Su-Lien Lu and Asia
Chau-Jung
PacificKuo/Asia
Management
Pacific
Review
Management
(2006) 11(2),
Review
111-122
(2006) 11(2), 111-122
The Default Probability of Bank Loans in Taiwan: An Empirical
Investigation by the Markov Chain Model
Su-Lien Lua,* and Chau-Jung Kuob
a
Department of Finance, National United University, No. 1, Lien Da, Kung-Ching Li, Miao-Li, Taiwan 36003, Republic of China
b
Department of Finance, National Sun Yat-sen University, 70 Lien-Hai Road, Kaohsiung, Taiwan
Accepted in January 2006
Available online
Abstract
This paper presents the Markov chain methodology to derive the default probability of bank loans. The model is more elaborate than
that of the previous model, which was developed by Jarrow, Lando and Turnbull (1997). In our model, through relaxing the assumption of
the Jarrow, Lando and Turnbull (1997), we estimate the risk premium only when the default state has not occurred in the stochastic process
of non-homogeneous Markov. In our model, the default probability for each borrower and its risk premium will be recursively endogenous.
We estimate default probabilities of thirty-one banks in Taiwan. On the other hand, we also compare default probabilities of banks whether
participate in financial holding companies or not. The empirical result indicates that banks participated in financial holding companies do
not have better credit risk management. Consequently, in facing the Basel II Accord, we hope that this paper will be helpful for Taiwan’s
financial institutions.
Keywords: Credit risk; Default probability; Markov chain model
veloped very rapidly. Many new banks were set up after
government deregulated financial restrictions in Taiwan.
Therefore, Taiwan’s banking industry has been evolving
from an oligopoly industry to a completely competitive
industry. Unfortunately, Taiwan’s banks have suffered serious problem, such as an increase in the credit risk. For
example, there were tremendous problems with bank loans
in Taiwan such as the Chung Shing Bank credit quality
problems. In the worst case, these problems can cause a
financial institution to become insolvent or result in such a
significant drain on capital and net worth that they adversely affect its growth prospects and ability to compete
with other domestic and international financial institutions.
On the other hand, the Taiwan’s Finance Ministry also
planned to consolidate weak financial institutions into
large, stable banks through mergers or acquisitions by the
new Finance Act. As a result, in Taiwan, the Merger Law
of financial institutions and the Financial Holding Company Act were signed into law on November 2000 and
June 2001, respectively. The later Act allowed the financial institutions to diversify their services such as banks,
insurance companies and securities firms. There are some
advantages such as informational advantages, economy of
scope and operational effectively, may be applicable to
banks that engage in those non-banking activities. But, we
do not know whether the credit risk management of a bank
1. Introduction
Financial institutions, such as banks, credit unions,
insurance companies, and mutual funds, perform the essential function of channeling funds from those with surplus funds (supplier of funds) to those with shortages of
funds (users of funds). Although we might separately
categorize or group financial institutions as life insurance
companies, banks, finance companies, and so on, they face
many common risks such as credit, market, and operational risks. In particular, financial institutions accept the
credit risk on these loans in exchange for a fair return sufficient to cover the cost of funding (e.g., covering the costs
of borrowing, or issuing deposits) to household savers and
the credit risk involved in lending. Therefore, the measurement of the credit risk on individual loans is crucial if a
financial institution manager want to price a loan correctly
and set appropriate limits on the amount of credit extended
to any one borrower or the loss exposure it accepts from
any particular counterpart.
For the past three decades, Taiwan has enjoyed the
kind of growth that many countries envied, especially
when most East Asian and Southeast Asian countries has
experienced economic downturns. In the early 1990s, the
Taiwan’s government decided to liberalize the financial
market. Subsequently, the financial market in Taiwan de*
Corresponding author, e-mail: lotus-lynn@nuu.edu.tw
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Su-Lien Lu and Chau-Jung Kuo/Asia Pacific Management Review (2006) 11(2), 111-122
participating in a financial holding company is better than
others. Consequently, for the topic of this paper, we not
only consider the default probability of banks loans, but
also analyze the credit risk management of banks taking
part in financial holding companies and others.
provide an appropriate model, the Markov chain model, to
measure the default probability of bank loans. We also
want to ensure that credits are made in accordance with
bank policy and to provide an independent judgment of
asset quality that is unaffected by relationships with the
borrower. A model that provides an effective credit risk
review not only helps to detect borrowers in difficulty, but
also helps present weak credits from being granted. This is
due to the reason that credit officers are likely to be more
diligent when they know their work will be subject to review. The Markov chain model in this paper represents the
core of credit risk management, which has been applied to
valuing the default probability of bonds in Jarrow, Lando
and Turnbull (1997), but never applied to the bank loans.
Since the method and model to assess the default probability of bonds and loans is similar, we extended the Markov
chain model to bank loans. 2 For Jarrow, Lando and
Turnbull (1997), they assume that the risk premium is independent, including default state. However, it is not necessary to estimate the risk premium in default state. As a
result, we relax the assumption of Jarrow, Lando and
Turnbull (1997) and estimate risk premium only before
going into default state. In particular, the default probability is endogenously determined for bank loans in this paper.
Over a decade ago, the Basel Committee on Banking
Supervision (“the Committee”)1 issued “the International
Convergence of Capital Standard” (or “Capital Accord of
1988”) in 1988, which established regulations regarding
the amount of capital banks should hold against credit risk.
Furthermore, the treatment of market risk was incorporated in the Committee’s 1996 amendment, i.e., “Amendment to the Capital Accord to Incorporate Market Risks”.
Therefore, Basel I was drawn up and provides a standard
approach for measuring credit and market risks to determine minimum capital requirements.
More than a decade has passed since the Committee
introduced its 1988 Capital Accord and the businesses of
banking, risk management practices, supervisory approach
and financial markets have seen significant changes over
this period. 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. For
the first time, the Committee addressed operational risk
and emphasized the alignment of actual risk with the allocation of capital. The Committee sought comments from
interested parties by 31 May 2001 and finalized the new
accord by the fourth quarter of 2003. The final version of
the new Accord was published in 2004 and will be implemented by the end of 2006.
In this paper, we contribute to the literature in the following aspect. First, this paper serves as one of the first
studies to adopt a Markov chain model estimating the default probability of bank loans3. Second, the model relaxes
the assumption of Jarrow, Lando and Turnbull (1997) and
estimate the risk premium of every borrower’s credit rating process before entering default state. Third, we also
compare differences of the default probability as to
whether banks take part in financial holding companies or
not. The empirical results indicate that credit risk management is indifferent between banks participate in financial holding companies and others. The issue is never discussed in previous studies. On the whole, dealing with the
The new framework focuses on improvements in the
measurement of credit risk, which are more elaborate than
those in the 1988 Capital Accord. Specifically, the new
framework proposes the internal rating based (IRB) approach, by which banks will be allowed to use their internal estimates of borrowers creditworthiness, including
probability of default (PD), loss given default (LGD), and
exposure at default (EAD) to assess the credit risk of loans,
subject to strict methodological and disclosure standards.
2
From the viewpoint of the Committee, it encourages
banks to construct an internal rating approach to measuring credit risk endogenously. Generally, lending policies
that ignore basic principles of analysis are those responsible for debacles as in the case of Chung Shing Bank. We
should form the essential components of credit risk management structures as fundamental analysis, or return to
basics and ongoing stress testing that help to predict the
borrower’s ability to repay. The purpose of this paper is to
1
3
The Basel Committee on Banking Supervision is a committee of banking supervision authorities that was established by a group of banking
regulators from leading industrialized nations, i.e., the central bank
governors of the G-10 countries. Originally intended to apply only to
internationally active banks based in G-10 nations, the Capital Accord
quickly became the standard for regulatory bodies worldwide.
112
In essence, both loans and bonds are contracts that promise fixed
payments in the future. Loans and bonds stand before the claims of a
firm’s equity holders if the firm goes into default. In addition, the rates
of loans are similar to bond yields, usually reflecting risk premiums
that vary with the perceived quality of the borrowers and the collateral
or security backing of the debt. As a result, we can use similar methods
and models to assess the credit risk on a bank’s loans and bonds.
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), Chang, Tu and Tsai (2005) and Kuo and Lu (2005). 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. For
Taiwan’s cases, Li and Lin (2000) adopt Markov chain model to examine the return variability for major East Asian market indices. Kuo
and Lu (2005) investigate the stock behavior of financial holding
companies via Markov chain model. Despite this method has been applied to estimate the default risk of bonds, it never applied to evaluate
the default probability of bank loans in Taiwan.
Su-Lien Lu and Chau-Jung Kuo/Asia Pacific Management Review (2006) 11(2), 111-122
new of Basel Capital Accord, we expect that the model
proposed in this paper will be helpful for Taiwan’s financial institutions4.
Longstaff and Schwartz (1995) recall Jones, Mason and
Rosenfeld (1984) have shown this kind of model produces
credit spreads are lower than the actual credit spreads in
the market. Therefore, Zhou (1997) proposes a framework
with a jump diffusion process. He supposes that a firm can
suddenly default because of a downward drop in its value.
There are also many researches based on the Merton’s
assumptions, such as Balck and Cox (1976), Brennan and
Schwartz (1978), Vasicek (1984), Ogden (1987), Kim,
Ramaswamy and Sundaresan (1989), and Titman and
Torous (1989). This class of models imposes assumptions
on the evolution of the value of a firm’s underlying assets.
The liability structure of the firm in conjunction with the
firm’s asset value determines the occurrence of bankruptcy
and the payoffs of default. Despite these improvements to
Merton’s original framework, these models still suffer
from some significant drawbacks. First, they still require
estimates for the parameters of the unobservable aspects of
a firm’s asset value. Second, structural-form models cannot incorporate credit-rating changes, although most corporate bonds undergo credit downgrades before they actually default.
This paper is organized as follows. Section 1 provides
motivation. In Section 2 reviews literature concerning
models of credit risk. Section 3 presents the formal methodology of this paper, i.e., the Markov chain model. Section 4 describes the sample data used in this paper. Section
5 shows the main empirical results. In this section, we
estimate default probability of bank loans and also compare credit risk management of banks whether participate
in financial holding companies or not. Finally, Section 6
includes a discussion of our findings and a conclusion.
2. Literature Review
Many credit risk models have been developed over the
last thirty years. These models can be divided into two
main categories: the structural-form model and the reduced-form model5. Historically, the first class of credit
models adopted was the structural-form model, which was
based on the original framework developed by Merton
(1974)6,using the principles of option pricing in Black and
Scholes (1973). Subsequently, many generations of structural-form models have tried to refine the original Merton
framework by removing one or more of Merton’s assumptions as above. Shimko, Tejima and van Deventer (1993)
have extended the Merton model to include stochastic interest rates. Jones, Mason and Rosenfeld (1984) attempted
an empirical test of the Merton technology. However,
The attempt to overcome these shortcomings gave rise
to reduced-form models, including Jarrow and Turnbull
(1995), Jarrow, Lando and Turnbull (1997), Duffie and
Singleton (1997) and Lando (1998) and others. Unlike
structural-form models, reduced-form models are not conditioned on the value of the firm. They impose their assumptions directly on the prices of the firm’s traded liabilities, primarily its debt, and on the risk-free term
structure of interest rates. Intuitively, the assumptions of
the firm’s debt prices are related to the credit spread,
which is decomposable into the probability of bankruptcy
multiplied by the loss in the event of bankruptcy. In addition, reduced-form models can extract credit risks from
actual market data and do not depend on asset volatility
and leverage. Therefore, parameters that are related to the
firm’s value need not be estimated in order to implement
them.
4
As noted by the Basel Committee on Banking Supervision in 1999:
In order to maintain a sound credit portfolio, a bank must have an established formal evaluation and approval process for the granting of
credits. Approval should be made in accordance with the bank’s written guidelines and granted by the appropriate level of management.
There should be a clear audit trail of documenting where the approval
process was complied with and individual(s) and/or committee(s) providing input as well as making the credit decision are identified.
5
In fact, there are other approaches to measure credit risk such as Lee
and 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
market-based approach.
6
To develop the Black-Scholes type pricing model, Merton (1974)
making the following assumption:
(A1) there are no transitions costs, taxes, or problems with individual of
assets.
(A2) there are sufficient numbers of investors with comparable wealth
levels so that each investor believes that he can buy and sells as
much of an asset as he wants at the market price.
(A3) there are exists an exchange market for borrowing and lending at
the same rate of interest.
(A4) short-sales of all assets, with full use of the proceeds, is allowed.
(A5) trading in assets takes place continuously in time.
(A6) the Modigliani-Miller theorem that the value of the firm is invariant
to its capital structure obtains.
(A7) the term structure is flat and known with certainty.
(A8) the dynamics for the value of the firm through time can be described by a diffusion-type stochastic process with stochastic differential equation.
The next step is the review of the literature of recovery.
Recovery is the overall amount reimbursed at default,
while the recovery rate is the fraction of the remaining
quantity or security that is given back to the creditor. Most
of the models substantially summarize this process
through a straight formula for the recovery rate at default.
The approach still allows for uncertain behavior if designed as a function of the assets value. Crouhy and Galai
(1997), whose study stresses some very useful interpretations of Merton’s model, shown that the recovery implicit
in Merton’s model is in fact stochastic. However, the
firm’s value is nonobservable. On the other hand, Longstaff and Schwartz (1995) assume a constant recovered
amount equal to a constant time the price of a risk-free
zero-coupon bond. Briys and de Varenne (1997) propose a
constant recovery rate. Consequently, we simulate the re-
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Su-Lien Lu and Chau-Jung Kuo/Asia Pacific Management Review (2006) 11(2), 111-122
cover rate from 0.1 to 0.9 as the exogenous variable in this
paper.
notes ending rating class, j=1,2,…,C+1. On the other hand,
each of the m ij probabilities represents the probability of
3. Model Specification
getting from class i to class j in one period of time. Thus,
the summation of each row is equal to 1, that is,
C
, ∀ i . Since state space is S={1, 2,…, C, C+1}.
m = 1
Credit ratings of firms are published in a timely manner by rating agencies, such as Standard & Poor’s or
Moody.7 They provide investors with invaluable information to assess firms’ abilities to meet their debt obligations.
For many reasons, credit ratings change from time to time
to point out firms’ with unpredictable credit risks. In recent years, it has become common to use a Markov chain
model to describe the dynamics of firm’s credit ratings.
∑
ij
j =1
If the borrower’s initial rating class in state 1 (i=1), then
state 1 will transfer to state j (j=1,2,…,C+1) at next period.
Therefore, the summation of first row in equation (1) is
equal to 1, that is
m 11 + m 12 + ... + m 1 C + m 1 , C + 1 = 1
.
The submatrix A defined on non-absorbing states
(C ×C )
In general, rating agencies categorize corporate issues
into several classes according to their perceived credit
quality. Different quality ratings are reflected in the degree
to which a corporate bond yield exceeds government’s
yield, i.e., the risk-free rate. As a result, if sufficient issues
are available, it may be possible to extract the required
credit risk premiums and implied probability of default
from actual market data on the rate of bank’s loans.
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
( C ×1 )
the column vector with components m i ,C +1 , which represent the transition probability of banks’ borrower for any
credit class, i.e., i=1, 2, …,C, switch to 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
To be more specific, let xt 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, which is independent of time on the state space
S={1, 2,…, C, C+1}, where state 1 represent the highest
credit class; and state 2 the second highest, …, state C the
lowest credit class; and state C+1 designates default. It is
usually assumed the sake of simplicity that the default
state
C+1
is
absorbing.
Furthermore,
let
m ij = P (x t +1 = j x t = i ) , i , j ∈ S , t=0,1,2,… denote
( 1× C )
the zero column vector giving a transition probability from
the default state at initial time until final time. Once the
process enters the default state, it would never return to
credit class state, so that m C + 1 , C + 1 = 1 . In such a case, we
would say that default state C+1 is an absorbing state.
The risk premium is not time-invariant but is actually
always time-variant. Because of this, for the pricing of the
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 or
equivalent martingale measure. An equivalent martingale
measure is a probability measure in which today’s fair (or
arbitrage-free) price of a derivative security is equal to the
discount expected value of the future payoff of the derivative security. This is in contrast to the actual probability
measure where more risky asset has a greater expected rate
of return than less risky asset.
the probability of state i transiting to state j through the
actual probability measure. That is, m ij and P represent
the one-step transition probability and actual probability
measure, respectively. When the Markov chain is
time-homogeneous, we have that the transition probabilities are time invariant and independent of time. Therefore,
the discrete time and the regime-switching of credit class i
transiting to credit class j can be represented by a
time-homogeneous transition matrix is given by
 m11
m
 21
M = M

mC1
 0
m12
L
m1C
m22
L
m2C
M
mC 2
O
L
M
mCC
0
L
0
m1,C +1 
m2,C +1  (CA
 ×C )
M =

mC ,C +1   (1Ο
 ×C )
1 
As a result, the matrix of equation (1) needs to be
transformed into a risk-neutral transition matrix under the
~
equivalent martingale measure where we let M denote
such a matrix. Without further assumptions, it is well
known that the transition matrix under the new measure
need not be Markovian. However, we assume that it is an
absorbing Markov chain, which may not be
time-homogeneous and is called non-homogeneous
Markov chain. That is, the transition matrix under the
equivalent martingale measure is time-varying and influenced by risk premium. This matrix is written as
D


1 
(1×1) 

( C ×1)
(1)
where mij > 0 ∀ i, j . The row of equation (1) denotes
initial rating class, i=1,2,…,C+1, and
7
the column de-
The rating agencies are Taiwan rating and Taiwan Economic Journal in
Taiwan.
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Su-Lien Lu and Chau-Jung Kuo/Asia Pacific Management Review (2006) 11(2), 111-122
~ (t , t + 1)
m
11
m
~ (t , t + 1)
21

~
M (t , t + 1) = 
M
~
 mC1 (t , t + 1)

0
~
A
(t , t + 1)
 (C × C )
=

~
 (1Ο
×C )

L
L
~ (t , t + 1)
m
1C
~
m2 C (t , t + 1)
O
M
~
mCC (t , t + 1)
L
L
~
D (t , t + 1) 

( C ×1)


1

(1×1)

0
where m~ ij (t , t + 1 ) = P~ { ~x t + 1 = j ~x t = i } ,
~
m
1, C +1 ( t , t + 1) 
~
m2 , C +1 (t , t + 1) 

M

~
mC , C +1 (t , t + 1) 

1
As shown by Jarrow, Lando and Turnbull (1997), under the assumptions that the Markov processes and the
interest rate are independent under the equivalent martingale measure. The value of the loan is equal to
= V0
(2)
= V0
i, j ∈ S .
ity and risk-neutral probability measure. The conditions
for equation (1) must be satisfied here, together with the
equivalence condition that m~ ij ( t , t + 1 ) > 0 if and only
τ>T
t
τ≤ T
i
t
T
)]}
(4)
i
t
~
V ( t , T ) − δ V0 ( t , T )
Q it ( τ > T ) = i
(1 − δ ) V0 ( t , T )
if m ij > 0 . For equation (2), the risk-neutral probability
measure is equal to equation (1) multiplied by risk pre~ ( t , t + 1) = λ ( t ) ⋅ m , i , j ∈ S ,
mium. That is, m
ij
ij
ij
(5)
C
~ (t, T ) = 1 − m
~
= ∑m
ij
i,C +1 ( t , T )
j=1
and λ ij denotes risk premium that will be discussed
which holds for time t ≤ T , including the current time,
t=0. Similarly, the probability that default occurs before
time T as
later.
Secondly, to gauge the default probability of the
bank’s borrower, we have to consider the zero risk rate, i.e.,
risk-free rate, in order to assess the value of the bank’s
loans for every rating class with the risk-neutral probability measure. Let V 0 ( t , T ) be the time-t price of a
~
V0 (t, T ) − Vi (t, T )
Q it ( τ ≤ T ) =
(1 − δ ) V 0 ( t , T )
, for i=1,….,C and
T=1,2,…
risk-free unit discount bond maturing at time T, and let
V i ( t , T ) be higher risk than V 0 ( t , T ) , i.e., risky coun-
(6)
Consequently, risk premium plays an important role in
gauging the credit risk of banks. As shown by Jarrow,
Lando and Turnbull (1997), they assume that
~ ( t , t + 1) = λ ( t ) ⋅ m and λ ( t ) = λ ( t ) , for j ≠ i ,
m
ij
i
ij
ij
i
i, j=1,2,…,C+1. The procedure for risk premium in Jarrow,
Lando and Turnbull (1997) is given by
terpart for the rating class, i. That is, if i=1, then V 1 ( t , T )
denotes the time-t value of the loan maturing at time T,
whose borrower in highest credit class or state 1. Similarly,
V 2 ( t , T ) denotes the time-t value of bank’s loan maturing at time T in second highest credit class or state 2. Thus,
the index “i” represents the credit class of borrower that in
state space as above.
λ i (0 ) =
Since a bank’s loan does not lose all interest and principal if the borrower defaults, we 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, 0 < δ ≤ 1 , where in general δ will be referred to as the recovery rate. If there is no collateral or
asset backing, then δ =08
V 0 ( 0 ,1 ) − V i ( 0 ,1 )
(1 − δ ) V 0 ( 0 ,1 ) m i , C + 1
C +1
~
λ i(t) = ∑ m
j=1
−1
ij
(0, t)
V 0 ( 0 , t + 1) − V i ( 0 , t + 1)
(1 − δ ) V 0 ( 0 , t + 1 ) m i , C + 1
(7)
(8)
The rationale behind equation (7) and (8) are a zero or
near-zero default probability would cause the risk premium to explode. That is, if m i , C + 1 = 0 , then equation
(7) and (8) will be unidentified. Furthermore, the implication of equation (7) and (8) indicate that the credit rating
process we have to estimate risk premium of every borrower, including the default state, C+1.
We assume the indicator function to be
8
t
i
t
~i
where Q
is the probability under the risk neut (τ > T )
tral probability measure that the loan with rating i will not
be in default before time T. It is clear that
~ and P~ represent the risk-neutral transition probabilm
ij
1{ I} = 

{ E~ [1 { } ] + E~ [δ { } ] }
~
~
( t , T ) {Q (τ > T ) + δ [1 − Q (τ >
~
( t , T ) {δ + (1 − δ )Q (τ > T ) }
V i ( t, T ) = V 0 ( t, T )
1 , if I ∈ {τ > T } (not default before time T ) (3)
δ , if I ∈ {τ ≤ T } (default before time T )
However, if the borrower defaults, then we should
never estimate the default probability in the future. As a
~ ( t , t + 1) = λ ( t ) ⋅ m , for
result, we assume that m
ij
i
ij
The best borrowers do not need to post collateral because they are good
credit risks, whereas higher risk borrowers need to post collateral. That
is, posting collateral may be a signal of more rather than less credit risk.
In this paper, we assume the recovery rate in term of exogenous variables from 0.1 to 0.9. The same assumption of the recovery rate also
has been discussed in Lu and Kuo (2005).
~
j=1,2,…,C, j ≠ C + 1 and m
,
i , C +1 ( t , t + 1) = 1 − λ i (1 − m i , C +1 )
for j=C+1. Then, we redefine the risk premium as
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Su-Lien Lu and Chau-Jung Kuo/Asia Pacific Management Review (2006) 11(2), 111-122
l i (0 ) =
and t=0
l i (t) =
(9)
C
1
~ −1 ( 0 , t ) Vi ( 0 , t ) − δ V0 ( 0 , t )
∑m
ij
1 − m i ,C + 1 j=1
(1 − δ ) V 0 ( 0 , t )
i=1,2,…,C and t=1,…,T
,
(10)
~
~
~
A ( 0 , t + 1) = A ( 0 , t ) A ( t , t + 1)
(11)
~
where m
~
trix A
Second, short- and long-term bank loan database of
TEJ records all debts of corporations in Taiwan, including
lender names, borrower names, rate of debt, and debt issuance dates, etc. From the viewpoint of banks, we can
analyze the credit rating class of borrowers to investigate
the credit risk of each bank.
1
V 0 ( 0 ,1 ) − δ V i ( 0 ,1 ) , for i=1,2,…,C
1 − m i,C +1
(1 − δ ) V 0 ( 0 ,1 )
−1
Finally, we take the government bonds’ yield as a
proxy for the risk-free rate. The yields of government
bonds for various maturities are obtained as published by
the Central Bank in Taiwan. The period of the government
bond’s yield is from 1998 to 2003. Because the maturity of
loans and government bonds are different, we have to adjust the yields of government bonds. Hence, we interpolate
the yields of government bonds, whose maturity are closest, and take them as the risk-free rates. Additionally, we
analyze default risk for at least a one-year horizon and
therefore exclude observations whose loan-terms are short
or whose data are incomplete. We also exclude the loans
that have an overly low rate, since they are likely to have
resulted from aggressive accounting politics and will bias
the estimated results. Since the data without posting collateral is insufficient for gauging credit risk, we don’t consider these bank loans. Therefore, we analyze the credit
risk of mid- and long-term loans, which are have posted
collateral in this paper. In other words, the recovery rate is
greater than zero and from 0.1 to 0.9.
( 0 , t ) are the components of the inverse ma~
( 0 , t ) and A ( 0 , t ) will be invertible. For
−1
ij
~
equation (10), 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 ∈ Sˆ . The above
framework assumes that the average, empirical transition
matrix under actual probability measure as equation (1)
remains constant over time. However, the risk-neutral
transition probability matrix as equation (2), varies over
time to accompany the changes in the risk premium by
equation (9) and (10).
4. Data
The sample data come from two databases of the Taiwan Economic Journal (TEJ) includes the Taiwan Corporate Risk Index (TCRI) and long and short-term bank
loans. The sample period is between 1997 and 2003. It
should be noted that TCRI database of TEJ is a complete
history of their short- and long-term rating assignments for
Taiwan’s corporations. The definitions of the ratings categories of TCRI for long-term credit are similar to Standard
& Poor’s and Moody. TEJ applies numerical definition
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. Obligations
rated as the four highest (i.e., numbers from 1 to 4) are
generally considered as being the lowest in terms of default risk, which is the investment grade for Standard &
Poor’s and Moody. Obligations rated from number 5 to 6
are regarded as having significant speculative characteristics. Obligations rated from number 7 to 9 are the most
risky and the rating class D denotes the default borrower.
Since borrowers’ obligation rated numbers of banks are
not consistent, we have to redefine rating classification.
According to the aforementioned, we combine the four
highest (i.e., number from 1 to 4) as a new rating class,
which is denoted by “ 1* ”. Similarly, we combine number
5 to 6 and number 7 to 9 as two new rating classes, “ 2 * ”
and “ 3 * ”, respectively. Therefore, we redefine a new numerical class from 1* to 3 * and D. As a whole, the
rating categories used by TEJ, Standard & Poor’s and
Moody are quite similar, though differences of opinion can
lead in some cases to different ratings for specific debt
obligations.
5. Empirical Results
5.1 The Default Probability for Bank Loans
The non-performing loan (NPL)9 refers to loan accounts the principle and/or interest of which have become
past due or those which exceeded the due date, and the
NPL ratio is equal to NPL divided by the total loan portfolio. Table 1 and 2 shows the NPL ratio and amount of
Taiwanese financial institutions, including domestic banks,
local branches of foreign banks, credit cooperatives, and
credit departments of farmer’s and fishermen’s associations from 1998 to 2003, respectively. From Table 1 and 2,
we find that Taiwan’s NPL problem has increasingly become a serious question of the financial institutions recently10.To effectively reduce the NPL ratio, the Ministry
of Finance (MOF) of Taiwan reduced the Gross Business
Receipt Tax (GBRT) for banks from 5% to 2% in 1999
and is planning to further reduce it to zero. Banks are required to use the resulting tax savings to write off bad
debts.
These efforts appear to have improved the health of
9
116
Loans under surveillance include: (i) Medium or long-term installment
loans overdue for more than 3 months but less than 6 months; (ii) Other
loans whose principal overdue for less than 3 months but with interests
repayment overdue for more than 3 months but less than 6 months; (iii)
Loans whose overdue period reaches NPL standard but need not be
classified as NPL, including restructured loans meeting specified performing standards, loans with approved compensation from the Credit
Guarantee Funds, loans with sufficient time deposits or demand deposits for repayment, and other specified loans granted exemption from
reporting as NPL.
Su-Lien Lu and Chau-Jung Kuo/Asia Pacific Management Review (2006) 11(2), 111-122
the system, indicating that the NPL ratio for banks in Taiwan had fallen to 5.7% by the end of November 2003.
Table 1. Non-performing Loan Ratios (%)
However, many banks have suffered serious credit risk
problem in Taiwan.
Credit
Cooperatives
Credit
departments
of farmer’s
and fishermen’s associations
Although Taiwanese financial institutions include domestic banks, local branches of foreign banks, credit cooperatives, and credit department of farmer’s and fishermen’s associations, this paper focus on the credit risk of
domestic banks, which not include Trust and Investment
corporations. Therefore, we estimate the default probabilities of thirty-one banks in Taiwan. For the model as above,
there are four steps to estimate default probability. First,
we calculate transition probability matrix by equation (1)
for thirty-one banks. We show the average transition
probability matrix of thirty-one banks from 1998 to 2003
in Table 3.
Total
Domestic
Banks
Local
Branches
of
Foreign
Banks
1998
4.93
4.37
1.65
7.55
13.10
1999
5.67
4.88
3.20
10.54
16.03
2000
6.20
5.34
3.22
12.45
17.91
2001
8.16
7.48
3.53
11.66
19.37
2002
6.84
6.12
2.36
10.34
18.62
2003
5.00
4.33
1.51
6.91
15.57
Note-1.Source: Bureau of Monetary Affairs, Financial Supervisory Commission, Executive Yuan, R.O.C.
2.Demostic banks do not include Trust and Investment Corporations
here.
3. The data are adopted the end, December, of this year.
Table 2. Non-performing Loan Amounts10
Total
Domestic
Banks
Local Branches
of Foreign Banks
Credit
Cooperatives
Credit Departments of
Farmer’s and Fishermen’s Associations
Other
1998
7303
5478
91
520
1075
139
1999
8833
6602
172
615
1273
171
2000
10211
7735
186
662
1380
248
2001
13274
10870
185
497
1316
406
2002
10747
8644
113
406
1134
450
2003
8028
6306
70
262
995
395
Note-1.Source: Bureau of Monetary Affairs, Financial Supervisory Commission, Executive Yuan, R.O.C.
2.Unit is NT$100 million.
3.Demostic banks do not include Trust and Investment Corporations here.
Table 3. Average Transition Probability Matrix, 1998-2003
Initial Rating
2*
*
0.707
0.195
0.024
0.075
2*
0.035
0.738
0.186
0.042
3*
0.004
0.066
0.820
0.110
1
10
Rating at the end of year
3*
1*
D
Over the past few years, there has been a steady increase in non-performing loans (NPL). At the end of December in 2000, the NPL ratio stood at 6.2%. If
we also include the rollover loans to households affected by the 1999 September 21 earthquake and those to traditional industries that have been operating normally but are experiencing financing difficulties, this ratio could increase by 1.5 to 2 percentage points.
117
Su-Lien Lu and Chau-Jung Kuo/Asia Pacific Management Review (2006) 11(2), 111-122
Second, we estimate risk premium to transform transition probability matrix into risk-neutral transition probability matrix by equation (9) and (10). Thus, we list average risk premium in Table 4. Third, owing to risk-neutral
probability measure, equation (2) is equal to equation (1)
multiplied by the risk premium. Table 5 represents average
risk-neutral probability matrix of thirty-one banks from
1998 to 2003 and recovery rate from 0.1 to 0.9.11
Finally, we estimate the default probability by equation (6) and show the empirical result in Table 6.
Table 4. Average Risk Premium
1*
1998
0.955
Maturity (Years)
1999
2000
0.967
0.977
2001
0.975
2002
0.968
2003
0.935
2*
0.951
0.918
0.944
0.944
0.871
0.773
3*
0.941
0.983
0.994
0.987
0.974
0.978
Rating
Table 5. Average Risk-Neutral Transition Probability Matrix11
Initial Rating
Rating at the End of Year
2*
3*
1*
D
*
0.679
0.189
0.022
0.110
2*
3*
0.031
0.667
0.164
0.138
0.004
0.000
0.064
0.000
0.801
0.000
0.131
1.000
1
D
Table 6. The Default Probability of Domestic Banks in Taiwan (%)
1998
4.401
8.538
35.989
4.490
6.669
2.511
3.623
27.232
34.639
13.608
28.867
14.472
5.870
7.604
23.206
7.300
4.266
9.357
20.494
7.303
11.447
5.513
22.748
2.983
4.846
10.660
19.371
7.532
4.163
5.852
8.832
11.402
Bank of Taiwan
Bank of Overseas Chinese
Bowa Bnak
Central Trust of China
Chang Hwa Commercial Bank
Chaio Tung Bank
Chinatrust Commercial Bank
Chinfon Commercial Bank
Cosmos Bank, Taiwan
EnTie Commercial Bank
E. Sun Commercial Bank
Far Eastern International Bank
First Commercial Bank
Fubon Commercial Bank
Fuhwa Commercial Bank
Grand Commercial Bank
Hua Nan Commercial Bank
Hsinchu International Bank
Jih Sun International Bank
Land Bank of Taiwan
Taipei International Bank
Taiwan Cooperative Bank
Taipei Bank
Taishin International Bank
Ta Chong Bank
The Chinese Bank
The Export-Import Bank of the Republic of China
The Farmers Bank of China
The International Commercial Bank of China
The Shanghai Commercial and Saving Bank
United World Chinese Commercial Bank
Average
1999
5.504
7.874
52.378
8.333
5.996
2.240
5.459
18.969
31.910
16.606
21.531
9.360
6.209
10.763
17.119
7.300
4.463
12.458
15.000
4.767
8.953
4.153
16.057
2.750
7.546
10.139
13.711
5.940
6.272
4.001
12.831
10.814
2000
4.578
10.073
30.210
9.909
4.457
2.639
8.526
18.620
40.858
15.114
7.450
9.579
5.490
5.721
16.437
8.546
10.127
5.388
14.940
5.773
7.568
5.081
17.321
7.869
9.136
9.643
3.300
5.911
4.924
15.294
8.501
9.082
2001
5.192
11.554
32.131
11.032
4.353
4.602
9.824
19.862
31.886
12.276
7.719
15.864
5.797
4.638
17.996
11.358
5.423
6.586
17.576
6.903
14.618
4.748
4.489
2.361
16.418
17.216
3.300
7.682
4.980
3.268
5.976
9.419
2002
10.353
14.781
30.270
19.744
7.469
4.543
9.312
21.666
38.532
12.626
14.813
15.221
5.146
11.554
17.164
19.117
13.963
20.551
16.597
11.223
9.417
8.006
6.827
6.241
15.738
23.073
4.122
12.223
6.919
13.817
9.228
12.500
2003
6.999
21.634
30.250
30.739
9.828
6.231
10.089
30.218
41.324
11.290
25.468
28.839
16.994
16.019
13.890
20.506
13.478
27.982
18.683
16.394
23.019
12.218
4.822
2.538
30.292
24.696
6.023
12.496
12.312
17.181
12.872
15.836
Note-This table was summarized by the author.
11
According to equation (9) and (10), we find that the recovery rate plays an important role for estimating risk-neutral probability matrix and risk premium.
Hence, Table 4 and 5 are average risk premium and risk-neutral transition probability matrix for recovery rate from 0.1 to 0.9.
118
Su-Lien Lu and Chau-Jung Kuo/Asia Pacific Management Review (2006) 11(2), 111-122
0.250
Default
Probability
0.200
0.150
0.7
0.100
0.4
0.050
0.000
0.1
1998 1999 2000 2001 2002 2003
Recovery
Rate
Year
Figure 1. The Average Default Probability of Domestic Banks in Taiwan
Table 6 shows the estimated results of the default
probability from 1998 to 2003 12 including thirty-one
banks in Taiwan. Since the recovery rate is given exogenous in this paper, we average the default probability of
each simulated recovery rate from 0.1 to 0.9 in Table 6.
Figure 1 reports the average default probability of domestic banks in Taiwan. In particular, many banks’ default
probabilities in 1998 are higher than others years because
of the Asian financial crisis which led to higher levels of
problem loans. According to the above findings, the model
in this paper for estimating default probability is accurate
and can reflect the quality of Taiwanese banks. The weak
banks should take care of the potential credit risk for their
lending policies and review their internal machines for
controlling risk to reduce their credit risk in the future.
The Taiwanese Government drafted the Financial
Holding Company Law to allow financial holding companies to be established and own such subsidiaries as banks,
insurance companies, and securities firms from the year
2001. Each financial holding company is likely to have
two or more wholly owned subsidiaries involved in banks,
insurance companies, securities firms, and other related
activities. Tax and non-tax incentives will be offered to
encourage local financial firms to set up financial holding
companies. Within a financial group, a certain amount of
cross-selling of financial products may be allowed if
proper precautions are established13. As of December 31,
2003, there were fourteen financial holding companies in
Taiwan including Fubon, Hua Nan, Cina Development,
Cathay, Esun, Mega, Fuhwa, Taishin, Shin Kong, Waterland, Sinopac, Chinatrust, and the Jihsun financial holding
company. A complete investigation of Taiwan’s financial
holding companies was carried out by Kuo and Lu (2005).
They presented a formal methodology using the two-state
Markov regime switching approach to allow for the uncertainty event-date of financial holding companies’ stock
behavior. They suggested that there are diversification
benefits accruing from the existence of financial holding
companies in Taiwan. However, they do not suggest that
whether banks taking part in a financial holding company
have better credit risk management.
Comparing Table 1 and Table 6, we find that there are
differences between the default probability and the NPL
ratio. Since 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. As a result, the NPL ratio and default probability are concepts of ex-ante and ex-post, respectively. On the other hand, some banks have insufficient data to analyze and we have excluded these banks.
Consequently, the NPL ratio and default probability are
not consistent.
First, we divided banks into two groups, including
twelve banks belonging to financial holding companies
5.2 The Default Probability of Banks belonging to Financial and Non-financial Holding Companies
13
12
Although the sample period runs from 1997 to 2003, the Markov chain
model has to use the data from last year in 1997 and the empirical result begin in 1998.
119
This bill represents a major reform of our financial system and is
consistent with international trends, especially following the enactment of the US Financial Modernization Act, i.e.,
Gramm-Leach-Blilely Act (“GLB Act”).
Su-Lien Lu and Chau-Jung Kuo/Asia Pacific Management Review (2006) 11(2), 111-122
Table 7. The Default Probability of Banks Belonging to Financial and Non-financial Holding Companies
2001
0.064
0.159
Banks belonging to Financial Holding Companies
Banks belonging to Non-financial Holding Companies
Default Probability
0.200
0.180
0.160
0.140
0.120
0.100
0.080
0.060
0.040
0.020
0.000
2002
0.137
0.161
2003
0.186
0.184
financial
non-financial
2001
2002
2003
Year
Note-The number in parentheses denotes the amount of banks have participated in financial holding companies. In 2001, there are two banks
participate in financial holding companies, including Fubon Commercial Bank and Hua Nan Commercial Bank. In 2002, there are nine
banks participate in financial holding companies, including E. Sun Commercial Bank, Fubon Commercial Bank, Jih Sun International
Bank, Taipei Bank, Chiao Tung Bank, The International Commercial Bank of China, Taishin International Bank, Chinatrust Commercial Bank and United World Chinese Bank. In 2003, there is only one bank, First Commercial Bank, participate in financial holding
company. Therefore, the amounts of banks have participated in financial holding companies from 2001 to 2003 are two, eleven and
twelve, respectively.
Figure 2. The Default Probability of Banks belonging to Financial and Non-financial Holding Companies
and others belonging to non-financial companies14.27.Then,
we consider the differences in the default risk between
banks belonging to financial and non-financial holding
companies, as reported in Table 7. According to Table 7,
we find that default probabilities for banks belonging to
financial holding companies are lower than others in 2001
and 2002. However, the phenomenon is reversed in 2003.
Although banks that do not take part in financial holding
companies may fall behind others for controlling credit
risk in the early stages, they will later catch up to others
belonging to financial holding companies. Consequently,
we recommend that credit risk management is not a significant advantage for banks to participate in financial
holding companies.
ernment deregulated financial restrictions in Taiwan.
Taiwanese banking environment was transformed
from an oligopoly industry to a completely competitive
industry. Although the banking environment improved in
the 1990s, banks faced more and more credit risks. This is
an important consideration for most financial transactions.
In particular, credit risk, or the probability of default, has
always been a major topic of concern for banks and other
financial institutions. What people remember most about
the banking environment of the 1990s was deterioration in
credit quality of the financial institutions in Taiwan. This
led to deteriorated lending decisions and many financial
institutions having no credit measuring function. Therefore,
bank regulators had to develop an effective credit review
process which was used to measure the credit risk for their
loans.
6. Conclusions
Recently, many new banks were set up after the gov-
This paper focused on providing an appropriate model,
the Markov chain model, to measure the default probability for bank loans in Taiwan. We analyzed the default
probability of banks in Taiwan as to whether there were
appropriate checks and balances and whether credit was
given appropriately. We also provide an effective method
to screen credit quality and to focus credit analysis where
14
There are no banks that have joined the Waterland financial holding
company. There are Cathay United Bank, China Development Industrial
Bank, Sinopac Bank and United Credit Commercial Bank have participated in Cathay, China Development, Sinopac and Shin Kong financial
holding company, respectively. But the sample period is too short and we
excluded these banks. Therefore, we consider twelve banks that participate in financial holding companies in this paper, which are shown in
Appendix.
120
Su-Lien Lu and Chau-Jung Kuo/Asia Pacific Management Review (2006) 11(2), 111-122
it can add the most value. On the other hand, the Financial
Holding Company Act was signed into law on June 2001
in Taiwan. The Act permit banks, insurance companies,
securities firms, and other financial institutions to affiliate
under common ownership and offer their customers a
complete range of financial services. Affiliation offers
some advantages for financial institutions. As a result, we
compare the differences in the default probability of banks
belonging to financial and non-financial holding companies. The empirical result indicates that banks have indifferent default probability whether participate in financial
holding companies or not.
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Acknowledgments
The authors thank anonymous reviewers and the editor for their helpful recommendations. Financial support
from
the
National
Science
Council
(NSC
93-2416-H-110-035) is gratefully acknowledged.
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(4) Hua Nan Commercial Bank belongs to the Hua Nan
financial holding company.
(5) Jih Sun International Bank belongs to the Jih Sun financial holding company.
(6) E. Sun Commercial Bank belongs to the Esun financial
holding company.
(7) Chiao Tung Bank and The International Commercial
Bank of China belong to the Mega financial holding
company.
Appendix
(8) Fuhwa Commercial Bank belongs to the Fuhwa financial holding company.
Listed below are the sample banks which belong to
financial holding companies in Taiwan.
(9) Taishin International Bank belongs to the Taishin financial holding company.
(1) First Commercial Bank belongs to the First financial
holding company.
(10) Chinatrust Commercial Bank belongs to the Chinatrust financial holding company.
(2) Fubon Commercial Bank and Taipei Bank belong to
the Fubon financial holding company.
122