Asia Pacific Management Review (2007) 12(2), 73-84
An Approach to Condition the Transition Matrix on Credit Cycle:
An Empirical Investigation of Bank Loans in Taiwan
Su-Lien Lu*
Department of Finance, National United University, Republic of China
Accepted date: February 2007
Available online
Abstract
Credit risk models for credit rating transitions are important ingredients for credit risk management. This paper propose a
formal model, conditional Markov chain model, for gauging credit risk and apply it to Taiwan’s bank loans from 1998-2003. The
model’s central feature is to incorporate credit cycle and time-varying risk premium into transition matrices. There are three main
contributions. First, we apply the methodology to bank loans in Taiwan, which is more elaborate than previous studies. Second, the
empirical results show that credit cycle and risk premium have significant influence on default probabilities. That is, if we ignore
credit cycle and risk premium, default probabilities may be underestimated. Third, the estimation procedures for assessing the
credit risk of financial institutions are easy to follow and implement. On the whole, the proposed model can provide more reliable
estimated results for credit risk of bank loans.
Keywords: Default probability; Bank loan; Credit cycle; Risk premium
1. Introduction
been described as a “one size fit all”. Now, the Committee allows banks using “internal models” rather
than the alternative regulatory (“standardized”) model
to assess their credit risks. As a result, most new models and technologies were emerged applying to analyze
credit risk. An accurate credit risk analysis becomes
even more important today.
Over a decade ago, the Basel Committee on Banking Supervision (“The Committee”) produced guidelines for determining bank regulatory capital. The objective of this accord was to level the global playing
field for financial institutions and protect all risks in
the financial system. In 1988, the Committee issued
“the International Convergence of Capital Standard”
(or “Capital Accord of 1988”), which established regulations regarding the amount of capital that banks
should hold against the credit risk. Furthermore, the
treatment of market and the operational risk were incorporated in 1996 and 2001, respectively. The final
version of the Accord was published in 2004 and will
be implemented after 2006.1
Although credit risk models have been developed
rapidly such as Jarrow, Lando and Turnbull (1997);
Kijima and Komoribayashi (1998) and Wei (2003),
they do not consider the credit risk of the bank loans.
For bank industry, credit risk is associated with the
quality of individual assets (or loans) and the likelihood of default. Credit risk is the potential variation in
net income and market value of assets resulting from
this nonpayment or delayed payment. Different types
of loans have different default probabilities. On the
other hand, changes in economic conditions and a
bank’s operating environment alter the cash flow of
their assets, such as loans. Therefore, it is extremely
difficult to assess credit risk of bank loans.
Most statistics showed a significant increase in the
credit risk, such as Japan, Korea and United States of
America etc.2 In the past, the Committee approach has
*
1
2
E-mail: [email protected]
Basel I was providing a standard approach, including measuring
credit risk and market risk to determine minimum capital requirements. In June 1999, the Committee released a proposal to replace
the 1988 Capital Accord with Basel II, a new Accord that has a
more risk-sensitive framework. The Committee sought comments
from interested parties and finalized the new accord by the fourth
quarter of 2003. The final version of the new Accord was published in 2004. The Basel II is based on three pillars: (i) minimum
capital requirements; (ii) supervisory review; (iii) market discipline.
For Japan, there are Daiwa Bank failure and Yamaichi Securities
failure in 1995 and 1997, respectively. Long Term Credit Bank of
Japan (LTCB) was one of the two biggest bank failures of the
twentieth century, with Nippon Credit Bank being the other. For
The purpose of this paper is to provide an effective
model to measure the credit risk. The model is more
elaborate than previous studies such as Jarrow, Lando
and Turnbull (1997) and Wei (2003) that ignore the
Korea, the Korean Development Bank is a famous case. There are
many factors that cause banks to default; however, credit risk is
number one.
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Su-Lien Lu/Asia Pacific Management Review (2007) 12(2), 73-84
credit cycle and time-varying risk premium. In this
paper, we contribute to the literature in the following
aspect. First, the model incorporates credit cycle in
transition matrices. Although Lu and Kuo (2006) have
applied Markov chain model to bank loans, they does
not consider the credit cycle, which is an important
factor. Therefore, this paper serves as one of the first
studies to adopt a conditional Markov chain model for
assessing the credit risk of bank loans, which has never
been discussed in previous studies.3 In addition, we
compare the difference between the observed and fitted
transition matrices. The observed transition matrix is
calculated from the original Markov chain model that
doesn’t consider credit cycle. On the other hand, we
incorporate the credit cycle with the transition matrix,
so-called fitted transition matrix.
difference between these two categories of models is
the implicit assumption they make about managerial
decisions regarding their capital structure. The structural-form model is also called the asset value model
for assessing credit risk, typically of a corporation’s
debt. It was based on the principle of pricing option in
Black and Scholes (1973) and a more detailed model
developed by Merton (1974).
Furthermore, the basic Merton model has subsequently been extended by removing one or more of his
assumptions. For instance, Black and Cox (1976) and
Kim, Ramaswamy and Sundaresan (1989) suggest that
capital structure is explicitly considered and default
occurs if the value of total assets is lower than the
value of liabilities. Brennan and Schwartz (1978) and
Longstaff and Schwartz (1995) investigate the stochastic interest rate correlated with the firm process. Leland
(1994) endogenizes the bankruptcy while accounting
for taxes and bankruptcy costs.
Second, when gauging credit risk of bank loans,
risk premium plays a crucial role. However, previous
researches handle the risk premium as time-invariant
(Jarrow, Lando and Turnbull, 1997; Wei, 2003). In fact,
the risk premium is not time-invariant but is actually
always time-variant (Kijima and Komoribayashi, 1998;
Lu and Kuo, 2006). Therefore, we will also relax assumptions of previous researches by incorporating
time-variant risk premium, which is more elaborate
than previous researches. Third, the estimated procedures are easy to follow and implement. The model can
apply to value the credit risk of other financial institutions. On the whole, we expect that the proposed
model not only provides an effective credit risk review
for financial institutions but also help to face the Basel
Capital Accord.
Over the last few years, KMV Corporation5 has
developed a credit risk methodology to assess default
probabilities and the loss distribution related to default
risks. KMV derives the default probability for each
obligor based on Merton model. KMV best applies to
publicly traded companies for which the equity’s value
is market determined. The information contained in the
firm’s stock price and balance sheet can then be translated into an implied risk of default. Thus, the default
probability is a function of the firm’s capital structure,
the volatility of the asset returns and the current asset
value. Consequently, structural-form models rely on
the balance sheet of the borrower and the bankruptcy
code in order to derive the probability of default.
This paper is organized as follows. Section 1 provides motivation. 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 the main empirical
results. Finally, Section 6 includes a discussion of our
findings with a conclusion.
In spite of the extensions of Merton’s original
framework, these models still suffer some drawbacks.
First, since the firm’s value is not a tradable asset, nor
it is easily observable, the parameters of the structuralform model are difficult to estimate consistently. 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 undergo default, but structural-form models
cannot incorporate these credit-rating changes.
2. Literature Review
Over the last few years, credit risk modeling and
credit derivatives valuation have received tremendous
attention worldwide. These credit risk models can be
grouped into two main categories: the structural-form
model and the reduced-form model.4 One important
3
4
Reduced-form models attempt to overcome these
shortcomings of structural-form models, such as Jarrow and Turnbull (1995), Duffie and Singleton (1997),
Jarrow, Lando and Turnbull (1997) and Lando (1998).
Unlike structural-form models, reduced-form model
Despite there have been extensive literatures on Markov chain model
such as Hamilton (1989), Turner, Startz and Nelson (1989), Engle
and Hamilton (1990), Kaminsky and Peruga (1990), Li and Lin
(2000), Chae and Lee (2005) and Chang, Tu and Tsai (2005), they
also don’t consider the credit cycle. Chae and Lee (2005) propose a
method for stationary probabilities of the embedded Markov chain of
a class of GI/M/1 queueing systems. Chang, Tu and Tsai (2005)
suggest a numerical method that relies on the Markov chain approximation to compute the optimal strategies. Li and Lin (2000)
adopt Markov chain model to examine the return variability for
major East Asian market indices.
There are other approaches to measure credit risk such as Lee and
5
74
Chang (1998) and Lu and Kuo (2005). Lee and Chang (1998) investigate the advanced officers of credit rating department on the bank
by questionnaire that used to establish the credit grading standard for
middle and small business in Taiwan. On the other hand, Lu and Kuo
(2005) assess the credit risk of a Taiwanese bills finance company by
a marketed-based approach.
KMV Corporation is a firm specialized in credit risk analysis.
Su-Lien Lu/Asia Pacific Management Review (2007) 12(2), 73-84
long-term interest rates, foreign exchange rates, government expenditures and the aggregate saving rate.
make no assumptions at all about the capital structure
of the borrowers. The calibration of this probability of
default is made with respect to the data of ratings
agencies. The original Jarrow and Turnbull (1995)
model, which was perhaps the first reduced-form
model to experience widespread commercial acceptance, was worked out through the use of matrices of
historical transition probabilities from the original ratings and recovery values at each terminal state. In contrast, reduced-form models can extract credit risks
from the actual market data and are not dependent on
asset value and leverage. Therefore, parameters that
are related to the firm’s value need not be estimated in
implementation.
From a credit risk modeling perspective, variation
in transition matrices attribute to credit cycle is potentially important. Belkin, Suchower and Forest (1998)
employ a parameter to measure the credit cycle, which
means the values of both default rates and end-ofperiod risk ratings are not predicted by the initial mix
of credit grades. Kim (1999) builds a credit cycle index,
which is similar to that of Belkin, Suchower and Forest
(1998). As well-known, credit ratings changes play a
crucial role in many credit risk models.
Wilson (1997) has shown that transition probabilities change over time as the state of the economy
evolves. In this approach, default probabilities are a
function of macrovariables such as unemployment,
interest rate, the growth rate, government expenses,
and foreign exchange rates. He uses macrovariables to
drive the credit cycle. Unfortunately, variables as proxies for macrovariables are seldom completely satisfactory. Whereas some of the contributions in the literature introduce observed macrovariables to capture covariation in default intensities between firms and economy environment, an alternative approach is to estimate the common components of default risk directly
from data. Unlike Wilson (1997), we will estimate
conditional default probabilities directly from the data.
The Basel Committee on Banking Supervision also
emphasizes the importance of the credit cycle, which
may improve the accurate assessments of credit risk.
Therefore, we will estimate credit cycle to construct a
fitted transition matrix that is first contribution in this
paper.
On the other hand, the current proposed industry
also sponsored methodologies for measuring credit risk,
such as CreditMetrics, KMV, CreditRisk+ and CreditPortfolio View. CreditMetrics was introduced in 1997
by J. P. Morgan and its co-sponsors (Bank of America,
Union Bank of Switzerland, Swiss Bank Corporation,
BZW, Deutsche Morgan Grenfell, etc.) as a value at
risk framework to apply to the valuation and risk of
nontradable assets such as loans and privately placed
bonds. The approach of CreditMetric is based on credit
rating transition analysis, i.e. the probability of moving
from one credit rating to another, including default,
within a given horizon, which is often taken arbitrarily
as 1 year. CreditMetrics’ approach applies primarily to
bonds and loans and it can be easily extended to any
type of financial claims.
KMV’s methodology, which is described previously, differs somewhat from CreditMetrics as it relies
upon the “Expected Default Frequency”, or EDF, for
each issuer, rather than upon the average historical
transition frequencies produced by the rating agencies,
for each credit rating. That is, KMV does not use
Moody’s or Standard & Poor’s statistical data to assign
a default probability which only depends on the rating
of the obligor.
On the other hand, although Wilson (1997) attempt
to incorporate the credit cycle into transition matrices,
they don’t consider the risk premium. Recently, Jarrow,
Lando and Turnbull’s (1997) model matches the
Committee’s opinion reasonably well, and represents a
major step forward in credit risk modeling. The model
of Jarrow, Lando and Turnbull (1997) is based on the
risk-neutral probability valuation model, also called the
Martingale approach to pricing of securities, which
derives a risk premium for the dynamic credit rating
process from the Markov chain process and then estimates the default probability by transition matrix.
CreditRisk+ is a model developed by Credit Suisse
Financial Products (CSFP) at the end of 1997.
CreditRisk+ assumes that default for individual bonds,
or loans, follows a Poisson process and attempts to
estimate the expected loss of loans and the distribution
of those losses, with a focus on calculating the required
capital reserves of financial institutions.
The risk-neutral probabilities use to value risky assets has been in the financial literature at least as far
back as Arrow (1953) and has been subsequently developed by Harrison and Kreps (1979), Harrison and
Pliska (1981), and Kreps (1982). In finance, it has been
traditional to value risky assets by discounting cash
flows. Risk-neutral probabilities can be used in setting
the required spread or risk premium on a loan. The
risk-neutral valuation framework provides valuable
CreditPortfolio View was proposed McKinsey,
which is a consulting firm. CreditPortfolio View is a
multi-factor model which is used to simulate the joint
conditional distribution of default and migration probabilities for various rating groups in different industries.
The default probability for CreditPortfolio View is
condition on the value macroeconomic factors like the
unemployment rate, the rate of growth in GDP, the
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Su-Lien Lu/Asia Pacific Management Review (2007) 12(2), 73-84
the second highest, …, state C the lowest credit class,
state C+1 designates the default. It is usually assumed
for the sake of simplicity that the default state C+1 is
absorbing. Furthermore, let f ij = P x t +1 = j x t = i ,
i, j ∈ S , t=0,1,2,… denote the probability of state i
transiting to state j through the actual probability
measure. That is, f ij and P represent the one-step
transition probability and actual probability measure,
respectively. Then, the discrete time and the regimeswitching of credit class i transiting to credit class j can
be represented by a time-homogeneous transition matrix according to the following, and is called as the
observed transition matrix.
tools for both default prediction and loan valuation.
Compared to history-based (or data-based) transition
probabilities, the risk-neutral model gives a forwardlooking prediction of default. In general, the riskneutral default probability will exceed the historybased transition default probability over some horizon
because it contains a risk premium reflecting the unexpected probability of default.
(
Furthermore, Wei (2003) proposes the multi-factor,
Markov chain model that considers the credit cycle and
risk premium on transition matrix. He allows the transition matrix to evolve according to the credit cycle.
Although his model is more general than previous
studies, he assumes the risk premium is kept constant
over time. Kijima and Komoribayashi (1998) and Lu
and Kuo (2006) propose a procedure to estimate the
risk premium, which is not time-invariant but is actually always time-variant. As a result, there are two important factors, the credit cycle and the risk premium,
for assessing the credit risk. The paper relaxes the assumptions of previous researches by incorporating the
credit cycle and time-variant risk premium into
Markov chain model for assessing the credit risk of
bank loans, which is more elaborated than previous
studies. Consequently, we expect that this work can
improve the accuracy of the assessment of credit risk
and be helpful for all banks with an internal rating system, including Taiwan’s banking industry.
⎡ f11
⎢f
⎢ 21
F=⎢ M
⎢
⎢ f C1
⎢⎣ 0
f12
L
f1C
f 22
L
O
L
L
f 2C
M
fC 2
0
where f ij > 0
submatrix
A
( C× C )
M
f CC
0
)
f1,C +1 ⎤
f 2 ,C +1 ⎥⎥ ⎡( CA
⎢ ×C )
M ⎥=⎢
⎥
f C ,C +1 ⎥ ⎢ (1Ο
⎣ ×C )
1 ⎥⎦
∀ i, j and
C
∑f
j =1
ij
D⎤
⎥ (1)
⎥
1 ⎥
(1×1) ⎦
( C ×1)
= 1 ∀ i . The
is defined on non-absorbing states
Ŝ = {1, 2, ..., C} . The components of submatrix A
denote the regime-switching of credit classes for the
bank’s borrower, however, it excludes default state
C+1. D is the column vector with components
3. Model Specification
( C×1)
3.1 Observed Transition Matrix
f i , C +1 , which represent the transition probability of
Credit ratings of firms are published in a timely
manner by rating agencies, such as Standard & Poor’s
or Moody. The importance of credit ratings has increased significantly with the introduction of the Basel
II. They provide investors with invaluable information
to assess the firms’ abilities to meet their debt obligations. If a company’s credit quality has improved or
deteriorated significantly over time, such a review will
prompt the agency to raise or lower its rating.
banks’ borrowers for any credit class, i.e., i = 1, 2,
…,C, switching to a default class, i.e., j = C+1. We
assume for the sake of simplicity that bankruptcy (state
C+1) is an absorbing state, so that Ο is the zero
It is obvious that the present rating of an obligor is
a predictor for its rating in the nearest future. A cardinal feature of any credit ratings is past and present ratings influencing the evolution. Therefore, the Markov
chain is a stochastic process of this kind. In recent
years, it has become common to use a Markov chain
model to describe the dynamics of firm credit ratings
such as Jarrow and Turnbull (1995) and Jarrow, Lando
and Turnbull (1997).
we would say that the default state C+1 is an absorbing
state.
(1×C )
column vector giving a transition probability from the
default state from initial to the final time. Once the
process enters the default state, it would never return to
credit class state, so that f C +1, C +1 = 1 . In such a case,
3.2 Fitted Transition Matrix
For considering the credit cycle, there are some
procedures to calculate the fitted transition matrix. The
first step is to devise a mapping through which the
transition probability can be translated into credit
scores. The paper employs the normal distribution
which is easy to calculate. Since the summation of
each row in a transition matrix is always equal to 1, we
can invert the cumulative normal distribution function
starting from the default function as in Belkin,
Suchower and Forest (1998), Kim (1999) and Wei
(2003). Therefore, the observed transition matrix, ma-
To be more specific, let x t represent the credit
rating at time t of a bank’s borrower. We assume that
x = {x t , t = 0,1,2,...} is a time-homogeneous
Markov chain on the state space S={1, 2,…, C, C+1},
where state 1 represents the highest credit class, state 2
76
Su-Lien Lu/Asia Pacific Management Review (2007) 12(2), 73-84
trix (1), can be converted as
⎡ y12
⎢y
22
Y=⎢
⎢ M
⎢
⎣ yC, 2
y13
y 23
y14
y 24
M
M
yC,3
yC, 4
factor
y1C ⎤
y 2 C ⎥⎥
O
M ⎥
⎥
L yC, C ⎦
L
L
Matrix (2) is ( C × C ) because there is no need to
covert the row for the absorbing state, i.e., default state.
If we convert matrix (2) into a probability transition,
then we get a transition matrix as in matrix (1). As in
Belkin, Suchower and Forest (1998), we decompose Y
of time-t into two factors:
3.3 Risk Premium and Default Probability
In addition, this paper uses the risk-neutral probability approach to assess the credit risk of bank loans.
For higher credit ratings, the historical default probability is almost zero, but the observed loan rates almost always imply a non-zero default probability in the
risk-neutral world. Moreover, risk-neutral framework
could contain unexpected default probability by estimating risk premium. Ideally, the implication is that
we should match loan rates and observed transition
matrices to obtain the risk premium. Accordingly, we
conclude that a risk-neutral probability approach can
also be applied to bank loans.
(3)
where L t is the systematic component shared by all
borrowers, meaning “credit cycle”. The credit cycle
will be positive in good year, which implies that for
each initial credit rating, a rate lower than average default and higher than the average of upgrades to downgrades. On the other hand, the credit cycle will be
negative in bad year. In any year, the observed transition matrix as shown above will deviate from the norm,
that is L t = 0 . Here ε t is a non-systematic, and the
idiosyncratic factor is unique for a borrower. We also
assume that L t and ε t are unit normal variables
and are mutually independent. The coefficient α is
an unknown coefficient, which represents the correlation between Yt and credit cycle, L t .
For the pricing of defaultable borrower, we need
to consider the corresponding stochastic process
~
x = {~
x t , t = 0,1, 2,L} of credit rating under the
risk-neutral probability measure. For valuation purposes, the fitted transition matrix needs to be transformed into a risk-neutral fitted transition matrix under
~
an equivalent martingale measure where we let M
denote such a matrix. The transition matrix under the
new measure need not be Markovian if it is an absorbing Markov chain, which may not be timehomogeneous. Thus the fitted transition matrix under
the risk-neutral probability measure is given by
We find the coefficient of each row of matrix (2) to
minimize the weight, mean-squared discrepancies between the observed and fitted transition probabilities.
We define
Pt (i, j) = Φ( yi , j+1 ) − Φ( yi , j )
(4)
⎛ y − αˆ Lt ⎞ ⎛ yi, j − αˆ Lt ⎞
⎟ (5)
⎟ − Φ⎜⎜
P(yi, j+1, yi, j Lt ) = Φ⎜⎜ i, j+1
2 ⎟
2 ⎟
⎝ 1− αˆ ⎠ ⎝ 1− αˆ ⎠
~ (t, t + 1) L m
~ (t, t + 1)
⎡m
11
1C
⎢m
~ (t, t + 1) L m
~ (t, t + 1)
2C
⎢ 21
~
M (t, t + 1) = ⎢
M
O
M
⎢~
~
⎢mC1 (t, t + 1) L mCC (t, t + 1)
⎢⎣
L
0
0
~
~
⎡A(t, t + 1) D(t, t + 1)⎤
( C×1)
⎢ (C×C)
⎥
⎥
=⎢
⎢ ~
⎥
Ο
1
⎢⎣ (1×C)
⎥⎦
(1×1)
where Φ (⋅) represents the standard normal cumulative distribution function, and equations (4) and (5)
represent the observed and fitted transition probabilities of state i transferred to j observed in time-t, respectively. The least square problem takes the form as in
Belkin, Suchower and Forest (1998).
[
n t , i P t ( i , j) − P t ( y i , j + 1 , y i , j L t )
min ∑ ∑ P ( y
Lt
i
j
i , j+1
]
2
, y i , j L t )[1 − Pt ( y i , j + 1 , y i , j L t ) ]
n t ,i
Pt ( y i , j +1 , y i , j , L t )[1 − Pt ( y i , j +1 , y i , j , L t )]
Therefore, we can get the fitted transition probability
via equation (6). Then, we construct the fitted transition matrix as
⎡ m 11 m 12 L m 1C m 1, C + 1 ⎤
⎢m
⎥
⎢ 21 m 22 L m 2 C m 2 , C + 1 ⎥
(7)
M =⎢ M
M
O
M
M ⎥
⎢
⎥
⎢ m C1 m C 2 L m CC m C , C + 1 ⎥
⎢⎣ 0
0
L
0
1 ⎥⎦
(2)
Yt = αL t + 1 − α 2 ε t
is
(6)
~ (t , t + 1) = P{~
where m
x t +1 = j ~
x t = i} , i, j ∈ S .
~ij
~
mij and P represent the risk-neutral fitted transition
probability and risk-neutral probability measure, respectively. The conditions for equation (1) must be
~
where
~ (t, t + 1) ⎤
m
1,C+1
~
m2,C+1 (t, t + 1) ⎥⎥
⎥
M
⎥ (8)
~
mC,C+1 (t, t + 1)⎥
⎥⎦
1
n t ,i is the number of borrowers from initial
state i transferred to state j. In addition, the weighting
77
Su-Lien Lu/Asia Pacific Management Review (2007) 12(2), 73-84
satisfied here, together with the equivalence condition
~ ( t , t + 1) > 0 if and only if m > 0 .
that m
ij
ij
l i (0) =
Note that the risk premium plays a crucial role for
assessing the credit risk. The zero risk rate (risk-free
rate) and risky rate (loan’s rate) can capture the credit
risk of the bank loans for every rating class with the
risk-neutral probability measure. First, let V0 ( t , T )
be the time-t price of a risk-free bond maturing at time
T, and Vi ( t , T) be its higher risk, that is, risky counterpart for the rating class, i. However, a loan does not
lose all interest and principal when the borrower defaults. Realistically, we assume that a bank will receive
some partial repayment even if the borrower goes into
bankruptcy. Let δ be the proportions of the loan’s
principal and interest, which is collectable on default,
where in general, δ will be referred to as the recovery rate. If there is no collateral or asset backing, then
δ =0. On the contrary, the recovery rate is 0 < δ ≤ 1 .
for i=1,2,…,C
1{I} = ⎧⎨
⎩
{
{
{
~
V ( t , T ) − δ V0 ( t , T )
Q it ( τ > T ) = i
(1 − δ ) V0 ( t , T )
(14)
(15)
C
~ (t, T ) = 1 − m
~
= ∑m
ij
i , C +1 ( t , T )
j =1
which holds for time t ≤ T , including the current
time, t = 0. Similarly, the default probability occurs
before time T as
~
V ( t , T) − Vi ( t , T)
Qit (τ ≤ T ) = 0
,
(1 − δ)V0 ( t , T)
~ − 1 ( 0 , t ) Vi ( 0 , t ) − δ V 0 ( 0 , t )
m
∑
ij
(1 − δ ) V0 ( 0 , t )
j=1
(11)
}
]}
~i
C
~
~
~
A ( 0 , t + 1) = A ( 0 , t ) A ( t , t + 1)
[
where Q t (τ > T ) is the probability under the riskneutral probability measure that the loan with rating i
will not be in default before time T. It is clear that
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. However, if the
borrower defaults, then we should never estimate the
default probability in the future. As a result, we modify
the assumption that every borrower’s credit rating class
is independent only before entering the default state.
We redefine the risk premium as
(10)
}
~
~
Vi (t ,T ) = V0 (t , T ) Et [1{τ >T } ] + Et [δ {τ ≤T } ]
~
~
= V0 (t ,T ) Qti (τ > T ) + δ 1 − Qti (τ > T )
~
= V0 (t ,T ) δ + (1 − δ )Qti (τ > T )
(9)
i=1,2,…,C and t=1,…,T
1 , if I ∈ {τ > T} (not default before time T ) (13)
δ , if I ∈ {τ ≤ T} (default before time T )
Since the Markov processes and the interest rate
are independent under the equivalent martingale measure, the value of the loan is equal to
In equation (9), it is apparent that a zero or nearzero default probability, i.e., m i , C +1 ≈ 0 , would cause
1
l i (t) =
1 − m i , C +1
(12)
Therefore, we can estimate the risk premium by a
recursive method for all loan periods, t = 0, 1,…, T. On
the whole, we also find that risk-neutral transition matrix varies over time to accompany the changes in the
risk premium by equations (10) and (12). Then, we
assume the indicator function to be
As shown by Jarrow, Lando and Turnbull (1997),
~ ( t , t + 1) = λ ( t ) ⋅ m ,
it can be assumed that m
i, j
ij
ij
i, j ∈ S , and λ ij ( t ) = λ i ( t ) , for j ≠ i and their procedure for risk premium as
V0 (0,1) − Vi (0,1)
λ i (0) =
(1 − δ)V0 (0,1)mi , C +1
1
V ( 0 ,1) − δ V i ( 0 ,1) ,
× 0
1 − m i,C +1
(1 − δ ) V 0 ( 0 ,1)
for i=1,….,C and T=1,2,…
(16)
There are four steps to assess credit risk of bank
loans. First, we construct observed transition matrices
based on the reports of rating agencies. Second, we get
fitted transition probabilities by equation (5) to construct fitted transition matrix. Third, we extract risk
premium via equation (10) and (12). Finally, we construct the transition matrix, matrix (8), condition on
credit cycle and time-varying risk premium. Consequently, we estimate the default probability of bank
loans by conditional Markov chain model that incorporates credit cycle and time-varying risk premium.
~ − 1 ( 0 , t ) are the components of the inverse
where m
ij
~ −1
~
matrix A
( 0 , t ) and A ( 0 , t ) will be invertible.
Note that m i , C +1 is the transition probability of fitted
transition matrix by procedure as above. The denominator of equation (10) is not m i , C +1 , but
(1 − mi , C +1 ) , which can avoid the problem in equa~
tion (9). For equation (11), A
( t , t + 1) = Λ ( t ) ⋅ A
and Λ ( t ) is the ( C × C ) diagonal matrix with diagonal components being the risk premium adjusted
to l j ( t ), j ∈ Ŝ . In particular, the risk premium of t = 0
is
However, there are some limits to apply the proposed model. First, the model limit to application of
borrowers with credit ratings. If a borrower without
78
Su-Lien Lu/Asia Pacific Management Review (2007) 12(2), 73-84
credit rating, then the proposed model can’t work. Second, if the credit ratings mass in a rating class overly,
the model may be failure. To exclude these limits, we
expect that the proposed model could obtain more reliable estimate results for credit risk management.
On the other hand, the risk-free rate is published
by the Central Bank in Taiwan. We take the government bond’s yield as a proxy for the risk-free rate. The
yields of government bonds for various maturities are
published by the Central Bank in Taiwan. Since the
maturity of bank loans and government bonds are different, we have to adjust the yields of government
bonds, so we interpolate the yield of government bond
whose maturity is the closest and take as the risk-free
rate.
4. Data
In Taiwan, there are two rating agencies, the Taiwan Rating and the Taiwan Economic Journal. The
sample data come from two databases of the Taiwan
Economic Journal (TEJ), including the Taiwan Corporate Risk Index (TCRI) and long and short-term bank
loans.
Finally, the recovery rate plays an important role
for making lending decisions that serves as security for
bank loans. In general, banks will set a recovery rate
according to kinds, liquidity, and value of collateral
before lending. Fons (1987) assumed a constant recovery rate of 0.41 according to the historic level. Longstaff and Schwartz (1995), and Briys and de Varenne
(1997) also assumed a constant recovery rate. Carty
and Lieberman (1996) assessed the recovery rate on a
small sample of defaulted bank loans and found that it
averaged over 71%. Copeland and Jones (2001) assumed that the recovery rate was equal to zero in all
sample years. On the other hand, Lu and Kuo (2005,
2006) suggested taking the recovery rate as the exogenous variable from 0.1 to 0.9.
The risk horizon is usually set at one year (Carty
and Lieberman, 1996; Wei, 2003; Lu and Kuo, 2006).
But this horizon is arbitrary and it mostly dictated by
the availability of the data and financial reports processed by rating agencies. Therefore, we choose yearly
data in the paper.6 The sample period is between 1997
and 2003.
The TCRI is a complete history of short and longterm rating assignments for Taiwan’s corporations.
The definitions of the rating categories of TCRI for
long-term credit are similar to Standard & Poor’s and
Moody. 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 being the lowest in
terms of default risk, which is similar to the investment
grade for Standard & Poor’s and Moody. Obligation
rated number 9 is the most risky and the rating class D
denotes the default borrower. Therefore, the rating
categories used by TEJ, Standard & Poor’s and Moody
are quite similar, though differences in opinions can
lead in some cases to different ratings for specific debt
obligations. On the other hand, since the borrowers’
obligation rated numbers are not consistent 7 every
year, we combine the number 1~4 as a new rating class,
denoted as 1*. Similarly, we combine numbers 5~6 and
7~9 as two new rating classes, denoted as 2* and 3 * ,
respectively. Thus, there are four rating classes, 1* , 2*,
3* and D.
In particular, Altman, Resti and Sironi (2004) present a detailed review of default probability, recovery
rate and their relationship. They find that most credit
risk models treat recovery rate as an exogenous variable either 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 1993; Hull and
White, 1995; Longstaff and Schwartz, 1995). On the
other hand, 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, and the data on recoveries on defaulted loans is clearly incomplete. Consequently, we
assume the recovery rate as exogenous variables from
0.1 to 0.9 in this paper.
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. We can analyze the credit rating class
of borrowers to investigate the credit risk of bank loans.
6
7
In conclusion, we analyze the default risk for at
least a one-year horizon and therefore exclude observations for short-term loans and incomplete data. We also
exclude loans that have an overly low rate because
they are likely to have resulted from aggressive accounting politics and bias the estimated results. Since
the data without posting collateral are insufficient for
gauging credit risk, we do not consider these loans.8
There are some shortcomings of our data. For example, the information about within-year rating transition is ignored by yearly data.
We analysis borrowers’ credit rating changes during their lending
periods to construct transition matrices. However, lending periods of
every borrower are not the same that may induce some rating classes
have no borrowers and the transition probabilities will be zero. Because of limit* sample size, we take credit ratings 1 through 4 is
grouped into 1 .
8
79
In general, a bank exposes itself to higher risks if it lends without
Su-Lien Lu/Asia Pacific Management Review (2007) 12(2), 73-84
Lending percentile
That is, we analyze the credit risk of mid-and longterm loans with posting collateral for thirty-one domestic banks in Taiwan.
5. Empirical Results
In this paper, we estimate the credit risk of thirtyone domestic banks in Taiwan.9 We compare differences in credit risk between observed and fitted transition matrices.
1
Table 1 is summary statistic of our data. Visual inspections of Table 1, average loan rates and their corresponding government bond yields are 6.721% and
5.410%, respectively. In general, the risky rates are
higher than risk-free rates. Furthermore, loan rates
have greater volatility than that of risk-free rates. The
average lending period is 6.488 years. Finally, loan
rates and risk-free rates are not normality from the
skewness and kurtosis. Figure 1 shows lending percentile for every industry in Taiwan. Averagely, the highest lending percentile is textile, which is the traditional
industry in Taiwan.
12.085%
0.939%
6.721%
Corresponding
government
bond
7.907%
1.206%
5.410%
Maximum
Minimum
Mean
Standard
deviation
Kurtosis
Skewness
Correlation
1.820%
1.313%
1.076
0.960
-0.516
1.394
-0.356
0.353
-0.339
0.201
5
7
9
11 13 15 17 19
Note: We divide loan industries of thirty-one banks into categories,
namely 1.Cement;
2.Foods; 3.Plastic; 4.Textile; 5.Electrical; 6.Wire&Cable;
7.Chemicals; 8.Glass/Ceramics;
9.Pulp/Paper;
10.Steel;
11.Rubber;
12.Automobile;
13.Electronics;
14.Construction,
15.Transportation,
16.Tourism; 17.Banking & Securities; 18.Conglomerate;
19.Others.
the default probabilities are estimated by rating agencies’ data that don’t consider risk premium and credit
cycle. It is interesting to note that the default prob*
ability is higher in rating class 1 than that in rating
*
class 2 that violate the monotonicity of transition
matrix. Therefore, we estimate the credit cycle and risk
premium in further steps.
Lending period
9.146
4.713
6.488
Then, we estimate the fitted transition probabilities
that consider credit cycle by equation (6) and construct
fitted transition matrix. Thus, we list the average fitted
transition matrix from 1998 to 2003 in Table 3. From
Tables 2 and 3, we find that the credit cycle has significant effect on transition matrix. The credit cycle
can explain the violation of monotonicity in Table 1.
There are four steps for assessing credit risk of
bank loans in this paper. First, we calculate the observed transition matrix as matrix (1). We show the
average observed transition matrix of thirty-one banks
from 1998 to 200310 in Table 2 without considering
risk premium and credit cycle. In other words, Table 2
is generated by history-based transition matrix. We
will focus our attention on the last column of Table 2,
which denotes the default probability. However,
9
3
Figure 1. Lending Percentile for Industry
Table 1. Summary Statistics
Loan rate
0.14
0.12
0.10
0.08
0.06
0.04
0.02
0.00
The default probability will be overestimated and
underestimated of less risk rating class to higher risk
rating class. On the other hand, the default probability
will be underestimated in the observed transition ma*
trix, especially for credit rating 3 . Furthermore, we
estimate the time-varying risk premium and incorporate into fitted transition matrix to get default probability under risk-neutral probability measure. On the other
hand, the coefficient α is the correlation between
Yt and credit cycle L t in equation (3). The estimated value of α is 0.0934 in Table 3.
collateral. As a result, banks always request borrowers to post collateral.
The thirty-one domestic banks include:(1) Bank of Taiwan; (2) Bank
of Overseas Chinese; (3) Bowa Bnak; (4) Central Trust of China; (5)
Chang Hwa Commercial Bank; (6) Chaio Tung Bank; (7) Chinatrust
Commercial Bank; (8) Chinfon Commercial Bank; (9) Cosmos Bank,
Taiwan; (10) EnTie Commercial Bank; (11) E. Sun Commercial
Bank; (12) Far Eastern International Bank; (13) First Commercial
Bank; (14) Fubon Commercial Bank; (15) Fuhwa Commercial Bank;
(16) Grand Commercial Bank; (17) Hua Nan Commercial Bank; (18)
Hsinchu International Bank; (19) Jih Sun International Bank; (20)
Land Bank of Taiwan; (21) Taipei International Bank; (22) Taiwan
Cooperative Bank; (23) Taipei Bank; (24) Taishin International
Bank; (25) Ta Chong Bank; (26) The Chinese Bank; (27) The Export-Import Bank of the Republic of China; (28) The Farmers Bank
of China; (29) The International Commercial Bank of China; (30)
The Shanghai Commercial and Saving Bank; (31) United World
Chinese Commercial Bank.
Table 2. Observed Transition Matrix, 1998-2003
Rating at the end of year
Initial Rating
*
1
2*
3*
80
*
1
2*
3*
D
0.707
0.195
0.024
0.075
0.035
0.738
0.186
0.042
0.004
0.066
0.820
0.110
Su-Lien Lu/Asia Pacific Management Review (2007) 12(2), 73-84
Table 3. Fitted Transition Matrix, 1998-2003
Finally, we assess the average default probability
of thirty-one banks by equation (16). Table 6 and Figure 2 show default probabilities of observed and riskneutral fitted transition matrices, respectively. The observed default probabilities are estimated via historical
credit ratings. In 1998, the observed default probability
is higher resulted from Asian crisis and many firms are
downgraded. However, if we consider credit cycle and
risk premium, we find that default probabilities in 2000
and 2001 are higher than default probabilities in other
years.
Rating at the end of year
Initial Rating
1*
2*
3*
1*
2*
3*
α̂
0.781
0.188
0.014
0.016
0.034
0.743
0.187
0.036
0.016
0.124
0.605
0.255
D
0.0934
Third, we estimate the time-varying risk premium
to transform fitted transition matrix into risk-neutral
fitted transition matrix via equations (10) and (12).
Therefore, we list the average risk premium in Table 4.
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*.10 We find
that when the risk premium is exactly unity, the default
probability will remain unchanged when the riskneutral probability measurement is performed, i.e.,
~ , ∀ i . A risk premium smaller than 1.0
m ij = m
ij
~ , ∀ i . From equation (10) and (12),
means m ij < m
ij
the smaller risk premium estimate compensates for the
bigger discrepancy between default probabilities under
the real and the risk-neutral world. Thus we extract risk
premium and incorporate that into fitted transition matrix. The fitted transition matrix under risk-neutral
probability measure is shown in Table 5.
Although Lu and Kuo (2006) have been estimate
credit risk of bank loans in Taiwan, they did not consider credit cycle. They adopt unconditional Markov
chain model. Comparing with results of Lu and Kuo
(2006), our results are somewhat differences. Our empirical results are all higher than Lu and Kuo (2006)
and we conjecture that differences may be resulted
from credit cycle.
In addition, we include the non-performing loan
(NPL) ratio in Table 6. The NPL refers to loan accounts the principle and interest of which have become
past due or those which exceeded the due date. The
NPL ratio is equal to NPL divided by the total loan.
Comparing with the NPL ratio, estimated results and
the NPL ratio are all highest in 2001. Since the NPL
ratio and default probability are concepts of ex-ante
and ex-post, respectively. As a result, the NPL ratios
are always lower than estimated results. In spite of the
estimated results are not the same as the NPL ratios,
we find that risk-neutral fitted default probabilities
have more power to assess credit risk. For example,
observed default probability in 2002 is 3.7% lower
than the NPL ratio that is unreasonable and may be
underestimated. Then, we compare empirical results
and business cycle in Taiwan that is explained below.
Averagely, the credit rating 2* have smaller risk
premium in Table 4. Ideally, we can conjecture that
there is larger discrepancy between default probabilities in real and neutral world. From Table 3 and 5, default probabilities in credit ratings 2* have bigger differences.
Table 4. Average Risk Premium
Rating
*
1
2*
3*
1998
Maturity (Years)
1999 2000 2001 2002
One contribution of this paper is that it incorporates the credit cycle and time-varying risk premium
into the transition matrix, which has never been discussed in previous studies. The credit cycle will be
positive in good years, and negative in bad years. Figure 3 represents the historical movement of credit cycle
that describes past credit conditions not evident in the
observed transition matrices. On the other hand, Figure
4 shows the total scores of monitoring indicators of
Taiwan11 from 1998 to 2003. The cyclical trough occurs in 2000-2001 that was the period in economic
recession of Taiwan.
2003 average
0.955
0.968
0.984 0.975 0.956 0.973
0.969
0.951
0.870
0.723 0.780 0.793 0.885
0.834
0.941
0.994
0.999 0.995 0.993 0.999
0.986
Table 5. Fitted Transition Matrix under Risk-neutral
Probability Measure, 1998-2003
Rating at the end of year
10
Initial Rating
*
1
2*
3*
D
1*
2*
3*
0.752
0.187
0.013
0.047
0.028
0.612
0.156
0.205
0.016
0.120
0.599
0.264
11
This results is similar to Kijima and Komoribayashi (1998) and Wei
(2003), who using a different set of data.
81
The total scores of monitoring indicators of Taiwan is composed of
monetary aggregates M1b, direct and indirect finance, bank clearing,
remittance, stock price index, manufacturing new order index, exports, industrial production index, manufacturing inventory to sale
ratio, nonagricultural employment, export price index, and manufacturing output price index.
Su-Lien Lu/Asia Pacific Management Review (2007) 12(2), 73-84
Table 6. Default Probability
Year
1998
Default probability
Observed default probability
Risk-neutral fitted default
probability
NPL
1999
2000
2001
2002
2003
0.151
0.066
0.079
0.080
0.037
0.040
0.134
0.137
0.167
0.171
0.160
0.136
0.0437
0.0488
0.0534
0.0748
0.0612
0.0433
Note: NPL is non-performing loan ration of domestic banks in Taiwan. The NPL data is obtained from Bureau of Monetary Affairs, Financial
Supervisory Commission, Executive Yuan, R. O. C.
0.180
Default probability
0.160
0.140
0.120
0.100
0.080
risk-neutral fitted
0.060
observed
0.040
0.020
0.000
1998
1999
2000
2001
2002
2003
Year
Figure 2. Average Default Probability
Ditrisk in 2000-2001
0.3
0.2
0.1
0
-0.1
-0.2
-0.3
-0.4
-0.5
-0.6
1998
1999
2000
2001
2002
2003
Year
Figure 3. Credit Cycle from 1998 to 2003
From Figures 3 and 4, we find that the credit cycle
drops below zero while Taiwan’s business cycle from
peak to trough in 2000-2001. From Table 6 and Figure
2, we find that default probabilities in 2000-2001 are
higher than other years. On the whole, the relative high
proportion of borrowers together with 2000-2001
credit slump accounts for a high number of defaults
that may be due to the business cycle. That is, the
credit risk in 2000-2001 is higher than other years. On
the other hand, credit cycle has stayed positive and
redit conditions have remained benign, and the default
probabilities are low during other periods. Consequently, the model for estimating the default probability is accurate and reliable.peak to trough in 2000-2001.
From Table 6 and Figure 2, we find that default probabilities in 2000-2001 are higher than other years. On
the whole, the relative high proportion of borrowers
together with 2000-2001 credit slump accounts for a
high number of defaults that may be due to the business cycle. That is, the cre dit risk in 2000-2001is hig
82
Su-Lien Lu/Asia Pacific Management Review (2007) 12(2), 73-83
40
35
30
25
20
15
10
5
2003/7/1
2003/1/1
2002/7/1
2002/1/1
2001/7/1
2001/1/1
2000/7/1
2000/1/1
1999/7/1
1999/1/1
1998/7/1
1998/1/1
0
Year/Month
Figure 4. Total Scores of Monitoring Indicators
effective tool for any credit review process of financial
institutions.
er than other years. On the other hand, credit cycle has
stayed positive and credit conditions have remained
benign, and the default probabilities are low during
other periods. Consequently, the model for estimating
the default probability is accurate and reliable.
For subsequent research, the availability of data on
recovery rate over time can be the way to develop a
dynamic measure of credit risk for financial institutions. On the other hand, further studies also can develop satisfactory way to derive the exposure and loss
distribution when default occurs.
6. Conclusion
This paper focuses on providing a formal model to
assess the credit risk of bank loans for thirty-one domestic banks in Taiwan over the period 1998 to 2003.
We extend Markov chain model by incorporating
credit cycle and risk premium, which so-called conditional Markov chain model. The proposed model will
get more reliable estimate results for risk management.
Acknowledgments
The authors thank anonymous reviewers and the
editor for their helpful recommendations. Financial
support from the National Science Council (NSC 942416- H- 239- 006) is gratefully acknowledged.
The main conclusions and contributes can be
drawn in the following aspects. First, the model incorporates the credit cycle into the observed transition
matrix, so-called fitted transition matrix. In addition,
we compare the differences between the observed and
fitted transition matrix. The empirical result shows that
the observed transition matrix will underestimate the
default probability of risky rating class may be resulted
from ignoring credit cycle.
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