HITOTSUBASHI

UNIVERSITY

Center for Financial Engineering Education (CFEE)CFEE Discussion Paper Series No. 2011-1

Estimating Default Probabilities Using Corporate
Bond Price Data

Reiko Tobe

Center for Financial Engineering Education (CFEE)
Department of Economics, Hitotsubashi University
2-1 Naka, Kunitachi Tokyo, 186-8601 Japan
http://www.econ.hit-u.ac.jp/~finmodelEstimating Default Probabilities Using Corporate
Bond Price Data
Reiko Tobe ∗March 13, 2012

Abstract
This paper discusses estimating method of implied default probability from theinterest spread. Instead of using Duffie & Singleton(1999) model directly, we use
statistical estimating procedure of Takahashi(2011). We shall provide a complete
proof of the consistency of the estimator in the Takahashi(2011) model employing
the method of Wald(1949) which gives the proof of consistency of maximum likelihood estimate. We also verify regularity conditions for asymptotic properties of theestimator in the Weibull survival function case. Furthermore, we provide an empirical analysis of implied survival probability estimating. In the empirical analysis, we
discuss asymptotic properties using bootstrap method. An interesting finding is thatthe number of the data and the existence of the outlier data have a major effect on
the bootstrap estimating results. When we analyze the impact of the bankruptcy ofLehman Brothers, we observe that the influences differ with the 5 individual companies.
Keywords: Implied default probability; Statistical model; Parametric model; Pseudo maximum
likelihood estimator; Consistency; Asymptotic normality; Bootstrap;

1 Introduction
In this paper we study estimating method of default probabilities of individual companies and the representative default probabilities of rating classes. Many rating agencies
publish bond issuer’s rating using their own methodology in measuring creditworthinessand a specific rating scale. Typically, ratings are expressed as letter grades that range,for example, from ’AAA’ to ’D’ to communicate the agency’s opinion of relative levelof credit risk. For example, a corporate bond that is rated ’AA’ is viewed by some rating
agency’s as having a higher credit quality than a corporate bond with a ’BBB’ rating. But
the ’AA’ rating isn’t a guarantee that it will not default, only that, in their opinion, it isless likely to default than the ’BBB’ bond. Some rating agencies also announce annual
default probabilities of each rating class.
For modeling credit risk, two classes of models exists; structural model and reducedform model. Reduced-form model was originally introduced by Jarrow & Turnbull(1992), and subsequently studied by Jarrow and Turnbull (1995), Duffie and Singleton(1999) among others. By using Duffie and Singleton(1999) model, we may calculate”implied default probability” by calibration. ”Implied default probability” means default∗ Graduate School of Economics, Hitotsubashi University

1

probability which is incorporated by the bond prices and it gives us view and expectations in the market. Takahashi(2011) introduced statistical model for estimating implieddefault probability. They assumed observed corporate bond price contains some error. As
a result of containing error term, we can discuss asymptotic properties of the estimator.
In Takahashi(2011), they gave a sketchy proof for consistency and a complete proof forasymptotic normality. They also recommended using bootstrap method for examining
asymptotic properties when we apply their method to real data.
The first objective of this paper is to give the complete proof of the consistency ofthe pseudo maximum likelihood estimator which is proposed by Takahashi(2011). We
employ the method of Wald(1949) which gives the proof of consistency of maximum
likelihood estimate. His proof assumes the compact property and uses the strong law oflarge numbers. Similarly, we prove consistency of our estimator under some regularly
conditions using the strong law of large numbers.
Furthermore we shall verify regularity conditions for consistency and asymptotic normality of our estimator. When we estimate the statistical model of Takahashi(2011), we
may use parametric model for estimating survival function. The parametric models weconsider are exponential model and Weibull model. Needless to say, exponential distribution is a special case of Weibull distribution, we need only to verify the conditions in
the Weibull model. That is the second objective of this paper.
The last objective is to apply our model to real data to estimate default probabilities(survival probabilities). We present several numerical results in this paper. First,
in order to set known constants of our model, we examine comparative study varying
the recovery rate and the coefficient of error term. Second, we compare the results on
two parametric survival functions; exponential survival function and Weibull survival
function. The sum of squared residuals from the exponential survival function turnedout much larger than that of the Weibull survival function. Third, we compare different
recovery models; RM recovery model proposed by Duffie & Singleton(1999) and RT recovery model proposed by Jarrow & Turnbull(1995). The examination showed that there
is no great difference between the two. Fourth, we show estimating results and bootstrap
results using 8 individual companies’ data and 5 rating classes’ data on April 16, 2004.
There is a close relation between the distribution of bootstrap estimates and the numberof data. For example, the number of Toyota’s data was only 3 and the estimated bootstrap
distribution of Toyota was quite different from normal distribution. It seems the larger
amount of data(bonds) being used, the more like the bootstrap distribution is normal distribution. The existence of outlier data also influence upon the distribution of bootstrap
estimates. For example, when we excluded 3 outlier data from the original ”AA” classdata, the warps are reduced and the distributions of bootstrap parameter estimates look
more like normal distribution. Kolmogorov-Smirnov test also supported the influence ofoutlier data on the estimating results. Finally we apply our statistical estimating method
to time series data. Since we are interested in the impact of the bankruptcy of Lehman
Brothers(September 15, 2008), we use the data from January 4, 2008 to March 31, 2009and select 5 companies, which might be affected by the bankruptcy. In fact, all 5 companies’ survival probabilities were declining after the bankruptcy, but the influences differ
with the individual company in some points.
This paper is organized as follows. Section 2 provides a review of statistical estimating method of implied survival probabilities proposed by Takahashi(2011). The proof ofasymptotic property of our estimator is discussed in Section 3. Some empirical results
are presented in Section 4, and the last Section is devoted to the summary and concluding
remarks.

2

2 Review of the model and some development
In this section, we provide a brief explanation of Duffie and Singleton(1999), which derive the relationship between the default probabilities and the interest yield. We review
Takahashi(2011) model which is the main model of this paper. We also show the application of the model to the different recovery type case.

2.1 Setup and results on implied default probabilities
Throughout this chapter, W = {W (t), t ≤ 0} denotes standard Brownian motion, on aprobability space (Π, F , Pr ) . The filtration {Ft }t≥0 is generated by W . We suppose
there are no arbitrage opportunities in the market and the market is assumed to be complete.
Let P(0,t) be time 0 price of the defaultable zero coupon bond (corporate bond) maturing at t and P∗ (0,t) be time 0 price of the corresponding default free bond (governmentbond). The spread (difference of the yield to maturity between corporate bond and government bond) ϒ is thus defined by·¸
1
P(0,t)
ϒ(0) = − log ∗tP (0,t)
Let Q be the unique equivalent Martingale measure, then,
½
· Zt
¸¾
P∗ (0,t) = E Q exp − r(v)dv0

where r(v) is an instantaneous risk free spot rate and E Q denotes the expectation underthe measure Q (Harrison and Pliska[1981]). Let τ be the time of default, then the survival
function G(t) is given byG(t) = Q{τ > t}
½ Zt¾
= exp − λ (u)du0

where λ (t) is the hazard function. We also let δt be the recovery rate in the event ofdefault at t.

The formula between the survival function and the interest spread varies with thedefinition of recovery model. We consider the following two recovery models which arein common use.

RM Type Recovery (cf. Duffie & Singleton(1999)) By assuming that δt ×P(τ −,t)is recovered at the time of default, we have
½
· ZT
¸
¾
P(t, T ) = E Q exp −(r(v) + (1 − δv )λ (v)) dv |Fttwhere P(τ −,t) is the price of corporate bond just before default. If λ is assumed to be
non random function and δ to be constant, we have
½
· Zt
¸¾
Q
P(0,t) = E
exp − (r(v) + (1 − δ )λ (v)) dv
0

3

½
= E

Q

· Zt
¸¾
¸
·
Z t
exp − r(v)dv exp −(1 − δ ) λ (v)dv00= P∗ (0,t)Q {τ > t}1−δ= P∗ (0,t)G(t)1−δUnder RM type recovery, spread becomes½¾
P(0,t)
1
ϒ(t) = − logtP∗ (0,t)no1= − log G(t)1−δ ,tand implied survival function1 is
³

´ 1
1−δ
e−tϒ
µ
¶ 1
P(0,t) 1−δ
=P∗ (0,t)G(t) =

RT Type Recovery (cf. Jarrow & Turnbull(1995))
recovery rate δt is formulated as

Under RT type recovery,δt = (1 − Lt )P∗ (t, T )for a given fractional recovery process (1 − Lt). We assume default hazard rate process λbe independent (under Q) of short rate r and recovery process be constant. With constantδ and non-random λ , the payoff at maturity in the event of default is δ × 1. Then we have
½
· Zs
¸
¸
¾
· Zs
P(u, s) = EuQ exp − r(v)dv δ · I{τ ≤s} + exp − r(v)dv · 1 · I{τ >s} |τ > u
u
u
½
· Zs
¸¾
©
ª
= EuQ exp − r(v)dv EuQ δ · I{τ ≤s} + 1 · I{τ >s} |τ > u
u
¡
©
ª¢
∗
= P (u, s) δ + (1 − δ )EuQ I{τ >s} |τ > u= δ P∗ (u, s) + (1 − δ )P∗ (u, s)Qu {τ > s}
By setting u = 0, s = t, it follows that
P(0,t) = δ P∗ (0,t) + (1 − δ )P∗ (0,t)Qu {τ > s}
= δ P∗ (0,t) + (1 − δ )P∗ (0,t)G(t)Under RT type recovery, spread becomes½¾
P(0,t)
1
ϒ (t) = − logtP∗ (0,t)1= − log {δ + (1 − δ )G(t)} ,t]

1 In this paper, implied survival function means survival function which is incorporated by the bond price.

4and implied survival function isG] (t) ==´
1 ³ −tϒ
e −δ
1−δ
µ
¶
1
P(0,t)
1 − δ P∗ (0,t)

where we define ϒ] (t) is the spread under RT type recovery and G] is survival function
under RT type recovery.

In both type of recovery model, the formula is derived which presents the relationshipof the default probabilities and the interest spread. Takahashi(2011) proposed the statistical model of estimating procedure of implied default probabilities under this relationship.
We review the results in the next subsection.

2.2 Review of the Statistical Model by Takahashi(2011)
We have an exact estimate of the ”true” survival function or each t and the distributionfunction of default time τ under the equivalent martingale measure Q.

τ ∼ 1 − G(t : θ )θ ∈ Θ ⊂ Rdd≥1

Since there may be many sources of errors such as a lack of our information and a noisein the market, we suppose observed P(0,ti ) contains some error.
P̂(0,ti ) = P(0,ti ) + σ hti εti

i = 1, 2, . . . , n

where σ is unknown constant and ht is a known function with h0 = 0 and ht is an increasing in t. We consider
ht = [1 − P∗ (0,t)]αfor some α . On the other hand, P∗ (0,ti ) is supposed to be observed without any error.
After having observed P̂(0,ti ), i = 1, . . . , n , we calculate ”observed spread”½¾
1
p(0,ti )ϒ̂(ti ) = − log
+
σ
ρ
ε
ti ti
t
p∗ (0,ti )htwhere ρti is p∗ (0,t
. Using the observed spread ϒ̂(ti ), implied survial function S(t) ini)stead of G(t) becomes³
S(t) =e−t ϒ̂

´ 1

1−δ

n
o¶ 1
µ
p(0,t )1−δ−t·− 1t log p∗ (0,ti ) +σ ρti εti
i
=
e
o¶ 1
µ n p(0,t )1−δ
log p∗ (0,ti ) +σ ρti εti
i=e

It is convenient to rewrite S(t) as
S1−δ (t) = Gt1−δ + σ ρt εt5Then we assume the following parametric model. Let Θ be a subset of Rd , where d denotethe dimension of the parameter space. We also let
G(t : θ ) = Q{τ > t : θ }be a survival function parameterized by d-dimensional vector θ = (θ1 , . . . , θd )0 ∈ Θ. After
having observed {st]i , i = 1, . . . , n}, the pseudo maximum likelihood estimate ϑ̂ of θ isdefined by

) 
(
δ 2
n
G(ti , θ )1−δ − st1−i.
ϑ̂ = argθ ∈Θ min ∑
ρti
i=1
Let
Ξ(t, θ ) =andX(t) =St1−δρt

G(t, θ )1−δρtandx(t) =st1−δ,ρtthen, ϑ̂ may be rewritten as"nϑ̂ = argθ ∈Θ min ∑ {Ξ(ti , θ ) − xti }

#
2

.

i=1

and X(t) as

X(t) = Ξ(t, θ ) + σ εt .

2.3 Statistical Model under RT Type Recovery
We shall next derive statistical model under RT type recovery. We set the same assumptions to P(0,ti ), P∗ (0,ti ) and ϒ̂(ti ) as those under RM type recovery. Using the observed
spread ϒ̂(ti ), implied survival function S] (t) under RT type recovery in stead of G] (t)becomes´
1 ³ −t ϒ̂S] (t) =
e −δ
1−δ
n
o
µ
¶
p(0,t )
1−t·− 1t log p∗ (0,ti ) +σ ρti εti
i
−δ
=
e
1−δ
µ
¶
1
p(0,t)1−δσ ρti εti=+
1 − δ p∗ (0,t)
1−δ
1= G] (t) +σ ρti εti1−δAfter having observed {st]i , i = 1, . . . , n}, the pseudo maximum likelihood estimate ϑ̂ of
θ is defined by³´ 2 
]n  (1 − δ ) G] (ti , θ ) − st iϑ̂ = argθ ∈Θ min ∑
.


ρ
ti
i=1

6

Let

and
X ] (t) =Ξ] (t, θ ) =(1 − δ )G] (t, θ )ρt(1 − δ )St]ρtandx] (t) =(1 − δ )st],ρtthen, ϑ̂ may be rewritten as"n nϑ̂ = argθ ∈Θ min ∑ Ξ] (ti , θ ) − xt]i

o2

#
.

i=1

and X ] (t) as

X ] (t) = Ξ] (t, θ ) + σ εt .

2.4 RM Type Recovery and RT Type Recovery
The pseudo maximum likelihood estimate under RM type recovery ϑ̂RM of θ is defined
by

(
) 
δ 2
n
G(ti , θ )1−δ − st1−iϑ̂RM = argθ ∈Θ min ∑
ρti
i=1


2
µ³
´ 1 ¶1−δ 


1−
δ


1−δ −


 
e−ti ϒ̂(ti )n  G(ti , θ )= argθ ∈Θ 
min
∑



ρti




i=1 





(

n

= argθ ∈Θ min ∑i=1

G(ti , θ )1−δ − e−ti ϒ̂(ti )
ρti

)2 


The pseudo maximum likelihood estimate ϑ̂RT of θ is defined by³´ 2 ]]n(1 − δ ) G (ti , θ ) − sti  ϑ̂RT = argθ ∈Θ min ∑



ρti
i=1




n  (1 − δ )= argθ ∈Θ min ∑

i=1



³



³
´´ 2 G] (ti , θ ) − 1−1 δ e−ti ϒ̂(ti ) − δ  


ρti



n  (1 − δ )G] (ti , θ ) −= argθ ∈Θ min ∑

i=1


n

= argθ ∈Θ min ∑

i=1

ρti


(

³

´ 2 
e−ti ϒ̂(ti ) − δ  

(1 − δ )G] (ti , θ ) + δ − e−ti ϒ̂(ti )
ρti

7

)2 


Statistical RM Statistical RT
RM-RT
G1−δ
(1 − δ )G + δ
G1−δG(δ = 0.3)(δ = 0.3)−(1 − δ )G + δ
1
1
1
0
0.99
0.992989
0.993-0.000011
0.98
0.985958
0.986-0.000042
0.97
0.978904
0.979-0.000096
0.96
0.971829
0.972-0.000171
0.95
0.964732
0.965-0.000268
0.94
0.957612
0.958-0.000388
0.93
0.950469
0.951-0.000531
0.92
0.943304
0.944-0.000696
0.936115
0.937-0.000885
0.91
0.9
0.928902
0.93-0.001098
Table 1: Simulation of RM model and RT model
Difference between ϑ̂RM and ϑ̂RT is the term
RM
RT

:
:

G(ti , θ )1−δ
(1 − δ )G] (ti , θ ) + δ

We consider Weibull distribution for the survival function G(ti , θ ) and G] (ti , θ ). Table 1
is a simulation of RM model and RT model for G = 1, 0.99, . . . , 0.9.
From Table 1 we can see the difference G1−δ − {(1 − δ )G + δ } is negative in allcases. The smaller the survival probability G gets, the larger the difference G1−δ −{(1 − δ )G + δ } becomes.

2.5 Calibration
We may also suppose a model in which error term is not contained to the price of defaultable bond P(0,t) (and of course P(0,t)∗ ). It means we assume that there is no noisein the market and P(0,t) is determined with no error in the market. Under RM type recovery and the assumption of no error term, we can estimate the survival function by thefollowing equation;
G(t) = G(t, θ ) θ ∈ Θ ⊂ Rd d ≥ 1.
Observe ϒ(ti ) i = 1, . . . , n and then estimate θ from#" ½³
´ 1 ¾2n−ti ϒ(ti ) 1−δϑ̂ = min ∑ G(ti , θ ) − e.
θ ∈Θ i=1

We can estimate the survival function under RT type recovery in the similar way;G(t) = G(t, θ )θ ∈ Θ ⊂ Rdd ≥ 1.

Observe ϒ(ti ) i = 1, . . . , n and then estimate θ from
#
" ½
´ ¾2
n
1 ³ −ti ϒ(ti )ϑ̂ = min ∑ G(ti , θ ) −e−δ.
θ ∈Θ i=1
1−δ

83 Asymptotic Properties
The aim of this section is to justify the estimating method described in the previous
section. We shall first give a complete proof of the consistency of the pseudo maximum
likelihood estimator. We employ the method of Wald(1949) which gives the proof of
consistency of maximum likelihood estimator. Furthermore, we give a verification of theconditions for the consistency and the asymptotic normality of our estimator in the caseof Weibull survival function.

3.1 Proof of the Consistency
In this subsection, we prove that under certain regularity conditions, our estimator ϑ̂ isstrongly consistent.
Let,
Q(t, Xt , θ ) = {Ξ(t, θ ) − Xt }2 .
For any ρ > 0, letQ(t, Xt , θ , ρ ) =and, for any r > 0, let,inf|θ 0 −θ |<ρ

Q(t, Xt , θ 0 ),ϕ (t, Xt , r) = inf Q(t, Xt , θ ).
|θ |>rWe shall now impose some regularity conditions for the consistency:[Regularity Condition C2] (conditions for the consistency)
1. If limi→∞ θi = θ ,lim Q(t, Xt , θi ) = Q(t, Xt , θ )i→∞for all (t, Xt ) except on a set of measure zero under the true parameter vector θ 0 .
The null set may depend on θ , but not on the sequence {θi , i = 1, 2, . . .}.
2. For any pairs θ 6= θ 0 ,Ξ(t, θ ) 6= Ξ(t, θ 0 ),at least one t

3. For sufficiently small ρ , the expected values
Z ∞−∞Q(t, Xt , θ , ρ )dF(Xt , θ 0 ) < ∞,where θ 0 denotes the true parameter.
4. For the true parameter vector θ 0 , we haveZ ∞−∞Q(t, Xt , θ 0 )dF(Xt , θ 0 ) < ∞.

5. If limi→∞ |θi | = ∞, then limi→∞ Q(t, Xt , θi ) = ∞ for any (t, Xt ) except perhaps on a
fixed set of measure zero according to the true parameter vector θ 0 .

9Let, θ 0 be the true parameter vector, then it follows that,ª©
Eθ 0 {Q(t, Xt , θ 0 )} = Eθ 0 (Ξ(t, θ 0 ) − Xt )2 = σ 2

(1)

and,
Eθ 0 {Q(t, Xt , θ )}on
= Eθ 0 (Ξ(t, θ ) − Xt )2n¡¢2 o= Eθ 0 Ξ(t, θ 0 ) − Xt + Ξ(t, θ ) − Ξ(t, θ 0 )
n¡
¢2 ¡
¢2 o
= Eθ 0 Ξ(t, θ 0 ) − Xt + Ξ(t, θ ) − Ξ(t, θ 0 )¡¢2= σ 2 + Ξ(t, θ ) − Ξ(t, θ 0 )(2)

We consider the case where Θ is compact. Let Θ̃ be a compact subset of Θ, which
does not contain the true parameter vector θ 0 .

[Lemma 1] For any θ ∈ Θ̃,Eθ 0 {Q(t, Xt , θ )} > Eθ 0 {Q(t, Xt , θ 0 )}.

(3)

Proof. It follows from [C2]-3 and [C2]-4 that the expectated values in (3) exsist.
Lemma1 follows easily from (2) and [C2]-2.

[Lemma 2]

lim Eθ 0 {Q(t, Xt , θ , ρ )} = Eθ 0 {Q(t, Xt , θ )}

ρ →0

(4)

Proof. It follows from [C2]-1 that,lim Q(t, Xt , θ , ρ ) = Q(t, Xt , θ ).

ρ →0

(5)

except on a set whose probability measure is zero. Since Q(t, Xt , θ , ρ ) is a decreasing
function of ρ , and Q(t, Xt , θ , ρ ) > 0, there exists an integrable function ψ such that
|Q(t, Xt , θ , ρ )| ≤ ψ (t, Xt , θ )(6)for all ρ . It follows from (5) and Lebesgue’s Convergence Thorem2 , that
Z ∞

lim

ρ →0 −∞
R∞

Q(t, Xt , θ , ρ )dF(Xt , θ 0 ) =
R∞

Z ∞
−∞

Q(t, Xt , θ )dF(Xt , θ 0 ).

(7)

2 To show lim00ρ →0 −∞ Q(t, Xt , θ , ρ )dF(Xt , θ ) = −∞ Q(t, Xt , θ )dF(Xt , θ ) is equivalent to show, for any
R∞R∞0
sequence {ρn ; ρn → 0}, limn→∞ −∞ Q(t, Xt , θ , ρn )dF(Xt , θ ) = −∞ Q(t, Xt , θ )dF(Xt , θ 0 ). Since Q(t, Xt , θ , ρ1 )is integrable from [C2]-3, then,(6) means for any sequense {ρn ; ρn → 0}, |Q(t, Xt , θ , ρn )| ≤ Q(t, Xt , θ , ρ1 ).

10Thus, Lemma2 is proved.

[Lemma 3]

lim E 0 {ϕ (t, Xt , r)} = ∞
r→∞ θ

(8)

Proof. According to [C2]-5,
lim ϕ (t, Xt , r) = lim inf Q(t, Xt , θ ) = ∞r→∞ |θ |>r

r→∞

(9)

for any (t, Xt ) except on a set of measure zero under the true parameter vector θ 0 .
ϕ (t, Xt , r) is an increasing function of r. If there exists r > 0 such that the measure F
of a set {(t, Xt ) ∈ X ; ϕ (t, Xt , r) = ∞} is positive, then, limr→∞ ϕ (t, Xt , r) = ∞ on a set{(t, Xt ) ∈ X ; ϕ (t, Xt , r) = ∞}. So, Lemma3
Z

lim

r→∞

ϕ (t, Xt , r)dF(t, Xt , θ 0 ) = ∞obviously holds. Thus, we shall merely consider the case when ϕ (t, Xt , r) < ∞ on {(t, Xt ) ∈X . Since ϕ (t, Xt , r) is an increasing function of r, we define, for any sequence ri ; r1 <r2 < . . . < ri < . . .

ϕ 0 (t, Xt , r1 ) = ϕ (t, Xt , r1 )ϕ 0 (t, Xt , ri ) = ϕ (t, Xt , ri ) − ϕ (t, Xt , ri−1 )Since

i = 2, 3, . . .

iϕ (t, Xt , ri ) = ∑ ϕ 0 (t, Xt , r j )
j=1

and

ϕ 0 (t, Xt , r j ) ≥ 0,
Z

lim

ri →∞

Z

ϕ (t, Xt , ri )dF(t, Xt , θ 0 ) =i∑ ϕ 0 (t, Xt , r j )dF(t, Xt , θ 0 )

lim

ri →∞

j=1
i

=

lim ∑

ri →∞

∞ Z

=

Z

ϕ 0 (t, Xt , r j )dF(t, Xt , θ 0 )j=1∑

ϕ 0 (t, Xt , r j )dF(t, Xt , θ 0 )

j=1

Z ∞

∑ ϕ 0 (t, Xt , r j )dF(t, Xt , θ 0 )

=
Z

=
= ∞

11

j=1

lim ϕ (t, Xt , ri )dF(t, Xt , θ 0 )

ri →∞

(10)

The second and fifth equality are from the change the order of infinite summation andintegration and the 6th equal is from (9). Thus, Lemma3 is proved.

[Theorem 1] (The consistency theorem)onPθ 0 lim ϑ̂n = θ 0 = 1.

(11)

n→∞

Proof. Let r0 be a positive number chosen such thatEθ 0 {ϕ (t, Xt , r0 )} > Eθ 0 {Q(t, Xt , θ 0 )}.

(12)

The existence of such a positive number follows from Lemma3.
Let Θ̆ be the subset of Θ̃ consisting of all parameter vector θ of Θ̃ for which |θ | ≤ r0 .
With each parameter vector θ ∈ Θ̆, we associate a positive value ρθ such that©
ªEθ 0 {Q(t, Xt , θ , ρθ )} > Eθ 0 Q(t, Xt , θ 0 ) .
(13)
The existence of such ρθ follows from Lemma1 and Lemma2.
To prove that θ̂n is strongly consistent we have to show that½¾
infθ ∈Θ̃ ∑ni=1 Q(ti , Xti , θ )Pθ 0> 1 = 1−ε∑ni=1 Q(ti , Xti , θ 0 )(14)

for all n sufficiently large. This means that for all n sufficiently large the values of θ 0
which minimizes ∑ni=1 Q(ti , Xti , θ 0 ), θ̂n , belongs to neighborhood of θ 0 with probability
one.
Since Θ̆ is compact, there exists a finite open covering of Θ̆(l){O(θ (l) , ρθ ), l = 1, . . . , L},(1)

where, O(θ , ρθ ) is a sphere with center θ and radius ρθ ; Θ̆ ⊂ O(θ (1) , ρθ ) + . . . +(L)O(θ (L) , ρθ ).

Clearly,n

inf ∑ Q(ti , Xti , θ ) ≥ minθ ∈Θ̃ i=1n

∑ Q(ti , Xti , θ (l) , ρθ ).
l=1,...,L
(l)

i=1

Hence
(
)
½
¾
(l)
minl=1,...,L ∑ni=1 Q(ti , Xti , θ (l) , ρθ )infθ ∈Θ̃ ∑ni=1 Q(ti , Xti , θ )Pθ 0> 1 ≥ Pθ 0>1
∑ni=1 Q(ti , Xti , θ 0 )∑ni=1 Q(ti , Xti , θ 0 )(15)
Thus from (15) we deduce½¾
infθ ∈Θ̃ ∑ni=1 Q(ti , Xti , θ )>1
1 − Pθ 0∑ni=1 Q(ti , Xti , θ 0 )()(l)
minl=1,...,L ∑ni=1 Q(ti , Xti , θ (l) , ρθ )≤ 1 − Pθ 0>1
(16)
∑ni=1 Q(ti , Xti , θ 0 )()(l)
minl=1,...,L ∑ni=1 Q(ti , Xti , θ (l) , ρθ )= Pθ 0≤1∑ni=1 Q(ti , Xti , θ 0 )12

By the strong law of large numbers, (12) and (13), the right-hand side of (16) is smaller
than ε for all n sufficiently large. This completes the proof of Theorem1.

3.2

Verification of the Regularity Conditions for consistency

We shall now verify the regularity conditions for consistency and the asymptotic normality of our estimator when we apply Weibull distribution to the survival function. When
we estimate the statistical model of Takahashi(2011), we consider exponential model andWeibull model for the parametric survival function. Needless to say, exponential distribution is the special case of Weibull distribution, we need only to verify the regularity
conditions in the Weibull model. Weibull survival function is defined in the form ofnot ≥ 0.
GWei (θ1 , θ2 ,t) = exp −(t θ1 )θ2 ,We now verify the conditions of [C2] which is proved in the previous subsection. Verificationof the conditions of [C2].
We shall verify the conditions for consistency [C2] inthe Weibull distribution case. Condition [C2]-1 holds clearly. To show [C2]-2, let(θ1 , θ2 ) 6= (θ10 , θ20 ) and define ∆θ2 = θ2 − θ20 . Then,00(t θ1 )θ2 − (t θ10 )θ2= (t θ1 )θ2 +∆θ2 − (t θ10 )θ2
θ 0 +∆θ2= t θ20 (t ∆θ2 θ1 2

0θ 0

− θ1 2 )6= 0 at least one t,
θ20

so that (t θ1 )θ2 6= (t θ10 )θ2 at least one t. Since G(t, θ ) = e−(t θ 1) is strictly monotonic,0(t θ1 )θ2 6= (t θ10 )θ2 implies G(t, θ ) 6= G(t, θ 0 ), and hence that Ξ(t, θ ) 6= Ξ(t, θ 0 ) at least onet.
We next show the existence of E[Q(t, Xt , θ 0 )] ([C2]-4). The integrand of the expectation
Z ∞E[Q(t, Xt , θ 0 )] =

−∞
Z ∞

=

−∞

Q(t, xt , θ 0 )dF(xt , θ 0 ){Ξ(t, θ 0 ) − xt }2 dF(xt , θ 0 ).

is written as follows.
({Ξ(t, θ ) − xt }
0

2

=
(
=
(
=
(
=

G(t, θ 0 )1−δ
− xt
ρt
θ

)2

(e−(t θ1 ) 2 )1−δ
− xt
ρt
θ2

θ

θ2

θ

)2

(e−θ1 )1−δ (e−t 2 )1−δ st1−δ
−
ρt
ρt

)2

(e−θ1 )1−δ (e−t 2 )1−δ P∗ (0,t) st1−δ P∗ (0,t)
−
ht
ht

13

)2

θ2

θ

Note that (e−θ1 )1−δ is positive constant. From 0 ≤ t , θ2 > 0 and 0 < δ < 1 , (e−t 2 )1−δ ≤1. It follows that st1−δ ≤ 1 and 0 < P∗ (0,t) ≤ 1. Therefore {Ξ(t, θ 0 )−xt }2 → ∞ as ht → 0; ht 6= 0 implies E[Q(t, xt , θ 0 )] < ∞. So [C2]-4 holds. [C2]-3 can be verified in the sameway. For [C2]-5, if limi→∞ |θi | = ∞, Ξ(t, θi ) → 0. [C2]-5 holds clearly.

3.3 Verification of the Regularity Conditions for Asymptotic Normality
The asymptotic normality is proved completely by Takahashi(2011), so we describe theconditions for the asymptotic normality without proof.

[Regularity Condition C3] (conditions for the asymptotic normality)
1. The pseudo data will be sampled in a way that the time sequence {ti , i = 1, 2, . . .}are selected so that there is a positive definite matrix Σ(θ 0 ) and B(θ 0 ) for which
1 nlim σ 2 Σ(θ 0 , n) = σ 2 Σ(θ 0 )∑ Var{ψ (θ 0 ,ti , Xti )} = n→∞
n→∞ n
i=1
lim

(17)

where (l, m)th element of Σ(θ 0 , n) = nσ1 2 ∑ni=1 Var{ψ (θ 0 ,ti , Xti )} is defined by
1 n ∂∂∑ ∂ θl Ξ(ti , θ 0 ) ∂ θm Ξ(ti , θ 0 )
n i=1
and,

·
¸
1 n ∂ 2 (l) 0ψ (θ ,ti , Xti ) = lim B(θ 0 , n) = B(θ 0 )
lim ∑
2
n→∞ n
n→∞
i=1 ∂ θih 2where (k, m)th element of 1n ∑ni=1 ∂∂θ 2 ψ (l) (θ 0 ,ti , Xti ) is given by

+
+
+

·
¸
©
ª
1 n
∂3
0
Ξ(t
,
θ
)
Ξ(ti , θ 0 ) − Xti∑n i=1 ∂ θk ∂ θm ∂ θl
¸·
¸
·
∂2
∂
1 n
0
0
∑ ∂ θm ∂ θl Ξ(ti , θ ) ∂ θk Ξ(ti , θ )
n i=1
¸·
¸
·
∂2
∂
1 n
0
0
)
Ξ(t
,
)
Ξ(t
,
θ
θ
i
∑ ∂ θl ∂ θ k i
n i=1
∂ θm
·
¸·
¸
1 n
∂2
∂
0
0
Ξ(t
,
θ
)
Ξ(t
,
θ
)
i
∑ ∂ θk ∂ θm i
n i=1
∂ θl

and the (k, m)th element of B(θ 0 , n) is defined by
¸·
¸
·
1 n
∂2
∂
0
0
∑ ∂ θm ∂ θl Ξ(ti , θ ) ∂ θk Ξ(ti , θ )
n i=1
·
¸·
¸
1 n
∂2
∂
0
0
Ξ(t
,
θ
)
Ξ(t
,
θ
)
+
i
∑ ∂ θ l ∂ θk i
n i=1
∂ θm
·
¸·
¸
1 n
∂2
∂
0
0
Ξ(t
,
θ
)
Ξ(t
,
θ
)
.
+
i
i
∑ ∂ θk ∂ θm
n i=1
∂ θl
14

(18)

2. For every θ ∈ Θ, the time sequence {ti , i = 1, 2, . . .} are selected so that
¸2
·
1 n
∂2
Ξ(t
,
θ
)
converges as n → ∞, f or all l and m∑ ∂ θl ∂ θ m i
n i=1
¸2
·
∂3
1 n
Ξ(t
,
θ
)
∑ ∂ θ k ∂ θ l ∂ θm i
n i=1

(19)

converges as n → ∞, f or all k, l and m . (20)

3. The sequence
¾
½
h
i0
0
(1) 0
(d) 0
ψ (θ ,ti , Xti ) = ψ (θ ,ti , Xti ), . . . , ψ (θ ,ti , Xti ) , i = 1, 2, . . .
obeys the multivariate central limit theorem;
"
#− 1
2 n
n
©
ª
∑ Var ψ (θ 0 ,ti , Xti )∑ ψ (θ 0 ,ti , Xti ) → N(0, Id ) as n → ∞
i=1

(21)

i=1

where 0 = (0, . . . , 0) is the d dimensional zero vector and Id is the d × d identity
matrix.

Next we shall verify the conditions of [C3] in the case of Weibull survival function.

Verification of the conditions [C3]. We next show the conditions of the asymptotic normality in the Weibull distribution case. To show condition[C3]-1, consider
α = (α 1, α 2)0 6= 0.
Ã
!
∂
Ξ
∂ θ1
(α 1, α 2) ·∂∂ θ2 Ξ

 µ
¶1−δ
θ

e(−t θ1 ) 2θ(1−δ )(−t θ )θ2 2

1
θ1




ρt
¶1−δ
= (α 1, α 2) ·  µ e(−t θ1 )θ2
θ2(1−δ )(−t θ1 ) ln[−t θ1 ] 

ρt

=

α1
³

=

³
´1−δ
θ
e(−t θ1 ) 2(1 − δ ) (−t θ1 )θ2 θθ21(−t θ1 )θ2

e

´1−δ

ρt

+ α2

³
´1−δ
θ
e(−t θ1 ) 2(1 − δ ) (−t θ1 )θ2 ln [−t θ1 ]ρt¶
(1 − δ ) (−t θ1 )θ2 µ θ
2
α 1 + α 2 ln [−t θ1 ]
ρt
θ1

³
´1−δ
θ
Since e(−t θ1 ) 2> 0 , 1− δ > 0, (−t θ1 )θ2 < 0, θ2 /θ1 > 0, ln[−t θ1 ] < 0 and ρt > 0Ã!
∂∂ θ1 Ξ
is nonzero on an interval. Therefore,
by definition, (α 1, α 2) ·
∂
∂θ Ξ
2

(
(α 1, α 2) ·Ã∂∂ θ1 Ξ
∂∂ θ2 Ξ

!)2= α 0 Σ(θ 0 )α > 0

15

Since this conclusion is true for any α 6= 0 , Σ must be positive definite matrix. The
existence of a positive matrix B(θ 0 ) can be showed in the similar way .
³ 2
´2
³
´2
3
Condition [C3]-2 holds by inspection of 1n ∑ni=1 ∂ θ∂∂ θm Ξ(ti , θ ) and 1n ∑ni=1 ∂ θ ∂∂θ ∂ θm Ξ(ti , θ ) ,lkl(l, m, k = 1, 2). In the case of k = 1, l = 1 and m = 1, it follows that
µ
¶2
1 n
∂2
Ξ(t
,
θ
)
∑ ∂ θ1 ∂ θ1 i
n i=1
³
´
µ e(−t θ1 )θ2 1−δ t 2 (1 − δ ) (−t θ )−2+θ2 (−1 + θ )θ
i 1
2 2
i
1 n
=
∑
n i=1
ρt
³
´
1−
δ
θ
e(−t θ1 ) 2
ti2 (1 − δ )2 (−ti θ1 )−2+2θ2 θ22 ¶2
,
+
ρt
and
¶2
µ
∂3
1 n
∑ ∂ θ1 ∂ θ1 ∂ θ1 Ξ(ti , θ )
n i=1
³
´1−δ
θ
µ
e(−t θ1 ) 2
ti3 (1 − δ ) (−ti θ1 )−3+θ2 (−2 + θ2 )(−1 + θ2 )θ2
1 n
=
∑ −
n i=1
ρt
³
´
1−
δ
θ
e(−t θ1 ) 2
ti3 (1 − δ )3 (−ti θ1 )−3+2θ2 (−1 + θ2 )θ22
+
ρt
³
´1−δ
θ
ti3 (1 − δ )3 (−ti θ1 )−3+3θ2 θ23e(−t θ1 ) 2
−
ρt
³
´1−δ
θ
e(−t θ1 ) 2
ti3 (1 − δ )2 (−ti θ1 )−3+2θ2 θ22 (−2 + 2θ2 ) ¶2
+
ρt
´2
³ 2
∗ (t ))α∂∗ (t ) < 1 and α > 0 , 1 n
i
,
0
<
P
Ξ(t
,
θ
)
Since 0 < ti < T < ∞, ρti = (1−P∑iin i=1 ∂ θ1 ∂ θ1P∗ (ti )
³
´2
∂3
1 n
and n ∑i=1 ∂ θ ∂ θ ∂ θ Ξ(ti , θ ) converges as n → ∞. The rest case can be verified in the
1 1 1
similar way.
By the equation (17), condition [C3]-3"ª∑ Var ψ (θ 0 ,ti , Xti )
©

n

#− 1
2

i=1

"
=

1
√
n

©
ª
∑ Var ψ (θ 0 ,ti , Xti )
n

i=1

n

∑ ψ (θ 0 ,ti , Xti )

i=1

#− 1
2

n

∑ ψ (θ 0 ,ti , Xti ) → N(0, Id ) as n → ∞i=1

is rearranged to
1 n√ ∑ ψ (θ 0 ,ti , Xti ) → N(0, σ 2 Σ(θ 0 )) as n → ∞
n i=1
16

(22)

We have already shown that
1 n
∑ Var{ψ (θ 0 ,ti , Xti )} → σ 2 Σ(θ 0 )n i=1
is non zero, so that only condition
Z

1 n
2
∑
√ kψ k dFi (ψ ) = 0n→∞ ni=1 kψ k>ε n
lim

(23)

for any ε > 0 is needed, where k · k is the Euclidean norm. Sinceqkψ k =|ψ (1) |2 + |ψ (2) |2s∂∂=
Ξ(ti , θ 0 ) (Ξ(ti , θ 0 ) − x) |2 + |Ξ(ti , θ 0 ) (Ξ(ti , θ 0 ) − x) |2 ,
|
∂ θ1
∂ θ2
it follows that for any ε > 0,Z

1 n
∑ kψ k>ε √n kψ k2 dFi (ψ ) → 0
n i=1
as n → ∞, from which the condition (23) follows in the Weibull case.

4 Numerical Results
In this section, we present some numerical results. We apply our model to real data toestimate default probabilities(survival probabilities). We describe data sources at first,
and then show the results of comparative study varying α and δ . We also compare results
on different recovery models and on two parametric distributions. Then we show estimating results and bootstrap results using 8 individual companies’ data and 5rating classes’
data on April 16, 2004. Finally we apply our statistical estimating method to time seriesdata(January 2008 to March 2009).

4.1

Data Description

Corporate bond prices and government bond prices were obtained from Bloomberg Financial Markets and Japan Securities Dealers Association covering the period April 2003
to March 2009. Specifically, we collected daily price data for OTC Bond Transactions.
We also obtained bond issuer’s rating information from Bloomberg. Rating and Investment Information, Inc. (R & I) is one of the leading rating companies in Japan and it
grade for bond issuers by the rating categories from AAA to CCC. The data we used in
the comparative study is bond price data of Mitsui & Co., LTD. also known as Mitsui
Bussan on April 16, 2004. We use 1520 individual companies’ data for estimating survival probabilities on one day, and 5 individual companies’ data for time series analysis.

4.2

Selection of a constant α the recovery rate δIn this subsection, we show the results of comparative study varying α and δ . For the purpose of equalizing the other conditions, we use only Weibull distribution for the survival
function G(θ ,t) in this subsection.
17

0.000007
sum of squared residuals

0.000006
0.000005
0.000004
0.000003
0.000002
0.000001
01

1/2 1/3 1/4 1/5 1/6 1/7 1/8 1/9 1/10
alpha

Figure 1: Sum of the squared residuals of Mitsui Corporation as a function of α
A constant α is defined byρt =[1 − P∗ (0,t)]α.
P∗ (0,t)

The restriction at the maturity ρ0 = 0 suggests that α > 0. We first consider α = 1.
∗ (0,t)∗(It means ρt = 1−PP∗ (0,t) .) Since P (0,t) is near to 1 with small t and then ρt takes asmall value, the weight of the data becomes very large with small t. We get biased
estimating results and larger values of sum of squared residuals. Next, we calculate sum
1of the squared residuals with different α ; α = 1, 12 , . . . , 10
Figure 1 shows the sum of thesquared residuals of Mitsui Corporation as a function of α . In this case, we see that thesum of squared residuals is decreasing as α → 0. We get similar results in most individualcompany cases. But as mentioned above, α is supposed to take positive value. Since thesums of the squared residuals with α = 13 are enough small in most cases, we set α to be1
3 throughout the rest of this section.
We next consider about recovery rate. In our model, recovery rate, δ , is assumed to
be non-random. We use δ when we calculate the survival function
1

G(t) = exp {tϒ(t : P∗, P)} 1−δ
µ ¶ 1
P 1−δ
=
,
P∗
and of course, when calculate pseudo random sample(data), sti in the statistical model;
µ
½ ·
¸¾¶ 1
1−δ
P
st = exp log+ σ ρt εtP∗
Figure1 shows the variation of the survival probability of Mitsui Corporation with thetime to maturity for δ = 0.15, δ = 0.3, δ = 0.45. The plot ”Observed” is the values ofsurvival probabilities calculated from the bond prices. The line ”Model” is the values of
estimated survival probabilities. Table 2 shows the values of the estimated parametersof Weibull model for δ = 0.15, δ = 0.3, δ = 0.45. The recovery rate δ becomes larger,
survival probability tends to smaller. An interesting results is that the values of θ2 are
the same with different δ . Since θ2 is the shape parameter of Weibull distribution, we

18

1.005

St

1
0.995

Observed(delta=0.15)

0.99

Model(delta=0.15)

0.985

Observed(delta=0.3)

0.98

Model(delta=0.3)

0.975

Observed(delta=0.45)

0.97
Model(delta=0.45)
0.965
0

2

4

6

8

term to maturity

Figure 2: Variation of the survival probability with the time to maturity for different δ

δ
0.15
0.3
0.45

θ1
0.018250.02023
0.02298θ2
1.88807
1.88807
1.88807

Table 2: Estimated parameters of Weibull model for different δ
see that the shape of the survival function does not vary with the recovery rate δ in ourmodel.
The similar results are obtained for the statistical model under RT type recovery (Figure2, Table 3). Throughout the rest of this section, we set the recovery rate δ to be 0.3.

4.3 Selection of the parametric model
In this subsection, we consider the parametric model for the survival function of our estimating model. We use two parametric models for the survival function: exponential
distribution and Weibull distribution. We set the survival function with exponential distribution Gexp and the survival function with Weibull distribution GWei . Gexp (θ ,t) and
GWei (θ1 , θ2 ,t) is defined by
Gexp (θ ,t) = exp {−t θ } ,

t ≥0

1.005
1

0.99

Observed(delta=0.15)
Model(delta=0.15)

0.985

Observed(delta=0.3)

0.98

Model(delta=0.3)

0.975

Observed(delta=0.45)
Model(delta=0.45)

St

0.995

0.97
0.965
0

2

4

6

8

term to maturity

Figure 3: Variation of the survival probability with the time to maturity for different δ

19

δ
0.15
0.3
0.45

θ1
0.018250.02023
0.02298θ2
1.88807
1.88807
1.88807

Table 3: Estimated parameters of Weibull model for different δ Mitsui Corp
1.005
1
0.995

St

0.99

Observed

0.985

Exponential

0.98
Weibull
0.975
0.97
0

2

4

6

8

term to maturity

Figure 4: Estimated survival functions of Mitsui & Co. for the exponential and theWeibull modelnoGWei (θ1 , θ2 ,t) = exp −(t θ1 )θ2 ,t ≥0

respectively. Figure 4 shows the estimated survival functions of Mitsui & Co. for the
exponential and the Weibull model. Figure 5 shows the cumulative value of the sum ofsquared residuals for the exponential(dotted line) and the Weibull model(staggered line).
It is clear that the exponential survival function is estimated far from the ”Observed”
plot.(See Figure 4. ”Observed” plots indicate the survival probabilities calculated fromthe interest yields.) And the sum of squared residuals from the exponential survival
function is much larger than that of the Weibull survival function. The result is of course
from the fact that the exponential distribution is allowed less 1 parameter than Weibull
distribution. We shall thus apply Weibull distribution for the rest of our analysis.

Cumulative Value of Residuals

0.000045
0.00004
0.000035
0.000030.000025

Exponential

0.00002

Weibull

0.000015
0.00001
0.000005
0
1

2

3

4

5

6

7

Figure 5: Cumulative value of the sum of squared residuals for the exponential and theWeibull model

20

Cumulative Value of Squared Residuals

0.000003
0.0000025
0.000002
0.0000015

RM

0.000001

RT

0.0000005
0
1

2

3

4

5

6

7

Figure 6: Cumulative value of the sum of squared residuals under RM type recovery andunder RT type recovery

4.4 Two recovery models; RM type recovery & RT type recovery
As mentioned in chapter 2, the pseudo maximum likelihood estimate of statistical model
under RM type recovery and RT type recovery are defined by

(
) 
δ 2
n
G(ti , θ )1−δ − st1−iϑ̂ = argθ ∈Θ min ∑
ρti
i=1

³
´ 2 
]n  (1 − δ ) G] (ti , θ ) − st iϑ̂ = argθ ∈Θ min ∑



ρ
ti
i=1
respectively, where,
³

´ 1

1−δ

s(t) =e−tϒs] (t) =´
1 ³ −tϒe −δ .
1−δWe also defined the pseudo maximum likelihood estimate of calibration model under RM
type recovery and RT type recovery as follows:#" ½³´ 1 ¾2n−ti ϒ(ti ) 1−δϑ̂ = min ∑ G(ti , θ ) − eθ ∈Θ i=1

"

ϑ̂

= min

n ½

∑

θ ∈Θ i=1´
1 ³ −ti ϒ(ti )G(ti , θ ) −
e
−δ
1−δ

¾2 #
.

Figure 6 shows the cumulative value of the sum of squared residuals of Mitsui & Co.
for the statistical model under RM type recovery(solid line) and that for the statisticalmodel under RT type recovery(dotted line).
We see from Figure 6 that the lines are lying on top each other and that the difference
between the sum of squared residuals from RM type recovery and RT type recovery isvery small. Table 4 indicates the absolute value of residuals and the sum of the squared
residuals when we apply statistical and calibration model to 10 individual companies.
The above table is for the statistical model and the bottom is for the calibration model.
21

Statistical Model
RM
RT
RM/RT
0.0000127 0.0000127
RT
0.0016643 0.0015528
RT
0.0004898 0.0004917
RM
0.0007438 0.0006811
RT
0.0008121 0.0008242
RM
0.0009871 0.0009972
RM
0.0015136 0.0015379
RM
0.0006707 0.0006997
RT
0.0000993 0.0001037
RM
0.0015214 0.0015431
RM
Calibration
Company
RM
RT
RM/RT
Toyota0.0000120 0.0000118
RM
NTT Data
0.0015446 0.0015477
RM
0.0004964 0.0004982
RM
Mitsui & Co.
Asahi Breweries 0.0005521 0.0005585
RM
Kajima0.0006030 0.0006375
RM
NEC
0.0008421 0.0007921
RM
Kintetsu
0.0014051 0.0013742
RM
Sumitomo Realty 0.0006942 0.0007389
RM
Cosmo Oil
0.0000794 0.0000984
RMJAL
0.0011855 0.0012178
RM
Table 4: The mean of the absolute value of residuals for the statistical model and calibration under RM type recovery and RT type recovery. Residuals are calculated byĜ(t) − G(t). The rightmost item is the model name which take the smaller value of thesum of squared residuals.
Company
ToyotaNTT Data
Mitsui & Co.
Asahi Breweries
KajimaNEC
Kintetsu
Sumitomo Realty
Cosmo OilJAL

22

1.005
1
0.995

St

0.99
Observed

0.985

Statistical
RM

0.98
0.975
0.97
0

2

4

6

8

term to maturity

Figure 7: Estimated survival function of Mitsui Corporation for the statistical model andthe calibration model
The rightmost item is the model name which takes the smaller value of the sum of squared
residuals.
We see that RM model and RT model have little difference in residuals, but the following results are interesting. When we apply calibration estimating model, residuals of
the model under RM type recovery are calculated smaller than that under RT type recovery for all 10 companies, though when applying statistical model, we can’t say whichmodel has smaller residuals.

4.5 Statistical Model and Calibration
Our estimators of statistical model under RM type recovery and calibration model under
RM type recovery are written as follows

(
) 
δ 2
n
G(ti , θ )1−δ − st1−iϑ̂ = argθ ∈Θ min ∑
ρti
i=1
"
#

ϑ̂

n

∑ {G(ti , θ ) − sti }2
θ ∈Θ

= min

i=1

respectively, where

³
´ 1
1−δ
sti = e−ti ϒ(ti ).

Figure 7 shows the estimated survival function of Mitsui Corporation for the two models;
the statistical model (solid line) and the calibration model(dotted line) and Figure 8 shows
the cumulative values of the sum of squared residuals for the two models, where we take
RM type recovery model.
We find that the residuals calculated by the calibration model are smaller than that by
the statistical model for all 10 companies(See Table 4). We also find that the cumulativevalues of the sum of squared residuals for the statistical model are take smaller values thanfor the calibration model in the first and second times of cumulation, and the larger values
after the third cumulation(See Figure7). It is because we defined ht (or ρt ) as a functionof the price of default free bond P∗ (0,t) and a constant α for our statistical model. Whenthe data is close to the maturity, P∗ (0,t) becomes nearly 1 and so ρt gets near to 0. Thatis the reason of the behavior of the cumulative values of the sum of squared residuals.
23

0.000003

cumulative value of residuals

0.0000025
0.000002
0.0000015

Statistical
RM

0.000001

Calibration
RM

0.0000005
0
1

2

3

4

5

6

7

Figure 8: Cumulative values of the sum of squared residuals for the statistical model andthe calibration modelθ1θ̃(Boots)θ̂gBias(θ̂ )(Boots)g θ̂ )(Boots)Var(f (t)(Boots)
SD
θ̂

RM

0.020400.020230.000174
0.000010
0.00319

RT

θ2

θ1

1.89376
1.888070.00569
0.015940.126270.020510.020300.000211
0.000010
0.00321

θ2
1.89701
1.89005
0.006960.01600
0.12651

Table 5: The parameter estimate of the bootstrap estimation for Mitsui & Co. using
statistical model. The notations are given in Appendix.
The bootstrap analysis showed that the estimate of statistical model has some warp.
Table 5 and Table 6 indicate the parameter estimate of the bootstrap estimation for Mitsui& Co.. Table 5 is for the statistical model and Table 6 is for calibration.
Figure 9 shows the distribution of bootstrap parameter estimate of Mitsui & Co. forstatistical model and Figure 10 shows the distribution of bootstrap parameter estimate forcalibration. The left graph in Figure 9 and Figure 10 shows the histogram of the bootstrap(m)(m)
estimate ϑˆ1 and the right graph shows that of ϑˆ2 .
As seen from a comparison between Figure 9 and Figure 10, the distribution for
statistical model has some warp, although that for calibration doesn’t.

θ1θ̃(Boots)θ̂g θ̂ )(Boots)Bias(g θ̂ )(Boots)Var(f (t)(Boots)
SD
θ̂

RM

0.02567
0.025430.000242
0.000018
0.00423

RT

θ2

θ1

2.104422.09540.00907
0.028780.16966

0.02573
0.025540.000186
0.000018
0.00425

θ2
2.105992.09880.007240.029010.17033

Table 6: The parameter estimate of the bootstrap estimation for Mitsui & Co. using
calibration. The notations are given in Appendix.

24

350

300

300

250

250
frequency

frequency

350

200
150
100

200
150
100

50

50

0
0.00990.01150.01310.01480.01640.01810.01970.02140.02300.02470.02630.02800.0296

1.450
1.513
1.577
1.640
1.704
1.767
1.831
1.894
1.958
2.021
2.085
2.148
2.212

0

θ1

θ2

Figure 9: Distribution of bootstrap parameter estimate for statistical model
300

300

250

250
frequency

150
100

200
150
100
50

0

0
1.581
1.665
1.750
1.834
1.919
2.003
2.088
2.172
2.256
2.341
2.425
2.510
2.594

50

0.01290.01500.01710.01920.02130.02340.02550.02760.02970.03180.03390.03600.0381

frequency

200

θ1

θ2

Figure 10: Distribution of bootstrap parameter estimate for calibration

4.6 Estimating results for 8 individual companies
We now present estimating results for individual companies. We set α to be 13 and therecovery rate δ to be 0.3, and use statistical model of Weibull survival function under RM
type recovery. We analyze the following 8 individual companies; Toyota Motor Corporation, NTT Data Corporation, Mitsui & Co., Ltd., Asahi Breweries, Ltd., Kajima Corporation, Kintetsu Corporation, Japan Airlines Co., Ltd. and Cosmo Oil Co., Ltd.. Table
7 indicates the data information and the parameter estimates of 8 individual companies.
Figure 11 shows the estimated survival function of 8 individual companies.
Toyota’s estimated Weibull shape parameter θ̂2 is the smallest of the 8 companies
and it is nearly 1. That means straight survival function and the almost constant hazardrate(constant instantaneous default probabilities). When the shape parameter θ̂2 is larger
than 1, the instantaneous default probabilities decline as t → 0 .
Table 8 indicates the bootstrap parameter estimates of 8 individual companies. Figure
12 shows the distribution of bootstrap parameter estimate of 8 individual companies. The(m)upper graph shows the histogram of the bootstrap estimate ϑˆ1
and the lower graph(m)shows that of ϑˆ2 .
We see from Table7 and Figure12 that there is a close relation between the distribution of bootstrap estimates and the number of data. The number of Toyota’s bonds being
issued on April 15, 2004 was only 3. That explains why the estimated bootstrap distribution of Toyota is quite different from normal distribution. It seems the larger amountof data(bonds) being used, the more like the bootstrap distribution is normal distribution.
The bootstrap distribution of Mitsui & Co. looks like to have two peaks. The number ofdata and the existence of outlier data may explain this.

25

Data
Estimating results
Company
Rating Observationsθ̂1θ̂2
mean(|residuals |)ToyotaAAA
30.0011 1.0749
0.000013
NTT Data
AA80.0165 1.9923
0.001664
Mitsui & Co.
AA
7
0.0202 1.8881
0.000490
Asahi BreweriesA
10
0.0207 1.8545
0.000744
KajimaA
11
0.0197 1.5656
0.000812
Kintetsu
BBB
15
0.0218 1.5364
0.001514
JALBBB
90.0297 1.6017
0.001521
Cosmo OilBB
90.0242 1.3502
0.000099
Table 7: Information about the data and the parameter estimates of 8 individual companies. The items ”Rating” and ”Observations” are the issuer’s rating on April 15, 2004 andthe number of a company’s bonds being issued on April 15, 2004, respectively. ”θ̂1 ” and”θ̂2 ” are parameter estimates and the rightmost item is the mean of the absolute values ofresiduals.

Observed(Toyota)

1Model(Toyota)
0.99

Oberved(NTT)Model(NTT)

0.98

Observed(Mitsui)
0.97
St

Model(Mitsui)
Obseived(Asahi)

0.96Model(Asahi)
0.95
Observed(Kajima)
0.94

Model(Kajima)

0.93

Observed(Kintetsu)Model(Kintetsu)

0.92

Observed(JAL)
0

1

2

3

4

5

6

7

term to maturity

Figure 11: Estimated survival function and ’Observed’ survival probabilities of 8 individual companies. ”Observed” plots are survival probabilities calculated from interest yieldand ”Model” line is estimated survival function of our estimating model.

26

Toyota
0.001080.00108
0.00000
0.00000
0.00008

NTT Data
0.018240.016530.001710.000120.01113

Mitsui0.020400.020230.000170.00001
0.00319

Asahi
0.020780.020670.000110.00001
0.00353

1.074621.07492-0.00030
0.000210.01435

2.04300
1.99226
0.050740.30568
0.55291

1.89376
1.888070.00569
0.015940.12627

1.85723
1.85452
0.002710.01484
0.12182θ̂1
g θ̂1 )(Boots)Bias(g θ̂1 )(Boots)Var(f (t)(Boots)SDθ̂1

Kajima0.01968
0.019650.000030.00000
0.00182

Kintetsu
0.021810.021760.00005
0.00000
0.00195

JAL
0.029760.029710.00005
0.00001
0.00248

Cosmo Oil
0.035530.02418
0.011350.00000
0.00159θ̂2
g θ̂2 )(Boots)Bias(g θ̂2 )(Boots)Var(f (t)(Boots)SDθ̂2

1.56630
1.565630.00067
0.00309
0.055641.53760
1.536410.001190.00324
0.05697

1.60337
1.601710.00166
0.00430
0.06560

1.59484
1.350180.24466
0.00099
0.03145(Boots)θ̃1
θ̂1
g θ̂1 )(Boots)Bias(g θ̂1 )(Boots)Var(f (t)(Boots)SDθ̂1(Boots)θ̃2θ̂2g θ̂2 )(Boots)Bias(g θ̂2 )(Boots)Var(f (t)(Boots)SDθ̂2(Boots)θ̃1(Boots)θ̃2

Table 8: The parameter estimate of the bootstrap estimation for 8 individual companies.
The notations are given in Appendix.

27

1.386
1.435
1.483
1.532
1.580
1.629
1.677
1.726
1.774

1.348
1.404
1.461
1.517
1.573
1.630
1.686
1.742
1.799

1.427
1.475
1.524
1.572
1.621
1.670
1.718
1.767
1.816

1.472
1.497
1.522
1.547
1.572
1.598
1.623
1.648
1.673
1.698

350
300
250
200
150
100
50
0
NTT DATA(θ2)

400

300

200

100

0

Kajima(θ1)
400
Kinki Nippon Railway(θ1)

300

200

100

0

28
1.514
1.603
1.691
1.780
1.868
1.957
2.046
2.134
2.223

500
400
300
200
100
0
1.450
1.537
1.625
1.712
1.799
1.886
1.974
2.061
2.148
2.235

350
300
250
200
150
100
50
0

NTT DATA(θ1)
Mitsui & Co.(θ1)

400

300

200

100

0

300
250
200
150
100
50
0

300
250
200
150
100
50
0
0.01170.01410.01660.01900.02150.02390.02640.02880.03130.00990.01210.01440.01660.01890.02120.02340.02570.02800.03020.00090.00100.00100.00100.00110.00110.00110.00120.00120.0012
500
400
300
200
100
0

0.0296
0.03100.03230.03370.03510.03640.03780.03920.0405

Toyota(θ2)
0.00000.01100.02210.0331
0.04420.0552
0.06630.07730.08830.09940.1104

Toyota(θ1)

0.02260.02410.02570.02730.02880.03040.03200.03360.03510.03670.0383

350
300
250
200
150
100
50
0
0.420
0.903
1.386
1.869
2.352
2.835
3.318
3.801
4.284
4.767
5.250

1.047
1.053
1.060
1.066
1.072
1.078
1.084
1.090
1.096
1.102

1200
1000
800
600
400
200
0

0.01530.01690.01860.02020.02180.02350.02510.02680.02840.03000.01380.01510.01640.01760.01890.02020.02150.02270.02400.02530.0266

1200
1000
800
600
400
200
0
300
250
200
150
100
50
0

300
250
200
150
100
50
0
Asahi Breweries(θ1)

Mitsui & Co.(θ2)
Asahi Breweries(θ2)

350
300
250
200
150
100
50
0

JAL(θ1)
350
300
250
200
150
100
50
0
Cosmo Oil(θ1)

Figure 12: Distributions of bootstrap parameter estimates for 8 individual companies.
(m)
The upper graph shows the histogram of the bootstrap estimate ϑˆ1 and the lower graph(m)shows that of ϑˆ2 .
Kajima(θ2)
Kinki Nippon Railway(θ2)
JAL(θ2)
Cosmo Oil(θ2)

Data
Estimating results
Class Observationsθ̂1θ̂2
mean(|residuals |)AAA80.0018 1.1348
0.001123
AA
523
0.0010 0.9297
0.002211
A
563
0.0140 1.4778
0.003447
BBB
381
0.0121 1.2030
0.449742
BB
45
0.0263 1.2351
0.936139
Table 9: Information about the data and the representative parameter estimates of 5 ratingclasses. The items ”Class” and ”Observations” are the rating class and the number ofcorporate bonds belonging to the class on April 15, 2004, respectively. ”θ̂1 ” and ”θ̂2 ” areparameter estimates and the rightmost item is the mean of the absolute values of residuals.
1

0.98

0.96
AAA

St

0.94

AA
0.92

A
BBB

0.9

BB

0.88

0.86
0

1

2

3

4

5

6

7

term to maturity

Figure 13: Estimated representative survival function of 5 rating classes.

4.7 Estimating results of 5 rating classes
In this subsection, we present estimating results for the data of 5 rating classes. As in theprevious subsection, we set α to be 31 and the recovery rate δ to be 0.3, and use statistical model of Weibull survival function under RM type recovery. We analyze 5 ratingclasses; AAA, AA, A, BBB, BB. Table 9 indicates the information about the data andthe representative parameter estimates of 5 rating classes. Figure 13 shows the estimatedrepresentative survival function of 5 rating classes. Figure 14 contains 5 separate graphsof the estimated representative survival functions of rating classes. ”Observed” plots indicate the survival probabilities calculated from the interest yields and the ”Model” lineis the estimated representative survival function of rating classes.
Table 10 indicates the bootstrap parameter estimates of 5 rating classes. Figure 15
shows the distribution of bootstrap parameter estimates of 5 rating classes. The upper(m)graph shows the histogram of the bootstrap estimate ϑˆ1
and the lower graph shows(m)that of ϑˆ2 .
As seen in the individual companies’ cases, there seems to be a close relation between
the distribution of bootstrap estimates and the number of data. It seems the larger amountof data(bonds) being used, the more like the bootstrap distribution is normal distribution.
Though many bonds (523 bonds) were issued by the ”AA” class’s companies, the bootstrap distribution of ”AA” class looks like to have some warp. (See Figure 15.) Thismay be explained by the existence of outlier data. When we look Figure 14 carefully, we
29

1

1

1

0.98

0.98

0.996

0.96

0.992

0.96

0.988

0.94

0.984

0.92

0.9

0.98

0.9

0.88

0.94
0.92

0

1

2

3

4

5

6

0

7

1

2

3

AAA

4

5

6

7

0

1

2

1

1

0.98

0.95

0.96

3

4

5

6

7

A

AA

0.9

0.94
0.85

observed

0.8

model

0.92
0.9

0.75

0.88
0.86

0.7
0

1

2

3

4

5

6

7

0

1

2

3

4

5

6

7

BB

BBB

Figure 14: 5 separate graphs of the estimated representative survival functions of ratingclasses. ”Observed” plots indicate the survival probabilities calculated from the interest
yields and the ”Model” line is the estimated representative survival function of ratingclasses.

(Boots)θ̃1
θ̂1
g θ̂1 )(Boots)Bias(g θ̂1 )(Boots)Var(f (t)(Boots)SDθ̂1(Boots)θ̃2θ̂2gBias(θ̂2 )(Boots)g θ̂2 )(Boots)Var(f (t)(Boots)
SD
θ̂2

AAA
0.002380.001800.000590.00000
0.00223AA
0.001200.00100
0.000210.00000
0.00091A
0.01406
0.014020.000030.00000
0.00102

BBB
0.012220.012110.000110.00001
0.00235BB
0.02775
0.026340.001410.000220.01480

1.158101.134810.023290.04835
0.219910.934200.92966
0.004540.01479
0.121641.47868
1.477820.000860.00133
0.03650

1.203641.20300
0.000640.00482
0.069411.238481.23508
0.003400.082220.28676

Table 10: The parameter estimate of the bootstrap estimation for 5 rating classes. The
notations are given in Appendix.

30

0.970
1.016
1.063
1.109
1.156
1.203
1.249
1.296
1.342
1.389
1.436
1.482
1.529

0.000
0.357
0.715
1.072
1.429
1.786
2.144
2.501
2.858
3.215
3.573
3.930
4.287

0.472
0.560
0.648
0.735
0.823
0.910
0.998
1.086
1.173
1.261
1.348
1.436
1.524

1.413
1.452
1.491
1.531
1.570
1.609
1.648
1.687
1.726
1.766
1.805
1.844
1.883

700
600
500
400
300
200
100
0

400
350
300
250
200
150
100
50
0

400
350
300
250
200
150
100
50
0

31
0.00000.00880.01750.02630.03500.04380.0525
0.06130.07000.07880.0875
0.09630.10500.00500.00660.00820.00980.01140.01300.01460.01620.01780.01940.02100.02260.0242

0.375
0.662
0.948
1.234
1.521
1.807
2.093
2.380
2.666
2.953
3.239
3.525
3.812

0.00000.00310.00610.00920.01220.01530.01830.02140.02440.02750.03050.0336
0.03660.03970.04270.00000.00070.00140.00210.00270.00340.00410.00480.00550.00620.00690.00750.00820.01050.01120.01190.01260.01330.01390.01460.01530.01600.01670.01740.0181

1400
1200
1000
800
600
400
200
0
700
600
500
400
300
200
100
0
400
350
300
250
200
150
100
50
0

AAA(θ1)AA (θ1)A (θ1)

500

400

300

200

100

0
350
300
250
200
150
100
50
0

AAA(θ2)AA (θ2)A (θ2)

1800
1600
1400
1200
1000
800
600
400
200
0

BBB (θ1)
800
700
600
500
400
300
200
100
0
BB (θ1)Figure 15: Distributions of bootstrap parameter estimates for 5 rating classes. Blue(deep)
distribution is for statistical model and pink(pale) is for calibration
BBB (θ2)
BB (θ2)

Data
Data
Observations
Original
523

Estimating results
Methodθ̂1θ̂2mean(|residuals |)Statistical model 0.00100 0.930
0.002210
Calibration
0.00241 1.106
0.002076
No Outlier
520
Statistical model 0.00455 1.293
0.001750
Calibration
0.00431 1.275
0.001749
Table 11: The representative parameter estimates for ”AA” original data and for ”AA”
no outlier data. Original data are the same as ”AA” in Table9. Outlier data are excluded
3 bonds’ price data from the original data.
Original data
No outlier data
Statistical model Calibration Statistical model Calibration(Boots)θ̃10.001200.00257
0.00458
0.00439θ̂10.00100
0.002410.00455
0.00431g θ̂1 )(Boots)
Bias(
0.000210.000160.000030.00008g θ̂1 )(Boots)
Var(
0.00000
0.00000
0.00000
0.00000(Boots)fSDθ̂1 (t)0.000910.001150.000730.00100
0.12476
0.06334
0.025540.02927
K-S statistic
K-S test
rejectrejectrejectreject(Boots)θ̃20.93420
1.10797
1.29289
1.27692θ̂20.92966
1.10592
1.29266
1.27502g θ̂2 )(Boots)
Bias(0.004540.00205
0.000230.00190(Boots)g
Var(θ̂2 )0.01479
0.01287
0.002530.00568f (t)(Boots)
0.12164
0.113450.05034
0.07534SDθ̂2
K-S statistic
0.048100.019280.01272
0.00910K-S test
rejectrejectnot reject
not reject
Table 12: The parameter estimate of the bootstrap estimation for ”AA” original data andfor ”AA” no outlier data. The results of K-S test at 0.05 level are given in the bottom 2lines.
notice that ”Observed” plots of ”AA” class contains outlier.
We next show the influence of outlier data on our estimating method using data of”AA” class. In the ”AA” graph of Figure 14, we may see 3 outlier plots among ”Observed” plots. We excluded these 3 bonds’ price data from the original data. Table 11
indicates the representative parameter estimates for ”AA” original data and for ”AA” nooutlier data. Original data are the same as ”AA” in Table9 and outlier data are excluded
3 bonds’ price data from the original data.
Table 12 indicates the bootstrap parameter estimates for ”AA” original data and for”AA” no outlier data. Figure 16 shows the distribution of bootstrap parameter estimatesfor ”AA” original data and for ”AA” no outlier data. The upper graph shows the his(m)(m)
togram of the bootstrap estimate ϑˆ1 and the lower graph shows that of ϑˆ2 .
We see from Figure 16 that the warps are reduced by the exclusion of the outlierdata(Compare the left graphs and the middle graphs). We also see that the distributions ofbootstrap parameter estimates using no outlier data look like normal distribution(middle
graphs). We tested these results for normality of the distribution. One of the most useful
tests for normality of the distribution is Kolmogorov-Smirnov test. Given M estimatesϑ̂ (m) i = 1, 2, m = 1, . . . , M, the Kolmogorov-Smirnov statistic for cumulative normal
32

400
350
300
250
200
150
100
50
0
0.00000.00070.00140.00210.00270.00340.00410.00480.00550.00620.00690.00750.00820.00250.00290.00340.00380.00430.00470.00520.00570.00610.00660.00700.00750.0079

500

350
300
250
200
150
100
50
0

200
100

0.472
0.560
0.648
0.735
0.823
0.910
0.998
1.086
1.173
1.261
1.348
1.436
1.524

0

AA, Original data (θ2)

400
300
200
100
0
1.120
1.151
1.182
1.213
1.244
1.275
1.306
1.336
1.367
1.398
1.429
1.460
1.491

300

1.120
1.151
1.182
1.213
1.244
1.275
1.306
1.336
1.367
1.398
1.429
1.460
1.491

400

AA, Original and no outlier data (θ1)

AA, No outlier data (θ1)

AA, Original data (θ1)
500

700
600
500
400
300
200
100
0
0.00250.00290.00340.00380.00430.00470.00520.00570.00610.00660.00700.00750.0079

700
600
500
400
300
200
100
0

AA, No outlier data (θ2)

AA, Original and no outlier data (θ2)

Figure 16: Distributions of bootstrap parameter estimates for ”AA” original data andno outlier data. The left graphs are for original data. Blue(deep) distribution is for statistical model and pink(pale) is for calibration. The middle graphs are for no outlier
data. 3 bonds’ data are excluded. Green(deep) distribution is for statistical model andorange(pale) is for calibration. The right graphs are the distributions using statisticalmodel for original data and no outlier data. Blue(deep) distribution is for original dataand green(pale) is for no outlier data.
distribution function F(x) is
DM = supx |Fm (x) − F(x)|where supx is the supremum of the set of distances andFM (x) =1 M∑ Iϑ̂ (m) <x ,M i=1

where Iϑ̂ (m) <x is the indicator function, equal to 1 if ϑ̂ (m) < x and equal to 0 otherwise.
The null hypothesis that ϑ̂ (m) i = 1, 2, m = 1, . . . , M is normal distributed is rejected at
level 0.05 if√
MDM > 1.36.
M is 10000 in our case.
The Kolmogorov-Smirnov statistic D10000 and the results of K-S test are in Table12.
We got better results by excluding only 3 bonds’ price data from the original 523 bonds’
data,

4.8 Estimation for Time Series Data
So far, we have estimated the survival probabilities of one day. In this subsection, we
estimate the time series survival probabilities. The data we used is daily price data of thecorporate bond and the government bond.
Since we are interested in the impact of the bankruptcy of Lehman Brothers(September
15, 2008), we use the data from January 4, 2008 to March 31, 2009. U.S. house sales
prices peaked in mid-2006 and began their steep decline forthwith. The subprime mortgage crisis began to affect the financial sector in 2007. Although Japanese financial firms
33held a few securities backed with subprime mortgages, Japanese economy was affectedby subprime mortgage crisis. We select five companies, which might be affected by
the bankruptcy. Toyota Motor Corporation is a multinational automaker and one of theworld’s largest automobile manufacturers. Toyota has a rating of AAA throughout ouranalysis. Mitsui & Co., LTD. also known as Mitsui Bussan, is one of the largest sogo
shosha in Japan. Mizuho Corporate Bank, Ltd. is the corporate and investment banking
subsidiary of Mizuho Financial Group. Tokyu Land Corporation is the 4th biggest realestate agency in Japan. Japan Airlines Co., Ltd. is an airline in Japan. The airline filed
for bankruptcy protection on January 19, 2010, after losses of nearly 100 billion yen in asingle quarter.
Figure 17 shows the historical survival probabilities of the 5 companies. The survival
probabilities of Tokyu Land Corporation began to decline slowly before the bankruptcy
of Lehman Brothers. Although the risk of subprime mortgages was pointed before thebankruptcy, the decline is considered by the cause of its own. In 2008, housing values
in Japan decreased by 10 percent with the previous year and the stock price of Tokyu
Land declined all through 2008. The survival probability curves in the graph begin todrop sharply at November 2009. The bond buyers were likely to avoid the risk in the realestate market.
We next contrast the graphs of Toyota, Mitsui and Mizuho. We see that the all 3 companies’ survival probabilities were declining after the bankruptcy, but Mizuho’s graphbegan to drop first. Since Mizuho Corporate Bank is the corporate and investment banking subsidiary and was at the core of the financial crisis, the bankruptcy was seem tomake an immediate effect. One month later of the Mizuho’s survival probability curve
drop, the survival probabilities of Toyota and Mitsui began to decline more slowly. It
seems the financial crisis spreaded over gradually.
JAL’s steep curve stands out in the graph. The 5 year survival probability varied 80%
to 64% for only 3 months. It is because JAL was confronted not only with financial crisis,
but also with the rise of oil price and slow cost cut. JAL’s prospects seem to have been
considered gloomy by the bond buyers.

5 Conclusion
In this paper, we study estimating method of implied survival probability. We focus on
the statistical estimating model introduced by Takahashi(2011). The advantage of the statistical estimating model is its ability to discuss asymptotic properties. Our contributions
are the following three points. First, we provide a complete proof of the consistency ofthe estimator in the statistical estimating model. Second, we verify regularity conditions
for asymptotic properties of the estimator in the Weibull survival function case. Third,
we provide an empirical analysis of implied survival probability estimating. When we
estimate implied survival probability, instead of using Duffie & Singleton(1999) model
directly, as in previous studies, we apply statistical estimating model which contains someerror term. In the empirical analysis, we discuss asymptotic properties using bootstrap
method.
Several interesting observations can be made. When we apply Weibull distribution toparametric survival function, the estimating result is much better than that of exponential
distribution. The number of the data and the existence of the outlier data have a major effect on the bootstrap estimating results. When we analyze the impact of the bankruptcy ofLehman Brothers, we observe that the influences differ with the 5 individual companies.

34

35
0.9

0.85
0.85

0.8
0.8

0.75
0.75

0.75
0.8

0.65

0.6
2008/11/10

2008/09/25

2009/03/26

0.9

2009/03/26

0.95

2009/02/10

0.95
2008/12/24

Tokyu Land Corp.

2009/02/10

1

2008/12/24

1

2008/11/10

0.75

2008/09/25

0.75
2008/08/12

0.8

2008/06/30

0.8

2008/08/12

0.85

2008/06/30

0.85

2008/05/19

0.9

2008/04/02

Toyota Motor Corp.

2008/05/19

0.9

2008/04/02

0.95

2008/02/19

2008/01/04

0.95

2008/02/19

2008/01/04

2009/03/26

2009/02/10

2008/12/24

2008/11/10

2008/09/25

2008/08/12

2008/06/30

2008/05/19

2008/04/02

2008/02/19

2008/01/04

1

2009/03/26

2009/03/26

2009/02/10

2008/12/24

2008/11/10

2008/09/25

2008/08/12

Mizuho Corporate Bank

2009/02/10

2008/12/24

2008/11/10

2008/09/25

2008/08/12

2008/06/30

2008/05/19

1

2008/04/02

2008/02/19

2008/01/04

2008/06/30

2008/05/19

2008/04/02

2008/02/19

2008/01/04

1

Mitsui & Co.

Japan Airlines Co.

0.95

0.85
0.9

0.5Y

0.7
1Y

3Y

5Y

Figure 17: Time series default probabilities of Toyota, Mitsui & Co, Mizuho, Tokyu
Land, JAL

Acknowledgements The author is grateful to Hajime Takahashi for his much valuable advice. The author is also grateful to Center for Financial Engineering Education(CFEE) of Hitotsubashi University for their support. Data for this study were provided
through CFEE.

AppendixA Bootstrap Method
When analyzing the real data, Takahashi(2011) recommends to use the bootstrap method
rather than asymptotic theory. We introduce the bootstrap method of Takahashi(2011),
which is used on the empirical research in section 4.

[Bootstrap Method of Takahashi(2011)]
We consider bootstrap estimate of ϑ̂ .
We first consider the case of statistical model under RM type recovery. We let {xti =1δ

x(ti ) i = 1, . . . , n} be a sequence of the pseudo data, where xt = x(t) = sρt t . We estimate

the base line estimate ϑ̂ by

"

#

n

ϑ̂ = argθ ∈Θ min ∑ {Ξ(ti , θ ) − xti }

2

i=1

where
Ξ(t, θ ) =G(t, θ )1−δ.
ρt

We estimate {σ εti , i = 1, . . . , n} byet∗i = Ξ(t, ϑ̂ ) − xti

i = 1, . . . , n.

We define the set E of standerdized residuals, from which the bootstrap sampling will bemade (cf.Wu(1986)):
E = {et1 , . . . , etn }
where

r
eti =

¢
n ¡ ∗
ē − e∗n − 1 ti

and
ē∗ =

1 n ∗
∑ eti .
n i=1

0th step Set
m=1
1st step We choose n elements at random from E with replacement, denote it E (m)
(m)

(m)

E (m) = {et1 , . . . , etn }2nd step Constract a bootstrap pseudo sample by
(m)

xti

(m)

= Ξ(ti , ϑ̂ ) − eti36

i = 1, . . . , n.

(m)3rd step Estimate θ from the pseudo data {xti , i = 1, . . . , n} and denote it as ϑ̂ (M)
#
"
o2
n n
(m)
ϑ̂ (m) = argθ ∈Θ min ∑ Ξ(ti , θ ) − xti.
i=1

4th step Repeat 1st step to 3rd step M times, and denote the estimate byϑ̂ (1) , . . . , ϑ̂ (M)We define bootstrap estimate byϑ̃ (Boots) =1 M (m)∑ ϑ̂ .
M m=1

We also define the estimate of bias and variance of ϑ̃ byBias(Boots) (ϑ̂ ) = ϑ̃ (Boots) − ϑ̂andVar(Boots) (ϑ̂ ) =

´³
´0
M ³
1
(m)
(Boot)(m)(Boot)
−
−
ϑ̂
ϑ̃
ϑ̂
ϑ̃
∑
M − 1 m=1

respectively.

We next consider bootstrap estimate when using statistical model under RT type recovery. The baseline estimate ϑ̂ is given by
#
"
o2
n n
]
]
ϑ̂RT = argθ ∈Θ min ∑ Ξ (ti , θ ) − xti
i=1

where
x] (t) =andΞ] (t, θ ) =(1 − δ )st.
ρt(1 − δ )G(t, ϑ̂RT ).
ρt

We estimate {σ εti , i = 1, . . . , n} byet]∗i = Ξ] (t, ϑ̂RT ) − xt]ii = 1, . . . , n.

Then we constract standerdized residuals E ] = {et]1 , . . . , et]n } and bootstrap estimate ϑ̃ (Boots)in the same way as that of RM type recovery.

References[1] Duffie, D. and K. Singleton (1999) ”Modeling Term Structures of Defaultable
Bonds”, Review of Financial Studies, 12, 678-720
37

[2] Ito, S. (1963) ”Introduction to the Lebesgue Integral”, Shokabo Publishing (inJapanese)[3] Jarrow, R. and S. Turnbull (1995) ”Pricing Options on Financial Securities Subject
to Default Risk”, Journal of Finance, 50, 53-86[4] Newey, W. K. and D. McFadden (1994) ”Large Sample Estimation and Hypothesis
Testing”, Handbook of Econometrics, 4, 2111- 2245, Elsevier Science B.V.
[5] Takahashi, H. (2011) ”On a Statistical Analysis of Implied Data ”, Asia Pacific Financial Markets, 18, 231-266[6] Wald, A. (1949) ”Note on the Consistency of the Maximum Likelihood Estimate”,
Annals of Mathematical Statistics, 20, 595-601[7] Wu, C. F. J. (1986) ”Jackknife, Bootstrap and Other Resampling Methods in Regression Analysis”, Annals of Statistics 14, 1261-1295

38