数理解析研究所講究録
第 1886 巻 2014 年 60-81

60

Identification Problem in the Joint Estimation of Default

Intensity and Recovery Rate using only Credit Default Swap
Data*

Dara Sim
Graduate School of Economics
Osaka University

Masamitsu Ohnishi
Graduate School of Economics & Center for the Study of Finance and Insurance
Osaka University

1

Introduction

In recent years, a number of studies has been devoted to looking for ways to estimate marketimplied recover rates. This quantity is important for risk management as well as for credit
derivative pricing. Unlike historical recovery rates which are backward-looking, market-implied
recovery rates subsume information about market’s expectation of future economic condition.
Although this quantity varies significantly over time and is negatively correlated with default
rates (See e.g. Altman et al., 2005, Acharya et al., 2007), most studies assume constant, andffx at 40% for senior unsecured bonds. The reason is because in most credit risk approach,default probability and recovery rate, that enter int the formula of credit derivatives, aremultiplicatively linked. This fact makes separate identification difficult. This identification
problem is known since Duffie & Singleton (1999).
Many techniques has been considered in order to be able to separately estimate implieddefault probabilities and recovery rates from credit spreads (See Schl\"afer, 2011). One technique
is to specify a link between implied default and recovery rates. For example, Bakshi et al. (2006)
specify recovery rates as a function of default intensity. One other technique is to use credit
default swap term structure information. Pan& Singleton (2008) use term structure information
on sovereign CDS to estimate constant implied recovery rates of sovereign bonds. They showthat default risk and recovery rate can be identified under the assumption of recovery of facevalue. Christensen (2007) use CDS data of Ford Moto Corp., and concludes that default andrecovery risk can be jointly identified from CDS term structure data. He assumes both constantand stochastic recovery. Schneider et al. (2011) assume constant implied recovery rates and use
CDS on senior unsecured bonds of 278 U.S. corporates. Doshi (2011) use senior and subordinate
CDS of 46 firms to estimate jointly default intensity and stochastic recovery rates.
In this paper, we attempt to use CDS term structure information to decompose marketimplied default intensity and recovery rate. We want to investigate if this is really possible. In
order to do so, we first specify ajoint model of interest rates, default intensity, and recovery rates,
which incorporates negative correlation between default intensity and recovery rates while trying
to preserve tractability. We consider both constant and stochastic recovery models. For interest
rate term structure model, we consider the modified arbitrage-free Nelson-Siegel model proposedby Sim & Ohnishi (2012). In the stochastic recovery case, logistic model is assumed for loss
given default (LGD) (so that the recovery rate value is ensured between and 1), and negative
correlation between default intensity and recovery rate is captured via an interest rate factor.
Using generalized transforms of affine processes, CDS pricing formula can be explicitly obtainedeven under the logistic LGD assumption. Then, we investigate whether it is possible to jointly
estimate default intensity and recovery rate through an empirical estimation and a simulation
$0$

$0$

’This work was supported by JSPS KAKENHI$Gr_{\epsilon}^{r}t$

Number 25-6058.

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study. For interest rate data, we use JPY Libor and swap rates. For CDS data, we consider two
sets of data: Sony’s senior CDS and Mitsubishi UFJ’s subordinate and senior CDS data. Withtwo sets of data, and two types of recovery model, we have four cases: constant recovery modelwith one seniority CDS, constant recovery model with two seniorities CDS, stochastic recovery
model with one seniority CDS, and stochastic recovery model with two seniorities CDS. First,
interest rate model is estimated using the extended Kalman filter. Then, given the estimatesof interest rate parameters and state variables, default intensity and recovery rate are jointly
estimated using the square-root unscented Kalman filter.
From the results of the empirical estimations, we find that in all four cases when jointly
estimating default intensity and loss given default together, we tend to largely overestimate
implied default probabilities, and largely underestimate (or overestimate) implied loss given
default (recovery rates), the result which does not seem plausible at all. However, in the constantrecovery case, when fixing the loss given default, we are able to obtain much more plausible result.
We conclude that separate identification of default intensity and recovery rates is not possible.
This is verified by a simple simulation study on the four cases. From the simulation results, wefind that for the constant recovery model, when jointly estimated, the estimates of the default
intensity’s parameters distribute far away from the true values to the right, and the estimates ofthe loss given default’s parameters distribute far away from the true values to the left. However,when one of the unknown components to be estimated is given, the parameter estimates of theother components distribute very close to the true. This suggests that the default intensity
factor and the loss given default parameter cannot be separately identified. In addition, in thestochastic recovery model with one seniority included, even if we fix the loss given default’s
parameters, the estimates of default intensity’s parameters are far from the true values. This isan obvious evidence that the paths of default risk factor and that of the recovery risk factor are
not jointly identified by the CDS data.
The organization of the paper is as follow. Joint model specification is described in section
2. Specifically, the model for term structure of interest rates, the model for default intensity,
as well as the model for loss given default, are presented. Section 3 describes data, estimation
process, and estimation results. The simulation study is conducted in section 4. Finally, section
5 draws a conclusion.

2

Joint Model Specifications

2.1

Model Setup

In this paper, we follow the reduced-form approach. In reduced-form approach, default is notbased on the economic notion of bankruptcy. Instead it is treated as an event governed by
an exogenously specified counting process. Lando (1998) and Duffie & Singleton (1999) use acox process (also called doubly stochastic Poisson process), i.e., Poisson process with stochastic
intensity, in the modelling of default event. Duffie & Singleton (1999) considers “Recoveryof Market Value” just before default time. Though this assumption results in mathematically
elegant models, it does not allow $DI$ and LGD to be separately identified (See, Duffie & Singleton(1999) and Houweling & Vorst (2005) . We assume “Recovery of Par Value” paid at the timeof default, as is the case in CDS markets,
We consider a complete filtered probability space, and a risk-neutral probability measure . The information filtration
is assumed satisfying the usual condition.
Letbe a c\‘adl\‘ag-adapted process,be -algebra generated by
the process X, $h=h(X)$ :be non-negative and integrable on any finite time interval.
Also, let $N$ be a cox process with stochastic intensity , that is, conditional on the realization$)$$(\Omega, \mathcal{F}, (\mathcal{F}_{t})_{t\geq 0}, \mathbb{P})$$\mathbb{Q}$$X\in \mathbb{R}^{d}$$(\mathcal{F}_{t})_{t\geq 0}$$(\mathcal{F}_{t})$$\mathcal{G}_{t}\equiv\sigma(X_{s} :$\mathbb{R}^{d}arrow \mathbb{R}$$h$0\leq s\leq t)$$\sigma$62of the path of

$h,$

$N$

becomes an inhomogenous Poisson process,(1)$\mathbb{Q}(N(T)-N(t)=n|\mathcal{G}_{t})=\frac{1}{n!}(l^{\tau_{h_{s}ds)^{n}e^{-\int_{t}^{T}h_{s}ds}}}\cdot$

The default occurs the first time the process $N$ is no longer zero. We denote the default time by, i.e., $T_{d}= \inf\{t\in \mathbb{R}_{0}^{+}|N(t)>0\}$ , and the -algebra generate by the default indicator, which contains the information whether default occurs, i.e.,
byandby time or not. Also,
can be thought of as the smallest a-algebra containing both
, i.e.,. Thus,and , but not . This means thatis a stopping time w.r.
knowing the sample path of X does not tell when the default happens.
For $t\leq u$ , we have$T_{d}$$1_{\{T_{d}\leq s\}}$$\sigma$$\mathcal{H}_{t}=\sigma(1_{\{T_{d}\leq s\}} :$\mathcal{H}_{t}$$t$$\mathcal{G}_{t}$0\leq s\leq t)$$\mathcal{H}_{t}$$\mathcal{F}_{t}$$T_{d}$$\mathcal{F}_{t}=\mathcal{G}_{t}\vee \mathcal{H}_{t}$$t\mathcal{H}_{t}$$\mathcal{G}_{t}$$\mathcal{F}_{t}$$\mathbb{Q}(t<T_{d}\leq u|\mathcal{G}_{\infty}\vee \mathcal{H}_{t}) =1_{\{T_{d}>t\}}(1-e^{-\int_{t}^{u}h_{s}ds})$-conditional distribution ofentiate w.r. $tu$ yields its density function

which is the$\mathcal{G}_{\infty}\vee \mathcal{H}_{t}$1

2.2

given $\{T_{d}>t\}$ (See Appendix A.1). Differ-$\{T_{d}>t\}hue^{-\int_{t}^{u}h_{\theta}ds}$:
Finally, denote byof the latent state variable factors $X.$$\varphi\equiv\varphi(X)$$T_{d}$$\mathbb{R}^{d}arrow \mathbb{R}$

,

.

(2)

the loss given default (LGD) assumed to be a function

Interest Rates Model

We use the Modified Arbitrage-Free Nelson-Siegel (MAFNS) of Sim & Ohnishi (2012) because
it has go $od$ empirical fit and its factors have Nelson-Siegel interpretations, namely level, slope,and curvature. This model is a modified version of the Arbitrage-Free Nelson-Siegel (AFNS)
proposed by Christensen et al. (2011). Sim & Ohnishi (2012) show that MAFNS has betterfits to the yield curves than AFNS. MAFNS belongs to a 3-factor affine class of dynamic term
structure models of Duffie & Kan (1996) with the following specification.
The short rate is given by$r_{t}$$r_{t}=X_{1,t}+X_{2,t}$(3),

where the dynamics of the three interest rate factors $X_{r}=(X_{1_{\rangle}}X_{2}, X_{3})’$ under risk-neutral
measure is$\mathbb{Q}$$d(\begin{array}{l}X_{1,t}X_{2_{\rangle}t}X_{3,t}\end{array})=(\begin{array}{lll}k_{1}^{Q} 0 00 \lambda -\lambda 0 0 \lambda\end{array})[(\theta_{3}^{Q}\theta_{2}^{Q}\theta_{1}^{Q})-(\begin{array}{l}X_{1,t}X_{2,t}X_{3,t}\end{array})]dt+(\sigma_{1}\sqrt{X_{1,t}}00$

and the dynamics of$X_{r}$

under the physical probability measure$\mathbb{P}$$W_{1},$ $W_{2},$ $W_{3}$$\sigma 00_{3})d(W_{3,t}^{Q^{t}}W_{2}^{Q^{t}}W_{1}^{Q})\cdot(4)$can be specified as follow,$d(\begin{array}{l}X_{1,t}X_{2,t}X_{3,t}\end{array})=(\begin{array}{lll}k_{1}^{P} 0 00 k_{2}^{P} 00 0 k_{3}^{P}\end{array})[(\theta_{2}^{P}\theta_{1}^{P}\theta_{3}^{P})-(\begin{array}{l}X_{1,t}X_{2,t}X_{3,t}\end{array})]dt+(00$where$\sigma_{2}00$$\sigma_{2}00$$\sigma 0_{3}0)d(W_{2}^{1}W_{3,t}W_{1_{i}t}^{P}\neq^{t}),$are independent Brownian motions.

Under this affine specification, the price at time
explicitly obtained as follow.

$t$of a zero-coupon bond with maturity$P(X_{r,t}, \tau)=\exp(-y(X_{r,t}, \tau)\tau)$,$\tau$

is

(6)

$(5)$

63
where $y(X_{r,t}, \tau)$ is the zero-coupon yield at time of maturity , the explicit formula of which
is given in the Appendix A.2.
Also, since in this paper Japanese Yen (JPY) Libor and swap rates are used for risk-freeinterest rates, expressions for Libor and swap rates are needed. Having obtained the formula forzero-coupon bond prices, we can straightforwardly do the calculation as these can be expressedin terms of prices of zero-coupon bonds with different maturities. Let $m$ denote the number ofpayments per year for swap rates, then LIBOR and swap rates at time with maturity canbe expressed respectively as follow:$t$$\tau$$t$

LIBOR$(X_{r,t}, \tau)$$=$$SWAP (X_{r,t}, \tau) = 100m\frac{1-P(X_{r,t},\tau)}{\sum_{x’=1}^{m\tau}P(X_{r,t},\frac{i}{m})}$2.3,$\frac{100}{\tau}(P(X_{r,t}, \tau)^{-1}-1)$.

$\tau$(7)(8)

Credit Risk Model

To evaluate credit default swap or corporate bond in the reduced-form framework, we need to
specify models for $DI$ and REC (or LGD). In this section we specify these models for the case of
constant recovery and the case of stochastic recovery. Many studies assume constant recovery(and fix it at 40% for senior unsecured bond); however, many empirical evidences find that
REC is time-varying and negatively correlated with $DI$ . As already mentioned, the purpose of
our study is to verify if it is possible to separately identify $DI$ and REC’s parameters (and risk
factors) from CDS data of different maturities. Even in the case that REC or LGD’s parameterand $DI$ cannot be separately identified, it could be that it is possible under the stochastic recovery
case when correlation between the two is modelled, and even in the case the parameters can be
jointly estimated, it could be that the factors cannot be separately identified. Therefore, it isparamount to consider both cases.

Constant Recovery ModelIn this case, we assume that $DI$ is governed by only one risk factor which follows a CIR process,i.e., $h(X)=X_{4}$ , where the dynamics of the factor
under both risk-neutral and physicalprobability measures are assumed as follows:$X_{4}$$dX_{4,t} = k_{4}^{\mathbb{Q}}(\theta_{4}^{\mathbb{Q}}-X_{4,t})dt+\sigma_{4}dW_{4,t}^{\mathbb{Q}},$$dX_{4,t} = k_{4}^{\mathbb{P}}(\theta_{4}^{\mathbb{P}}-X_{4,t})dt+\sigma_{4}dW_{4,t}^{P}.$We assume essentially risk premium specification, i.e.,, in which case wehave
is a Brownian motion independent of
Needless to say, the loss given default LGD is assumed constant and takes on different values
for different seniorities.
Therefore, in this case we have 4 factors $X=(X_{1}, X_{2}, X_{3}, X_{4})’$ directly entered into ourcredit default swap model, three of which are those of the interest rates $X_{r}=(X_{1}, X_{2}, X_{3})’,$and one is the default risk factor$dW_{4}^{\mathbb{Q}}=dW_{4}^{\mathbb{P}}+\gamma dt$$k_{4}^{\mathbb{Q}}\theta_{4}^{\mathbb{Q}}=k_{4}^{P}\theta_{4}^{\mathbb{P}}.$$W_{4}$$W_{1},$$W_{2},$ $W_{3}.$$X_{4}.$

Stochastic Recovery ModelIn this case, we assume that $DI$ is governed by two factors: the interest rate factor corresponding
to the level factor, and another factor which can be considered as a default-risk-specific factor.
That is, $DI$ has the following form:$h(X)=\rho X_{1}+X_{4}$

,

(9)

64
where the dynamics of factor
assumed as follows:$X_{4}$

under both risk-neutral and physical probability measures are$dX_{4,t} = k_{4}^{\mathbb{Q}}(\theta_{4}^{\mathbb{Q}}-X_{4,t})dt+\sigma_{4}dW_{4,t}^{\mathbb{Q}},$$dX_{4,t} = k_{4}^{1P}(\theta_{4}^{1P}-X_{4,t})dt+\sigma_{4}dW_{4,t}^{1P}.$, in which case
Again, we assume essentially risk premium specification, i.e.,
is a constantis a Brownian motion independent of
we haveand is assumed to take value in the interval [-1, 1]. It should be noted that it is well-known that
$DI$ is negatively correlated with instantaneous short rate, but we do not assume thattakes
on only negative value. In practice, when we estimate our model, we consider two cases. First,
we restrict that $\rho\in[-1,0]$ , and then $\rho\in[0,1]$ . We choose the case that has better fit to thedata. Also note that when takes on negative value the $DI$ process will no longer be guaranteed
non-negative, which will violate the mathematical requirement that it is a non-negative process.
In that case we neglect this requirement. However, when takes on non-negative value, the $DI$
process is guaranteed non-negative because it is a linear function of two CIR processes. This is
as well another reason we prefer MAFNS to AFNS.
For LGD model, we assume Logistic function which can be considered one of the mostappropriate assumption and which ensures that the value of REC rate is guaranteed between
and 1. That is take the following form.
$dW_{4}^{\mathbb{Q}}=dW_{4}^{P}+\gamma dt$$k_{4}^{\mathbb{Q}}\theta_{4}^{\mathbb{Q}}=k_{4}^{\mathbb{P}}\theta_{4}^{P}.$$W_{1},$ $W_{2},$ $W_{3}.$$W_{4}$$\rho$$\rho$$\rho$$\rho$$0$$\varphi$$\varphi(X) = \frac{1}{1+\exp(-b_{0}-b_{1}X_{1,t}-b_{2}X_{2,t}-b_{3}X_{3,t}-b_{5}X_{5,t})}$

,

(10)

can be thought of as the recovery-risk-specific factor. Its dynamics under both riskwhere
neutral and physical probability measures are assumed as follows:$X_{5}$$dX_{4,t} = \sigma_{5}dW_{5,t}^{\mathbb{Q}},$$dX_{4,t} = k_{5}^{\mathbb{P}}(\theta_{5}^{\mathbb{P}}-X_{5,t})dt+\sigma_{5}dW_{5,t}^{P},$, and. Note that by
is a Brownian motion assumed independent ofwhere
specifying its dynamics under both measures, we implicitly specify the risk premium. The
are assumed different for different seniorities. As will be mentioned
parameters
bellow, in this paper, two cases are considered: the case when only one seniority are included,
to 1
and the case when two seniorities are included. Without losing generality, we restrict
when only one seniority CDS data are included, and restrict one of the two ’s to 1 when two
seniorities CDS data are used in the estimation. In addition, to capture the negative correlation
to have the same $sign$ as
between $DI$ and REC, we impose restriction on the parameterto be positive. This is equivalentMoreover, we further impose restriction on the parameteris corresponding to theto assume thatand LGD are positively correlated. Recall thatnegative slope of the yield curve, i.e., the difference between short- and long-term yields. Whenthe short-term rate exceeds the long-term rate (inverted yield curve), we may expect future
economic down-turn, in which case REC rate may be lower than that in the situation of normal
yield curves if the default event occurs.
To sum up, in the stochastic recovery model we have 5 factors $X=(X_{1}, X_{2}, X_{3}, X_{4}, X_{5})’$
directly entered into our credit default swap model –three of which are those of the interest, and the other one is the recoveryrates $X_{r}=(X_{1}, X_{2}, X_{3})’$ , one is the default risk factor
risk-specific factor$W_{1},$ $W_{2},$ $W_{3}$$W_{5}$$b_{0},$$b_{1},$$b_{2},$$b_{3},$$W_{4}$$b_{5}$$b_{5}$$b_{5}$$b_{1}$$\rho.$$b_{2}$$X_{2}$$X_{2}$$X_{4}$$X_{5}.$

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2.4

Credit Default Swap Pricing Formula

A CDS is a single-name credit derivative contract that involves 2 parties: the protection buyer
and the protection seller. It provides protection against default of a 3rd party called the reference
entity. A CDS contract consists of a time-t -maturity , premium payment dates$t_{n}=$, and the CDS premium $S(t, T)$ that need to be determined at time $t=t_{0}$ as the contract ismade. Thus, $S(t, T)$ is -measurable,
Let$i=1,2,$be the time between premium payments, then the protection
buyer needs to pay to the protection seller a premium amount equal toat time $i=$
1, 2, , if default does not yet occur and the accrued premium $(T_{d}-t_{i})S(t, T)$ if default occurs
between the interval. On the other hand, the protection seller is obligated to pay to theprotection buyer at time of default the amount equal to the LGD at default time
. Thus,under the arbitrage-free assumption, the CDS spread at time with maturity is given by thefollowing formula (See Appendix A.3).
$0$

$T$

$t_{1},$$t_{2},$$\ldots,$$T$$\mathcal{F}_{t}$$\delta_{i}=t_{i}-t_{i-1},$$\ldots,$$n$$\delta_{i}S(t, T)$$\cdots$$t_{i},$$n$$[t_{i-1}, t_{i}]$$\varphi\tau_{d}$$t$$\tau$$S(t, t+ \tau)=\frac{\int_{t}^{t+\tau}\mathbb{E}^{\mathbb{Q}}[e^{-\int_{t}^{9}(r_{u}+h_{u})du}\varphi_{s}h_{s}1\mathcal{G}_{t}]ds}{\sum_{i=1}^{n}\delta_{i}\mathbb{E}^{\mathbb{Q}[e^{-\int_{t}^{t_{i}}(r_{u}+h_{u})du|\mathcal{G}_{t}]+\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}(s-t_{i-1})\mathbb{E}^{\mathbb{Q}}[h_{s}e^{-\int_{t}^{s}(r_{u}+h_{u})du1\mathcal{G}_{t}]d_{S}}}}}.(11)$In the constant recovery case, we only need to evaluate$\mathbb{E}^{\mathbb{Q}}[h_{S}\exp\{-\int_{0}^{s}(r_{u}+h_{u})du\}]$, and$\mathbb{E}^{\mathbb{Q}}[\exp\{-\int_{0}^{S}(r_{u}+h_{u})du\}].$

These can be evaluated directly or as transforms of affine processes applying the results of Duffie
et al. (2000).
In the stochastic recovery case, in addition to the above expectations, we also need to obtain
the expression for$\mathbb{E}^{\mathbb{Q}}[h_{s}\varphi_{S}\exp\{-\int_{0}^{s}(r_{u}+h_{u})du\}]$.

Following the demonstration of Chen & Joslin (2012), first note that
Appendix A.5)

(12)$\varphi$can be written (See$\varphi_{t}=\frac{1}{4}\int_{-\infty}^{\infty}\frac{e^{(iv-1)Y_{t}}}{Cosh(\frac{\pi v}{2})}dv,$

where $Y_{t}=- \frac{1}{2}\{b_{0}+b_{1}X_{1,t}+b_{2}X_{2,t}+b_{3}X_{3,t}+b_{5}X_{5,t}\}$ . Then substituting this into (12), wehave$\mathbb{E}^{\mathbb{Q}}[h_{s}\varphi_{S}\exp\{(r_{u}+h_{u})du\}]$$=$$Re \{\frac{1}{2}\int_{0}^{\infty}\frac{1}{Cosh(\frac{\pi v}{2})}\mathbb{E}^{\mathbb{Q}}[h_{s}\exp\{-\int_{0}^{s}(r_{u}+h_{u})du+(iv-1)Y_{S}\}]\},$

where $Re\{.\}$ denotes the real part of a complex number.
Once again, the explicit expression for $\mathbb{E}^{\mathbb{Q}}[h_{s}\exp\{-\int_{0}^{s}(r_{u}+h_{u})du+(iv-1)Y_{s}\}]$ can be obtained as a transform of affine processes applying the results of Duffie et al. (2000). Thus, even in
the case of stochastic recovery model we still obtain semi-analytical expression for CDS spreads.

3

Empirical Estimation

In this section, we estimate our joint models using CDS data of different maturities. We consider
four different cases: a constant recovery model with only senior CDS data included, a constant

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recovery model with two seniorities of CDS data included, a stochastic recovery model with only
senior CDS data included, and finally a stochastic recovery model with two seniorities of CDS
data included. First, we describe the data used in the estimation, and then the estimation procedure and methods, as well as the estimation results. The purpose of this empirical estimation
is to see if $DI$ and REC can be separately identified. To do so, we compute implied default
probabilities and implied recovery rates, and examine if these values are plausible by comparing
our result with that of Schl\"afer (2011). Schl\"afer (2011) introduces a new approach for estimating market-implied recovery rates. The approach uses information of CDS on different-seniority
debt instruments (Senior unsecured bonds, subordinated bonds, and secured loans) as well as
firm’s capital structure.
3.1

Data

For risk-free interest rates data, we use JPY Libor rates and JPY swap rates. Included in theestimation of our term structure model are the Libor rates with maturity of 3 months, 6 months,
and 12 months, as well as the swap rates with maturity of 2, 3, 4, 5, 7, and 10 years. For CDS
data, we consider data of two Japanese firms: Sony Corp. and Mitsubishi UFJ Financial Group.
For Sony, we have senior CDS data with maturity of 1,3,5,7, and 10 years. For MUFG, we haveboth senior and subordinate CDS data with maturity of 1,3,5,7, and 10 years. The sample usedconsists of 100 weekly observations for the period from Jan. 2006 to Nov. 2007. This period
is just before the period of financial market turbulence, which started from the end of 2007 asconsequence of sub-prime loan crisis. For our purpose, we restrict to study only the period of
relatively stable market. Small sample of nearly 2 years of weekly data is considered for easeof estimation and in order to avoid impact of structural change. The data is acquired from
Thomson Reuters Datastream. It should be noted for the sample period considered, the Mood’s
rating of Sony was A2, and of MUFG was Al and was later upgraded to Aa2.

3.2

Estimation Procedure and Methods

Estimation procedure is divided into 2 steps. In the first step, we estimate the MAFNS modelusing JPY Libor rates and swap rates. The maximum likelihood estimation method based on
the Extended Kalman Filter (EKF) is used. We will need expressions for first derivatives ofLibor and Swap rates w.r. . interest rate factors. These are given in Appendix A.6. In step2,
given the parameter estimates and estimated paths of interest rate factors, we jointly estimate$DI$ and REC rates using CDS data.
The maximum likelihood estimation method based on
the Square-Root Unscented Kalman Filter ( $SR$-UKF) is used. The Unscented Kalman Filter(UKF) proposed by Julier & Uhlmann (1997) is derivative-free and outperforms EKF in terms
of prediction and estimation errors. The $SR$-UKF proposed by Van Der Merwe & Wan (2001)
is better than UKF in terms of numerical stability, and guarantees positive semi-definiteness ofthe state covariance. Note that since models to be estimated can be formulated as non-linear
dynamic systems with additive noises, state variables are not augmented,
$t$

3.3

Estimation Results

3.3.1

Results for the Term Structure ModelFirst, note that heteroskedasticity is assumed for the measurement errors, and the estimates of
their standard deviations are reported in Table 2.
Table 1 reports parameter estimates and standard errors for the term structure model. In
order that each factor be identified as meaningful factor, the parameter , which determines the$\lambda$

speed of decay of loadings on factor 2 and 3, is fixed at 0.57. Observing the speeds of mean

67
reversion rates, we can see that all the three factors are highly persistent, and the most persistentis the first factor followed by the second and the third. As expected, these factors have NelsonSiegel interpretations. The first factor is closely related to long term yields (level), whereas thesecond and third factors are respectively closely related to negative slope, and curvature of theyield curve. This is confirmed in Table 3, which reports correlations between the first factor
and 10-year yields (0.86 for their levels, and 0.91 for their first differences), between the second
factor and the difference between 3-month and 10-year yields (0.99 for their levels and 0.96 fortheir first differences), between the third factor and empirical curvature (two times of 2-yearyields less 3-month yields and less 10-year yield)( .98 for both levels and first differences).
To see the fitting performance of the $mo$ del, we refer to Table 2 which reports Root Mean
Square Errors (RMSE), Mean Absolute Percentage Errors (MAPE), as well as Measurement
Errors Standard Deviation ( $ME$ $SD$ ). RMSE and $ME$ $SD$ are reported in basis points, whereas
MAPE is reported in percents. Except for 3-month Libor rates for which our model is slightly
difficult to fit (with MPAE of 12.7%), our model is able to achieve almost perfect fit for mostLibor and swap rates of different maturities, with MAPE for 6-month Libor rate of 0.00%,
and MPAE for 3, 4, 5, 7, and 10-year swap rates of 0.38%, 0.02%, 0.14%, 0.12%, and 0.66%
respectively. Having difficulty fitting short-term rates is not restricted to our model; most term
structure models do. However, the overall performance is very good. Not surprisingly, this is
one of the most important feature of the Nelson-Siegel type models.
$O$

-

3.3.2

-

Results for Credit Default Swap model

Using the parameter estimates and estimated paths of risk factors, we are able to calculate thej-year-horizon implied default probabilities as follow:$\mathbb{Q}(t<T_{d}\leq t+j|\mathcal{F}_{t})=1-\mathbb{E}^{\mathbb{Q}}[\exp\{-l^{t+j_{h_{s}ds}}\}|\mathcal{F}_{t}].$Similarly, for the stochastic recovery, we calculate the j-year-horizon implied recovery rates as
expected recovery rates if the firm were to default years later from now. That is,$j$$REC(t,j)=1-\mathbb{E}^{\mathbb{Q}}[LGD_{t+j}|\mathcal{F}_{t}].$Note that the measurement errors are assumed homoskedastic for CDS data of the sameseniority through out the paper, so we have one measurement errors’ standard deviation parameter when one seniority data is included, and two standard deviation parameters when two
seniorities are included, to be estimated. This assumption is especially important for the below
simulation study which is computationally intensive.
As already mentioned above, we have the estimation results for 4 different cases. The resultsfor Casel-Constant recovery & Sony’s senior CDS data-are reported in Table 4, 5, and 6.
First of all, the fitting performance of the model can be seen from Table 5, which reports
RMSE and MPAE for the CDS spreads of the five maturities. Although the model has difficulty
fitting 1-year CDS spreads, it fits quite well for the 3-year, 5-year, 7-year, and 10-year CDS
spreads with MPAE of 7.32%, 5.79%, 9.79%, and 10.78% respectively. Table 4 reports theparameter estimates and standard errors. The estimated parameter of LGD is 0.0318, which isthen converted to an implied recovery rate of over 96%. This value is so much larger than theusually-assumed recovery rate for senior CDS, which is 40%, and is of course much larger than
the implied recovery rates estimated by Schl\"afer (2011) which range from 10.00% to 42.90% withan average of 24.00% for senior unsecured bonds. However, with this result alone, it is too early
to draw any conclusion that it is incorrect. Next, we examine the implied probabilities of default
which is reported in Table 6. The values reported are average values across times. The average

68values of 1 to 5-year-horizon implied $PD$ are respectively 1.93%, 5.21%, 10.36%, 17.81%, and27.62%. These values are too high to be plausible. According to Moody’sInvestorService (2012),’
firm are 0.061% and 0.598%
the average one and five-year historical $PD$ for a Moody’s-rated
respectively. Even with the risk-premium incorporated, we may not expect the implied $PD$ tobe so much higher than the historical $PD$ as this result shows. This result suggests that we may
have largely overestimated implied $PD$ and underestimated (or overestimated) implied LGD (orrecovery rates). This may be the consequence of inability of the CDS data to separately identify$DI$ and LGD (which is in fact the case as will be seen from the simulation results below). To
see what happens if we do not estimate LGD along with $DI$ , we fix it at 0.6 (i.e., assume 40%
constant recovery rate) and re-estimate the model. From Table 16, we see that in such a casethe average values of implied $PD$ of 1 to 5-year horizons are quite plausible.
The results for Case -Constant recovery & MUFG’s senior and subordinate CDS data-arereported in Table 7, 8, and 9, Table 8 reports RMSE in basis points and MPAE in percents forthe 2 seniorities and 5 maturities CDS spreads. The constant recovery model seems to have poor
fitting performance for the subordinate CDS spreads with an average MPAE of 22.93%. As we
can also see in Table 7 which shows parameter estimates and standard errors, the measurement
errors’ standard deviations are estimated to be 7.09 basis points for subordinate CDS spreads,
and 1.13 basis points for senior CDS spreads. The estimate of LGD is 0.0510 for the senior, and0.0955 for the subordinate CDS. These can be then converted to implied recovery rates of 94.90%
for the senior, and 90.45% for the subordinate CDS. These values are far from plausible sinceaccording to Moody’sInvestorService (2012), the average historical recovery rates are reported
to be from around 40% to around 60% for senior bonds, and around 20% for subordinate bonds.
This result implies a large negative recovery risk premium. In other words, the result suggeststhat investors prefer to take on a high recovery risk, which is implausible. Schl\"afer $(2011)$ ’s
estimates of implied recovery rates for subordinated bonds range from 1.40% to 11.20%, whichsuggests that investors demand positive recovery risk premium. Table 9 shows average values
across time of implied $PD$ for 1 to 5-year horizons, which are respectively 0.72%, 2.04%, 4.21%,
7.48%, and 11.97%. Once again, according to Moody’sInvestorService (2012), the average one
and five-year historical $PD$ for a Moody’s-rated ‘Aa’ firm are respectively 0.02%, and 0.22%.
Thus, the values of the implied $PD$ does not seem reasonable because since the implied $PD$
are probabilities under risk-neutral measure, this will imply a too high default risk premium
demanded by investors. Therefore, it appears that in the case when two seniorities are included,
the joint estimation of the three unknown components ( $DI$ , senior LGD, and subordinate LGD)
may result in an overestimation of $PD$ and recovery rates. Similar to Casel above, we consider
what happens when one of the LGD parameter is fixed. The implied $PD$ and implied recovery
rates of senior CDS after fixing subordinate LGD parameter at 1 (ie., assume zero recovery rate
for the subordinate contracts) are given in Table 17. The implied recovery rate for senior CDS
is 47%, and the implied $PD$ is 0,07% for the 1-year horizon, and 1.20% for the 5-year horizon.
This result looks much more reasonable than that of the joint estimation case, suggesting that
$DI$ and senior LGD may have been probably separately identified. In fact, this is the case as we
will see in the simulation result below.
The results for Case -Stochastic recovery & Sony’s senior CDS data-are reported in Table
10, 11, and 12. From Table 11, we can see that with the stochastic recovery model, there isquite an improvement in the fitting performance with average MAPE of 7.95% versus 11.53%of the constant recovery model. Nevertheless, even the stochastic recovery model has difficulty
fitting the 1-year CDS spreads. Just as in casel, the average values across time of the implied$PD$ and recovery rates, reported in Table 12, are too high to be plausible. Even in this case,when jointly estimate $DI$ and LGD together, we tend to largely overestimate implied $PD$ and
$A$

$2$

$3$

recovery rates.

69
Finally, the results for $Case4-$ Stochastic recovery & MUFG’s senior and subordinate CDS
data-are reported in Table 13, 14, and 15. There is significant improvement in fitting performance especially for subordinate CDS spreads. However, just as in Case2, the average values of
the implied $PD$ and implied recovery rates for both seniorities are far from plausible.

4

Simulation Study

From the estimation results above, we see that when jointly estimate $DI$ and REC, the implied
probabilities of default tend to be largely overestimated, whereas the implied LGD (or REC)
tend to be largely underestimated (or overestimated). Such failure to estimate plausible valuesof implied $PD$ and LGD (or REC) may be attributed to the fact that the information contained
within CDS data of different maturities alone is not sufficient for separately identifying $DI$ ’s
factor and LGD’s parameter in the constant recovery case, and $DI$ ’s parameters and factor and
LGD’s parameters and factor in the stochastic recovery case. In this section, simulation studies

are conducted for the four different cases in order to verify that this is really the case. We firstdescribe the methodology and then examine the simulation results.

4.1

Methodology

The simulation procedures for the four different cases are almost identical. They slightly differ
only in how to determine true parameters to be chosen. Below we describe how we determine
true parameters for the four different cases considered so far.

.

.
.
.

Casel: Constant recovery & only one seniority
Fixing LGD parameter at 0,6, we estimate the constant recovery model using Sony’s
senior CDS data. We consider the acquired parameter estimates and the fixed LGD as
true parameters, and consider the fitted values of CDS spreads as theoretical CDS spreads.
Case2; Constant recovery
two seniorities
Fixing the subordinate LGD parameter at 1, we estimate the constant recovery model using
MUFG’s senior and subordinate CDS data. We consider the acquired parameter estimates
and the fixed subordinate LGD as true parameters, and consider the fitted values of CDS
spreads as theoretical CDS spreads.
$\xi y$Case3: Stochastic recovery & $only$ one seniority
First, fixing LGD parameter at 0.6, we estimate a constant recovery model (but with
$DI$ ’s model as that in
the stochastic recovery model) using $Sony$ ’s senior CDS data. Afteracquiring the $DI$ ’s parameter estimates and its estimated path, we then fix these parametersand path and estimate the stochastic recovery model for the LGD’s parameters and forthe path of recovery risk factor, using once again the Sony’s senior CDS data. We consider
these $DI$ and LGD’s parameter estimates as true parameters, and the fitted values of CDS
spreads from the stochastic recovery model as theoretical CDS spreads,

Case4; Stochastic recovery E3; two seniorities
First, fixing subordinate LGD parameter at 0.8, we estimate a constant recovery model(but with $DI$ ’s model as that in the stochastic recovery model) using MUFG’s senior andsubordinate CDS data. After acquiring the $DI$ ’s parameter estimates and its estimatedpath, we then fix these parameters and path and estimate the stochastic recovery model forthe senior and subordinate LGD’s parameters and for the path of recovery risk factor, using
once again the MUFG’s senior CDS data. We consider these $DI$ and LGD’s parameters

70
estimates as true parameters, and the fitted values of CDS spreads from the stochastic
recovery model as theoretical CDS spreads.
To generate CDS data for simulation from the models, we simply take the generate data tothesum of theoretical CDS spreads and measurement errors, where measurement errors are
begenerated from independently identically normal distribution with zero mean and a homoskedastic variance. This is done times to obtain sets of CDS data, which will be used to estimate
the models’ parameters and factors times so as to finally obtain their distributions. Note thatfor simplicity, when generating artificial CDS data, we generate the random source only fromthe measurement errors. This means that we do not generate paths of default risk factor andpaths of recovery risk factor. For our purpose, we think that it is not necessary for doing so.
$n$

$n$

$n$

4.2

Simulation Results

The simulation results are reported in Table 18, 19, 20, and 21. The tables report the true values
along with mean and standard deviation of the parameter estimates. Note tha indicates thatthe true lies within 1 standard deviation from the mean, and ** indicates that the true lies
within 2 standard deviations from the mean.
The result for Casel for 1000 trials is reported in Table 18. In this case, we further consider
3 subcases. First, we fix LGD’s parameter and then estimate the model for $DI$ ’s parameters
together with its path as well as the measurement error standard deviation. The result isreported in Panel(A). For the $DI$ ’s parameters, the true value lies within 2 standard deviations
from the mean, and the standard deviation is so small compared to the true value. This suggeststhat the estimates distribute very close to the true values. Next, we consider fixing the $DI$ ’s
parameters at true values, and estimate the model for the LGD’s and $ME$ $SD$ ’s parameter as wellas the path of $DI$ . From Panel(B), we find that the mean of both parameter estimates are notsignificantly different (the absolute difference between the mean and the true value is less than
two standard errors, where the standard error is $SD/\sqrt{1000}$) from the true values. Finally, we
consider the joint estimation, the result of which is shown in Panel(C). All the $DI$ ’s and LGD’s
parameter estimates distribute far away from the true values. It is interesting to note that the
only parameter with a mean not significantly different from the true is the $ME$ $SD$ . This suggeststhat there are many combinations of different values (with different levels) of $DI$ ’s parametersand LGD’s parameter that give almost equivalent fitting performances. Also notice that while
the estimates of LGD distribute far to the left, the estimates of $DI$ ’s parameters distribute far
to the right. This is the reason why we tend to overestimate $PD$ and underestimate LGD. To
sum things up, in this case, attempting to jointly estimate the two components ( $DI$ and LGD)
together, we will end up with far incorrect parameter estimates. When information about oneof the two component is given, we are able to obtain quite accurate estimates of the parameters.
This is clear evidence of identification problem.
The result for Case2 for 1000 trials is shown in Table 19. Similar to Casel, we consider
four subcases. First, we fix subordinate LGD at the true, and estimate the model for $DI$ ’s
parameters, its path, senior LGD, and both $ME$-SDs. The result is in Panel(A), for which wefind that for all of the parameters, the true lies within one or two standard deviations fromthe mean. Also, the value of the deviations is relatively very small. This means that there ishigh probability of obtaining estimates that are very close to the true. Second, we fix senior
LGD at the true, and estimate the rests of the parameter along with $DI$ ’s path. As shown inPanel(B), the result is similar to the first subcase. Third, we fix $DI$ ’s parameters, and estimate
its path along with the two LGDs and the two $ME$ -SDs, the result of which is given in Panel(C).
From similar observation, we can draw the same conclusion as in the first and second subcases.
Finally, the joint estimation result is given in Panel(D). Except for the two $ME$-SDs, the other$t^{*}$--

71
parameter estimates distribute far away from the true value. Similar conclusion can be drawn
just as in Casel. When the three components ( $DI$ , senior LGD, and subordinate LGD) are
jointly estimated, due to identification problem we will not be able to obtain accurate results.
However, when one of the three unknown components is given, the identification problem isclear.
Next, the result for Case3 for 100 trials is reported in Table 20. We have two state variables
to estimate: the default risk factor, and the recovery risk factor. First, we flx the LGD’s
parameters, and estimate $DI$ ’s parameters, its path, and the path of the recovery risk factor.
The result is shown in Panel(A), for which we find that except for the risk-neutral parameterfor the other parameters, the true lies within one or two standard deviations. However, it should
be notice that the mean and standard deviation of the long-term mean parameterare many
times larger than the true value. This means that there is small probability of obtaining an
estimate that is close to its true value. That is we will greatly overestimate the default intensity.
What happens is that even though we fixed the parameters of the recovery risk factor, its pathcan still fluctuate in a range allowed by its parameters. This, together with the fact that we
can obtain accurate estimate of $ME$ $SD$ , suggests that many combinations of different values
with different levels of $DI$ ’s parameters and its path, and the path of recovery risk factor could
produce almost equivalent fitting performances. This is a clear evidence of factor identification
problem. Next, we consider fixing $DI$ ’s parameters without fixing its path. Examining result
shown in Panel(B), we find that except for , for all the other parameters, the true value lies
within one or two standard deviation, but since the standard deviation is not relatively so small,
it is likely that we will obtain estimates that are less accurate. However, compared to the firstsubcase when we fixed LGD’s parameters, the identification problem in this subcase is not so
serious. The reason is that in credit spreads the most prominent component is not the recovery
risk factor, but the default risk factor. So, when the default risk factor’s parameters are fixed,
to achieve equivalent fitting performances, its path does not have enough freedom to take on
different levels. Without having to consider the joint estimation, we can conclude in this casethat not only parameters but also factors can not be separately identified.
Finally, the result for Case4 for 95 trials is reported in Table 21. As from the analysis of
the results above, we are obvious about the existence of parameter and factor identification, in
this case we consider the subcase of joint estimation only. Although for all parameters, except, the true value lies within one or two standard deviations, the standard deviation for mostparameters, especially the long-term means and mean-reversion rates, is so large compared to
its true value. This means that the parameter estimates have disperse distributions. Onceagain, together with the fact that the estimates of both $ME$-SDs distribute very close to theirtrue values, this suggests that many combinations of different values with different levels of$DI$ ’s parameters, its path,
LGD’s parameters, and its path, could produce equivalent fitting
performance. This is obviously an evidence of parameter and factor identification problem.
$\kappa_{4}^{\mathbb{Q}},$$\theta_{4}^{\mathbb{P}}$-$\kappa_{5}^{\mathbb{P}}$$\rho$

5

Conclusion

In this paper, we examined if it is possible to jointly estimate default intensity and recovery
rates using only information embedded in the CDS term structure of different seniorities. We
first specified ajoint model of risk-free term structure of interest rates, default intensity, and loss
given default. We considered ajoint model with constant recovery assumption, and ajoint modelwith stochastic recovery assumption. We used JPY Libor and swap rates as risk-free interest
rates. Taking the parameter estimates and estimated path of the interest rate factors as given,
we estimated credit default swap model using two sets of data: Sony’s senior CDS, and MUFG’ssenior and subordinate CDS. We then divided our study into four cases: constant recovery

72& Sony’s senior CDS, constant recovery & MUFG’s senior and subordinate CDS, stochastic
recovery & Sony’s senior CDS, and stochastic recovery & MUFG’s senior and subordinate CDS.
From the results of the empirical estimations, we found that in all four cases when jointly
estimating default intensity and loss given default together, we tend to greatly overestimate
implied default probabilities, and greatly underestimate (or overestimate) implied loss given
default (recovery rates). We were not able to obtain plausible estimates of the model parameters
at all. However, in the constant recovery case, when we fixed the loss given default, we wereable to obtain much more plausible result. We conclude that separate identification between
default intensity and recovery rates is not possible. In order to verify this, we conducted asimple simulation study on the four cases. From the simulation results, we found that, forthe constant recovery model, when jointly estimated, the estimates of the default intensity
parameters distribute far away from the true values to the right, and the estimates of the loss
given default parameters distribute far away from the true values to the left. However, when
one of the unknown components to be estimated is given, the parameter estimates of the othercomponents distribute very close to the true. This suggests that default intensity factor and loss
given default parameter cannot be separately identified. In addition, in the stochastic recovery
model with one seniority included, even if we fixed the loss given default’s parameters, theestimates of default intensity’s parameters are far different from the true values. This is because
the path of default risk factor and that of the recovery risk factor could not be jointly identified
by the CDS data.
In conclusion, joint estimation of default intensity and recovery rates using only CDS data
is not possible. The information subsumed in the different-seniority CDS term structure may
not be enough to allow for such separation. Attempting to jointly estimate default intensity
and recovery rates using only term structure of CDS premia could result in overestimation of
implied recovery rates and probabilities of default. For example, Pan & Singleton (2008) and
Schneider et al. (2011) may have overestimated implied recovery rates. Pan & Singleton (2008)
estimate implied recovery rates for Mexico, Turkey at 78.9, 76.4% respectively. The market
convention for sovereign CDS is to fix implied recovery rates at 25%. Schneider et al. (2011)
obtain estimates of implied recovery rates ranging from 68.7% to 89.9% with average of 79.0%
much higher than the market convention which is 40%. Therefore, it might be important toconsider a more reliable approach for separately estimating default probabilities and recovery
rates.

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$\rangle$

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Graduate School of Economics
Osaka University, Toyonaka 560-0043, Japan$E$-mail address: simdara@gmail.com$*\beta R\star\not\cong\ovalbox{\tt\small REJECT} ffi\not\cong\Re ae\Re$Dara Sim

74
A

Appendices

A. 1

Show that$\mathbb{Q}(t<T_{d}\leq u|\mathcal{G}_{\infty}\vee \mathcal{H}_{t})=1_{\{T_{d}>t\}}(1-e^{-\int_{\ell}^{u}h_{s}ds})$

From (1), we obtain the following conditional probabilities.
$\mathbb{Q}(T_{d}>t|\mathcal{G}_{\infty})=\mathbb{Q}(T_{d}>t|\mathcal{G}_{t})$$=\mathbb{Q}(N(t)-N(0)=0|\mathcal{G}_{t})$$=e^{-\int_{0}^{t}h_{\hslash}ds}, t\geq 0,$$\mathbb{Q}(T_{d}>t|\mathcal{G}_{\infty}\vee \mathcal{H}_{t})=1_{\{T_{d}>t\}}\mathbb{Q}(T_{d}>T|\mathcal{G}_{\infty}\vee \mathcal{H}_{t})+1_{\{T_{d}\leq t\}}\mathbb{Q}(T_{d}>T|\mathcal{G}_{\infty}\vee \mathcal{H}_{t})$$=1_{\{T_{d}>t\}}\mathbb{Q}(T_{d}>T|\mathcal{G}_{\infty}\vee \mathcal{H}_{t})$$=1_{\{T_{d}>t\}} \frac{\mathbb{Q}(\{T_{d}>T\}\cap\{T_{d}>t\})}{Q(T_{d}>t|\mathcal{G}_{\infty})}$$=1_{\{T_{d}>t\}}e^{-\int_{t}^{\tau_{d}}h_{s}ds}.$

A.2

Zero-coupon yield curves$y(X_{r,t}, \tau)=\frac{2(1-e^{(}-\eta\tau))}{\tau(\eta+k_{1}^{Q}+(\eta-k_{1}^{Q})e^{-\eta\tau})}X_{1,t}+\frac{1-e^{-\lambda\tau}}{\lambda\tau}X_{2,t}+[\frac{1-e^{-\lambda\tau}}{\lambda\tau}-e^{-\lambda\tau}]X_{3,t}+\frac{A(\tau)}{\tau}$

, (1)

where
$\eta = \sqrt{(k_{1}^{Q})^{2}+2\sigma_{1}^{2}}$$\frac{A(\tau)}{\tau}$$=$$\sigma_{2}^{2}[\frac{1}{2\lambda^{2}}-\frac{1}{\lambda^{3}}\frac{1-e^{-\lambda\tau}}{\tau}+\frac{1}{4\lambda^{3}}\frac{1-e^{-2\lambda\tau}}{\tau}]$$+ \sigma_{3}^{2}[\frac{1}{2\lambda^{2}}+\frac{1}{\lambda^{2}}-\frac{1}{4\lambda}\tau e^{-2\lambda\tau}-\frac{3}{4\lambda^{2}}e^{-2\lambda\tau}-\frac{2}{\lambda^{3}}(1-e^{-\lambda\tau})+\frac{5}{8\lambda^{2}}\frac{1-e^{-2\lambda\tau}}{\tau}]$$- \kappa_{1}^{\mathbb{Q}}\theta_{1}^{\mathbb{Q}}[\frac{2}{\tau\sigma_{1}^{2}}\log(\frac{\kappa_{1}^{\mathbb{Q}}+\eta+(\eta-\kappa_{1}^{\mathbb{Q}})e^{-\eta\tau}}{2\eta})+\frac{2}{\eta+\kappa_{1}^{\mathbb{Q}}}].$

A.3

Derivation of the General CDS Spread Formula

From (2), the present values at time
calculated as follow.

$t$of the premium leg and default leg can be respectively

Default leg;$E^{\mathbb{Q}}[e^{-\int_{t}^{T_{d}}r_{u}du}\varphi_{T_{d}}1_{\{t\leq\tau_{d}\leq\tau\}}|\mathcal{F}_{t}]=1_{\{\tau_{d}\geq t\}}\mathbb{E}^{\mathbb{Q}}[E^{\mathbb{Q}}[e^{-\int_{t}^{T_{d}}r_{u}du}\varphi_{T_{d}}1_{\{t\leq\tau_{d}\leq\tau\}}|\mathcal{G}\tau\vee \mathcal{H}_{t}]|\mathcal{F}_{t}]$$=1_{\{T_{d}\geq t\}}\mathbb{E}^{\mathbb{Q}}[l^{\tau_{e^{-\int_{t}^{S}(r_{u}+h_{u})du}\varphi_{S}h_{s}ds1\mathcal{F}_{t}]}}$$=1_{\{T_{d}\geq t\}} \int_{t}^{T}\mathbb{E}^{\mathbb{Q}}[e^{-\int_{t}^{s}(r_{u}+h_{u})du}\varphi_{s}h_{s}|\mathcal{G}_{t}]ds.$

Premium leg:$S(t, T) \sum_{i=1}^{n}\mathbb{E}^{\mathbb{Q}}[\delta_{i}e^{-\int_{t}^{t}tr_{u}du}1_{\{T_{d}>T\}}+(T_{d}-t_{i-1}e^{-\int_{\ell}^{T}r_{u}du}1_{\{t_{i-1}<\tau_{d<t_{i}\}1\mathcal{F}_{t}]}}$$=1_{\{T_{d}\geq t\}}S(t, T)( \sum_{i=1}^{n}\delta_{i}E^{\mathbb{Q}}[e^{-\int_{t}^{t}t.(r_{u}+h_{u})du}|\mathcal{G}_{t}]+\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{i}}(s-t_{i-1})E^{\mathbb{Q}}[h_{s}e^{-\int_{t}^{n}(r_{u}+h_{u})du}|\mathcal{G}_{t}])$.

75
Since the default leg and premium leg must be equal in order to ensure arbitrage-free, we musthave
$S(t, t+ \tau)=\frac{\int_{t}^{t+\tau}\mathbb{E}^{\mathbb{Q}}[e^{-\int_{t}^{s}(r_{u}+h_{u})du}\varphi_{s}h_{s}1\mathcal{G}_{t}]ds}{\sum_{i=1}^{n}\delta_{i}\mathbb{E}^{\mathbb{Q}}[e^{-\int_{t}^{t_{i}}(r_{u}+h_{u})du|\mathcal{G}_{t}]+\sum_{i=1}^{n}\int_{t_{i-1}}^{t_{l}}(s-t_{i-1})\mathbb{E}^{\mathbb{Q}}[-\int_{t}^{s}(r_{u}+h_{u})du|\mathcal{G}_{t}]ds}h_{s}e}.$A.4

Explicit Formula for$\mathbb{E}_{t}^{\mathbb{Q}}[e^{-\int_{t}^{T}(r_{u}+h_{u})du}h_{T}e^{(iv-1)Y_{T}}]$

Applying the result of Duffie et al. (2000), we obtain$\psi(i, v, T-t, X_{t}) := \mathbb{E}_{t}^{\mathbb{Q}}[e^{-\int_{t}^{T}(r_{u}+h_{u})du}e^{(iv-1)Y_{T}}]=\exp\{\alpha+\sum_{j=1}^{5}\beta_{j}X_{j,t}\}$,(2)$\hat{\psi}(i, v, T-t, X_{t}) := \mathbb{E}_{t}^{\mathbb{Q}}[e^{-\int_{l}^{\tau_{(r_{u}+h_{u})du}}}h_{T}e^{(iv-1)Y_{T}}]$$= (A+B_{1}X_{1,t}+B_{4}X_{4,t})\psi(i, v, T-t, X_{t})$where$\alpha,$$\beta_{j},$$j=1,2,3,4,5$ , and $A,$

$B_{1},$ $B_{4}$,(3)satisfy the following Riccati equations:$\dot{\beta}_{1}(t) = -(1+\rho)-k_{1}^{\mathbb{Q}}+\frac{\sigma_{1}^{2}}{2}\beta_{1}^{2},$$\dot{\beta}_{2}(t) = -1-\lambda\beta_{2}(t)$,$\dot{\beta}_{3}(t) = -\lambda\beta_{3}(t)+\lambda\beta_{2}(t)$,,$\dot{\beta}_{4}(t) = -1-k_{4}^{\mathbb{Q}}\beta_{4}(t)+\frac{\sigma_{4}^{2}}{2}\beta_{4}^{2}(t)$$\dot{\beta}_{5}(t) = 0,$,$\dot{\alpha}(t) = k_{1}^{\mathbb{Q}}\theta_{1}\mathbb{Q}+\frac{\sigma_{2}^{2}}{2}\beta_{2}^{2}(t)+\frac{\sigma_{3}^{2}}{2}\beta_{3}^{2}(t)+k_{4}^{\mathbb{Q}}\theta_{4}^{\mathbb{Q}}\beta_{4}(t)+\frac{\sigma_{5}^{2}}{2}\beta_{5}^{2}(t)$$\dot{B}_{1}(t) = -k_{1}^{\mathbb{Q}}B_{1}(t)+\sigma_{1}^{2}B_{1}(t)\beta_{1}(t)$,$\dot{B}_{4}(t) = -k_{4}^{\mathbb{Q}}B_{4}(t)+\sigma_{4}^{2}B_{4}(t)\beta_{4}(t)$,with boundary conditions:$\beta_{j}(0)=\frac{1}{2}b_{j}(1-iv),$$j=1,2,3,5$ , and $\beta_{4}(0)=0,$$\alpha(0)=\frac{1}{2}b_{0}(1-$$iv),$ $B_{1}(0)=\rho,$ $B_{4}(0)=1,$ $A(O)=0.$

A.5

Show that$\varphi_{t}$can be written$\frac{1}{4}\int_{-\infty}^{\infty}\frac{e^{(iv-1)Y_{t}}}{Cosh(\frac{\pi v}{2})}dv$

First, it can be written$\varphi_{t}=\frac{1}{1+e^{2Y_{t}}}=\frac{e^{-Y_{t}}}{2}\frac{1}{Cosh(Y_{t})},$

where Cosh $(Y)= \frac{1}{2}(e^{y}+e^{-y})$ , and $Y=- \frac{1}{2}(b_{0}+b_{1}X_{1}+b_{2}X_{2}+b_{3}X_{3}+b_{5}X_{5})$ .
The Fourier transform ofis$\frac{1}{Cosh(y)}$$\hat{g}(v)=\frac{\pi}{Cosh(\frac{\pi v}{2})}.$

Then$\frac{1}{Cosh(y)}$can be recovered by the Fourier inversion as follow:$\frac{1}{Cosh(y)}=\frac{1}{2\pi}\int_{-\infty}^{\infty}\hat{g}(v)e^{ivy}dv=\frac{1}{2}\int_{-\infty}^{\infty}\frac{e^{ivy}}{Cosh(\frac{\pi v}{2})}dv.$

76First derivative of Libor and Swap rates w.r. . interest rate factors:

A.6

$t$

If $P(X_{r,t}, \tau)=\exp\{L(\tau)+m_{1}(\tau)X_{1,t}+m_{2}(\tau)X_{2,t}+m_{3}(\tau)X_{3,t}\}$ , then$\frac{\partial LIBOR}{\partial X_{j,t}}(X_{r,t}, \tau)$$=$$\frac{\partial SWAP}{\partial X_{j,t}}(X_{r,t}, \tau)$$=$$- \frac{100}{\tau}P(X_{r,t}, \tau)^{-1}m_{j}(\tau)$-$SWAP$$(X_{r,t}, \tau)[\frac{P(X_{r,t},\tau)}{1-P(X_{r,t},\mathcal{T})}m_{j}(\tau)-\sum_{l=1}^{h\tau}m_{j}(\frac{i}{h})\frac{(X_{r,t},\frac{l}{h})}{\sum_{l=1}^{h\tau}P(X_{r,t},\frac{l}{h})}],$

for $j=1,2,3.$
Table 1: Risk-free term structure parameter estimates$\frac{\theta_{1}^{P}\theta_{2}^{1P}\theta_{3}^{1P}\kappa_{1}^{P}\kappa_{2}^{P}\kappa_{3}^{\mathbb{P}}\sigma_{12}\sigma\sigma_{3}\kappa_{1}^{\mathbb{Q}}}{Estimates0.0042-0.0042-0.00380.69281.01994.84900.12480.00990.01310.0088}$

0.0048$SE$

0.00000.0000Note: In order that each factor is identified,$\lambda$

0.7986

0.76961.30480.01220.0000 0.0000 0.0072

is fixed at 0.57,

Table 2: Fitting performance of the term structure model

RMSE
MAPE
ME-SD

$LIB3M$$LIB6M$

LIB $12M$$SW2Y$$SW3Y$$SW4Y$$SW5Y$$SW7Y$$SW10Y$

4.70
12.17
4.7

0.00
0.00
0.01

4.03
7.05
4.02

1.21
1.15
1.23

0.52
0.38
0.54

0.04
0.02
0.11

0.25
0.14
0.29

0.24
0.12
0.39

1.61
0.66
1.68

Note: LIB $3M$ , LIB $6M$ , and LIB $12M$ denotes 3-month, 6-month, and 12-month Libor rates respectively, and3, 4, 5, 7, and 10-year Swap rates respectively.
RMSE is Ro $ot$ Mean Square Errors; MAPE is Mean Absolute Percentage Errors, and $ME$ $SD$ is the standard
deviation of the measurement errors. RMSE and $ME$ $SD$ are reported in basis points, whereas MAPE are reported
in percents.
$SW2Y,$ $SW3Y,$ $SW4Y,$ $SW5Y,$ $SW7Y$ , and $SW10Y$ denotes 2,--

Table 3: Correlations between interest rate factors and empirical level, slope, and curvature

Note: $10y$ ’ denotes 10-year zero-coupon yields (or zero rates). $\rangle[0.25y-10y]$ ’ is the difference between -month and 10-year
zero rates. $[2^{*}(2y)-0.25y-10y]$ means two times of 2-year zero rates less 3-month zero rates and less 10-year zero rates.
denotes changes (first differences).
‘‘$\triangle$’

77
Table 4: Parameter estimates and standard errors (Casel)$\frac{\theta_{4}^{\mathbb{P}}\kappa_{4}^{\mathbb{P}}\sigma_{4}\kappa_{4}^{\mathbb{Q}}LGD\sigma_{e}}{Estimates0.01060.26070.1037-0.46880.03181.6972}$$SE$

0.00390.08450.00320.00920.0024 0.1287

Note: Homoskedasticity is assumed for the measurement errors.
standard deviation.

$\sigma_{e}$is their

Table 5: Fitting performance (Casel)

RMSE
MAPE

1

3

5

7

10

Mean

Median

1.72
23.98

1.21
7.32

1.35
5.79

3.04
9.79

4.14
10.78

2.29
11.53

1.72
9.79

Note: This table reports Root Mean Square Errors (RMSE) and Mean Absolute Percentage Errors (MAPE) for 1,3,5,7,10-year CDS spreads, and its
mean and median. RMSE are in basis points, and MAPE are in percents.

Table 6. Implied Probabilities of Default ( $PD$ ) (Casel)$\frac{1-year2-year3-year4-year5-year}{ImpliedPD1.935.2110.3617.8127.62}$

Note: All values are average values across times and are in percents.

Table 7: Parameter estimates and standard errors (Case2)$\frac{\theta_{4}^{P}k_{4}^{\mathbb{P}}\sigma_{4}\kappa_{4}^{\mathbb{Q}}LGD^{SEN}LGD^{SUB}\sigma_{e}^{SEN}\sigma_{e}^{SUB}}{Estimates0.00140.94370.1310-0.50270.05100.09551.13447.0903}$$SE$0.00020.01830.00230.00790.0063

Note: Homoskedasticity is assumed for the measurement errors.
the measurement errors for senior CDS and subordinate CDS respectively,

0.0118$\sigma_{e}^{SEN},$ $\sigma^{SUB}$

0.0217are standard deviations ofTable 8: Fitting performance (Case2)$\frac{135710Me.anMedi.an}{RMSE0.891.101.512.113.19176151}$
Senior$\frac{MAPE19.428..894.426..415.878..996.41}{SubordinateRMSE5.735154.6953512.256645.35}$

MAPE

45.27

24.43

8.58

13.63

22.71

22.93

22.71

Note: This table reports Root Mean Square Errors (RMSE) and Mean Absolute Percentage
Errors (MAPE) for 1,3,5,7,10-year senior and subordinate CDS spreads, and its mean andmedian. RMSE are in basis points, and MAPE are in percents.

Table 9: Implied Probabilities of Default ( $PD$ ) (Case2)$\frac{1-year2-year3-year4-year5-year}{Implied-PD0.722.044.217.4811.97}$

Note: All values are average values across time and are in percents,

0.0803

78
Table 10:
(Case3)

Parameter estimates and standard errors$\frac{\theta_{4}^{\mathbb{P}}\kappa_{4}^{P}\sigma_{4}\kappa_{4}^{\mathbb{Q}}\rho}{Estimates0.05100.33160.3510-0.21620.6582}$$SE$

0.05870.01120.02730.02840.0406$\frac{\theta_{5}^{\mathbb{P}}\kappa_{5}^{\mathbb{P}}\sigma_{5}b_{0}b_{1}b_{2}b_{3}\sigma_{e}}{Estimates-2.28970.22760.9965-0.92456.74064.606632.00761.1916}$

0.2414$SE$

0.1001

0.05690.0802

1.2194

Note: Homoskedasticity is assumed for the measurement errors.

$\sigma_{e}$1.43812.8606

0.1127is their standard deviation.

Table 11: Fitting performance (Case3)

RMSE
MAPE

1

3

5

7

10

Mean

Median

1.61
23.07

1.04
5.02

1.48
6.79

1.35
3.04

1.13
1.83

1.32
7.95

1.35
5.02

Note: This table reports Root Mean Square Errors (RMSE) and Mean
Absolute Percentage Errors (MAPE) for 1,3,5,7,10-year senior CDS
spreads, and its mean and median. RMSE are in basis points, and MAPE
are in percents.

Table 12: Implied Probabilities of Default ( $PD$ ) and Implied
Recovery rates (REC) (Case3)$\frac{1-year2-year3-year4-year5-year}{Implied-PD8.3217.9827.6836.5844.37}$

Implied-REC

99.20

98.67

97.99

97.21

96.36

Note: All values are average values across time and are in percents.

Table 13: Parameter estimates and standard errors (Case4)$\frac{\theta_{4}^{P}\theta_{5}^{\mathbb{P}}\sigma_{e}^{SUB}}{Estimates0.00060.47940.1463-0.37770.12895.68794.99330.37471.26703.7832}$$\underline{\underline{SE0.00050.33700.02110.042S0.01711.32163.41450.06820.08070.1264}}$$\frac{b_{0}^{SEN}b_{1}^{SEN}b_{2}^{SEN}b_{3}^{SEN}b_{5}^{SEN}b_{0}^{SUB}b_{1}^{SUB}b_{2}^{SUB}b_{3}^{SUB}}{Estimates3.134129.853230.0584-4.6874-1.0438-7.787331.85200.0000-58.9594}$$\frac{SE2.07460.96856.96654.48770.20601.29750.24601.095015.5034}{Note:b_{5}^{SUB}isfixedat1.Homoskedasticityisassumedforthemeasurementerrors.\sigma_{e}^{SEN},\sigma_{e}^{SUB}arestandarddeviations}$of the measurement errors for senior CDS and subordinate CDS respectively.

79

Table 14: Fitting performance (Case4)

RMSE

1

3

5

7

10

Mean

Median

0.75

1.60

1.34

0.83

1.13

1.13

1.13

$\frac{SeniorMAPE14..1218..608..342..583..369..408.34}{SubordinateRMSE3702383291765533333.29}$

MAPE

25.89

9.99

5.54

3.47

5.26

10.03

5.54

Note: This table reports Root Mean Square Errors (RMSE) and Mean Absolute Percentage
Errors (MAPE) for 1,3,5,7,10-year senior and subordinate CDS spreads, and its mean andmedian. RMSE are in basis points, and MAPE are in percents.

Table 15: Implied Probabilities of Default ( $PD$ ) and Implied Recovery rates (REC) (Case4)
Implied-$PD$

Implied-REC-SEN
Implied-REC-SUB

1-year2-year3-year4-year5-year

0.47
91.84
78.89

1.13

89.71
76.94

2.02
87.63
75.22

3.17
85.71
73.62

4.57
83.96
72.11

Note. All values are average values across time and are in percents. Case4:
Stochastic recovery & MUFG’s senior and subordinate CDS

Table 16: Implied Probabilities of Default ( $PD$ ) (Fix LGD$=0.6)$ (Casel)

Implied-$PD$

1-year2-year3-year4-year5-year

0.10

0.28

0.58

1.02

1.60

Notes: All values are reported in percents.

Table 17: Implied Probabilities of Default ( $PD$ ) and Implied Re$=1$ ) (Case2)
covery rates of senior CDS (REC) (Fix LGD
Implied- $PD$
Implied-REC-SEN

1-year2-year3-year4-year5-year

0.07
47.00

0.20
47.00

0.42
47.00

0.74
47.00

1.20
47.00

Notes: All values are reported in percents.

80

Table 18: Simulation results for casel $(n$ trials $=1000)$$\frac{}{}\frac{(A)FixLGD--0.6\theta_{4}^{1P**}\kappa_{4}^{P**}\sigma_{4}^{**}\kappa_{4}^{\mathbb{Q}**}\sigma_{e}^{*}}{True1.13E-043.37150.1686-0.38732.1709}$

Mean

2.9894

1. $40E$-04

0. $1592$$-0.3446$

2.2017$\frac{SD2.64E-050.39330.00560.02210.0882}{(B)FixDI’ sparametersattrue}$$\overline{\frac{LGD^{*}\sigma_{e}^{*}}{iRue0.60002.1709}}$Mean

2.19920.5995$\frac{SD0.01400.0886}{(C)Jointestimation}$$\overline{\frac{\theta_{4}^{P}\kappa_{4}^{P}\sigma_{4}\kappa_{4}^{Q}LGD\sigma_{e}^{*}}{True1.13E-043.37150.1686-0.38730.60002.1709}}$

Mean
$SD$

0.0055 0.6134
0.0009 0.2723

0. $1408$$-0.3950$

0.00540.01640.06260.0393

2.17770.0759Note: * denotes the true lies within 1 standard deviation from the mean. $**$
denotes the true lies within 2 standard deviations from the mean.

Table 19: Simulation results for Case2 $(n$ trials $=1000)$$\frac{}{}\frac{(A)FixLGD^{SUB}=1}{True2.44E-054.49100.1341-0.48970.53001.34296.7101\theta_{4}^{JP*}\kappa_{4}^{P**}\sigma_{4}^{**}\kappa_{4}^{Q**}LGD^{SEN*}\sigma_{e}^{SEN*}\sigma_{e}^{SUB*}}$

Mean
$SD$

2. $39E$-05
2. $10E$-06

5.43420.60230.1368 -0.4760
0.0022 0.0097

1.34460.0494

0.52800.00506.70320.2206(B) Fix $LGD^{6bN}=0.53$$\overline{\frac{\theta_{4}^{P*}\kappa_{4}^{1P**}\sigma_{4}^{**}\kappa_{4}^{\mathbb{Q}**}LGD^{SUB*}\sigma_{e}^{SEN*}\sigma_{e}^{SUB*}}{True2.44E-054.49100.1341-0.48971.00001.34296.7101}}$Mean

2. $37E$-05

5.4652

0. $1368$$-0.4760$

1.3446

0.99686.7061$\frac{SD2.04E-06060410.00220.00970.00580.04940.2208}{(C)FixDIparameters}$$\overline{\frac{LGD^{5\Delta N*}BLGD^{5UB*}\mathfrak{l}\sigma_{e}^{6BN*}\sigma_{e}^{6UB*}}{True0.53001.00001.34296.7101}}$

Mean

0.52690.9932

1.33796.7003$\frac{SD0.00820.01370.04820.2205}{(D)Jointestimation}$$\overline{\frac{\theta_{4}^{P}\kappa_{4}^{1P}\sigma_{4}\kappa_{4}^{Q}LGD^{SEN}LGD^{SUB}\sigma_{e}^{SEN*}\sigma_{e}^{SUB*}}{True2.44E-054.49100.1341-0.48970.53001.00001.34296.7101}}$

Mean
$SD$

0.00120.00010.6947
0.06140.12500.0022-0.49360.00940.12420.0062

0,06580,0033

Note: * denotes the true lies within 1 standard deviation from the mean.
standard deviations from the mean.

**

1.32830.0475

6.69470.2204denotes the true lies within 2

81

Table 20: Simulation results for Case3 ( trials $=100$ )$n$$\frac{}{}\frac{(A)FixLGD_{\theta_{4}^{\mathbb{P}*}\kappa_{4}^{\mathbb{P}**}\sigma_{4}^{**}\kappa_{4}^{Q}\rho^{*}\sigma_{e}^{*}}sparameters}{\prime nue8.78E-053.45750.1651-0.43630.013716806}$

Mean

0.0054

1.7832

0. $1813$$-0.2312$

0,00561.7443$\frac{SD0.00931.04420.00780.06650.00630.0704}{(B)FixDIsparameters}$$\ovalbox{\tt\small REJECT}\frac{\theta_{5^{*}}\kappa_{5}\sigma_{5}^{**}b_{0}^{*}b_{1}^{*}b_{2}^{*}b_{3}^{*}\sigma_{e}^{*}}{rbue-0.82170.01380.5597-0.133431.78091.581586.74211.6806}$

Mean$-0.5671$

1.2246$SD$

0.31220.5464

0. $7437$
0.1196$-0.0797$

0.216431.7261

0.096390.33523.2082-1.6696
10.8787

Note: * denotes the true lies within 1 standard deviation from the mean.
within 2 standard deviations from the mean.

**

1,69380.0688denotes the true lies

Table 21: Simulation results for Case4 ( trials $=95$ )$n$Joint Estimation$\overline{\frac{\theta_{4}^{\mathbb{P}*}\kappa_{4}^{\mathbb{P}*}\sigma_{4}^{**}\kappa_{4}^{\mathbb{Q}*}\rho^{**}\theta_{5}^{\mathbb{P}*}k_{5}^{\mathbb{P}**}\sigma_{5}^{*}}{-0.53070.03461.93602.51051.3159}}$

True
Mean

6. $17E$ -05
1. $43E$ -04
1. $62E$ -04

2,57772.6675
1.09180.14050.14700.0048-0.51190.07552.0911

7.2707

1,4274$\frac{}{}\frac{SD0.03000.03362_{r}.\cdot 55174.63510.7552b_{0}^{5AN*}\mathfrak{l}b_{1}^{61N*}i\prime b_{2}^{5BN*}lb_{3}^{6t!\prime N*}b_{5}^{6AN*}\sigma_{e}^{bAN*}}{True-0.679818.877912.4192-5.4701-0.268012436}$

Mean,1.2520

13,4349

13. $7458$$-9.0068$$-0.5973$

1,2381$\frac{}{}\frac{SD1.47676.10037..921714.18210.\cdot 66230..0464}{True-2.128625.555400289-406.80291000042869b_{0}^{5U13*}b_{1}^{6UB*}b_{2}^{6UB*}b_{3}^{5U\mathcal{B}*}1b_{5}^{5U13}I\sigma_{e}^{bUt3*}}$

Mean
$SD$

-3.6328
2.632822.7463
8.540327.8052
29.2064-319.4820

1,0000127,8082

0.00004.21490.1733

Note: * indicates the true lies within 1 standard deviation from the mean. ** indicates the true lies
within 2 standard deviations from the mean.