Chia-Wu Lu
∗ Tsung-Kang Chen ** Hsien-Hsing Liao ∗∗∗
Abstract
Among the first studies, this research develops a structural-form credit risk model being able to integrate both “asset inadequacy risk” and “liquidity crunch risk” of a firm. The new model differs from traditional structural-form credit models in that it additionally considers a firm’s flow-based insolvency as well as default risk due to asset inadequacy. This model can endogenously generate a firm’s probability of default, resulting from either asset inadequacy, liquidity crunch, or both.
Our results of numerical analyses show that the model can catch the short-term default risk underestimated by Merton-type structural form models. Besides, an application to American banks also shows that this model performs better than the Merton-type structural form models in evaluating short-term credit risk.
Key words: Asset inadequacy, Liquidity crunch, Flow-based Credit Model, Stock-based Credit
Model,
JEL Classification Code : G32, M41
∗ Assistant Professor, Department of Finance, Providence University, Email:chiawu@pu.edu.tw.
** Assistant Professor, Department of International Trade and Finance, Fu Jen Catholic University, Email: r91723010@ntu.edu.tw.
∗∗∗
Corresponding Author: Professor, Department of Finance, National Taiwan University, Email address: hliao@ntu.edu.tw
, Phone/Fax:(886) 02-2363-8897, Address: Rm. 814, Building #2, College of
Management, National Taiwan University, 85, Sec. 4 Roosevelt Road, Taipei 106, Taiwan.

“Technical insolvency” is a phenomenon that a firm has surplus in financial statements but cannot fulfill its payment obligations. This phenomenon is especially worth noting during the recent market-wide financial crisis. In practice, liquidity crunch usually takes place before
“stock-based” default (i.e. asset inadequacy default) because asset inadequacy relies upon the information generated by a time-lagged financial reporting system or, in many cases, by a complicated asset valuation process. Therefore, information on the probability of a liquidity crunch and the expected liquidity shortfall is important for determining the required internal liquidity reserve that supports a specific credit quality target, and is especially important for the periods of market liquidity crunch. However, existing Merton-type structural-form credit models ignore flow-based insolvency risk and consider only the difference between values of a firm’s assets and its liabilities. Their most distinctive attribute is that they derive a firm’s asset value distribution from its equity market value and estimate its probability of default (PD) and recovery rate (RR) endogenously.
1 Although researchers have developed many varieties from the original Merton model to overcome several major challenges to these models both in theory and practice, these modified models still barely consider corporate flow-based insolvency risk due to liquidity crunch.
On the other hand, the reduced-form credit models, which are intensity-based, disregard any of a firm’s fundamental information, including internal liquidity, and rely on exogenous information
1 This line of study includes Black and Cox (1976), Geske (1977), Vasicek (1984), Jones, Mason and Rosenfeld
(1984), and Crouhy and Galai (1994), Kim, Ramaswamy and Sundaresan (1993), Hull-White (1995), Longstaff and Schwartz (1995), Leland and Toft (1996), and Collin-Dufresne and Goldstein (2001), and Duffie, Saita and

such as credit ratings, recovery rates, or other default-related proxies to estimate a firm’s default probability.
2 Consequently, they are limited in being able to provide the necessary information for credit risk management and to price liquidity-related credit assets and derivatives. To fill this gap, this study develops an integrated structural-form credit risk model combining both stock-based and flow-based corporate credit risk information.
Many previous studies pointed out the estimation errors of Merton-type models when applying to corporate default risk estimation. Ogden (1987) found that Merton (1974) model under-predicts spreads by 104 bps on average. Duffie and Lando (2001) showed that structural-form models usually underestimate the credit spreads of short-term bonds, especially for maturity less than three years.
3 Collin-Dufresne et. al. (2001) showed that traditional models of default risk only explain about 25% of the variation in credit spreads.
Empirical results of Eom et al. (2004) showed that all the following models, Merton (1974, later denoted as M), Leland and Toft (1996, later denoted as LT), Longstaff and Schwartz
(1995, later denoted as LS), and Collin-Dufresne and Goldstein (2001, later denoted as CDG), have substantial spread prediction errors. All the above empirical evidences reveal that the structural-from credit models miss catching some important information which is crucial for determining the variations of bond credit spreads.
2
Wang (2007).
These models include Jarrow and Turnbull (1995), Lando (1998), Duffie and Singleton (1999), Duffie (1998) Duffee
(1999) and Unal, Madan and Güntay (2003). The latter covers Wilson (1997a and 1997b), Guption, Finger and Bhatia
3
(1997), McQuown (1997), Crosbie (1999).
It is also called the transparency spreads in Yu (2005).

Different from the above structural-form (stock-based) credit models, Liao et al. (2009) employed a flow-based risk proxy, the corporate internal liquidity risk, to investigate the effects of internal liquidity risk on bond yield spreads. Their results showed that a firm’s internal liquidity risk can significantly explain the variations of bond yield spreads when controlling for variables well-known in the literature.
4 The controlling variables include a firm’s leverage ratio, equity volatility, maturity, coupon, external liquidity, credit rating and four proxies of information uncertainty. Among them, the leverage ratio and equity volatility are the proxies of stock-based credit risk. Therefore, both a firm’s flow-based and stock-based credit information are relevant for credit risk evaluations. Their research results enlighten this study to construct an integrated structural-form credit risk model being able to combine stock-based and flow-based corporate credit information. It differs from traditional structural-form credit models in that it considers not only stock-based default but also flow-based insolvency. This model has several characteristics. First, it considers not only information of balance sheet, but also that of cash flow statement, which is able to capture more operating information than stock-based ones do.
Stock-based models put most attentions on the historically accumulated information such as asset and liabilities values, and are less sensitive to the recent operating conditions. Second, the flow-based information is more sensitive to the short term credit condition and plays a key role in improving short term credit risk estimation which, as mentioned in Duffie and Lando (2001),
4 The controlling variables include a firm’s leverage ratio, equity volatility, maturity, coupon, external liquidity, credit rating and the four proxies of information uncertainty. The variables except the information uncertainty variables are used in Yu (2005).

is underestimated by traditional structural-form models. The current integrated model can also provide a simple way to evaluate corporate default probabilities, which are results of interactions of stock-based default and flow-based liquidity crunch.
Stock-based models assume that default happens when a firm’s assets value falls below the default threshold. Because many studies such as Opler and Titman(1997), Fischer et. al.(1989), Goldstein et. al.(2001), and Collin-Dufresne and Goldstein(2001), asserted that a firm tends to maintain a target leverage ratio, this study therfore models a firm’s leverage by a mean-revertig Gaussian process. For the flow-based risk, Liao et al. (2008) indicated that a firm tends to adopt an appropriate or optimal internal liquidity level. This study, therefore, also assumes that a firm’s internal liquidity follows a mean-reverting Gaussian process. As a result, the integrated model is built upon the combination of the two stochastic processes. The joint distribution of the two risk indicators (the internal liquidity and the leverage ratio) allows us to endogenously generate the probability of default for each point in time.
Numerical analyses of this study show that the integrated model can catch short-term default risk which is underestimated by stock-based models. Moreover, a preliminary application of the integrated model to evaluate the default probability of a sample of
American banks shows that the integrated model exhibits better performance than that of the four Merton-type models in evaluating short-term default probabilities.
5
5 This sample banks are similar to those of Liao et al. (2009). The four Merton-type models are Merton model (M,
1974), Leland and Toft (LT, 1996), Longstaff and Schwartz (LS, 1995) and Collin-Dufresne and Goldstein (CDG,
2001).

The rest of the study is organized as follows: Section II constructs the integrated model combining stock-based and flow-based credit risk information. Section III employs numerical analyses to investigate the properties of the model. Section IV empirically examines the model’s effectiveness using an American bank sample. Finally, Section V concludes this study.
This section constructs an integrated model being able to combine stock-based and flow-based credit risk information. We first define the structures of stock-based and flow-based models respectively, and then derive the integrated model.
A. The Stock-based Credit Risk Model
Stock-based credit models assume that the default event happens when a firm’s asset value falls below a default threshold, usually its liabilities. Suppose A t
and D t
represent the firm’s asset value and the default threshold (total liability) respectively, this study defines
M t
, which equals to the log value of the firm’s leverage as Eq.(1):
M t
= log
⎛
⎝
⎜⎜
A t
D t
⎠
⎟⎟
⎞
(1)
By definition, default happens when A t
< D t
, or equivalently M t
< 0 .
Most of previous studies provided evidences that a firm tends to maintain a target leverage ratio, including Opler and Titman (1997), Fischer et al. (1989), Goldstein et al. (2001),
Collin-Dufresne and Goldstein (2001). According to these results, the current study assumes

that M t
follows an Ornstein-Uhlenbeck stochastic process as Eq. (2), which contains the mean-reverting property: dM t
= a
M
( b
M
−
M t
) dt
+ σ
M dz
M
(2)
By solving the above stochastic differential equation, M t
follows the normal distribution as Eq. (3):
M t
~ N
⎝
⎜⎜
⎛
M
0 e
− a
M t + b
M
(
1 − e
− a
M t
),
σ 2
M
(
1 − e
− 2 a
M t
2 a
M
)
⎠
⎟⎟
⎞
=
N
(
µ
M
, s
2
M
)
(3)
B. The Flow-based Credit Risk Model
In contrast to the stock-based models, few studies evaluate corporate credit risk from the perspective of internal liquidity. Liao et al. (2008) developed a state-dependent internal liquidity credit risk model, which is flow-based and considers a firm’s net internal liquidity affected by both industrial and macroeconomic factors. Their flow-based credit risk model is able to incorporate both systematic and idiosyncratic shocks into corporate internal liquidity dynamics. The model defines a firm’s internal liquidity as the net balance of “available liquidity” and “payment obligations” shown as Eq.(4), which shows that the higher internal liquidity, the lower flow-based credit risk. Liquidity crunch happens when “available liquidity” falls below “payment obligation”.
Internal liquidity = available payment liquidity obligation
(4)
Liao et al. (2008) asserted that a firm’s internal liquidity may be influenced by the dynamics of the state of the economy, such as the business cycle, market-wide internal liquidity, and

money fund market liquidity risk. As a result, they designs the corporate internal liquidity model as state dependent and a firm’s internal liquidity level is a linear combination of a long-term average internal liquidity level, state innovations, and an idiosyncratic shock. Furthermore, the dynamics of state factors are assumed as weakly stationary and thus follow mean-reverting Gaussian processes according to most of the economic literature.
6 A firm’s internal liquidity dynamics therefore follow a mean-reverting Gaussian process. The structure of a firm’s internal liquidity process may alter according to changes in the business cycle. Following Liao et al. (2008), this study takes the natural log for internal liquidity, and defines internal liquidity as Eq. (5).
I t
= log
(
Internal liquidity
)
(5)
With the mean-reverting property, the diffusion process of Ornstein-Uhlenbeck is employed to model internal liquidity as Eq. (6). For simplicity, this study does not design a state-dependent process structure as did by Liao et al. (2008). dI t
= a
I
( b
I
− I t
) dt + σ
I dz
I
(6)
I t
has the distirbution as Eq.(7):
I t
~ N ⎜⎜
⎝
⎛
I
0 e
− a
I t + b
I
(
1 − e
− a
I t
),
σ
I
2
(
1 −
2 a
I e
− 2 a
I t
)
⎠
⎟⎟
⎞
=
N
(
µ
I
, s
I
2
)
(7)
C. The Integrated structural-form credit risk model
From the derivations in the previous two sections, a firm faces two kinds of default risk:
6 The most representative researches related are Vasicek (1977) and Cox, Ingersoll, Ross (1985a,b).

a stock-based one, resulting from M t
< 0 , and a flow-based one, resulting from I t
< 0 . The shocks of these two risk indicators both follow Wiener processes and have a constant correlation as shown in Eq. (8): dz
M dz
I
=
ρ dt (8)
The joint density of M t and I t
, f
M , I correlation coefficient ρ .
is a bivariate normal distribution with a f
M , I
( m , i )
=
2 π s
I s
M
1
1 ρ 2 e
1
2 ( 1 ρ 2 )
[
( m s
µ
2
M
M
) 2
+
( i s
µ
I
2
I
) 2
2 ρ
( m µ s
M
I s
)( i
M
µ
I
)
]
(9)
When either of M t
< 0 and I t
< 0 or both take place, a firm faces insolvency problem and the default probability is derived as Eq.(10):
1 −
0
∞ ∞ ∫ ∫
0 f
M , I
( ) dmdi = 1 −
0
∞ ∞ ∫ ∫
0 2 π s
I s
M
1
1 − ρ 2 e
1
− 2 ( 1 − ρ 2 )
[
( m − µ
M s 2
M
) 2
+
( i − µ s
I
2
I
) 2
− 2 ρ
( m − µ
M s
I
)( i − µ
I s
M
)
] dmdi
= 1 −
0 0
∞ f
I M i
( ) di f
M
( ) dm = 1 −
0
⎜
⎜
0
∞
2 π s
I
1
1 − ρ 2 e
−
⎡
⎢ i − µ
I
− ρ
2 s
I
2 s
I s
M
(
1 − ρ m − µ
M
)
⎤
⎥
2 di
⎟
⎟ f
M
( ) dm
= 1 − ∫
0
∞
⎜
⎜
⎜
1 − ∫
−
0
∞ 2 π s
I
1
1 − ρ 2 e
−
⎡
⎢
⎣ i − µ
I
− ρ
2 s
I
2 s
I s
M
1 −
( m − µ
M
)
⎤
⎥
⎦
2 di
⎟
⎟ f
M
( ) dm
= 1 − ∫
0
∞
⎛
⎜
⎜
⎜
⎝
1 − N
⎛
⎜
⎜
⎝
0 − µ
I
− s
I
ρ s
I s
M
1 −
( m
ρ 2
− µ
M
) ⎞
⎟
⎟
⎠
⎞
⎟
⎟
⎟
⎠ n
⎛ −
⎜⎜ s
M
µ
M
⎞
⎟⎟ dm

= 1 − ∫
0
∞ n ( m − s
M
µ
M ) dm + ∫
0
∞
N
⎛
⎜
⎜
⎜
⎝
0 − µ
I
− ρ s
I s
I s
M
1 −
( m
ρ 2
− µ
M
) ⎞
⎟
⎟
⎟
⎠ n
⎛ −
⎜⎜ m s
M
µ
M
⎞
⎟⎟ dm
= 1 −
⎛
⎜⎜
1 −
0 ∫
− ∞
= N
⎛ −
⎜⎜
0 s
µ
M
M n ( m − s
M
µ
M
⎞
⎟⎟
+ ∫
0
∞
) dm
⎞
⎟⎟
+
N
⎛
⎜
⎜
⎝
0 − µ
I
0
∫ ∞
N
⎛
⎜
⎜
⎝
0 − µ
I
− ρ s
I s
I s
M
1 −
( m
ρ 2
− µ
M
) ⎞
⎟
⎟
⎠ n
⎛ −
⎜⎜ m s
M
µ
M
− s
I
ρ s
I s
M
1 −
( m
ρ 2
− µ
M
) ⎞
⎟
⎟
⎠ n
⎛ −
⎜⎜ m s
M
µ
M
⎞
⎟⎟ dm
⎞
⎟⎟ dm
(10)
The first term of Eq.(10) is similiar to the traditional structural-form models’ default probability function; the second term is the interactive effects when additionally considering flow-based insolvency risk into the model.
This section investigates the properties for various types of firm by numerical analysis.
The study sets the parameters for numerical analysis referring to the average of the 31 sample banks which this study uses for preliminary model applications in section IV. This study calculates both the stock- and flow-based parameters respectively, and uses the average to be the base case parameters for numerical analyses. Table 1 shows the average of the parameters of the 31 sample banks and the base case parameters.
[Insert Table 1 here]
The base case parameter settings are a
M
=0.6, b
M
=0.2, σ
M
=0.06; a
I
=0.7, b
I
=1.4,
σ
I
=0.5, ρ =-0.04. M
0
and I
0
represent the credit conditions at the initial time (the

evaluating time); The M
0
is set as 0.2 . The study sets two vlaues of I
0
(1.4 and 0.2) to represent a “Good” and a “Bad” internal liquidity scenarios.
Table 2 demonstrates the generated default probabilities by the assumed stock-based and the integrated credit model under both a “Good” and a “Bad” scenarios of corporate internal liquidity. The results of Table 2 show that when I
0
is in the “Bad” scenario, indicating that a firm is in a tight liquidity condition, the generated default probability by the integrated model is high, especially in short term, mostly resulting from the internal liquidity shortfalls. The internal liquidity effects decrease in longer periods because, due to the mean-reverting property of the I t
, corporate internal liquidity tends to return back to its long-term average level in the long run. Therefore, the cumulative default probabilities due to liquidity crunch are expected to decline in a longer period.
[Insert Table 2 here]
Table 3 shows generated default probabilities under different I
0
settings. The results show that that under different settings of I
0
, the effects of flow-based risk reduce in the long term period. Because of the mean-reverting characteristic of internal liquidity indicator, as the credit risk evaluating period increase, I t
is expected to return to the long term level b
I
and the effects of low initial I
0
(i.e. the “Bad” initial internal liquidity condition) decrease. It idicates that a firm’s internal liquidity play an important role in evaluating short term default probabilities, which are underestimated by traditional structural-form models.
[Insert Table 3 here]

Table 4 demonstrates the sensitive analysis of the correlation coefficient ρ , which indicates the correlation between between the stock-based and the flow-based credit risk shocks. It shows that as ρ decreases, the generated default probabilities decrease. It is expected because a low (small positive or a large negative) correlation coefficient indicates that the levels of M
0
I credit risk are in opposite scenarios. This leads to mild default probabilities. The lower the correlation, the more significant decrease in default probability. However, this effect decays when the period of default probabilities estimated becomes longer due to the mean-reverting property of M t
I
[Insert Table 4 here]
Table 5, 6 and 7 implement the sensitive analysis of the effects of the parameters of internal liquidity indicators ( I t
) process shown in Eq.(6) on default probabilities. The parameters analyzed includes mean-reverting speed parameter a
I
, long term liquidity level b
I
, and, standard deviation parameter σ
I
. Regarding the mean-reverting speed parameter a
I
, as a
I
decreases, default probability goes up because a lower mean-reverting speed indiactes that when a firm face liquidity shortfalls, the speed of returning back to normal is slower than the higher mean-reverting speed cases. Therefore, default probability is larger.
[Insert Table 5 here]
Table 6 demonstrates the sensitive analysis of the effects of long term liquidity level b
I

on default probabilities. A low b
I
denotes a lower long term liquidity level of I t
, which indicates that a firm’s liquidity level is closer to the liquidity crunch threshold ( I t
= 0 ), and therefore has a higher default probability.
[Insert Table 6 here]
Table 7 shows how the change of standard deviation parameter σ
I
. A larger σ
I represents that I t
has more chance to is closer to the liquidity crunch threshold ( I t
= 0 ).
Therefore, the default probability will decrease as the σ
I
declines.
[Insert Table 7 here]
This section applies the integrated model to an American bank sample to examine the model’s appropriateness. This study compares the performance of the integrated model with four famous structural-form credit models in short term default probability estimation. We expected that the integrated model can better catch short term default risk than stock-based structural model.
Following Liao et al. (2009), this study employs the middle point of the default probability range implied by the sample bank’s credit rating as a benchmark to show the better performance of the integrated model than the structural-form models. This study uses the similar criteria to obtain the bank sample as did by Liao et al. (2009).
A. Sample Bank Selection Criteria
Liao et al. (2009) selected the pure lending and depository institutions (such as

commercial banks and savings and loans) as their sample banks. Sample banks are collected with the criteria: publicly traded, with credit ratings, with both stock price data and corporate financial data from COMPUSTAT BANK. Their final sample included thirty-eight banks.
Due to the lack of cash flow data in the database, this study has only thirty-one banks in the final sample. Table 8 shows the sample distribution and related characteristic information of these sample banks.
[Insert Table 8 here]
B. Model’s Proxies and Parameters Estimation
B.1. The Stock-based Model
In stock-based model, we use total liabilities to be the default threshold proxy. Firm’s asset, represented by A, is calculated by Eq.(11):
A = Face value of total liabilities + Market value of equities (11)
Market value of equities is the multiplication of outstanding shares and the close price of common stocks on the last trading day of every quarter, which can be obtained from the
Center for Research in Security Prices (CRSP) database. The data period is 1994~2005 and the evaluating date is the end of 2005. This study obtains quarterly balance sheet data from
COMPUSTAT BANK and IDEA system of Securities and Exchange Commission (SEC) of
US. We employ maximum likelihood estimation (MLE) method to estimate the parameters of a
M
, b
M
and σ
M
. Table 9 demostrates the parameter distributions of stock-based model of

the total sample banks.
[Insert Table 9 here]
B.2. The Flow-based model
Refer to Liao et al. (2008), the proxy of I can be constructed by Eq.(12):
I = log( Internal liquidity )
Internal liquidity
=
=
Available
Payment
Liquidity t
Obligation t
OCIF t
+ INCF t
OCOF t
+
+
Int
FNCF t
+ t
Tax t
+
+
C t − 1
DA t
+ SI t − 1
(12)
In Eq. (12), OCOF t denotes operating cash outflows; INCF t
represents investing net cash flow in period t ; FNCF t
indicates financing net cash flow excluding cash dividends paid and debt changes during time t . This is the net increase after equity and debt financing, representing a component of available liquidity for a firm. C t − 1
and SI t − 1
denote the beginning balance of cash and short-term investments in period t.
DA t
indicates the amortization of debt principals, calculated from the net decreases of both total short- and long-term debts in the period t . The
OCIF t
indicates the “adjusted” operating cash inflows, can be illustrated as Eq.(13):
OCIF t
=
OCF t
+
Int t
+
Tax t
+
NIAP t
(13)
In Eq. (13), OCF t
, Int t
and Tax t
stand for operating cash flow, interest expenses and income taxes paid during the t period respectively; NIAP t
indicates the net increases in accounts payable during the t period. In the bank case, this study uses the total deposits instead of accounts payable. Since interest payments, income taxes, and net decreases on accounts payable are

obligatory payments, they have to be added back into the original net operating cash flows.
Following Liao et al. (2008), the current study uses the four-quarter moving-average cash flows on flow-related items in Eq. (12) to eliminate the effects of seasonality and management manipulation of credit policy.
Since the COMPUSTAT BANK provides quarterly cash flow data starting 2004, we collect the data before 2004 from financial statements recorded on IDEA system of SEC4.
MLE method is employed to estimate the parameters of a
I
, and σ
I
. Table 10 shows the the distributions of the flow-based model parameters of the total sample banks. In addition, the correlation of the diffusion terms ρ , is calculated as the residuals’ correlation of M t and I t
processes.
[Insert Table 10 here]
C. Empirical Results
Table 11 shows the estimated one year default probabilities of the four structural-form models, the integrated model, and the Standard and Poor’s (S & P) cumulative average default rates (the middle point of the default probability range implied by a sample bank’s credit rating).
[Insert Table 11 here]
Using the S&P default rates as the benchmark, the current study finds that the four structural-form models underestimate the one year default probability in all sample banks. In fact, their estimated default probabilities are almost zero. These results are consistent with the

assertion of Duffie and Lando (2001).
On the other hand, the integrated model obviously performs much better than the structural-form models. The integrated model produces considerably accurate default probability estimation in most sample banks. It indicates that the integrated model is able to improve the effectiveness of structural-form model in evaluating short-term credit risk.
Specifically, the integrated model overestimates six sample banks and underestimate the other twenty-five ones. Among the overestimated banks, AF and NCC are most overestimated
This is the first study that develops an integrated structural-form credit risk model, which combines both stock-based and flow-based corporate credit risk information. Differing from traditional structural-form credit models, the new model considers not only stock-based insolvency but also flow-based insolvency. This model can endogenously generate a firm’s default probabilities, resulting from either asset inadequacy, liquidity crunch, or both. Our numerical analyses show that the new model can catch short-term default risk which is underestimated by Merton-type stock-based models. In addition, an application to American sample banks also shows that the integrated model is able to improve the effectiveness of structural-form model in evaluating short-term credit risk.

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Table 1. Parameters of the sample banks and the numerical analysis
The table presents the setting of parameters for simulations. They are referred to the 31 sample banks which this study uses for preliminary applications in section IV. The characters of the sample banks are depicted in table 8.
Stock- and flow- based parameters are calculated by the maximum likelihood estimation (MLE) method respectively. The correlation of the diffusion terms ρ , is calculated as the residuals’ correlation of stock- and flow- based processes. The evaluating time is the year end of 2005. For simulation, we set a “Good” case of internal liquidity 1.4, which is close to the average of the sample banks’ flow-based condition, and a “Bad” case internal liquidity 0.2, near to the insolvency threshold 0.
Parameters
Sample Bank’s
Average
Base case setting
Parameters Setting for Simulations a
M b
M
σ
M
Panel A. Stock-based model parameters
0.6661 0.6
0.1917 0.2
0.0546 0.06
Panel B. Flow-based model parameters a
I
0.7773 0.7 Base case parameter
± b
I
1.3822 1.4 Base case parameter
±
σ
I
ρ
M
I
0
0
0.4016 0.5 Base case parameter
±
Panel C. Correlation of the two Wiener processes
-0.0382 -0.04 -0.8 –0.5 -0.2 0
0.2 0.5 0.8
Panel D. Initial credit conditions at simulation time
0.1898 0.2
1.3802
Good 1.4
Bad 0.2

Table 2. Default probabilities generated by the hypothetical stock-based and integrated credit risk models
Table 2 demonstrates the simulated default probabilities (denoted as PD) by the assumed stock-based model and the integrated model. These simulations base upon the base case parameters in table 1. Default probabilities are simulated for periods from 1 to 10 years. The generated default probability of the integrated model
(C) is the sum of PD (stock-based model) (A) plus the second term of Eq. 10 (B). The default probabilities are generated under a “Good” internal liquidity scenario ( I
0
= 1.4) and a “Bad” one ( I
0
= 0.2).
PD
=
N
⎝
⎜⎜
µ
M s
M
⎠
⎟⎟
⎞
+
0
∞
N
⎜
⎜
⎛
⎜
⎜
⎝
0 − µ
I
− ρ s
I s
I
( m
− s
M
1 − ρ 2
µ
M
)
⎟
⎟
⎞
⎟
⎟
⎠ n
⎝
⎜⎜
⎛ − s
M
µ
M
⎠
⎟⎟
⎞ dm
(Equation 10)
Default probabilities ( %)
1-year 2-year 3-year 4-year 5-year 6-year 7-year 8-year 9-year 10-year
I
0
= 1.4 (“Good” internal liquidity scenario)
PD (stock-based) (A)
2 nd term of Eq.10 (B)
PD (Integrated model)
(C=A+B)
0.0006 0.0064
0.0107
0.0123
0.0128
0.0129
0.0130
0.0130
0.0130
0.0130
0.0068 0.0315
0.0422
0.0451
0.0459
0.0461
0.0461
0.0461
0.0461
0.0461
0.0074
0.0379
0.0528
0.0574
0.0587
0.0591
0.0592
0.0592
0.0592
0.0592
I
0
= 0.2 (“Bad” internal liquidity scenario)
PD (stock-based) (A)
2 nd term of Eq.10 (B)
PD (Integrated model)
(C=A+B)
0.0006 0.0064
0.0107
0.0123
0.0128
0.0129
0.0130
0.0130
0.0130
0.0130
1.4181 0.3509
0.1405
0.0827
0.0622
0.0536
0.0497
0.0479
0.0470
0.0466
1.4187
0.3573
0.1512
0.0950
0.0750
0.0666
0.0628
0.0609
0.0600
0.0596
22

Table 3. Default probabilities generated by the integrated model under different initial internal liquidity scenario (
I
0
) settings
Table 3 shows the default probabilities generated by the integrated model under different settings of I
0
. The model parameters are the base case parameters in table 1. Default probabilities are generated for periods from 1 to 10 years. I
0
is set to be from the “Bad” scenario 0.2, increase 0.3 for every internal liquidity case, to the “Good” scenario 1.4,
Default probabilities (%)
Initial Internal Liquidity level
1-year 2-year 3-year 4-year 5-year 6-year 7-year 8-year 9-year 10-year
I
0
= 0.2
1.4187 0.3573
0.1512
0.0950
0.0750
0.0666
0.0628
0.0609
0.0600
0.0596
I
0
= 0.5
I
0
= 0.8
0.4689
0.1336
0.2074
0.1181
0.1158
0.0836
0.0705
0.0888
0.0736
0.0663
0.0646
0.0618
0.0627
0.0609
0.0605
0.0598
0.0600
0.0596
0.0595
0.0594
I
0
= 1.1
I
0
= 1.4
0.0330
0.0074
0.0666
0.0379
0.0683
0.0528
0.0650
0.0574
0.0624
0.0587
0.0608
0.0591
0.0600
0.0591
0.0596
0.0592
0.0594
0.0592
0.0593
0.0592
23

Table 4. Generated default probabilities under different settings of correlation coefficient
ρ
between two credit risk indicators
Table 4 demonstrates the sensitive analysis of effects of the correlation coefficient ρ between the Wiener processes of credit risk indicators of the hypothetical stock-based and flow-based models. The other parameters of the models are the base case parameters stated in table 1. Default probabilities are generated for periods from 1 to 10 years. ρ is set to 0, ± 0.8, ± 0.5 and ± 0.2, under the “Good” scenario of I
0
(=1.4).
Default probabilities (%) correlation coefficient 1-year 2-year 3-year 4-year 5-year 6-year 7-year 8-year 9-year 10-year
ρ = 0.8 0.0089 0.0372
0.0501
0.0545
0.0557
0.0560
0.0561
0.0561
0.0562
0.0562
ρ
= 0.5 0.0036 0.0406
0.0511
0.0524
0.0535
0.0537
0.0538
0.0539
0.0539
0.0539
ρ = 0.2
ρ
= 0
0.0036
0.0036
0.0171
0.0100
0.0343
0.0151
0.0366
0.0174
0.0402
0.0404
0.0163
0.0165
0.0404
0.0404
0.0166
0.0166
0.0404
0.0166
0.0404
0.0166
ρ = - 0.2
ρ = - 0.5
ρ = - 0.8
0.0063
0.0038
0.0000
0.0119
0.0117
0.0000
0.0141
0.0143
0.0000
0.0163
0.0165
0.0000
0.0171
0.0174
0.0000
0.0173
0.0177
0.0000
0.0174
0.0177
0.0000
0.0174
0.0178
0.0000
0.0174
0.0178
0.0000
0.0174
0.0178
0.0000
24

Table 4. Generated default probabilities under different settings of correlation coefficient
ρ
between two credit risk indicators (Cont.)
Table 4 demonstrates the sensitive analysis of effects of the correlation coefficient ρ between the Wiener processes of credit risk indicators of the hypothetical stock-based and flow-based models. The other parameters of the models are the base case parameters stated in table 1. Default probabilities are generated for periods from 1 to 10 years. ρ is set to 0, ± 0.8, ± 0.5 and ± 0.2, under the “Good” scenario of I
0
(=1.4).
Default probabilities (%) correlation coefficient 1-year 2-year 3-year 4-year 5-year 6-year 7-year 8-year 9-year 10-year
ρ = 0.8 0.0026 0.3554
0.1468
0.0909
0.0720
0.0634
0.0595
0.0581
0.0571
0.0567
ρ = 0.5 1.4157 0.3581
0.1551
0.0949
0.0744
0.0637
0.0597
0.0579
0.0570
0.0566
ρ = 0.2 1.2764 0.3451
0.1447
0.0927
0.0714
0.0537
0.0500
0.0482
0.0473
0.0469
ρ
= 0 0.0043 0.1847
0.0332
0.0160
0.0165
0.0040
0.0038
0.0036
0.0036
0.0036
ρ = - 0.2 0.0041 0.0038
0.0142
0.0159
0.0039
0.0180
0.0174
0.0174
0.0172
0.0172
ρ = - 0.5 0.0036 0.0081
0.0149
0.0041
0.0172
0.0039
0.0181
0.0179
0.0178
0.0178
ρ = - 0.8 0.0000 0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
0.0000
25

Table 5. Sensitive analysis of the effects of mean-reverting speed parameter a
I
on default probabilities
Table 5 demonstrates the sensitive analysis of the effects of mean-reverting speed parameter a
I
on default probabilities. The other parameters of the model are the base case parameters stated in table 1. Default probabilities are generated for periods from 1 to 10 years.
a
I
is set to 0.7 (base case), ± 20%, ± 50%, under the “Good” and the “bad” scenarios of I
0
respectively.
Simulated default probabilities (%)
Changes Rates 1-year 2-year 3-year 4-year 5-year 6-year 7-year 8-year 9-year 10-year
+50%
+20%
0% ( a
I
=0.7)
-20%
-50%
I
0
=1.4 (“Good” internal liquidity scenario)
0.0006
0.0086
0.0131
0.0148
0.0153
0.0154
0.0155
0.0155
0.0155
0.0155
0.0035
0.0175
0.0243
0.0264
0.0270
0.0272
0.0272
0.0272
0.0272
0.0272
0.0074
0.0379
0.0528
0.0574
0.0587
0.0590
0.0591
0.0592
0.0592
0.0592
0.0159
0.0924
0.1387
0.1563
0.1623
0.1643
0.1649
0.1651
0.1652
0.1652
0.0487
0.3542
0.6303
0.7940
0.8811
0.9256
0.9480
0.9592
0.9648
0.9676
+50%
+20%
0%
( a
I
=0.7)
-20%
-50%
I
0
=0.2 (“Bad” internal liquidity scenario)
0.1220
0.0191
0.0153
0.0154
0.0155
0.0155
0.0155
0.0155
0.0155
0.0155
0.5635
0.1019
0.0456
0.0336
0.0298
0.0283
0.0277
0.0275
0.0273
0.0273
1.4187
0.3573
0.1512
0.0950
0.0750
0.0666
0.0628
0.0609
0.0600
0.0596
3.2700
1.2034
0.5741
0.3505
0.2574
0.2139
0.1918
0.1800
0.1735
0.1699
9.5539
6.0614
4.0105
2.8422
2.1585
1.7443
1.4851
1.3181
1.2081
1.1343
26

Table 6. Sensitive analysis of the effects of long-term level parameter b
I on default probabilities
Table 6 demonstrates the sensitive analysis of the long-term level parameter b
I
on default probabilities. The other parameters of the model are the base case parameters stated in table 1. Default probabilities are generated for periods from 1 to 10 years. b
I
is set to 1.4 (base case), ± 20%, ± 50%, under the “Good” and the “bad” scenarios of I
0
respectively.
Simulated default probabilities (%)
Changes Rates 1-year 2-year 3-year 4-year 5-year 6-year 7-year 8-year 9-year 10-year
+50%
+20%
0% ( b
I
=1.4)
-20%
-50%
I
0
=1.4 (“Good” internal liquidity scenario)
0.0006
0.0064
0.0107
0.0123
0.0128
0.0130
0.0130
0.0130
0.0130
0.0130
0.0006
0.0106
0.0150
0.0163
0.0166
0.0166
0.0166
0.0166
0.0166
0.0166
0.0074
0.0379
0.0528
0.0574
0.0587
0.0590
0.0591
0.0592
0.0592
0.0592
0.0305
0.1909
0.3066
0.3635
0.3902
0.4029
0.4090
0.4121
0.4135
0.4143
0.2150
1.6617
3.0608
3.9282
4.4007
4.6455
4.7696
4.8318
4.8629
4.8783
+50%
+20%
0% ( b
I
=1.4)
-20%
-50%
I
0
=0.2(“Bad” internal liquidity scenario)
0.0814
0.0098
0.0111
0.0123
0.0128
0.0130
0.0130
0.0130
0.0130
0.0130
0.4996
0.0725
0.0283
0.0205
0.0182
0.0173
0.0169
0.0167
0.0166
0.0166
1.4187
0.3573
0.1512
0.0950
0.0750
0.0666
0.0628
0.0609
0.0600
0.0596
3.5313
1.4660
0.8262
0.5946
0.4985
0.4551
0.4346
0.4246
0.4198
0.4174
10.9070
7.9595
6.3972
5.6319
5.2577
5.0737
4.9829
4.9379
4.9156
4.9045
27

Table 7. Sensitive analysis of the effects of standard deviation parameter
σ on default probabilities
I
Table 7 demonstrates the sensitive analysis of the standard deviation parameter
Default probabilities are generated for periods from 1 to 10 years.
σ
I
. The other parameters of the model are the base case parameters stated in table 1.
σ
I
is set to 1.5 (base case), ± 20%, ± 50%, under the “Good” and the “bad” scenarios of I
0
respectively.
Simulated default probabilities (%)
Changes Rates 1-year 2-year 3-year 4-year 5-year 6-year 7-year 8-year 9-year 10-year
+50%
+20%
0% (
σ
I
=0.5)
-20%
-50%
I
0
=1.4 (“Good” internal liquidity scenario)
0.5476
1.1396
1.3132
1.3579
1.3691
1.3719
1.3726
1.3727
1.3728
1.3728
0.0741
0.2258
0.2810
0.2961
0.2999
0.3009
0.3012
0.3013
0.3013
0.3013
0.0074 0.0379
0.0528
0.0574
0.0587
0.0591
0.0591
0.0592
0.0592
0.0592
0.0006
0.0074
0.0122
0.0139
0.0145
0.0147
0.0147
0.0148
0.0148
0.0148
0.0006
0.0064
0.0107
0.0123
0.0128
0.0130
0.0130
0.0130
0.0130
0.0130
+50%
+20%
0% (
σ
I
=0.5)
-20%
-50%
I
0
=0.2 (“Bad” internal liquidity scenario)
7.1946
3.6204
2.3300
1.8099
1.5806
1.4738
1.4224
1.3973
1.3849
1.3788
3.3866
1.2395
0.6497
0.4496
0.3694
0.3338
0.3171
0.3091
0.3051
0.3032
1.4187 0.3573
0.1512
0.0950
0.0749
0.0666
0.0628
0.0609
0.0600
0.0596
0.3075
0.0440
0.0201
0.0165
0.0155
0.0151
0.0150
0.0149
0.0148
0.0148
0.0006
0.0060
0.0107
0.0122
0.0128
0.0129
0.0130
0.0130
0.0130
0.0130
28

Table 8. Characteristics of the sample banks sorted by SIC codes
The table presents some market and credit risk characteristics of sample banks sorted by SIC codes.
The number for each characteristic is the average of each group banks. The bank’s rating, obtained from COMPUSTAT, is the Standard & Poor’s long-term domestic issuer credit rating in a point system.
The market leverage ratio is derived from the book value of total liabilities over the asset market value, which is the sum of the market value of a bank’s equity and its book value of total liabilities, evaluating on the year end of 2005.
Industry
National commercial banks
State commercial banks
Savings institutions, federally chartered
(SIC code: 6021) (SIC code: 6022) (SIC code: 6035)
Numbers 19 10
Percent of sample
Credit rating of the sample banks
61.29%
AA- : 3
A+ :4
A: 5
A-: 4
BBB+:2
BBB-:1
32.26%
AA- : 1
A+ :1
A: 2
A-: 3
BBB+:1
BBB:2
2
6.25%
BBB:2
Average market leverage ratio
82.71% 82.00% 87.62%
29

Table 9. The distribution of the stock-based model parameters of the sample banks
This table presents the distribution of the estimated parameters of the stock-base model.
Maximum likelihood estimation method is employed to estimate the Ornstein Uhlenbeck process.
Data used is from 1994~2005. M
0
is evaluated at end of 2005.
Quartiles of the Parameters of the Stock-based Model the of Total Sample Banks
(Grouped by Ratings)
25% quartile median 75% quartile
Panel A: Above (including) A+ : 9 firms a
M b
M
ρ
M
M
0
0.4372 0.7059 0.8294
0.1415 0.1818 0.2348
0.0464 0.0610 0.0808
0.1497 0.1810 0.2137
Panel B: A~A- : 14 firms a
M b
M
ρ
M
M
0
0.5022 0.6261 0.7870
0.1711 0.1851 0.2130
0.0404 0.0474 0.0505
0.1726 0.1987 0.2160
Panel C: Under (including) BBB+ : 8 firms a
M b
M
ρ
M
M
0
0.5040 0.5357 0.6513
0.1269 0.1443 0.1868
0.0327 0.0354 0.0396
0.1354 0.1606 0.2411
30

Table 10. The distribution of the flow-based model parameters of the sample banks
This table presents the distribution of the estimated parameters of the flow-base model.
Maximum likelihood estimation method is employed to estimated the Ornstein Uhlenbeck process.
Data used is from 1994~2005. I
0
is evaluated atr end of 2005.
Quartiles of the Parameters of the Flow-based Model the of Total Sample Banks
(Grouped by Ratings)
25% quartile median 75% quartile
Panel A: Above (including) A+ : 9 firms
ρ
I
I
0 a
I b
I
0.4233 0.7081 0.8219
1.4738 1.9687 2.1440
0.3294 0.3756 0.3899
1.3549 1.6806 2.1674
Panel B: A~A- : 14 firms a
I b
I
ρ
I
I
0
0.4180 0.6878 1.1153
1.0361 1.1745 1.3586
0.2228 0.3022 0.6072
0.8872 1.1502 1.4531
Panel C: Under (including) BBB+ : 8 firms
ρ
I
I
0 a
I b
I
0.5191 0.7614 1.2647
0.9654 1.0556 1.2647
0.2846 0.3567 0.6835
0.8279 1.1789 1.4170
31

Table 11. Comparisons of 1-year default probabilities estimated by structural-form models
The table presents the comparison of 1-year default probabilities estimated by different structural-form models, including Merton model (M, 1974), Leland and Toft (LT, 1996),
Longstaff and Schwartz(LS, 1995) , Collin-Dufresne and Goldstein (CDG, 2001), and the integrated one. Stock- and flow- based parameters are calculated by the maximum likelihood estimation
(MLE) method respectively. The correlation of the diffusion terms ρ , is calculated as the residuals’ correlation of stock- and flow- based processes. The evaluating time point is the year end of 2005. S&P default rate is the middle point of the default probability range implied by a sample bank’s credit rating.
Ticker S & P
Default Rate model
Rating ： AA-
BAC 2.29E-71 7.01E-31
STT
WFC
2.00E-04
1.11E-15 2.48E-50 2.93E-16 4.72E-20 4.75E-06
3.86E-37 3.17E-88 5.81E-31 1.17E-41 5.49E-09
NTRS 2.02E-14 2.51E-50 5.12E-17 3.64E-22 4.76E-04
Rating ： A+
USB 5.66E-23 1.22E-52 7.34E-21 3.37E-35 6.32E-02
WB 1.24E-38 7.56E-66 2.89E-19 1.51E-24 2.66E-06
STI
5.00E-04
2.50E-33
4.96E-34 2.84E-85 1.44E-23 2.05E-27 3.02E-08
BK 3.84E-15 5.64E-48 1.05E-17 1.04E-26 9.50E-05
Rating ： A
SNV
CMA
MI
NCC
6.00E-04
BBT
RF
WL
3.88E-22 1.50E-43 7.70E-19 2.19E-23 1.79E-05
1.82E-11 1.63E-54 3.82E-17 7.93E-24 6.75E-08
8.29E-29 2.70E-56 3.13E-20 1.04E-23 4.18E-11
7.45E-19 5.43E-53 1.15E-14 1.05E-19 1.42E-01
4.44E-26 1.00E-54 4.95E-19 2.42E-28 4.22E-07
7.50E-30 1.71E-48 2.04E-15 5.01E-20 1.06E-07
3.89E-18 5.36E-43 1.01E-16 9.60E-27 8.94E-10
32

Table 11. Comparisons of 1-Year default probabilities estimated by structural-form models (Cont.)
The table presents the comparison of 1-yr default probabilities estimated by different structural-form models, including Merton model (M, 1974), Leland and Toft (LT, 1996),
Longstaff and Schwartz(LS, 1995) , Collin-Dufresne and Goldstein (CDG, 2001), and the integrated one. Stock- and flow- based parameters are calculated by the maximum likelihood estimation
(MLE) method respectively. The correlation of the diffusion terms ρ , is calculated as the residuals’ correlation of stock- and flow- based processes. The evaluating time point is the year end of 2005. S&P default rate is the middle point of the default probability range implied by a sample bank’s credit rating
Ticker S&P Integrated
Default Rate model
Rating ： A-
7.00E-04
VLY 1.32E-20
MTB
Rating ： BBB+
BPOP
Rating ： BBB
AF
Rating ： BBB-
33

In this appendix we provide a brief discussion on the employed four structured-form credit models. We also discuss the estimation of default probabilities under physical measures.
1. Merton Model (the M model)
The process of firm value is assumed to follow a geometric Brownian motion: dV t
= µ V dt + σ V dZ t
(A.1)
Default occurs when asset value drops under total liabilities and the default probability is given by equation (A.2).
(
T
≤ K ) = (
T d
2
= ln
⎛
⎝
V
K
0 ⎟
⎛
⎠ ⎝
σ
µ
T
≤
−
σ
2
2 ⎞
⎠
T
V
0
K
σ
+ µ −
σ
2
2
) T
) = (N d
2
)
T
(A.2)
Where K is the he book value of total liabilities and µ is the mean of the asset return.
When under risk-neutral measure, µ would be replaced by risk-free rate. The difference between the physical measure and risk-neutral measure is risk premium ( λ ).
2. Leland and Toft Model (the LT model)
34

The process of firm value is assumed to follow a geometric Brownian motion: dV t
= [ µ − δ ]
V dt + σ V dZ t
(A.3)
Default occurs when asset value drops under the default boundary V
B
and f(s; V,
V
B
) denotes the density of the first passage time s to V
B from V.
The present value of debt can be defined as equation (A.4). d V V t = ∫
0 t
+ ∫
0 t
[ − ( ; , )
] + − rt
[ − ( ; , )
]
(A.4) e − rs ρ ( )
B
( ; , )
Where F(s) is the cumulative distribution function of the first passage time of bankruptcy, c is the interest payout ratio per year, P is the book value of total liabilities and ρ is the recovery rate
Integrating the equation (A.4):
Where = t ∫ s = 0
− rs
= r
+ e − rt
⎡
⎢
( ) − r
⎤
⎦
[
1 − ( )
] +
⎡
⎣
ρ t V
B
− r
⎤
G t (A.5)
Rubinstein and Reiner (1991) have defined
=
⎛ ⎞
⎝ ⎠
[ ] +
V
B
[
2
]
(A.6)
From Harrison (1990), we know that
( ) = [ ]
⎛ ⎞
− 2 a
V
B
[
2
]
(A.7)
35

Where ( ) =
( σ
σ t
) a = r
σ
−
2
δ
−
1
2
=
( − +
σ t
σ )
=
⎛ V ⎞
⎜
V
B
⎟
=
( − −
σ t
σ )
=
( a σ ) + 2 r σ 2
⎦
1 2
σ 2
=
( − +
σ t
σ )
(A.8)
The default boundary V
B
is given as
V
B
=
( )( ( ) − B ) − ( ) − τ
1 + α x (1 α ) B
Cx r
(A.9)
Where C is the interest payout per year, P is the book value of total liabilities,
1α represents the recovery rate, x = a + z , T is the maturity of liabilities, here, we assume is 10 years, τ is the tax rate and tax benefit accrue at rate τ C per year as long as V ＞ V
B
,
A = 2 ae − rT ( σ T
B = − (2 z +
σ
2
− ( σ T
) ( σ T ) −
σ
2
T
) −
σ
(
2
σ
T
T
( σ T ) +
2 e − rT
σ T
( σ T z a )
) ( z a )
σ
1
; = +
(A.10)
The default probability can be calculated from F(t, V, V
B
) by given t.
When under physical measure, the default probability in equation (A.7) will become:
F P ( t ) = N
[ h
1
P ( t )
]
+
V
V
B '
− 2 a P
N
[ h
2
P ( t )
]
(A.11)
Where h
1
P ( t ) =
− b '' − a P σ 2 t
σ t
; h
2
P ( t ) =
− b '' + a P σ 2 t
σ t
; a p = a +
σ
λ
2
; b '' = ln
V
V
B ''
The default boundary V
B '' is given as:
36

V
B
''
=
( )( P ( rT ) − B P ) − P ( )
1 + α x P (1 α ) B P
− τ
(A.12)
A P
B P
= 2 a e −
= − (2 z + rT
σ
2
( P σ
) (
T
σ
− ( σ T
T ) −
σ
2
T
) −
( σ
σ
2
T
T
( σ T
) ( z a P
) +
2 e − rT
σ T
) +
σ
1
( P σ T
; x P = a P + z z a P )
3. Longstaff and Schwartz Model (the LS model)
The LS model relaxes the assumption of constant interest rate under the M and
LT model. The interest rates under the LS model follow the stochastic process described by the Vasicek model as: dV t
= µ V dt + σ V dZ
1 dr t
= ( ζ β t
+ σ r dZ
2
2
= ρ dt
(A.13)
The default probability is defined as Q in the following:
= n ∑
(A.14) i = 1 q i
Where q
1
= N a i = q i
= N a a i
=
− ln
− i − 1 ∑ j = 1 q N b i
−
= b ij
= n
( / , ) − ( / , )
( / ) − ( / )
= ⎜
⎛
⎝
α ρσ σ
β r −
+ ⎜
⎛
⎝ r
−
α
2
+
σ
β β β r
2
3
σ
β
⎟
⎞
⎠
2 r
2
(
−
−
σ v
2
2
⎟
⎠
⎞ t + ⎜
⎝
⎛ ρσ σ r
β 2
β )
+
2
σ
β r
2
2
⎟
⎞
⎠ exp( − β T β t
−
⎛
⎜
⎝ 2
σ
β r
2
3
⎞
⎟
⎠ exp( − β T − − β t )),
−
= ⎜
⎛
⎝
ρσ σ r
β
+
σ
β
2 r
2
+ σ v
2
⎠ ⎝
ρσ σ
β 2 r +
2 σ
β 3 r
2
⎟
⎞
⎠
( − − β t )
)
+
⎛
⎜
⎝ 2
σ
β r
2
3
⎞
⎟
⎠
(
− ( − β t
) )
(A.15)
37

Where α represents the sum of the parameter ζ and a constant representing the market price of interest rate risk. And q i
represents the first passage density which default occurs at time i. The default probability Q can be derived by accumulated q i .
The discussion of transferring to estimating default probabilities under
P-measure for LS model illustrated in the next CDG part because LS mode is a special case of CDG model when its mean-reverting speed of leverage process is set to be zero.
4. Colline-Dufresne and Goldstein Model (the CDG model)
Colline-Dufresne and Goldstein (2001) propose a model with stochastic interest rate and mean reverting leverage ratio. dV t
= ( r t
− dr t
= ( α β
δ ) V dt
)
+ σ
+ σ r dZ
2
Q t
Q
1 t
(A.16)
1 t
Q
2 t
= ρ dt
By variable transformation y = log , =
⎛
⎝ r t
δ
σ
2
2 ⎞
⎠ dt + σ v dZ Q
1 t
(A.17)
The authors set k t
as the log default threshold and varies with time, dk t
= ( y t
− − t
) (A.18)
Therefore, if log leverage is l t
= − y t
and d l t
= dk t
− dy t
= ϑ
( l
Q
− l t
) dt − σ v dZ Q
1 t
(A.19)
It represents the default takes place if l t
≥ 0 . The CDG model estimates default
38

probability Q by two-dimensional Fortet method (1943) to deal with two stochastic process l t and r t
as follows:
Divide time and interest rate: t j
= / ≡ ∆ ; i
= + × ∆ r ( r and r represent interest rates down limit and up limit respectively). T is maturity time.
Then we can obtain the default probability Q as follows:
Q T
( r
( i
,
1
) l T
)
≡ nT nr j
∑ ∑
= 1 i = 1
= ∆ Ψ (
,
1
) (
1, 2,..., n r
)
Ψ
ψ
⎢
⎡
⎣
− v j − 1
∑ ∑
= 1 u nr
= 1 i
( v
) ψ
(
1,..., n r
)
(
, s
,
= π
(
,
0
, 0
)
N
)
= π (
, s
,
( i j u
, v
(
2,..., n
T
⎝
⎜
⎜
⎛
)
µ
∑
(
(
, l
0
, r
0 r t l
0 r
⎝
⎜
⎜
⎛ µ
∑
(
( t t
, 0
)
) l s
= l
⎠
⎟
⎟
⎞ l s
= l r s r s
)
)
⎠
⎟
⎟
⎞
)
)
⎥
⎤
⎦
(A.20)
Where π
(
, s
,
)
is the transition density which time passes from s to t and interest rate changes fro r s to r t
. µ
( r t l s r s
)
a nd
∑ (
, l s
, ,
) are conditional expectation and conditional variance of log leverage at time t.
Eom, Helwege and Huang (2004) derived the above log leverage threshold to be as d ln K t
= k l
⎡⎣ ln
( t
/ t
) − − φ ( r t
− θ ) ⎤⎦ dt (A.21)
It degenerates to the LS model when k l
= 0 .
The default probability Q can be expressed as:
39

Q FT
(
,
) = i n
∑
= 1
⎛ q t
⎝ i −
1
2
; t
0
⎞
⎟
⎠
, t
0
= 0, t i
= iT n
⎛
⎜
⎝ i −
1
2
; t
0
⎞
⎟
⎠
=
(
( i
;
0
) = −
=
( i
(
( i
( tj tj
)
)
X
0
≡ /
0
,
(
,
( u
)
≡
(
( i 0
) )
0
,
0
0
,
0
)
)
0
,
− ∑ i − 1 j = 1
⎛
⎜
⎛
⎜
⎝
⎛
⎝ j −
2
1
;
0
⎞
⎠
⎛
⎜
⎛
⎝ i
, i r
0
)
≡ E
0
FT
[ ln X t
] (
0
,
0
,
0
)
−
(
, j −
1
2
⎞
⎟
⎠
⎞
⎟
0
,
0
) j −
1
2
)
⎞
⎠
⎞
⎟
≡ var
0
F
T
[ ln X t
] cov F
T
0
[ ln X t
,ln X
(
0
) u
]
, u ∈
( u
) (
0
)
−
(cov F
T
0
[ ln X t
,ln X u
]
)
(
0
)
2
, u ∈
(A.22)
Under T-forward measure, e ln X t
= ln X
0
+
(
− 1
)
+ ∫ t
0
⎣
⎡ (
1 + φ ) u
− ρσ σ r r t
= r e − β t +
⎝
⎜
⎛ α σ
β β r
2
2
⎟
⎞
⎠
(
1 − e − β t
)
+
2
σ
β r
2
2
⎦
⎤ − ∫ t
0
σ v e dZ F
T
1 u e − β T
( e β t − e − β t
)
+ σ r e − β t t e dZ
0
∫ F
T
2 u
(A.23) where v ≡
( v − φθ )
−
(
δ + σ 2 v
2
) k l
, B =
1
β
(
1 − e − β (
T − t
)
)
And taking expectation under T-forward measure will make equation (A.23) become:
40

E
0
FT
[ ] = r e − β t e E
0
F
T
[ ln X t
] =
+
⎛ α σ
β β ln r
2
2
⎞
⎠
(
1 − e − β t
)
+
2
σ
β r
2
2 e − β T
( e β T − e
X
0
+
( ) ∫
0 t (
1 + φ )
0
FT
[ ]
− β t
)
− cov F
0
T
[ ln X t
, ln
]
ρσ σ r
β l
+ )
⎡
⎢
⎣ e k l
− 1
− e β T e
( k l k l
+ β
+
) t
β
− 1
⎤
⎥
⎦
I
1
I I
3
I
4
(A.24)
Which,
I
1
=
σ v
2
2 k l
( e 2 − 1
)
I
2
( φ k l
)
ρσ σ r k l
+ β
⎡
⎢
⎣ e 2
2 k l
− 1
− e
( k l k l
− β
−
) u
β
− 1
⎤
⎥
⎦
I
3
( φ k l
)
ρσ σ r k l
+ β
⎢
⎣
⎡
1 − k l e
( k l
− β ) t
− β
+ e 2
2 k l
− 1
+ e
( k l
+ β ) u e
( k l
− β ) t k l
−
− e
( k l
− β ) u
β
⎤
⎥
⎦
I
4
( φ k l
) 2
σ
2 r
β
2
⎡
⎢
⎣
−
( e
( k l
− β ) t − 1
) ( e
( k l
− β )
( k l
− β ) 2 u − 1
)
+
( e
( k l
+ β ) u − 1
) e
( k l
− β ) t k l
2
− e
( k l
− β ) u
− β 2
− k l
2
β
− β 2 e 2 k l
− 1
+ k l
2
1
− β 2
(
− e
( k l
− β ) u + e 2
) ⎥
⎦
⎤
⎥
(A.25)
For each period,
(
,
0
) (
0
,
0
)
,
(
, u
) ( u
)
,
(
0
)
,
( ) and
⎛
⎜
⎝ i −
1
2
; t
0
⎞
⎟
⎠
could be calculated by above equations and then we can get
Q F
T
(
,
)
.
If we want to estimate default probabilities under physical measure (P-m easure), we have to adjust the probability measure and modify in the following steps.
Step 1. The model setting in equation (A.16) is modified by adding a risk premium
41

parameter ( λ ) in the drift term of equation (A.17).
Step 2. Adju sting from T-for ward measure to P measure in equation (A.
22) so that
(
0
)
and
(
0
,
0
)
will become as equation (A.26)
(
,
0
,
0
)
≡ E
0
[ ln X t
] (
0
)
≡ [
X t
]
(A.26)
Therefore equation (A.23) will become equation (A.27) when it is under P measure. e ln X t r t
= r e − β t
= ln
+
α
β
X
0
+
( )
λ e k l
− 1
+ t
(
1 +
0
∫ φ
(
1 − e − β t
)
+ σ r e − β t t e dZ
0
∫
2 u l
) − ∫ t
0
σ v e dZ
1 u
(A.27)
It follows that e E
0
[ ln X t
] = ln X
(
1
0
[
X t
, ln
]
(
+
+ u )
( k l
φ )
⎛
⎜ r
0
= + I
2
)
−
α
β
⎞
⎠
λ e − 1 e
( k l k l
− β ) t k l
− β
+
− 1
( e −
) ( + k l
)
β k l
+ + I
4
(A.28)
Where, the param eters I
1
, I
2
, I
3
, I
4 in equation (A.28) are defined as those in equa tion (A.25).
Based upon the above adjustments, the default probability Q , expressed in equation (A.22), can be recalculated to a new defau lt probability P under the physical meas ure by considering the asset risk premium ( λ ).
probability P under the physical measure could b e obtained by setting k l
to zero.
42

43