measurement and comparison of credit risk by a markov chain an empirical investigation of bank loans in taiwan

International Research Journal of Finance and Economics
ISSN 1450-2887 Issue 30 (2009)
© EuroJournals Publishing, Inc. 2009


Measurement and Comparison of Credit Risk by a Markov
Chain: An Empirical Investigation of Bank Loans in Taiwan


Su-Lien Lu
Assistant Professor,Department of Finance,National United University
No. 1, Lien-Da, Kung-Ching Li, Miao-Li, 360 Taiwan R.O.C
E-mail:
Tel: 886-37-381859; Fax: 886-37-338380

Kuo-Jung Lee
Assistant Professor,Department of Commerce Automation and Management
National Pingtung Institute of Commerce
E-mail:


Abstract

Transition matrices are at the center of modern credit risk management. In this
paper, the estimation of transition matrices based on discrete- and continuous-time Markov
chain models is presented. These different models were applied to bank loans, including
secured and unsecured loans, for twenty-eight banks in Taiwan. Furthermore, the
differences between discrete- and continuous-time methods are compared with a statistics,
mobility estimator. Substantial differences between the credit risks of the two methods
were found. The continuous-time Markov chain model can hood up nicely with rating-
based term structure modeling. The empirical results indicate that the discrete-time Markov

chain model may underestimate default probabilities when the dynamic rating process is
not taken into consideration. Consequently, the conclusion is made that care has to be taken
when discrete- and continuous-time Markov chain models are employed for dynamic credit
risk management.


Keywords: Credit risk, discrete-time Markov chain model, continuous-time Markov chain
model, bank loans
JEL Classification Codes: G10, G21

1. Introduction
In the past ten years, major developments in financial markets have led to a more sophisticated
approach to credit risk management. The origination of credit is still based on the relationship between
the banker and his client. In the banking industry, the classic risk is credit risk that may cause a
financial institution to become insolvent or result in a significant drain on capital and net worth that
may adversely affect its growth prospects and ability to compete with other financial institutions.
Therefore, credit risk management has become a major concern for the banking industry and other
financial intermediaries. This is also stated by the Basel Committee on Banking Supervision (“the
Committee”) that formalizes a universal approach to credit risk in financial institutions.
In fact, the intention of the Committee is to assure the safety and soundness of the financial
system. To achieve this goal, the Committee issued the “International Convergence of Capital
International Research Journal of Finance and Economics - Issue 30 (2009) 109
Measurement and Capital Standards” document, published in July 1988. Furthermore, the treatment of
market and operational risk were incorporated in 1996 and 2001, respectively. In June 2004 the
Committee published the final draft of the revised framework for capital measurement and capital
standards.
Traditional credit analysis is an expert system that relies on the subjective judgment of trained
professionals, implying that credit decisions are the reflection of personal judgment about a borrower’s
ability to repay. However, traditional credit analysis has often lulled banks into a false sense of
security, failing to protect them against the many risks embedded in their business. In contrast with the

traditional approach, the other approach is primarily based on statistical methods, such as presented by
Jarrow and Turnbull (1995) and Jarrow, Lando and Turnbull (1997). Jarrow and Trunbull (1995) used
matrices of historical transition probabilities from original ratings and recovery values at each terminal
state. The Jarrow, Lando and Turnbull (1997) method is based on the risk-neutral probability valuation
model for pricing securities by transition matrices. Consequently, a crucial element in such models is
the transition matrix.
Theoretically, transition matrices can be estimated for any desired transition horizon. Generally,
transition matrices are estimated by a yearly time horizon, such as Carty and Lieberman (1996), Wei
(2003) and Lu and Kuo (2006). The computation of such transition matrices estimated from yearly data
implies the assumption that the underlying process is a discrete-time Markov chain model. However,
large movements are often achieved via some intermediary steps implying that there are no transitions
from AAA to default, but there are transitions from AAA to AA and from AA to default. Thus,
transitions of the intermediate state ( defaultAAAAA →→ ) contribute to the estimation of the
transition matrices. In other words, the shorter the measurement interval, the fewer rating changes are
omitted. Therefore, transition matrices estimated over short time periods best reflect the rating process
and the underlying process is a continuous-time Markov chain model. The information is gained using
the full information of exact transitions, which the discrete-time model ignores, but the continuous-
time model does not.
The purpose of this paper is to assess the credit risk of bank loans using two different Markov
chain models, the discrete- and continuous-time models. The different Markov chain models depend on
the generation of transition matrices. The discrete-time Markov chain model uses the discrete
multinomial (or cohort) method and the continuous-time Markov chain model is estimated with
continuous hazard rate (or duration) methods. Therefore, we also compare the estimated results of
different methods. Since continuous-time methods incorporate full information of rating transitions, it
seems that continuous-time approaches bring more efficient results than discrete-time methods.
There are four contributions in this paper. First, the credit risk of bank loans is discussed;
including secured and unsecured loans, both from analytical and empirical perspectives. To our
knowledge, not much research has been done on the estimation of loans’ transition matrices
considering both discrete- and continuous-time approaches. This paper adopts discrete- and
continuous-time Markov chain models for measuring the credit risk of bank loans from a more

comprehensive perspective than previous studies.
Second, although risk premium plays a crucial role in gauging the credit risk of bank loans,
previous research has handled the risk premium as a time-invariant (Jarrow, Lando and Turnbull, 1997;
Wei, 2003). In fact, the risk premium is actually always a time-variant parameter (Kijima and
Komoribayashi, 1998; Lu and Kuo, 2006). Therefore, the assumptions made in previous research were
relaxed by incorporating the time-variant risk premium into the transition matrices making it more
elaborate.
Third, a comparison of the estimated results of discrete- and continuous-time Markov chain
models is given. It was found that the continuous-time approach has a more reliable default probability
than the discrete-time approach. The discrete-time estimator may underestimate the default
probabilities due to the neglect of some rating transitions whereas the continuous-time estimator
incorporates all information on the exact timing of rating transitions.
110 International Research Journal of Finance and Economics - Issue 30 (2009)
Fourth, a statistics mobility estimator was extended for investigating the migration size of
discrete- and continuous-time transition matrices. The mobility estimator was designed to give a
measure of the transition matrices propensity, which had been developed by Jafry and Schuermann
(2004). It was found that the discrete-time method had a higher mobility estimator than the continuous-
time method, implying that a higher off-diagonal probability is concentrated in a discrete-time
transition matrix rather than diluted in a continuous-time transition matrix. On the whole, credit risk
modeling is crucial for bank regulators in providing an effective credit risk review, not only in helping
to detect borrowers in difficulty, but also in facilitating to the Basel Capital Accord. We expect that
this study can provide a suitable model to gauge the credit risk for financial institutions.
This paper is organized as follows: Section 1 provides the motivation for this study. Section 2
reviews literature concerning models of credit risk. Section 3 presents the formal methodology, and
Section 4 describes the sample data used in this paper. Section 5 shows empirical results and
robustness tests. Finally, Section 6 includes a discussion of our findings with a conclusion.


2. Literature Review
Credit risk research has gained considerable momentum over the last decade. Many different classes of

models have been put forward to measure, manage, and price credit risk. In general, these models can
be divided into two main categories: (a) structural-form models and (b) reduced-form models. Both
categories have advantages and limitations in valuing credit risk. One important difference between
these two categories of models is the implicit assumption they make about managerial decisions
regarding capital structure. The structural-form approach imposes assumptions on the evolution of the
value of the firm’s underlying assets. The reduced form approach does not make this implicit
assumption.
The structural-form models include the original work of Black and Scholes (1973) and Merton
(1974). In such a framework, the securities issued by a firm as contingent claims on its own value and,
therefore, the credit risk is driven by the value of the company’s assets. The basic intuition behind the
Merton model is that default occurs when the value of a firm’s asset is lower than that of its liabilities.
Furthermore, the basic Merton model has subsequently been extended by removing one or more of
Merton’s assumptions. Black and Cox (1976) suggest that bondholders can force the reorganization or
the bankruptcy of the firm if its value falls to a specific value. Kim, Ramaswamy and Sundaresan
(1989) and Collin-Dufresne and Goldstein (2001) propose a model similar to the Black and Cox (1976)
model, suggesting that capital structure is explicitly considered and default occurs if the value of total
assets is low enough to reach a trigger value, which is assumed to be exogenous. Leland (1994)
endogenizes the bankruptcy while accounting for taxes and bankruptcy costs. Leland and Toft (1996)
propose a Barrier option model, suggesting that expected default probabilities depend on the
endogenously defined bankruptcy threshold.
In spite of these improvements on Merton’s original framework, structural-form models still
suffer some drawbacks, which are the main reasons behind their relatively poor empirical performance
(Altman, Resti and Sironi, 2004; Emo, Helwege and Huang, 2004). First, since the firm’s value is not a
tradable asset, the parameters of the structural-form model are difficult to estimate consistently. In
other words, unlike the stock price in Black and Scholes model for valuing equality options, the current
market value of a firm is not easily observable. Second, the inclusion of some frictions like tax shields
and liquidation costs would break the last rule. Third, corporate bonds undergo credit downgrades
before they actually default, but structural-form models cannot incorporate these credit-rating changes.
Finally, most structural-form models assume that the value of the firm is continuous in time and,
consequently, the time of default can be predicted just before it happens.

Reduced-form models attempt to overcome the above mentioned shortcomings of structural-
form models. These include Jarrow and Turnbull (1995), Jarrow, Lando and Trunbull (1997), Lando
(1998), Duffie (1998) and Duffie and Singleton (1999). Unlike structural-form models, reduced-form
models do not default on the firm’s value, and parameters related to the firm’s value need not be
International Research Journal of Finance and Economics - Issue 30 (2009) 111
estimated to implement them. These variables related to default risk are modeled independently from
the structural features of the firm, its asset value and leverage.
The calibration of the credit risk for reduced-form models is made with respect to rating
agencies’ data. Therefore, rating systems have become increasingly important for reduced-form
models. Their key purpose is to provide a simple qualitative classification of the solidity, solvency and
prospects of a debt issuer. The importance of credit ratings has increased significantly with the
introduction of the Basel II. It is obvious that the present rating of an obligor is a strong predictor of his
rating in the nearest future. A cardinal feature of any credit rating is the past and present rating
influencing the evolution. Therefore, the Markov chain is a stochastic process, in which the transition
probabilities, given all past ratings, depend only on the present state. It allows all transition
probabilities for a specific time-horizon to be collected in a so-called transition matrix, such as
presented by both Jarrow, Lando and Turnbull (1997) and Lando (1998) using transition matrices to
determine credit risk.
In most applications, transition matrices are estimated by discrete-time observations with a
yearly time horizon. For example, Lu and Kuo (2006) have applied the discrete-time Markov chain
model to assess the credit risk of bank loans by yearly transition matrices. However, if borrowers
seldom change their rating, then transition matrices typically concentrate along the main diagonal. That
is, the most probability mass resides along the diagonal and most of the time there is no migration. The
low occurrence of certain transitions may be a problem when estimating default probabilities. For the
lowest risk grade, such as AAA in Standard and Poor’s rating, defaulting in a given period is a rare
event. Although there are no transitions from AAA to default, there are transitions from AAA to AA
and from AA to default. As a result, the estimator for transitions from AAA to default should be non-
zero and these rare events are ignored by the discrete-time Markov chain model. In order to avoid the
embedding problem for discrete-time observations, the continuous-time Markov chain model has been
adopted to estimate meaningful default probabilities.

On the other hand, differences between discrete- and continuous-time Markov chain models
were compared to determine the credit risk of bank loans. Bangia et al. (2002) found that only the
diagonal elements were estimated with high precision, since transition matrices are dominated
diagonally. They found that if it was one transition away from the diagonal, then the degree of
estimated precision decreases. However, Jafry and Schuermann (2004) also suggest a criterion,
distribution discriminatory, which is particularly relevant for transition matrices that are sensitive the
distribution of off-diagonal probability mass. They also propose statistics to compare the differences
between the discrete- and continuous-time methods. Consequently, Jafry and Schuermann’s (2004)
estimation was used to compare the differences of transition matrices generated by two distinct Markov
chain models.


3. Model Specification
3.1. The discrete-time Markov chain model
Let
t
x represent the credit rating of a bank’s borrower at time t. Assume that , }2,1,0t,x{x
t
== is a
Markov chain on the finite state space S={1, 2,…, C, C+1}, where state 1 represents the highest credit
class; and state 2 the second highest, …, state C the lowest credit class; and state C+1 designates the
default. It is usually assumed for the sake of simplicity that the state C+1 is the absorbing state.
Furthermore, let P(s, t) denote the )1C()1C(
+
×
+
transition matrix generated by a Markov
chain model with transition probability as
()
ixjxP)t,s(p

stij
=== , Sj,i
∈
, ts
<
, t=0, 1, 2,… (1)
Equation (1) is the probability that a borrower rated i at time s migrates to rating j at time t. Let
P
~
andP denote transition matrices for the discrete-time estimator and continuous-time estimator.
112 International Research Journal of Finance and Economics - Issue 30 (2009)
Hereafter, let “P” and “
ij
p ” be generally termed the transition matrices and transition probabilities,
respectively.
The first Markov chain model applied to the transition matrix is the discrete-time Markov chain
model based on annual migration frequencies. Generally, estimation in a discrete-time Markov chain
can be viewed as a multinomial experiment since it is based on the migration away from a given state
over a one-year horizon. Let
)t(N
i
denote the number of firms in state i at the beginning of the year
and )t(N
ij
represent the number of firms with rating i at date t migrated to state j at time t+1. Thus, the
one-year transition probability is estimated as
)t(N
)t(N
)t(p
i

ij
ij
= (2)
If the rating process is assumed to be a time-homogeneous Markov chain, i.e., time-
independent, then the transitions for different borrowers away from a state can be viewed as
independent multinomial experiments. Therefore, the maximum likelihood estimator (MLE) for time-
independent probability is defined as
∑
∑
=
=
=
T
1t
i
T
1t
ij
ij
)t(N
)t(N
p (3)
where T is the number of sample years. For a special case, the number of firms are the same over the
sample period,
ii
N)t(N = , the estimator for the transition probabilities is the average of the year-on-
year transition matrices, such as Bangia et al. (2002) and Hu, Kiesel and Perraudin (2002). However,
the special case is implausible. Accordingly, the estimator of transition probabilities is always modified
by the number of firms during the sample years. If
P denotes transition matrix for a Markov chain

over a year horizon, then the discrete-time transition matrix is as
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
=
+
+
+
1000
pppp
pppp
pppp
P
1C,CCC2C1C
1C,2C22221
1C,1C11211
L

L
MMOMM
L
L
(4)
where
,j,i,0p
ij
∀≥ and
∑
+
=
∀=
1C
1j
ij
i,1p .
Since the information concerning within-year rating transition is ignored in the discrete-time
Markov chain model, the continuous-time Markov chain model had to be used to determine the
additional migration within the year. According to Christensen, Hansen and Lando (2004), the
advantages of the continuous-time Markov chain model can be summarized as: (i) The duration
method can obtain non-zero estimates for probabilities of rare events whereas the cohort method
estimates to zero. (ii)The duration method uses all available information in the data including
information of a firm even when it enters a new state. In the discrete-time estimator, the exact date
within the year that a firm changed its rating cannot be distinguished. Therefore, the continuous-time
Markov chain models were also adopted for valuing the credit risk of bank loans.

3.2. The continuous-time Markov chain model
As for continuous-time models, the non-parametric method of Aalen and Johansen (1978) was adopted
to replace the cohort methods. The Aalen-Johansen estimator imposes fewer assumptions on the data

generating process by allowing for time heterogeneity while fully accounting for all movements within
International Research Journal of Finance and Economics - Issue 30 (2009) 113
the sample period. In other words, the Aalen-Johansen estimator can be applied to an extremely short
time interval and observe a borrower’s rating movement during the sample period.
Let )t,s(P
~
be the transition matrix over the horizon [s, t] and take the Aalen-Johansen
estimator (or product-limit estimator) for the transition matrix. The estimator for the transition matrix,
)t,s(P
~
, from time s to time t is given by
[]
∏
=
Δ+=
m
1i
i
)T(A
ˆ
I)t,s(P
~
(5)
where
i
T represents the transition point over the sample period [s, t] and m is the total number of
transitions over the sample period from s to t. The estimator is clearly a duration approach, which
allows for time non-homogeneous while fully accounting for all movements with the sample period
(estimated horizon). The matrix
)T(A

ˆ
i
Δ is given by
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
Δ
Δ
−
ΔΔ
ΔΔ

Δ
−
Δ
ΔΔ
ΔΔ
−
=Δ
+
000
)T(Y
)T(N
)T(Y
)T(N
)T(Y
)T(N
)T(Y
)T(N
)T(Y
)T(N
)T(Y
)T(N
)T(Y
)T(N
)T(Y
)T(N
)T(Y
)T(N
)T(Y
)T(N
)T(Y

)T(N
)T(Y
)T(N
)T
(A
ˆ
iC
i1C,C
iC
iC
iC
i2,C
iC
i1,C
i2
iC2
i2
i23
i2
i2
i2
i21
i1
iC1
i1
i13
i1
i12
i1
i1

i
LL
L
MLOMM
L
L
(6)
where )T(N
ihj
Δ denotes the number of transitions observed from state h to j at date
i
T
1
. The diagonal
element
)T(N
iK
Δ counts the total number of transitions away from state k at date
i
T and )T(Y
ik
is the
number of firms in state k prior to date
i
T
. Hence, the off-diagonal elements
{
}
hj
i

)T(A
ˆ
Δ ,jh
≠
denote
the fraction of the firms at state h just before date
i
T that migrate to state j at date
i
T . The bottom row
is zero since firms leaving the default state, the absorbing state, were not taken into consideration. Note
that the sum of each row of )T(A
ˆ
i
Δ is zero and the rows of ))T(A
ˆ
I(
i
Δ+ automatically sum to one. In
summary, the Aalen-Johansen estimator is equal to the cohort method for short time intervals. For a
short time horizon, one could neglect the differences between the discrete- and continuous-time
estimators. However, as the time horizon extends, differences between the two estimators increase,
because of the higher migration potential for longer time horizons.

3.3. Risk premium
Consider the corresponding stochastic process },2,1,0t,x
~
{x
~
t

L
=
=
of credit rating under the risk-
neutral probability measure. For valuation purposes, the transition matrices, P, need to be transformed
into a risk-neutral transition matrix under the equivalent martingale measure. Therefore, let M
~
denote
the risk-neutral transition matrix.
2
Thus, the transition matrix under the risk-neutral probability
measure is given by


1
)1t,t(N
hj
+ counts the total number of transitions from state h to state j from date t to t+1, and )T(N
ihj
Δ is an increment
of this process
2
The assumption is also made in Copeland and Jones (2001) and Lu and Kuo (2005, 2006).
114 International Research Journal of Finance and Economics - Issue 30 (2009)
()
⎥
⎥
⎥
⎥
⎦

⎤
⎢
⎢
⎢
⎢
⎣
⎡
Ο
++
=
⎥
⎥
⎥
⎥
⎥
⎥
⎦
⎤
⎢
⎢
⎢
⎢
⎢
⎢
⎣
⎡
+++
+++
+++
=+

×
×
××
+
+
+
)11(
)C1(
)1C()CC(
1C,CCC1C
1C,2C221
1C,1C111
1
~
)1t,t(D
~
)1t,t(A
~
100
)1t,t(m
~
)1t,t(m
~
)1t,t(m
~
)1t,t(m
~
)1t,t(m
~
)1t,t(m

~
)1t,t(m
~
)1t,t(m
~
)1t,t(m
~
1t,tM
~
L
L
MMOM
L
L
(7)
where 0m
~
ij
> the risk-neutral transition probability and
∑
+
=
=
1C
1j
ij
1m
~
, i
∀

. The submatrix
)CC(
A
~
×
is defined
on non-absorbing states }C ,,2,1{S
ˆ
= The components of submatrix A
~
denote the regime-switching
of credit classes for the bank’s borrower. However, it excludes default state C+1.
)1C(
D
~
×
is the column
vector with components
1C,i
m
~
+
, which represent the transition probability of banks’ borrowers in any
rating class, i.e., i=1, 2, …,C, transiting to default, i.e., j=C+1. Assume for the sake of simplicity that
bankruptcy (state C+1) is an absorbing state, so that
)C1(
~
×
Ο
is the zero row vector giving a transition

probability from the default state at initial time until the final time. Once the process enters the default
state, it does not return to the credit class state, so that 1m
~
1C,1C
=
++
.In such a case, it can be said that
default state C+1 is an absorbing state.
If transition matrix, P, is multiplied by the corresponding risk premium, then the transition
matrix will be a risk-neutral transition matrix as equation (7). Therefore, risk premium is the risk
adjustment that transforms the actual probability into the risk-neutral probability. First, let )T,t(V
0
be
the time-t price of a risk-free bond maturing at time T, and let
)T,t(V
i
be its higher risk, i.e., riskier
counterpart for the rating class, i. Since a loan does not lose all interest and principal if the borrower
defaults, one has to realistically consider that a bank will receive some partial repayment even if the
borrower goes into bankruptcy. Let
δ be the proportions of the loan’s principle and interest, which is
collectable on default,
10 ≤δ< , where in general
δ
will be referred to as the recovery rate. If there is
no collateral or asset backing, then
δ =0.
As shown by Jarrow, Lando and Turnbull (1997), it can be assumed that
ijijj,i
p)t()1t,t(m

~
⋅μ=+ , Sj,i ∈ , and )t()t(
iij
μ
=
μ , for ij
≠
and their procedure for risk premium is
1C,i0
i0
i
p)1,0(V)1(
)1,0(V)1,0(V
)0(
+
δ−
−
=μ
(8)
In equation (8), it is apparent that a zero or near-zero default probability, 0p
1C,i
≈
+
, would
cause the risk premium estimate to explode and it is also implied that the credit rating process
(including default state) of every borrower is independent, which is inappropriate and irrational for
bank loans. If the borrower defaults, the default probability for the future is not to be estimated.
Consequently, the assumption that every borrower’s credit rating class is independent only before
entering the default state has to be modified. Redefine the risk premium as
)t,0(V)1(

)t,0(V)t,0(V
)t,0(m
~
p1
1
)t(
0
0i
C
1j
1
ij
1C,i
i
δ−
δ−
−
=
∑
=
−
+
l , i=1, 2,…,C and t=1,…,T (9)
)1t,t(A
~
)t,0(A
~
)1t,0(A
~
+=+ (10)

International Research Journal of Finance and Economics - Issue 30 (2009) 115
where
)t,0(m
~
1
ij
−
are the components of the inverse matrix
)t,0(A
~
1−
and
)t,0(A
~
will be invertible. The
denominator of equation (9) is not that
1C,i
p
+
, but that )p1(
1C,i +
−
, the estimation problem in equation
(8) is avoided this way. For equation (10), A)t()1t,t(A
~
⋅Ω=+ and )t(
Ω
is the )CC(
×
diagonal

matrix with diagonal components being the risk premium, which is adjusted to S
ˆ
j),t(
j
∈l . In
particular, the risk premium of t=0 is
)1,0(V)1(
)1,0(V)1,0(V
m1
1
)0(
0
i0
1C,i
i
δ−
δ−
−
=
+
l
, for i=1,2,…,C (11)
Therefore, estimate risk premium using a recursive method for all loan periods, t=0, 1,…, T. On
the whole, it is found that the risk-neutral transition matrix varies over time to accompany the changes
in the risk premium by equation (9) and (11).
Then, assume the indicator function to be
{}( )
{}( )
⎩
⎨

⎧
≤∈
>∈
=
TtimebeforedefaultTIif,0
TtimebeforedefaultnotTIif,1
1
}I{
τ
τ
(12)
Since the Markov processes and the interest rate are independent under the equivalent
martingale measure, the value of the loan is equal to
{}
[]
{
}
() ()
[]
{}
()( )
{}
TQ
~
1)T,t(V
TQ
~
1TQ
~
)T,t(V

11E
~
)T,t(V)T,t(V
i
t0
i
t
i
t0
}T{Tt0i
>−+=
>−+>=
+=
≤>
τδδ
τδτ
δ
ττ
(13)
where
()
TQ
~
i
t
>τ is the probability under the risk-neutral probability measure that the loan with rating
class i will not be in default before time T. It is clear that
)T,t(m
~
1)T,t(m

~
)T,t(V)1(
)T,t(V)T,t(V
)T(Q
~
1C,i
C
1j
ij
0
0i
i
t
+
=
−=
∑
=
δ−
δ−
=>τ
(14)
which holds for time
Tt ≤ , including the current time, t=0. Similarly, the default probability occurs
before time T as
)T,t(V)1(
)T,t(V)T,t(V
)T(Q
~
0

i0
i
t
δ−
−
=≤τ , for i=1,….,C and T=1,2, (15)
Consequently, the default probability of bank loans under a risk-neutral probability measure
can be estimated by incorporating time-varying risk premium. Furthermore, for three different Markov
chain models to generate transition matrices, their risk premiums also need to be estimated to construct
transition matrices under risk-neutral probability measurement.

3.4. Mobility
Jafry and Schuermann (2004) propose a statistics, mobility estimator, to compare the differences in the
estimated models. Let P be the transition matrix and the dynamic part be measured by mobility matrix
asP
~
~
3

IPP
~
~
−= (16)
where I is an identity matrix, i.e., the static (no migration) matrix. That is, the state vector of the matrix
is unchanged from one period to the next. Thus, subtract the identity matrix, I, leaving only the


3
Decompose the transition matrix into a static and dynamic component, whereas Geweke, Marshall and Zarkin (1986) use the original transition matrix,
P.

116 International Research Journal of Finance and Economics - Issue 30 (2009)
dynamic part of the original matrix, which reflects the “magnitude” of the matrix in terms of the
implied mobility. Therefore, the mobility estimator as
1C
)P
~
~
P
~
~
(
)P(m
1C
1i
i
+
′
=
∑
+
=
λ
(17)
where
i
λ
denotes the i-th eigenvalue of P
~
~
P

~
~
′
. An interpretation can be contributed to equation (17),
)P(m , in term of “average migration rate”, as it would yield exactly the average probability of
transition if such probability were constant across all possible states.
Let two matrices
dis
P
~
~
and
con
P
~
~
are mobility matrices of discrete- and continuous-time
observations, respectively. Then, we use the difference,
)P
~
~
,P
~
~
(m
condis
Δ , to take into account estimation
uncertainty and measurement errors in the transition matrices.
)P
~

~
(m)P
~
~
(m)P
~
~
,P
~
~
(m
condiscondis
−=Δ (18)
The continuous-time transition matrix spreads the transition probability mass more off-diagonal
which implies a considerable decrease in the )P
~
~
(m metric. In the absence of any theory on the
asymptotic properties of equation (18), a resampling technique of bootstrapping is a reasonable and
feasible alternative. Therefore, the mobility estimator is adopted to measure the dispersion in transition
matrices by a bootstrapping.


4. Data
The sample data come from two databases of the Taiwan Economic Journal (TEJ), namely the Taiwan
Corporate Risk Index (TCRI) and long and short-term bank loans. The sample period is from Quarter
1, 1997 to Quarter 4, 2005.
The TCRI is a complete credit rating record for Taiwan’s corporations. TEJ applies a numerical
class from 1 to 9 and D for each rating classification. The categories are defined in terms of default risk
and the likelihood of payment for each individual borrower. Obligation rated number 1 is generally

considered as the lowest in terms of default risk, which is similar to the investment grade for Standard
& Poor’s and Moody. Obligation number 9 is the most risky and rating class D denotes the default
borrower. The definitions of the rating categories of TCRI for long-term credit are similar to Standard
& Poor’s and Moody. The TEJ also define rating classes 1-4 as investment grade and 7-9 as
speculative grade. Therefore, we group credit ratings 1 through 4 into
*
1
4
. Similarly, numbers 5-6 and
7-9 are grouped into
*
2 and
*
3
, respectively. Thus, there are four rating classes
*
1,
*
2,
*
3
and D.
The long- and short-term bank loan database records all debts of corporations in Taiwan,
including lender names, borrower names, rate of debt, and debt issuance dates. The credit risk of bank
loans was investigated according to every borrower’s lending structure.
The government bond yield was taken as a proxy for the risk-free rate which is published by the
Central Bank in Taiwan. Since the maturity of bank loans and government bonds differ, the yields of
government bonds had to be adjusted by interpolating the yield of the government bond whose
maturity was the closest and used as the risk-free rate.
The recovery rate served as a security for bank loans that had influence on credit risk. In

general, banks set a recovery rate according to the kind, liquidity, and value of collateral prior to
lending. Altman, Resti and Sironi (2004) present a detailed review of default probability, recovery rate
and their relationship. They found that most credit risk models treated recovery rate as an exogenous
variable either as structural-form models or reduced-form models. For structure-form models, recovery
rate is exogenous and independent from the firm’s asset value (Kim, Ramaswamy and Sundaresan


4
The motive of this work is also due to a limit in the sample size.
International Research Journal of Finance and Economics - Issue 30 (2009) 117
1989; Hull and White, 1995; Longstaff and Schwartz, 1995). Reduced-form models also assume an
exogenous recovery rate that is either a constant or a stochastic variable independent from default
probability (Litterman and Iben, 1991; Madan and Unal, 1995; Jarrow and Turnbull, 1995; Jarrow,
Lando and Turnbull, 1997; Lando, 1998; Duffee, 1999). According to previous studies, there is no
clear definition of the recovery rate. Fons (1987), Longstaff and Schwartz (1995), Briys and de
Varenne (1997) and assumed a constant recovery rate according to the historic level. Therefore, for
secured loans, recovery rates were taken from 0.1-0.9 (Lu and Kuo, 2005, 2006). For unsecured loans,
the recovery rate was zero (Copeland and Jones, 2001).
Finally, the default risk for at least a one-year horizon was analyzed and therefore excluded
observations for short-term loans and incomplete data. Loans that had an overly low rate were also
excluded because they were likely to have resulted from aggressive accounting politics and would have
biased the results. Consequently, the credit risk of mid- and long-term loans, including secured and
unsecured loans, were analyzed for 28 domestic banks in Taiwan.


5. Empirical Results
5.1. Summary statistics
In this paper, the credit risk of 28 domestic banks in Taiwan was estimated.
5
Since a key feature in any

lending and loan-pricing decision is the degree of collateral of the loan, we consider secured
(collaterialized) and unsecured (uncollaterialized) loans in this paper. Table 1 presents the summary
statistics of the sample. From Panel A, average collateral loan rates and their corresponding
government bond yields (risk-free rates) were 6.392% and 5.344%, respectively. From Panel B, the
average rates of unsecured loans and their corresponding government bond yields were 5.451% and
4.625%, respectively. Generally, the risky rates were higher than the risk-free rates as can be seen from
the results of both Panel A and Panel B. Furthermore, loan rates had greater volatility than risk-free
rates. The average lending periods for secured and unsecured loans were 5.603 and 4.226 years,
respectively. This phenomenon may be due to unsecured loans give a bank a more risky claims to this
debt. In general, a loan with collateral had longer lending periods than an unsecured loan. Finally, the
kurtosis is excess implying that loan rates and risk-free rates were not normal.


5
The 28 domestic banks include: (1) Agricultural Bank of Taiwan; (2) Bank of Taiwan; (3) Bank of Overseas Chinese; (4)
Bank of Sinopac Company Ltd.; (5) Bowa Bank; (6) Cathay United Bank; (7) Chang Hwa Commercial Bank; (8) Chaio
Tung Bank; (9) China Development Industrial Bank Inc.; (10) Chinatrust Commercial Bank; (11) Chinfon Commercial
Bank; (12) Cosmos Bank, Taiwan; (13) EnTie Commercial Bank; (14) E. Sun Commercial Bank; (15) Far Eastern
International Bank; (16) First Commercial Bank; (17) Fuhwa Commercial Bank; (18) Hua Nan Commercial Bank; (19)
Jih Sun International Bank; (20) Land Bank of Taiwan; (21) International Bank of Taipei; (22) Ta Chong Bank Ltd.; (23)
Taiwan Cooperative Bank; (24) Taipei Fubon Commercial Bank; (25) Taishin International Bank; (26) The Chinese
Bank; (27) The International Commercial Bank of China; (28) Taiwan Business Bank.
118 International Research Journal of Finance and Economics - Issue 30 (2009)
Table 1: Summary statistics
This table was summary statistics for bank loans and government bonds in Taiwan. The government
bond yield was taken as a proxy for the risk-free rate. In general, the loan rates (or risky rates) were
higher than risk-free rates. For example, the average loan rates of secured loans and their
corresponding government bonds’ rates were 6.392% and 5.344%, respectively. In Taiwan, the
average lending periods of secured loans always longer than unsecured loans.

Loan rate Risk-free rate Lending period (Year)
Panel A. Secured loans
Maximum 13.452% 11.024% 7.796
Minimum 0.466% 1.111% 2.684
Mean 6.392% 5.344% 5.603
Standard deviation 2.279% 1.794% 1.180
Kurtosis -0.045 -0.043 -0.028
Skewness -0.214 -0.186 -0.271
Panel B. Unsecured loans
Maximum 12.190% 8.371% 6.354
Minimum 0.467% 0.861% 2.684
Mean 5.451% 4.625% 4.226
Standard deviation 2.482% 2.020% 0.923
Kurtosis -0.543 -0.975 -0.782
Skewness 0.153 -0.278 0.209

5.2. The transition matrix
First, the transition matrices estimated from discrete- and continuous-time Markov chain models in
Table 2 were compared. Both transition matrices were diagonally dominant, meaning that they had a
heavy concentration around the diagonal. Interestingly, the default probabilities of unsecured loans
estimated by discrete-time method were not observed in Panel B of Table 2. However, the continuous-
time method could provide an estimate of such loans. Regarding the discrete-time method, there were
many default probabilities close to zero. On the other hand, most default probabilities in discrete-time
estimator were lower than those in the continuous-time estimator. This is because the dynamic rating
process is ignored in the discrete-time estimator result in lower estimated default probabilities.
International Research Journal of Finance and Economics - Issue 30 (2009) 119
Table 2: Average transition matrix, 1997-2005
This table shows the average 9 one-year transition matrices in the period 1997-2005. Panel A and B
present average transition matrices based on discrete-time Markov chain model that are estimated by

cohort method as equation (4). On the other hand, Panel C and D show average transition matrices
using continuous-time Markov chain model, which are estimated by equation (5) and (6), including
all movements within the sample period. However, whether incorporating the dynamic rating
process results in different estimated transition matrices between discrete- and continuous-time
estimators. Rather than extracting default rates from the discrete-time transition matrix, the
continuous-time transition matrix analyze default probabilities by incorporating all information of
migration within the sample periods, which is more elaborate.

Rating at the end of year
Initial Rating
*
1
*
2
*
3
D
Panel A. Secured loans based on discrete-time model
*
1
0.854 0.135 0.009 0.002
*
2
0.052 0.796 0.149 0.004
*
3
0.005 0.132 0.855 0.009
Panel B. Unsecured loans based on discrete-time model
*
1

0.894 0.099 0.006 0.000
*
2
0.051 0.761 0.188 0.000
*
3
0.004 0.112 0.883 0.001
Panel C. Secured loans based on continuous-time model
*
1
0.822 0.151 0.024 0.004
*
2
0.045 0.774 0.153 0.028
*
3
0.008 0.180 0.714 0.097
Panel D.Unsecured loans based on continuous-time model
*
1
0.840 0.075 0.012 0.002
*
2
0.046 0.712 0.137 0.034
*
3
0.010 0.174 0.626 0.119

However, the continuous-time transition matrix is estimated using Anlen-Johansen estimator
which takes into account all movements within the sample period. Therefore, the average movements,

A
ˆ
Δ , is shown in Table 3. For the discrete-time estimator, this information would have been ignored.
120 International Research Journal of Finance and Economics - Issue 30 (2009)
Table 3: Average rating movements, A
ˆ
Δ
This table was average all movements within the sample period, 1997-2005. The continuous-time
method is Aalen and Johansen non-parametric method, which take into account all movement
within the sample period as equation (6). The diagonal element is the fraction of exposed borrowers
which leave the state. The off-diagonal entries represent the fraction of transitions away from the
state divided by the number of exposed borrowers. For the discrete-time estimator, this information
of rating movements would have been ignored.

Rating at the end of year
Initial Rating
*
1
*
2
*
3
Panel A. Secured loans
*
1
-0.0400 0.0378 0.0022
*
2
0.0149 -0.0663 0.0514
*

3
0.0004 0.0931 -0.0935
Panel B. Unsecured loans
*
1
-0.0257 0.0247 0.0010
*
2
0.0134 -0.0645 0.0511
*
3
0.0008 0.1014 -0.1022

Then, we give the density of the default probabilities for three rating grades. Figure 1 and 2
give the density of default probabilities for secured and unsecured loans, respectively. Both from
Figure 1 and 2, it can be seen that some default observations were ignored with the discrete-time
method giving it zero default probability estimates.

Figure 1: Default frequency of discrete- and continuous-time Markov chain models for secured loans
Figure 1 A: Default frequency of rating class
*
1

0
40
80
120
160
200
240

0.000
0.132
0.266
0.399
0.533
0.666
0.800
0.933
default probability
frequency
continuous
discrete


Figure 1 B: Default frequency of rating class
*
2

0
40
80
120
160
200
240
0.000
0.066
0.132
0.199
0.266

0.333
0.399
0.466
0.533
0.600
0.666
0.733
0.800
0.867
0.933
0.990
default probability
frequency
continuous
discrete

International Research Journal of Finance and Economics - Issue 30 (2009) 121
Figure 1 C: Default frequency of rating class
*
3

0
40
80
120
160
200
240
280
0.000

0.066
0.133
0.199
0.266
0.333
0.400
0.466
0.533
0.600
0.666
0.733
0.800
0.867
0.933
0.990
default probability
frequenc
y
conti nuous
di screte

Note: This figure gives the density of default probabilities of secured loans for three rating classes,
*
1 ,
*
2 and
*
3
Comparing with two approaches, the discrete-time approach has higher density of low default probabilities. That
is, the discrete-time approach clusters in the group of low default probabilities. On the other hand, many default

events are neglected in discrete-time approach. Therefore, we infer that default probabilities have been
underestimated in the discrete-time Markov chain model.

Figure 2: Default frequency of discrete- and continuous-time Markov chain models for unsecured loans

Figure 2 A: Default frequency of rating class
*
1

0
40
80
120
160
200
240
280
0.000
0.067
0.133
0.200
0.267
0.333
0.400
0.467
0.533
0.600
0.667
0.733
0.800

0.867
0.933
0.990
default probability
frequency
continuous
discrete


Figure 2 B: Default frequency of rating class
*
2

0
40
80
120
160
200
240
280
0.000
0.067
0.133
0.200
0.267
0.333
0.400
0.467
0.533

0.600
0.667
0.733
0.800
0.867
0.933
0.990
default probability
frequency
continuous
discrete

122 International Research Journal of Finance and Economics - Issue 30 (2009)
Figure 2 C: Default frequency of rating class

0
40
80
120
160
200
240
280
0.000
0.067
0.133
0.200
0.267
0.333
0.400

0.467
0.533
0.600
0.667
0.733
0.800
0.867
0.933
0.990
default probability
frequenc
y
continuous
discrete

Note: This figure gives the density of default probabilities of unsecured loans for three rating classes,
*
1 ,
*
2 and
*
3 .
Comparing with two approaches, the discrete-time approach has higher density of low default probabilities. That
is, the discrete-time approach clusters in the group of low default probabilities. On the other hand, many default
events are neglected in discrete-time approach. Therefore, we infer that default probabilities have been
underestimated in the discrete-time Markov chain model.

From Table 2, Figure 1 and 2, the transition matrices obtained from the discrete-time Markov
chain model were not consistent with continuous-time transition matrices that were due to the
concentration probability mass around the main diagonal. The continuous-time Markov chain model

probably incorporated full information of the rating process within the sample and captured the rare
events or non-zero estimates for probability that were ignored in the discrete-time Markov chain
model. The discrete-time method apparently underestimated the true probability of default making the
continuous-time Markov chain model an improvement on it.
Generally, the default probabilities estimated from historic data as Table 2 is not based on risk-
neutral valuation method. If assume that the financial institution manager is not risk averse, we have to
take a risk-neutral valuation method into account when analyze the credit risk. That is, we have to
estimate the risk premium to construct risk-neutral transition matrices. Therefore, the time-varying risk
premium to transform the transition matrix into a risk-neutral transition matrix was estimated. The
average risk premium estimated using equation (9) and (11) is shown in Table 4.
International Research Journal of Finance and Economics - Issue 30 (2009) 123
Table 4: Average risk premium
This table was average risk premium estimated by equations (9) and (10). Panel A and C represent
bank loans with collateral and therefore, we set the recovery rate is from 0.1 to 0.9. On the other
hand, Panel B and D denote unsecured loans with zero recovery rate. If the original transition
matrices, equation (4) or (5), multiply by the corresponding risk premium, then the transition
matrices will be risk-neutral discrete-time or continuous-time transition matrices as equation (7).

Maturity (Years)
Rating 1997 1998 1999 2000 2001 2002 2003 2004 2005
Panel A. Secured loans based on discrete-time model
*
1
0.880 0.991 0.979 0.990 0.988 0.970 0.977 0.968 0.957
*
2
0.880 0.979 0.945 0.952 1.003 0.967 0.930 0.933 0.937
*
3

0.879 0.950 0.972 0.987 0.969 0.964 0.968 0.970 0.973
Panel B. Unsecured loans based on discrete-time model
*
1
0.969 1.010 0.990 0.993 0.980 0.991 0.979 0.996 0.971
*
2
0.964 0.989 0.986 0.989 0.976 0.982 0.976 0.977 0.950
*
3

0.957 0.993 0.991 0.991 0.969 0.987 0.978 0.997 0.977
Panel C. Secured loans based on continuous-time model
*
1
0.879 0.982 0.958 0.983 0.981 0.970 0.983 0.973 0.976
*
2
0.879 0.970 0.904 0.954 0.909 0.875 0.815 0.763 0.687
*
3

0.862 0.989 0.991 0.990 0.992 0.993 0.996 0.987 0.994
Panel D. Unsecured loans based on continuous-time model
*
1
0.967 1.010 0.983 0.963 0.983 0.989 0.976 0.983 0.985
*
2
0.961 0.987 0.914 0.914 0.829 0.775 0.663 0.783 0.793

*
3

0.955 0.998 0.998 0.999 0.999 0.999 0.999 0.999 0.999

If the risk premium were to be plotted against the ratings, a skewed, U-shaped curve would
emerge, with the trough corresponding to rating class
*
2
6
It was found that when the risk premium
was equal to one, the default probability remained unchanged after taking the risk-neutral probability
into consideration. A risk premium is smaller than one means that default probabilities in the risk-
neutral framework are higher than those in the real world. The smaller risk premium estimate
compensates for the discrepancy between the default probabilities under the real probability and risk-
neutral probability measure. Thus, the risk premium was extracted and incorporated into the transition
matrix based on discrete- and continuous-time methods as shown in Table 5.
Table 5 is the transition matrix estimated by discrete- and continuous-time methods using the
risk-neutral probability. According to Table 5, a common characteristic was row monotonicity, which
is a general feature of transition matrices. This means that the risky credit rating class had a higher
default probability than the rating class with lower risk. Panel A-C of Table 5 are all following this
rule, i.e., monotonicity. However, a violation of monotonicity occurs for Panel D of Table 5 so that
there is a higher default probability in rating class
*
2 than that in rating class
*
3
. A possible
explanation could be the noisy nature of the data for credit rating
*

2
7
.


6
These results are similar to Kijima and Komoribayashi (1998) and Wei (2003), who used a different set of data
7
The same phenomenon is also found in Fuertes and Kalotychou (2005).
124 International Research Journal of Finance and Economics - Issue 30 (2009)
Table 5: Risk-neutral transition matrix, 1997-2005
This table was risk-neutral discrete- and continuous-time transition matrices that incorporate time-
varying risk premium by averaging 9 one-year risk-neutral transition matrices in the period 1997-
2005. Panel A and C represent bank loans with collateral and therefore, we set the recovery rate is
from 0.1 to 0.9. On the other hand, Panel B and D denote unsecured loans with zero recovery rate.

Rating at the end of year
Initial Rating
*
1
*
2
*
3
D
Panel A. Secured loans based on discrete-time model
*
1
0.825 0.131 0.008 0.036
*

2
0.049 0.764 0.140 0.047
*
3
0.005 0.127 0.818 0.050
Panel B. Unsecured loans based on discrete-time mode
*
1
0.882 0.097 0.006 0.015
*
2
0.050 0.745 0.182 0.024
*
3
0.008 0.110 0.764 0.118
Panel C. Secured loans based on continuous-time mode
*
1
0.798 0.148 0.027 0.027
*
2
0.046 0.671 0.137 0.147
*
3
0.016 0.185 0.698 0.101
Panel D. Unsecured loans based on continuous-time mode
*
1
0.894 0.073 0.012 0.021
*

2
0.042 0.660 0.115 0.183
*
3
0.012 0.201 0.562 0.225

On the other hand, comparing Table 2 and 5, it was found that default probabilities were higher
after incorporating the risk premium. This was because the risk neutral model gave a forward-looking
prediction of the default. Thus, the risk-neutral default probabilities in Table 5 exceeded the history-
based transition default probabilities in Table 2 resulting in a risk premium reflecting the unexpected
probability of the default.
After incorporating the time-varying risk premium into the transition matrix, we also show the
density of default probabilities under risk-neutral probability measurement. Figures 3 and 4 represent
the density of default probabilities of discrete- and continuous-time methods using risk-neutral
probability measurement. The former and latter are secured and unsecured loans, respectively.
Although the risk premium is considered in Figures 3 and 4, the discrete-time method still shows many
zero default probabilities. Comparing discrete- and continuous-time methods, it was found that the
transition matrices estimated by continuous-time approach exhibit a greater default probability than the
discrete-time approach. One reason was that the continuous estimators better captured the rating
dynamics. For example, if a borrower was in
*
1 at year beginning and was then downgraded to
*
2 and
default at the end of the year, the discrete-time estimator recorded a migration from
*
1 to default.
However, the migration D21
**
→→ was recorded by a continuous-time estimator.

International Research Journal of Finance and Economics - Issue 30 (2009) 125
Figure 3: Risk-neutral default frequency of discrete- and continuous-time Markov chain models for secured
loans
Figure 3 A: Risk-neutral default frequency of rating class
*
1

0
40
80
120
160
200
0.000
0.037
0.075
0.114
0.152
0.191
0.229
0.267
0.306
0.344
0.383
0.421
0.460
0.498
default probability
frequency
continuous

discrete


Figure 3 B: Risk-neutral default frequency of rating class
*
2

0
40
80
120
160
0.000
0.054
0.108
0.162
0.216
0.270
0.324
0.378
0.432
0.486
0.540
0.594
0.648
0.702
0.756
0.810
default probability
frequency

continuous
discrete


Figure 3 C: Risk-neutral default frequency of rating class
*
3

0
40
80
120
160
0.000
0.026
0.051
0.077
0.103
0.129
0.154
0.180
0.206
0.232
0.257
0.283
0.309
0.335
0.360
0.386
default probability

frequency
continuous
discrete

Note: This figure gives the density of risk-neutral default probabilities of secured loans for three rating classes,
*
1
*
2 ,
and
*
3 . Although the risk premium is considered, the discrete-time approach still has many zero default
probabilities. In other words, many default events are neglected in the discrete-time approach. Therefore, we infer
that default probabilities have been underestimated in the discrete-time Markov chain model.
126 International Research Journal of Finance and Economics - Issue 30 (2009)
Figure 4: Risk-neutral default frequency of discrete- and continuous-time Markov chain models for unsecured
loans
Figure 4 A: Risk-neutral default frequency of rating class
*
1

0
30
60
90
120
150
180
210
0.000

0.014
0.079
0.144
0.208
0.273
0.338
0.402
0.467
0.531
0.596
0.661
0.725
0.790
default probability
frequency
continuous
discrete


Figure 4 B: Risk-neutral default frequency of rating class
*
2

0
30
60
90
120
150
0.000

0.063
0.127
0.190
0.254
0.317
0.381
0.444
0.508
0.571
0.634
0.698
0.761
0.825
0.888
0.925
default probability
frequency
continuous
discrete


Figure 4 C: Risk-neutral default frequency of rating class
*
3


0
30
60
90

120
150
180
0.000
0.021
0.042
0.063
0.084
0.105
0.127
0.148
0.169
0.190
0.211
0.232
0.253
0.274
0.295
0.316
0.332
default probability
frequency
continuous
discrete

Note: This figure gives the density of risk-neutral default probabilities of unsecured loans for three rating classes,
*
1
*
2 ,

and
*
3 . Although the risk premium is considered, the discrete-time approach still has many zero default
probabilities. In other words, many default events are neglected in the discrete-time approach. Therefore, we infer
that default probabilities have been underestimated in the discrete-time Markov chain model.

5.3. Robustness test
5.3.1 Comparing LGD with NPL ratio
We use the loss given default (LGD) and non-performing loans (NPL) to prove the effectiveness of
estimated results for discrete- and continuous-time Markov chain models. The NPL refers to loan
accounts the principal and interest of which have become past due or those exceeded the due date. The
NPL ratio is equal to NPL divided by the total loan portfolio. The LGD is the proportion of exposure
International Research Journal of Finance and Economics - Issue 30 (2009) 127
that is lost in the event of a default. Since the recovery rate is the proportion of the loan’s principal or
interest that is collectable on default, we assume that
)rateerycovre1(yprobabilitdefaultLGD −×
=
.
8

Therefore, the estimated LGD is shown in Table 6 with recovery rate from 0 to 0.9. For unsecured
loans, the recovery rate is zero, while secured loans with recovery rates from 0.1 to 0.9. For last
column in Table 6, we find that LGD is lower in the discrete-time method than that estimated by the
continuous-time approach. We infer that due to default probabilities is always underestimated in
discrete-time approach.

Table 6: LGD of discrete- and continuous-time Markov chain models
We assume the
)cov1( rateeryreyprobabilitdefaultLGD
−

×
= with recovery rate from 0 to 0.9. For
unsecured loans, the recovery rate is equal to zero, while secured loans with recovery rate from 0.1
to 0.9. For last column, we find that LGD is lower in the discrete-time method than that estimated
by the continuous-time approach. The phenomenon is due to default probabilities are
underestimated in discrete-time approach that ignores the dynamic rating process.

Recovery rate 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
Average
Panel A. LGD of discrete-time Markov chain model
1997 0.037 0.035 0.040 0.040 0.040 0.038 0.040 0.037 0.038 0.037 0.038
1998 0.003 0.018 0.014 0.014 0.013 0.015 0.012 0.014 0.014 0.008 0.009
1999 0.011 0.017 0.017 0.017 0.016 0.014 0.016 0.015 0.014 0.006 0.013
2000 0.010 0.019 0.016 0.014 0.013 0.013 0.011 0.012 0.010 0.005 0.011
2001 0.026 0.008 0.007 0.006 0.006 0.005 0.005 0.006 0.012 0.003 0.016
2002 0.014 0.013 0.013 0.012 0.012 0.012 0.013 0.013 0.016 0.007 0.013
2003 0.023 0.017 0.018 0.018 0.018 0.017 0.019 0.018 0.022 0.006 0.020
2004 0.011 0.015 0.022 0.022 0.022 0.014 0.022 0.018 0.021 0.007 0.014
2005 0.035 0.030 0.018 0.018 0.018 0.030 0.021 0.022 0.018 0.010 0.028
Panel B. LGD of continuous-time Markov chain model
997 0.089 0.077 0.073 0.069 0.061 0.055 0.053 0.049 0.044 0.040 0.073
1998 0.054 0.044 0.040 0.036 0.034 0.031 0.026 0.023 0.020 0.012 0.042
1999 0.086 0.077 0.070 0.062 0.049 0.043 0.032 0.026 0.019 0.007 0.064
2000 0.092 0.057 0.051 0.045 0.044 0.037 0.031 0.026 0.017 0.007 0.063
2001 0.113 0.068 0.061 0.054 0.047 0.039 0.033 0.027 0.019 0.008 0.076
2002 0.128 0.083 0.075 0.066 0.058 0.049 0.042 0.030 0.019 0.009 0.088
2003 0.169 0.092 0.083 0.073 0.063 0.052 0.043 0.035 0.028 0.011 0.111
2004 0.129 0.117 0.103 0.091 0.079 0.068 0.058 0.041 0.028 0.012 0.098
2005 0.125 0.135 0.121 0.108 0.094 0.081 0.067 0.048 0.033 0.014 0.101

For proving the effectiveness of two approaches, Figure 5 shows LGD of discrete- and
continuous-time methods and NPL ratio. Since NPL concerns loans for which collection in full
improbable after the realization of collateral or the institution of legal proceeding have exhausted, the
NPL ratio is ex-post concept. However, the LGD is ex-ante perspective here. Thus, we infer the LGD
is always higher than NPL ratio. From Figure 5, we find that the LGD of the continuous-time approach
is higher than NPL ratio. However, the LGD of the discrete-time approach is lower than NPL ratio. We
conclude that the discrete-time approach may underestimate default probabilities result in lower LGD.
On the other hand, in Taiwan, banks accelerated the write-off of bad loans, such as selling non-
performing loans to asset management companies (AMC) in 2002. Thus, the NPL ratio has decreased
sharply from 2002. Consequently, we recommend that continuous-time Markov chain model is more
elaborate than discrete-time Markov chain model that ignore the dynamic rating process.


8
The assumption is also made in Lu and Kuo (2005, 2006).
128 International Research Journal of Finance and Economics - Issue 30 (2009)
Figure 5: LGD of discrete- and continuous-time Markov chain models and NPL Ratio


0
0.02
0.04
0.06
0.08
0.1
0.12
1997 1998 1999 2000 2001 2002 2003 2004 2005
LGD_c on
LGD_dis
NPL

The dotted line is NPL ratio. The top and bottom lines are LGD of the continuous- and discrete-
time Markov chain mode, respectively. Since the NPL refers to loan accounts the principal and interest
of which have become past due or those exceeded the due date. The NPL ratio is ex-post concept.
Oppositely, the LGD is ex-ante perspective. Thus, we infer the LGD is always higher than NPL ratio.
However, the LGD of discrete-time approach is always lower than NPL ratio. Consequently, we
conclude that the default probabilities are underestimated in the discrete-time Markov chain model.

5.3.2 Mobility estimator
The continuous-time estimator captured the frequent short duration transitions from the low categories
resulting in comparatively higher estimated default probabilities. The continuous-time method spread
the off-diagonal probability mass over almost all ratings whereas the discrete-time method
concentrated the probability mass around the diagonal. Therefore, it had to be determined whether the
differences between the discrete- and continuous-time default probabilities were significant. For this
purpose, the ratings mobility estimator was applied to the discrete- and continuous-time methods and
was shown in Table 7.
Now we address the issue of whether there are significant differences in the overall rating
mobility implied by a discrete-and continuous-time Markov chain model. Regarding the mobility
metric, it was found that the discrete-time matrix needed a higher mobility estimator than the
continuous-time matrix for either secured or unsecured loans. This suggests that the continuous-time
transition matrix spread the default probability mass off-diagonal. For the discrete-time method, the
presence of a few large off-diagonal terms inflated the mobility estimator. Since the absence of
asymptotic distribution of the mobility estimator, a bootstrap method is adopted for a formal
comparison between different methods. The difference is statistics significant as suggested by the 95%
confidence interval not containing aero. That is, the null hypothesis, namely that the difference
between the discrete- and continuous-time methods are rejected either secured or unsecured loans.
These findings are just a reflection of a shortcoming of discrete- versus continuous-time
transition probability estimators, since the discrete-time method does not take into account rating
duration and may underestimate the default probability. Since transition matrices are sensitive to off-

diagonal probability, the process for generating transition matrices has a significant effect on
measuring the credit risk. In general, credit matrices are cardinal inputs into many risk management
applications; therefore, the accurate estimation of the transition matrix is vital.
International Research Journal of Finance and Economics - Issue 30 (2009) 129
Table 7: Bootstrap tests for difference between a discrete- and continuous-time Markov chain model
This table was bootstrap tests for differences between discrete- and continuous-time transition
matrices. The mobility differential,
m
Δ
, are computed as the )P
~
~
(m
dis
minus
)P
~
~
(m
con
The null
hypotheses are rejected either transition matrices of secured or unsecured loans. Therefore, the two
estimators give different migration risk measures, which calls for caution in practical applications.


m
Δ
Mean(
m
Δ

) Standard
Deviation
m
Δ

95% confidence
interval
0m:H
0
=
Δ

Secured loans 0.3333 0.0469 0.0004 [0.0357,0.0717] reject
Unsecured loans 0.4178 0.0531 0.0005 [0.0363,0.0821] reject


6. Conclusion
This study focused on developing an effective credit review process to measure the credit risk of bank
loans, including secured and unsecured loans, for 28 banks in Taiwan. The credit risk of bank loans
was estimated and compared using discrete- and continuous-time Markov chain models. Four
conclusions are presented in this paper. First, investigating the credit risk of bank loans using discrete-
and continuous-time Markov chain models was found to be more comprehensive than previous studies.
Second, the assumption of risk premium that was taken as time-invariant in previous researches
was relaxed. This study took risk premium as a time-varying parameter and incorporated that into
default probabilities. Since risk premium reflects the unexpected probability of default, the risk-neutral
default probability exceeded the history-based default probability.
Third, the importance of estimating transition matrices based on full information from the
rating transition process was shown. It was found that the discrete-time Markov chain model contained
the embedding problem whereas the continuous-time Markov chain model did not. Finally, default
probabilities were apparently underestimated in discrete-time models as a result of not taking within-

year rating transitions into consideration. In other words, the continuous-time model enables us to
estimate default probabilities meaningfully. On the whole, this paper proposes a comprehensive
investigation of bank loans that is expected to be helpful for financial institutions.


Acknowledgement
The authors thank anonymous reviewers and the editor for their helpful recommendations. Financial
support from the National Science Council (NSC96-2416-H-239-004-MY2) is gratefully
acknowledged.
130 International Research Journal of Finance and Economics - Issue 30 (2009)
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