3 [ Var { ψ(θ,t i,x ti ) }] 2 ψ(θ,t i,x ti ) [ = Var { ψ(θ,t i,x ti ) }] 2 ψ(θ,t i,x ti ) N(,I d ) as is rearraged to ψ(θ,t i,x ti ) N(,σ 2 Σ(θ )) as (22) 6

18 We have already show that is o zero, so that oly coditio lim Var{ψ(θ,t i,x ti )} σ 2 Σ(θ ) ψ >ε ψ 2 df i (ψ) = (23) for ay ε > is eeded, where is the Euclidea orm. Sice ψ = ψ () 2 + ψ (2) 2 = Ξ(t i,θ θ )(Ξ(t i,θ ) x) 2 + Ξ(t i,θ θ )(Ξ(t i,θ ) x) 2, 2 it follows that for ay ε >, ψ >ε ψ 2 df i (ψ) as, from which the coditio (23) follows i the Weibull case. 4 Numerical Results I this sectio, we preset some umerical results. We apply our model to real data to estimate default probabilities(survival probabilities). We describe data sources at first, ad the show the results of comparative study varyig α ad δ. We also compare results o differet recovery models ad o two parametric distributios. The we show estimatig results ad bootstrap results usig 8 idividual compaies data ad 5ratig classes data o April 6, 24. Fially we apply our statistical estimatig method to time series data(jauary 28 to March 29). 4. Data Descriptio Corporate bod prices ad govermet bod prices were obtaied from Bloomberg Fiacial Markets ad Japa Securities Dealers Associatio coverig the period April 23 to March 29. Specifically, we collected daily price data for OTC Bod Trasactios. We also obtaied bod issuer s ratig iformatio from Bloomberg. Ratig ad Ivestmet Iformatio, Ic. (R & I) is oe of the leadig ratig compaies i Japa ad it grade for bod issuers by the ratig categories from AAA to CCC. The data we used i the comparative study is bod price data of Mitsui & Co., LTD. also kow as Mitsui Bussa o April 6, 24. We use 52 idividual compaies data for estimatig survival probabilities o oe day, ad 5 idividual compaies data for time series aalysis. 4.2 Selectio of a costat α the recovery rate δ I this subsectio, we show the results of comparative study varyig α ad δ. For the purpose of equalizig the other coditios, we use oly Weibull distributio for the survival fuctio G(θ,t) i this subsectio. 7

19 sum of squared residuals /2 /3 /4 /5 /6 /7 /8 /9 / alpha Figure : Sum of the squared residuals of Mitsui Corporatio as a fuctio of α A costat α is defied by = [ P (,t)] α P. (,t) The restrictio at the maturity ρ = suggests that α >. We first cosider α =. (It meas = P (,t) P (,t).) Sice P (,t) is ear to with small t ad the takes a small value, the weight of the data becomes very large with small t. We get biased estimatig results ad larger values of sum of squared residuals. Next, we calculate sum of the squared residuals with differet α ; α =, 2,..., Figure shows the sum of the squared residuals of Mitsui Corporatio as a fuctio of α. I this case, we see that the sum of squared residuals is decreasig as α. We get similar results i most idividual compay cases. But as metioed above, α is supposed to take positive value. Sice the sums of the squared residuals with α = 3 are eough small i most cases, we set α to be 3 throughout the rest of this sectio. We ext cosider about recovery rate. I our model, recovery rate, δ, is assumed to be o-radom. We use δ whe we calculate the survival fuctio G(t) = exp{tϒ(t : P,P)} δ = ( P P ) δ, ad of course, whe calculate pseudo radom sample(data), s ti i the statistical model; ( { [ ]}) P s t = exp log P + σρ δ tε t Figure shows the variatio of the survival probability of Mitsui Corporatio with the time to maturity for δ =.5, δ =.3, δ =.45. The plot Observed is the values of survival probabilities calculated from the bod prices. The lie Model is the values of estimated survival probabilities. Table 2 shows the values of the estimated parameters of Weibull model for δ =.5, δ =.3, δ =.45. The recovery rate δ becomes larger, survival probability teds to smaller. A iterestig results is that the values of θ 2 are the same with differet δ. Sice θ 2 is the shape parameter of Weibull distributio, we 8

20 St Observed(delta=.5).99 Model(delta=.5).985 Observed(delta=.3).98 Model(delta=.3).975 Observed(delta=.45).97 Model(delta=.45) term to maturity Figure 2: Variatio of the survival probability with the time to maturity for differet δ δ θ θ Table 2: Estimated parameters of Weibull model for differet δ see that the shape of the survival fuctio does ot vary with the recovery rate δ i our model. The similar results are obtaied for the statistical model uder RT type recovery (Figure2, Table 3). Throughout the rest of this sectio, we set the recovery rate δ to be Selectio of the parametric model I this subsectio, we cosider the parametric model for the survival fuctio of our estimatig model. We use two parametric models for the survival fuctio: expoetial distributio ad Weibull distributio. We set the survival fuctio with expoetial distributio G exp ad the survival fuctio with Weibull distributio G Wei. G exp (θ,t) ad G Wei (θ,θ 2,t) is defied by G exp (θ,t) = exp{ tθ}, t.5 St Observed(delta=.5 ) Model(delta=.5) Observed(delta=.3) Model(delta=.3) Observed(delta=.45 ) Model(delta=.45) term to maturity Figure 3: Variatio of the survival probability with the time to maturity for differet δ 9

21 δ θ θ Table 3: Estimated parameters of Weibull model for differet δ Mitsui Corp St Observed Expoetial Weibull term to maturity Figure 4: Estimated survival fuctios of Mitsui & Co. Weibull model for the expoetial ad the { } G Wei (θ,θ 2,t) = exp (tθ ) θ 2, t respectively. Figure 4 shows the estimated survival fuctios of Mitsui & Co. for the expoetial ad the Weibull model. Figure 5 shows the cumulative value of the sum of squared residuals for the expoetial(dotted lie) ad the Weibull model(staggered lie). It is clear that the expoetial survival fuctio is estimated far from the Observed plot.(see Figure 4. Observed plots idicate the survival probabilities calculated from the iterest yields.) Ad the sum of squared residuals from the expoetial survival fuctio is much larger tha that of the Weibull survival fuctio. The result is of course from the fact that the expoetial distributio is allowed less parameter tha Weibull distributio. We shall thus apply Weibull distributio for the rest of our aalysis. Cumulative Value of Residuals Expoetial Weibull Figure 5: Cumulative value of the sum of squared residuals for the expoetial ad the Weibull model 2

22 Cumulative Value of Squared Residuals RM RT Figure 6: Cumulative value of the sum of squared residuals uder RM type recovery ad uder RT type recovery 4.4 Two recovery models; RM type recovery & RT type recovery As metioed i chapter 2, the pseudo maximum likelihood estimate of statistical model uder RM type recovery ad RT type recovery are defied by ˆϑ = arg θ Θ respectively, where, mi ˆϑ = arg θ Θ mi s(t) = s (t) = { } G(ti,θ) δ st δ 2 i i ) ( δ) (G (t i,θ) st 2 i (e tϒ) δ i ( ) e tϒ δ. δ We also defied the pseudo maximum likelihood estimate of calibratio model uder RM type recovery ad RT type recovery as follows: [ { ˆϑ = mi G(t i,θ) (e ) } 2 ] t iϒ(t i ) δ θ Θ ˆϑ = mi θ Θ [ { G(t i,θ) ( e t iϒ(t i ) δ) } ] 2. δ Figure 6 shows the cumulative value of the sum of squared residuals of Mitsui & Co. for the statistical model uder RM type recovery(solid lie) ad that for the statistical model uder RT type recovery(dotted lie). We see from Figure 6 that the lies are lyig o top each other ad that the differece betwee the sum of squared residuals from RM type recovery ad RT type recovery is very small. Table 4 idicates the absolute value of residuals ad the sum of the squared residuals whe we apply statistical ad calibratio model to idividual compaies. The above table is for the statistical model ad the bottom is for the calibratio model. 2

23 Statistical Model Compay RM RT RM/RT Toyota RT NTT Data RT Mitsui & Co RM Asahi Breweries RT Kajima RM NEC RM Kitetsu RM Sumitomo Realty RT Cosmo Oil RM JAL RM Calibratio Compay RM RT RM/RT Toyota.2.8 RM NTT Data RM Mitsui & Co RM Asahi Breweries RM Kajima RM NEC RM Kitetsu RM Sumitomo Realty RM Cosmo Oil RM JAL RM Table 4: The mea of the absolute value of residuals for the statistical model ad calibratio uder RM type recovery ad RT type recovery. Residuals are calculated by Ĝ(t) G(t). The rightmost item is the model ame which take the smaller value of the sum of squared residuals. 22

24 St Observed Statistical RM term to maturity Figure 7: Estimated survival fuctio of Mitsui Corporatio for the statistical model ad the calibratio model The rightmost item is the model ame which takes the smaller value of the sum of squared residuals. We see that RM model ad RT model have little differece i residuals, but the followig results are iterestig. Whe we apply calibratio estimatig model, residuals of the model uder RM type recovery are calculated smaller tha that uder RT type recovery for all compaies, though whe applyig statistical model, we ca t say which model has smaller residuals. 4.5 Statistical Model ad Calibratio Our estimators of statistical model uder RM type recovery ad calibratio model uder RM type recovery are writte as follows ˆϑ = arg θ Θ mi [ ] ˆϑ = mi {G(t i,θ) s ti } 2 θ Θ { } G(ti,θ) δ st δ 2 i i respectively, where s ti = (e ) t iϒ(t i ) δ. Figure 7 shows the estimated survival fuctio of Mitsui Corporatio for the two models; the statistical model (solid lie) ad the calibratio model(dotted lie) ad Figure 8 shows the cumulative values of the sum of squared residuals for the two models, where we take RM type recovery model. We fid that the residuals calculated by the calibratio model are smaller tha that by the statistical model for all compaies(see Table 4). We also fid that the cumulative values of the sum of squared residuals for the statistical model are take smaller values tha for the calibratio model i the first ad secod times of cumulatio, ad the larger values after the third cumulatio(see Figure7). It is because we defied h t (or ) as a fuctio of the price of default free bod P (,t) ad a costat α for our statistical model. Whe the data is close to the maturity, P (,t) becomes early ad so gets ear to. That is the reaso of the behavior of the cumulative values of the sum of squared residuals. 23

25 .3 cumulative value of residuals Statistical RM Calibratio RM Figure 8: Cumulative values of the sum of squared residuals for the statistical model ad the calibratio model RM RT θ θ 2 θ θ 2 θ (Boots) ˆθ Bias( ˆθ) (Boots) Ṽar( ˆθ) (Boots) SD ˆθ (t)(boots) Table 5: The parameter estimate of the bootstrap estimatio for Mitsui & Co. usig statistical model. The otatios are give i Appedix. The bootstrap aalysis showed that the estimate of statistical model has some warp. Table 5 ad Table 6 idicate the parameter estimate of the bootstrap estimatio for Mitsui & Co.. Table 5 is for the statistical model ad Table 6 is for calibratio. Figure 9 shows the distributio of bootstrap parameter estimate of Mitsui & Co. for statistical model ad Figure shows the distributio of bootstrap parameter estimate for calibratio. The left graph i Figure 9 ad Figure shows the histogram of the bootstrap (m) (m) estimate ϑˆ ad the right graph shows that of ϑˆ 2. As see from a compariso betwee Figure 9 ad Figure, the distributio for statistical model has some warp, although that for calibratio does t. RM RT θ θ 2 θ θ 2 θ (Boots) ˆθ Bias( ˆθ) (Boots) Ṽar( ˆθ) (Boots) SD ˆθ (t)(boots) Table 6: The parameter estimate of the bootstrap estimatio for Mitsui & Co. usig calibratio. The otatios are give i Appedix. 24

26 frequecy frequecy θ Figure 9: Distributio of bootstrap parameter estimate for statistical model θ2 frequecy frequecy θ Figure : Distributio of bootstrap parameter estimate for calibratio θ2 4.6 Estimatig results for 8 idividual compaies We ow preset estimatig results for idividual compaies. We set α to be 3 ad the recovery rate δ to be.3, ad use statistical model of Weibull survival fuctio uder RM type recovery. We aalyze the followig 8 idividual compaies; Toyota Motor Corporatio, NTT Data Corporatio, Mitsui & Co., Ltd., Asahi Breweries, Ltd., Kajima Corporatio, Kitetsu Corporatio, Japa Airlies Co., Ltd. ad Cosmo Oil Co., Ltd.. Table 7 idicates the data iformatio ad the parameter estimates of 8 idividual compaies. Figure shows the estimated survival fuctio of 8 idividual compaies. Toyota s estimated Weibull shape parameter ˆθ 2 is the smallest of the 8 compaies ad it is early. That meas straight survival fuctio ad the almost costat hazard rate(costat istataeous default probabilities). Whe the shape parameter ˆθ 2 is larger tha, the istataeous default probabilities declie as t. Table 8 idicates the bootstrap parameter estimates of 8 idividual compaies. Figure 2 shows the distributio of bootstrap parameter estimate of 8 idividual compaies. The (m) upper graph shows the histogram of the bootstrap estimate ϑˆ ad the lower graph (m) shows that of ϑˆ 2. We see from Table7 ad Figure2 that there is a close relatio betwee the distributio of bootstrap estimates ad the umber of data. The umber of Toyota s bods beig issued o April 5, 24 was oly 3. That explais why the estimated bootstrap distributio of Toyota is quite differet from ormal distributio. It seems the larger amout of data(bods) beig used, the more like the bootstrap distributio is ormal distributio. The bootstrap distributio of Mitsui & Co. looks like to have two peaks. The umber of data ad the existece of outlier data may explai this. 25

27 Data Estimatig results Compay Ratig Observatios ˆθ ˆθ 2 mea( residuals ) Toyota AAA NTT Data AA Mitsui & Co. AA Asahi Breweries A Kajima A Kitetsu BBB JAL BBB Cosmo Oil BB Table 7: Iformatio about the data ad the parameter estimates of 8 idividual compaies. The items Ratig ad Observatios are the issuer s ratig o April 5, 24 ad the umber of a compay s bods beig issued o April 5, 24, respectively. ˆθ ad ˆθ 2 are parameter estimates ad the rightmost item is the mea of the absolute values of residuals. St Observed(Toyota) Model(Toyota) Oberved(NTT) Model(NTT) Observed(Mitsui) Model(Mitsui) Obseived(Asahi) Model(Asahi) Observed(Kajima) Model(Kajima) Observed(Kitets u) Model(Kitetsu) Observed(JAL) term to maturity Figure : Estimated survival fuctio ad Observed survival probabilities of 8 idividual compaies. Observed plots are survival probabilities calculated from iterest yield ad Model lie is estimated survival fuctio of our estimatig model. 26

28 Toyota NTT Data Mitsui Asahi θ (Boots) ˆθ Bias( ˆθ ) (Boots) Ṽar( ˆθ ) (Boots)..2.. SD ˆθ (t) (Boots) θ (Boots) ˆθ Bias( ˆθ 2 ) (Boots) Ṽar( ˆθ 2 ) (Boots) SD ˆθ 2 (t) (Boots) Kajima Kitetsu JAL Cosmo Oil θ (Boots) ˆθ Bias( ˆθ ) (Boots) Ṽar( ˆθ ) (Boots).... SD ˆθ (t) (Boots) θ (Boots) ˆθ Bias( ˆθ 2 ) (Boots) Ṽar( ˆθ 2 ) (Boots) SD ˆθ 2 (t) (Boots) Table 8: The parameter estimate of the bootstrap estimatio for 8 idividual compaies. The otatios are give i Appedix. 27

29 Toyota(θ) NTT DATA(θ) Mitsui & Co.(θ) Asahi Breweries(θ) Toyota(θ2) NTT DATA(θ2) Mitsui & Co.(θ2) Asahi Breweries(θ2) Kajima(θ) Kiki Nippo Railway(θ) JAL(θ) Cosmo Oil(θ) Kajima(θ2) Kiki Nippo Railway(θ2) JAL(θ2) Cosmo Oil(θ2) Figure 2: Distributios of bootstrap parameter estimates for 8 idividual compaies. (m) The upper graph shows the histogram of the bootstrap estimate ϑˆ ad the lower graph (m) shows that of ϑˆ 2. 28

30 Data Estimatig results Class Observatios ˆθ ˆθ 2 mea( residuals ) AAA AA A BBB BB Table 9: Iformatio about the data ad the represetative parameter estimates of 5 ratig classes. The items Class ad Observatios are the ratig class ad the umber of corporate bods belogig to the class o April 5, 24, respectively. ˆθ ad ˆθ 2 are parameter estimates ad the rightmost item is the mea of the absolute values of residuals St AAA AA A BBB BB term to maturity Figure 3: Estimated represetative survival fuctio of 5 ratig classes. 4.7 Estimatig results of 5 ratig classes I this subsectio, we preset estimatig results for the data of 5 ratig classes. As i the previous subsectio, we set α to be 3 ad the recovery rate δ to be.3, ad use statistical model of Weibull survival fuctio uder RM type recovery. We aalyze 5 ratig classes; AAA, AA, A, BBB, BB. Table 9 idicates the iformatio about the data ad the represetative parameter estimates of 5 ratig classes. Figure 3 shows the estimated represetative survival fuctio of 5 ratig classes. Figure 4 cotais 5 separate graphs of the estimated represetative survival fuctios of ratig classes. Observed plots idicate the survival probabilities calculated from the iterest yields ad the Model lie is the estimated represetative survival fuctio of ratig classes. Table idicates the bootstrap parameter estimates of 5 ratig classes. Figure 5 shows the distributio of bootstrap parameter estimates of 5 ratig classes. The upper (m) graph shows the histogram of the bootstrap estimate ϑˆ ad the lower graph shows (m) that of ϑˆ 2. As see i the idividual compaies cases, there seems to be a close relatio betwee the distributio of bootstrap estimates ad the umber of data. It seems the larger amout of data(bods) beig used, the more like the bootstrap distributio is ormal distributio. Though may bods (523 bods) were issued by the AA class s compaies, the bootstrap distributio of AA class looks like to have some warp. (See Figure 5.) This may be explaied by the existece of outlier data. Whe we look Figure 4 carefully, we 29

31 AAA AA A BBB BB observed Figure 4: 5 separate graphs of the estimated represetative survival fuctios of ratig classes. Observed plots idicate the survival probabilities calculated from the iterest yields ad the Model lie is the estimated represetative survival fuctio of ratig classes. model AAA AA A BBB BB θ (Boots) ˆθ Bias( ˆθ ) (Boots) Ṽar( ˆθ ) (Boots) SD ˆθ (t) (Boots) θ (Boots) ˆθ Bias( ˆθ 2 ) (Boots) Ṽar( ˆθ 2 ) (Boots) SD ˆθ 2 (t) (Boots) Table : The parameter estimate of the bootstrap estimatio for 5 ratig classes. The otatios are give i Appedix. 3

32 AAA(θ) AAA(θ2) AA (θ) AA (θ2) A (θ) A (θ2) BBB (θ) BB (θ) BBB (θ2) Figure 5: Distributios of bootstrap parameter estimates for 5 ratig classes. Blue(deep) distributio is for statistical model ad pik(pale) is for calibratio BB (θ2) 3

33 Data Estimatig results Data Observatios Method ˆθ ˆθ 2 mea( residuals ) Origial 523 Statistical model Calibratio No Outlier 52 Statistical model Calibratio Table : The represetative parameter estimates for AA origial data ad for AA o outlier data. Origial data are the same as AA i Table9. Outlier data are excluded 3 bods price data from the origial data. Origial data No outlier data Statistical model Calibratio Statistical model Calibratio θ (Boots) ˆθ Bias( ˆθ ) (Boots) Ṽar( ˆθ ) (Boots).... SD ˆθ (t) (Boots) K-S statistic K-S test reject reject reject reject θ (Boots) ˆθ Bias( ˆθ 2 ) (Boots) Ṽar( ˆθ 2 ) (Boots) SD ˆθ 2 (t) (Boots) K-S statistic K-S test reject reject ot reject ot reject Table 2: The parameter estimate of the bootstrap estimatio for AA origial data ad for AA o outlier data. The results of K-S test at.5 level are give i the bottom 2 lies. otice that Observed plots of AA class cotais outlier. We ext show the ifluece of outlier data o our estimatig method usig data of AA class. I the AA graph of Figure 4, we may see 3 outlier plots amog Observed plots. We excluded these 3 bods price data from the origial data. Table idicates the represetative parameter estimates for AA origial data ad for AA o outlier data. Origial data are the same as AA i Table9 ad outlier data are excluded 3 bods price data from the origial data. Table 2 idicates the bootstrap parameter estimates for AA origial data ad for AA o outlier data. Figure 6 shows the distributio of bootstrap parameter estimates for AA origial data ad for AA o outlier data. The upper graph shows the histogram of the bootstrap estimate ϑˆ ad the lower graph shows that of ϑˆ 2 (m) (m). We see from Figure 6 that the warps are reduced by the exclusio of the outlier data(compare the left graphs ad the middle graphs). We also see that the distributios of bootstrap parameter estimates usig o outlier data look like ormal distributio(middle graphs). We tested these results for ormality of the distributio. Oe of the most useful tests for ormality of the distributio is Kolmogorov-Smirov test. Give M estimates ˆϑ (m) i =,2,m =,...,M, the Kolmogorov-Smirov statistic for cumulative ormal 32

34 AA, Origial data (θ) AA, No outlier data (θ) AA, Origial ad o outlier data (θ) AA, Origial data (θ2) AA, No outlier data (θ2) AA, Origial ad o outlier data (θ2) Figure 6: Distributios of bootstrap parameter estimates for AA origial data ad o outlier data. The left graphs are for origial data. Blue(deep) distributio is for statistical model ad pik(pale) is for calibratio. The middle graphs are for o outlier data. 3 bods data are excluded. Gree(deep) distributio is for statistical model ad orage(pale) is for calibratio. The right graphs are the distributios usig statistical model for origial data ad o outlier data. Blue(deep) distributio is for origial data ad gree(pale) is for o outlier data. distributio fuctio F(x) is D M = sup x F m (x) F(x) where sup x is the supremum of the set of distaces ad F M (x) = M M I ˆϑ (m) <x, where I ˆϑ (m) <x is the idicator fuctio, equal to if ˆϑ (m) < x ad equal to otherwise. The ull hypothesis that ˆϑ (m) i =,2,m =,...,M is ormal distributed is rejected at level.5 if MDM >.36. M is i our case. The Kolmogorov-Smirov statistic D ad the results of K-S test are i Table2. We got better results by excludig oly 3 bods price data from the origial 523 bods data, 4.8 Estimatio for Time Series Data So far, we have estimated the survival probabilities of oe day. I this subsectio, we estimate the time series survival probabilities. The data we used is daily price data of the corporate bod ad the govermet bod. Sice we are iterested i the impact of the bakruptcy of Lehma Brothers(September 5, 28), we use the data from Jauary 4, 28 to March 3, 29. U.S. house sales prices peaked i mid-26 ad bega their steep declie forthwith. The subprime mortgage crisis bega to affect the fiacial sector i 27. Although Japaese fiacial firms 33

35 held a few securities backed with subprime mortgages, Japaese ecoomy was affected by subprime mortgage crisis. We select five compaies, which might be affected by the bakruptcy. Toyota Motor Corporatio is a multiatioal automaker ad oe of the world s largest automobile maufacturers. Toyota has a ratig of AAA throughout our aalysis. Mitsui & Co., LTD. also kow as Mitsui Bussa, is oe of the largest sogo shosha i Japa. Mizuho Corporate Bak, Ltd. is the corporate ad ivestmet bakig subsidiary of Mizuho Fiacial Group. Tokyu Lad Corporatio is the 4th biggest real estate agecy i Japa. Japa Airlies Co., Ltd. is a airlie i Japa. The airlie filed for bakruptcy protectio o Jauary 9, 2, after losses of early billio ye i a sigle quarter. Figure 7 shows the historical survival probabilities of the 5 compaies. The survival probabilities of Tokyu Lad Corporatio bega to declie slowly before the bakruptcy of Lehma Brothers. Although the risk of subprime mortgages was poited before the bakruptcy, the declie is cosidered by the cause of its ow. I 28, housig values i Japa decreased by percet with the previous year ad the stock price of Tokyu Lad declied all through 28. The survival probability curves i the graph begi to drop sharply at November 29. The bod buyers were likely to avoid the risk i the real estate market. We ext cotrast the graphs of Toyota, Mitsui ad Mizuho. We see that the all 3 compaies survival probabilities were decliig after the bakruptcy, but Mizuho s graph bega to drop first. Sice Mizuho Corporate Bak is the corporate ad ivestmet bakig subsidiary ad was at the core of the fiacial crisis, the bakruptcy was seem to make a immediate effect. Oe moth later of the Mizuho s survival probability curve drop, the survival probabilities of Toyota ad Mitsui bega to declie more slowly. It seems the fiacial crisis spreaded over gradually. JAL s steep curve stads out i the graph. The 5 year survival probability varied 8% to 64% for oly 3 moths. It is because JAL was cofroted ot oly with fiacial crisis, but also with the rise of oil price ad slow cost cut. JAL s prospects seem to have bee cosidered gloomy by the bod buyers. 5 Coclusio I this paper, we study estimatig method of implied survival probability. We focus o the statistical estimatig model itroduced by Takahashi(2). The advatage of the statistical estimatig model is its ability to discuss asymptotic properties. Our cotributios are the followig three poits. First, we provide a complete proof of the cosistecy of the estimator i the statistical estimatig model. Secod, we verify regularity coditios for asymptotic properties of the estimator i the Weibull survival fuctio case. Third, we provide a empirical aalysis of implied survival probability estimatig. Whe we estimate implied survival probability, istead of usig Duffie & Sigleto(999) model directly, as i previous studies, we apply statistical estimatig model which cotais some error term. I the empirical aalysis, we discuss asymptotic properties usig bootstrap method. Several iterestig observatios ca be made. Whe we apply Weibull distributio to parametric survival fuctio, the estimatig result is much better tha that of expoetial distributio. The umber of the data ad the existece of the outlier data have a major effect o the bootstrap estimatig results. Whe we aalyze the impact of the bakruptcy of Lehma Brothers, we observe that the iflueces differ with the 5 idividual compaies. 34