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

Keywords: Credit risk, discrete-time Markov chain model, continuous-time Markov chain model, bank loans JEL Classification Codes: G10, G21 1.. The computation of such transition matric

Assistant Professor,Department of Finance,National United University

No 1, Lien-Da, Kung-Ching Li, Miao-Li, 360 Taiwan R.O.C

E-mail: lotus-lynn@nuu.edu.tw Tel: 886-37-381859; Fax: 886-37-338380

Kuo-Jung Lee

Assistant Professor,Department of Commerce Automation and Management

National Pingtung Institute of Commerce

based term structure modeling The empirical results indicate that the discrete-time Markov

chain model may underestimate default probabilities when the dynamic rating process is not taken into consideration Consequently, the conclusion is made that care has to be taken

when discrete- and continuous-time Markov chain models are employed for dynamic credit

risk management

Keywords: Credit risk, discrete-time Markov chain model, continuous-time Markov chain

model, bank loans JEL Classification Codes: G10, G21

1 Introduction

In the past ten years, major developments in financial markets have led to a more sophisticated approach to credit risk management The origination of credit is still based on the relationship between the banker and his client In the banking industry, the classic risk is credit risk that may cause a financial institution to become insolvent or result in a significant drain on capital and net worth that may adversely affect its growth prospects and ability to compete with other financial institutions Therefore, credit risk management has become a major concern for the banking industry and other financial intermediaries This is also stated by the Basel Committee on Banking Supervision (“the Committee”) that formalizes a universal approach to credit risk in financial institutions

In fact, the intention of the Committee is to assure the safety and soundness of the financial system To achieve this goal, the Committee issued the “International Convergence of Capital

Measurement and Capital Standards” document, published in July 1988 Furthermore, the treatment of market and operational risk were incorporated in 1996 and 2001, respectively In June 2004 the Committee published the final draft of the revised framework for capital measurement and capital standards

Traditional credit analysis is an expert system that relies on the subjective judgment of trained professionals, implying that credit decisions are the reflection of personal judgment about a borrower’s ability to repay However, traditional credit analysis has often lulled banks into a false sense of security, failing to protect them against the many risks embedded in their business In contrast with the traditional approach, the other approach is primarily based on statistical methods, such as presented by Jarrow and Turnbull (1995) and Jarrow, Lando and Turnbull (1997) Jarrow and Trunbull (1995) used matrices of historical transition probabilities from original ratings and recovery values at each terminal state The Jarrow, Lando and Turnbull (1997) method is based on the risk-neutral probability valuation model for pricing securities by transition matrices Consequently, a crucial element in such models is the transition matrix

Theoretically, transition matrices can be estimated for any desired transition horizon Generally, transition matrices are estimated by a yearly time horizon, such as Carty and Lieberman (1996), Wei (2003) and Lu and Kuo (2006) The computation of such transition matrices estimated from yearly data implies the assumption that the underlying process is a discrete-time Markov chain model However, large movements are often achieved via some intermediary steps implying that there are no transitions from AAA to default, but there are transitions from AAA to AA and from AA to default Thus, transitions of the intermediate state (AAA→AA→default) contribute to the estimation of the transition matrices In other words, the shorter the measurement interval, the fewer rating changes are omitted Therefore, transition matrices estimated over short time periods best reflect the rating process and the underlying process is a continuous-time Markov chain model The information is gained using the full information of exact transitions, which the discrete-time model ignores, but the continuous-time model does not

The purpose of this paper is to assess the credit risk of bank loans using two different Markov chain models, the discrete- and continuous-time models The different Markov chain models depend on the generation of transition matrices The discrete-time Markov chain model uses the discrete multinomial (or cohort) method and the continuous-time Markov chain model is estimated with continuous hazard rate (or duration) methods Therefore, we also compare the estimated results of different methods Since continuous-time methods incorporate full information of rating transitions, it seems that continuous-time approaches bring more efficient results than discrete-time methods

There are four contributions in this paper First, the credit risk of bank loans is discussed; including secured and unsecured loans, both from analytical and empirical perspectives To our knowledge, not much research has been done on the estimation of loans’ transition matrices considering both discrete- and continuous-time approaches This paper adopts discrete- and continuous-time Markov chain models for measuring the credit risk of bank loans from a more comprehensive perspective than previous studies

Second, although risk premium plays a crucial role in gauging the credit risk of bank loans, previous research has handled the risk premium as a time-invariant (Jarrow, Lando and Turnbull, 1997; Wei, 2003) In fact, the risk premium is actually always a time-variant parameter (Kijima and Komoribayashi, 1998; Lu and Kuo, 2006) Therefore, the assumptions made in previous research were relaxed by incorporating the time-variant risk premium into the transition matrices making it more elaborate

Third, a comparison of the estimated results of discrete- and continuous-time Markov chain models is given It was found that the continuous-time approach has a more reliable default probability than the discrete-time approach The discrete-time estimator may underestimate the default probabilities due to the neglect of some rating transitions whereas the continuous-time estimator incorporates all information on the exact timing of rating transitions

Fourth, a statistics mobility estimator was extended for investigating the migration size of discrete- and continuous-time transition matrices The mobility estimator was designed to give a measure of the transition matrices propensity, which had been developed by Jafry and Schuermann (2004) It was found that the discrete-time method had a higher mobility estimator than the continuous-time method, implying that a higher off-diagonal probability is concentrated in a discrete-time transition matrix rather than diluted in a continuous-time transition matrix On the whole, credit risk modeling is crucial for bank regulators in providing an effective credit risk review, not only in helping

to detect borrowers in difficulty, but also in facilitating to the Basel Capital Accord We expect that this study can provide a suitable model to gauge the credit risk for financial institutions

This paper is organized as follows: Section 1 provides the motivation for this study Section 2 reviews literature concerning models of credit risk Section 3 presents the formal methodology, and Section 4 describes the sample data used in this paper Section 5 shows empirical results and robustness tests Finally, Section 6 includes a discussion of our findings with a conclusion

The structural-form models include the original work of Black and Scholes (1973) and Merton (1974) In such a framework, the securities issued by a firm as contingent claims on its own value and, therefore, the credit risk is driven by the value of the company’s assets The basic intuition behind the Merton model is that default occurs when the value of a firm’s asset is lower than that of its liabilities Furthermore, the basic Merton model has subsequently been extended by removing one or more of Merton’s assumptions Black and Cox (1976) suggest that bondholders can force the reorganization or the bankruptcy of the firm if its value falls to a specific value Kim, Ramaswamy and Sundaresan (1989) and Collin-Dufresne and Goldstein (2001) propose a model similar to the Black and Cox (1976) model, suggesting that capital structure is explicitly considered and default occurs if the value of total assets is low enough to reach a trigger value, which is assumed to be exogenous Leland (1994) endogenizes the bankruptcy while accounting for taxes and bankruptcy costs Leland and Toft (1996) propose a Barrier option model, suggesting that expected default probabilities depend on the endogenously defined bankruptcy threshold

In spite of these improvements on Merton’s original framework, structural-form models still suffer some drawbacks, which are the main reasons behind their relatively poor empirical performance (Altman, Resti and Sironi, 2004; Emo, Helwege and Huang, 2004) First, since the firm’s value is not a tradable asset, the parameters of the structural-form model are difficult to estimate consistently In other words, unlike the stock price in Black and Scholes model for valuing equality options, the current market value of a firm is not easily observable Second, the inclusion of some frictions like tax shields and liquidation costs would break the last rule Third, corporate bonds undergo credit downgrades before they actually default, but structural-form models cannot incorporate these credit-rating changes Finally, most structural-form models assume that the value of the firm is continuous in time and, consequently, the time of default can be predicted just before it happens

Reduced-form models attempt to overcome the above mentioned shortcomings of form models These include Jarrow and Turnbull (1995), Jarrow, Lando and Trunbull (1997), Lando (1998), Duffie (1998) and Duffie and Singleton (1999) Unlike structural-form models, reduced-form models do not default on the firm’s value, and parameters related to the firm’s value need not be

structural-estimated to implement them These variables related to default risk are modeled independently from the structural features of the firm, its asset value and leverage

The calibration of the credit risk for reduced-form models is made with respect to rating agencies’ data Therefore, rating systems have become increasingly important for reduced-form models Their key purpose is to provide a simple qualitative classification of the solidity, solvency and prospects of a debt issuer The importance of credit ratings has increased significantly with the introduction of the Basel II It is obvious that the present rating of an obligor is a strong predictor of his rating in the nearest future A cardinal feature of any credit rating is the past and present rating influencing the evolution Therefore, the Markov chain is a stochastic process, in which the transition probabilities, given all past ratings, depend only on the present state It allows all transition probabilities for a specific time-horizon to be collected in a so-called transition matrix, such as presented by both Jarrow, Lando and Turnbull (1997) and Lando (1998) using transition matrices to determine credit risk

In most applications, transition matrices are estimated by discrete-time observations with a yearly time horizon For example, Lu and Kuo (2006) have applied the discrete-time Markov chain model to assess the credit risk of bank loans by yearly transition matrices However, if borrowers seldom change their rating, then transition matrices typically concentrate along the main diagonal That

is, the most probability mass resides along the diagonal and most of the time there is no migration The low occurrence of certain transitions may be a problem when estimating default probabilities For the lowest risk grade, such as AAA in Standard and Poor’s rating, defaulting in a given period is a rare event Although there are no transitions from AAA to default, there are transitions from AAA to AA and from AA to default As a result, the estimator for transitions from AAA to default should be non-zero and these rare events are ignored by the discrete-time Markov chain model In order to avoid the embedding problem for discrete-time observations, the continuous-time Markov chain model has been adopted to estimate meaningful default probabilities

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

3 Model Specification

3.1 The discrete-time Markov chain model

Let x represent the credit rating of a bank’s borrower at time t Assume that t x= xt t=0,1,2, } is a Markov chain on the finite state space S={1, 2,…, C, C+1}, where state 1 represents the highest credit class; and state 2 the second highest, …, state C the lowest credit class; and state C+1 designates the default It is usually assumed for the sake of simplicity that the state C+1 is the absorbing state

Furthermore, let P(s, t) denote the (C+1)×(C+1) transition matrix generated by a Markov chain model with transition probability as

(x jx i)P

)t

Hereafter, let “P” and “p ” be generally termed the transition matrices and transition probabilities, ij

respectively

The first Markov chain model applied to the transition matrix is the discrete-time Markov chain

model based on annual migration frequencies Generally, estimation in a discrete-time Markov chain

can be viewed as a multinomial experiment since it is based on the migration away from a given state

over a one-year horizon Let Ni(t) denote the number of firms in state i at the beginning of the year

and )Nij(t represent the number of firms with rating i at date t migrated to state j at time t+1 Thus, the

one-year transition probability is estimated as

)t(N

)t(N)

If the rating process is assumed to be a homogeneous Markov chain, i.e.,

time-independent, then the transitions for different borrowers away from a state can be viewed as

independent multinomial experiments Therefore, the maximum likelihood estimator (MLE) for

time-independent probability is defined as

i

T

1 t

ij ij

)t(N

)t(N

where T is the number of sample years For a special case, the number of firms are the same over the

sample period, Ni(t)=Ni, the estimator for the transition probabilities is the average of the

year-on-year transition matrices, such as Bangia et al (2002) and Hu, Kiesel and Perraudin (2002) However,

the special case is implausible Accordingly, the estimator of transition probabilities is always modified

by the number of firms during the sample years If P denotes transition matrix for a Markov chain

over a year horizon, then the discrete-time transition matrix is as

100

0

ppp

p

ppp

p

ppp

p

P

1 C , C CC 2

C 1 C

1 C , 2 C 2 22

21

1 C , 1 C 1 12

11

LL

MMOMM

LL

ij 1, i

Since the information concerning within-year rating transition is ignored in the discrete-time

Markov chain model, the continuous-time Markov chain model had to be used to determine the

additional migration within the year According to Christensen, Hansen and Lando (2004), the

advantages of the continuous-time Markov chain model can be summarized as: (i) The duration

method can obtain non-zero estimates for probabilities of rare events whereas the cohort method

estimates to zero (ii)The duration method uses all available information in the data including

information of a firm even when it enters a new state In the discrete-time estimator, the exact date

within the year that a firm changed its rating cannot be distinguished Therefore, the continuous-time

Markov chain models were also adopted for valuing the credit risk of bank loans

3.2 The continuous-time Markov chain model

As for continuous-time models, the non-parametric method of Aalen and Johansen (1978) was adopted

to replace the cohort methods The Aalen-Johansen estimator imposes fewer assumptions on the data

generating process by allowing for time heterogeneity while fully accounting for all movements within

the sample period In other words, the Aalen-Johansen estimator can be applied to an extremely short

time interval and observe a borrower’s rating movement during the sample period

Let )P~(s t be the transition matrix over the horizon [s, t] and take the Aalen-Johansen

estimator (or product-limit estimator) for the transition matrix The estimator for the transition matrix,

= m1 i

i)T(

AˆI)

transitions over the sample period from s to t The estimator is clearly a duration approach, which

allows for time non-homogeneous while fully accounting for all movements with the sample period

(estimated horizon) The matrix ΔAˆ(Ti) is given by

−Δ

Δ

ΔΔ

Δ

−Δ

ΔΔ

ΔΔ

−

=Δ

+

00

0

)T(Y

)T(N)T(Y

)T(N)

T(Y

)T(N)

T(Y

)T(N

)T(Y

)T(N)

T(Y

)T(N)

T(Y

)T(N)

T(Y

)T(N

)T(Y

)T(N)

T(Y

)T(N)

T(Y

)T(N)

T(Y

)T(N

)T

(

Aˆ

i C

i 1 C , C i

C

i C i

C

i 2 , C i

C

i 1 , C

i 2

i C 2 i

2

i 23 i

2

i 2 i

2

i 21

i 1

i C 1 i

1

i 13 i

1

i 12 i

1

i 1

i

LL

L

ML

OM

M

L

L

(6)

where )ΔNhj(Ti denotes the number of transitions observed from state h to j at date Ti 1 The diagonal

element ΔNK(Ti) counts the total number of transitions away from state k at date Ti and Yk(Ti) is the

number of firms in state k prior to date Ti Hence, the off-diagonal elements {ΔAˆ(Ti)}hj,h≠ denote j

the fraction of the firms at state h just before date Ti that migrate to state j at date Ti The bottom row

is zero since firms leaving the default state, the absorbing state, were not taken into consideration Note

that the sum of each row of ΔAˆ(Ti) is zero and the rows of I+ΔAˆ(Ti)) automatically sum to one In

summary, the Aalen-Johansen estimator is equal to the cohort method for short time intervals For a

short time horizon, one could neglect the differences between the discrete- and continuous-time

estimators However, as the time horizon extends, differences between the two estimators increase,

because of the higher migration potential for longer time horizons

3.3 Risk premium

Consider the corresponding stochastic process ~x={~xt t=0,1,2,L} of credit rating under the

risk-neutral probability measure For valuation purposes, the transition matrices, P, need to be transformed

into a risk-neutral transition matrix under the equivalent martingale measure Therefore, let M~ denote

the risk-neutral transition matrix 2 Thus, the transition matrix under the risk-neutral probability

++

+

++

+

=+

) 1 ( )

C 1 (

) 1 C ( )

C C (

1 C , C CC

1 C

1 C , 2 C

2 21

1 C , 1 C

1 11

1

~

)1tt(

D~)1tt(

A~

10

0

)1tt(

m~)1tt(

m~)

1tt(

m~

)1tt(

m~)1tt(

m~)

1tt(

m~

)1tt(

m~)1tt(

m~)

1tt(

m~

1t

t

M~

LL

MM

OM

LL

ij 1

m~ , i∀ The submatrix

) C C (A~

× is defined

on non-absorbing states Sˆ= 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

) 1 C (D~

× is the column vector with components m~ ,C+1, which represent the transition probability of banks’ borrowers in any

rating class, i.e., i=1, 2, …,C, transiting to default, i.e., j=C+1 Assume for the sake of simplicity that

bankruptcy (state C+1) is an absorbing state, so that

) C 1 (

~

×

Ο is the zero row vector giving a transition probability from the default state at initial time until the final time Once the process enters the default

state, it does not return to the credit class state, so that m~C+1,C+1=1.In such a case, it can be said that

default state C+1 is an absorbing state

If transition matrix, P, is multiplied by the corresponding risk premium, then the transition

matrix will be a risk-neutral transition matrix as equation (7) Therefore, risk premium is the risk

adjustment that transforms the actual probability into the risk-neutral probability First, let V0( ,T) be

the time-t price of a risk-free bond maturing at time T, and let Vi(t,T) be its higher risk, i.e., riskier

counterpart for the rating class, i Since a loan does not lose all interest and principal if the borrower

defaults, one has to realistically consider that a bank will receive some partial repayment even if the

borrower goes into bankruptcy Let δ be the proportions of the loan’s principle and interest, which is

collectable on default, 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 =0

As shown by Jarrow, Lando and Turnbull (1997), it can be assumed that

ij ij

j(t t 1) (t) p

m~ + =μ ⋅ , Si j∈ , and μij(t)=μi(t), for j≠ and their procedure for risk premium is i

1 C , 0

i 0

i

p)1,0(V)1(

)1,0(V)1,0(V)

In equation (8), it is apparent that a zero or near-zero default probability, p,C+1 ≈0, would

cause the risk premium estimate to explode and it is also implied that the credit rating process

(including default state) of every borrower is independent, which is inappropriate and irrational for

bank loans If the borrower defaults, the default probability for the future is not to be estimated

Consequently, the assumption that every borrower’s credit rating class is independent only before

entering the default state has to be modified Redefine the risk premium as

)t0(V)1(

)t0(V)t0(V)t0(

m~p

1

1)

t

(

0

0 i

C 1 j

1 ij 1 C ,

)1tt(

A~)t0(

A~)1t

0

(

where m~ij−1(0,t) are the components of the inverse matrix A~−1(0 t) and A~(0 t) will be invertible The

denominator of equation (9) is not that p,C+1, but that (1−p,C+1), the estimation problem in equation

(8) is avoided this way For equation (10), A~(t t+1)=Ω(t)⋅A and Ω is the (t) (C×C) diagonal

matrix with diagonal components being the risk premium, which is adjusted to lj(t),j∈Sˆ In

particular, the risk premium of t=0 is

)1,0(V)1(

)1,0(V)1,0(Vm1

1)

0

(

0

i 0

1 C , i

Therefore, estimate risk premium using a recursive method for all loan periods, t=0, 1,…, T On

the whole, it is found that the risk-neutral transition matrix varies over time to accompany the changes

in the risk premium by equation (9) and (11)

Then, assume the indicator function to be

Iif,0

Ttimebeforedefault

notT

Iif,1

1{I}

τ

τ

(12) Since the Markov processes and the interest rate are independent under the equivalent

martingale measure, the value of the loan is equal to

T

Q~1T

Q~)T,t(V

11

E~)T,t(V)T

i t i

t 0

} T { T t 0

i

>

−+

=

>

−+

τδ

τ

τ

(13)

where Q~it(τ>T) is the probability under the risk-neutral probability measure that the loan with rating

class i will not be in default before time T It is clear that

)T,(

m~1)T,(

m~

)T,(V)1(

)T,(V)T,(V)T(

Q~

1 C , C

1

j ij

0

0 i

)T,(V)T,(V)T(

Q~

0

i 0

Consequently, the default probability of bank loans under a risk-neutral probability measure

can be estimated by incorporating time-varying risk premium Furthermore, for three different Markov

chain models to generate transition matrices, their risk premiums also need to be estimated to construct

transition matrices under risk-neutral probability measurement

3.4 Mobility

Jafry and Schuermann (2004) propose a statistics, mobility estimator, to compare the differences in the

estimated models Let P be the transition matrix and the dynamic part be measured by mobility matrix

as P~3

IP

P

~

−

where I is an identity matrix, i.e., the static (no migration) matrix That is, the state vector of the matrix

is unchanged from one period to the next Thus, subtract the identity matrix, I, leaving only the

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

P

dynamic part of the original matrix, which reflects the “magnitude” of the matrix in terms of the

implied mobility Therefore, the mobility estimator as

1C

)P

~P

~()

P

(

m

1 C 1 i i+

m , in term of “average migration rate”, as it would yield exactly the average probability of

transition if such probability were constant across all possible states

Let two matrices P~dis and P~con are mobility matrices of discrete- and continuous-time

observations, respectively Then, we use the difference, Δm(P~dis,P~con), to take into account estimation

uncertainty and measurement errors in the transition matrices

)P

~(m)P

~(m)

P~,P

~

(

The continuous-time transition matrix spreads the transition probability mass more off-diagonal

which implies a considerable decrease in the m(P~) metric In the absence of any theory on the

asymptotic properties of equation (18), a resampling technique of bootstrapping is a reasonable and

feasible alternative Therefore, the mobility estimator is adopted to measure the dispersion in transition

matrices by a bootstrapping

4 Data

The sample data come from two databases of the Taiwan Economic Journal (TEJ), namely the Taiwan

Corporate Risk Index (TCRI) and long and short-term bank loans The sample period is from Quarter

1, 1997 to Quarter 4, 2005

The TCRI is a complete credit rating record for Taiwan’s corporations TEJ applies a numerical

class from 1 to 9 and D for each rating classification The categories are defined in terms of default risk

and the likelihood of payment for each individual borrower Obligation rated number 1 is generally

considered as the lowest in terms of default risk, which is similar to the investment grade for Standard

& Poor’s and Moody Obligation number 9 is the most risky and rating class D denotes the default

borrower The definitions of the rating categories of TCRI for long-term credit are similar to Standard

& Poor’s and Moody The TEJ also define rating classes 1-4 as investment grade and 7-9 as

speculative grade Therefore, we group credit ratings 1 through 4 into 1 * 4 Similarly, numbers 5-6 and

7-9 are grouped into 2 and * 3 , respectively Thus, there are four rating classes * 1 , * 2 ,* 3 and D *

The long- and short-term bank loan database records all debts of corporations in Taiwan,

including lender names, borrower names, rate of debt, and debt issuance dates The credit risk of bank

loans was investigated according to every borrower’s lending structure

The government bond yield was taken as a proxy for the risk-free rate which is published by the

Central Bank in Taiwan Since the maturity of bank loans and government bonds differ, the yields of

government bonds had to be adjusted by interpolating the yield of the government bond whose

maturity was the closest and used as the risk-free rate

The recovery rate served as a security for bank loans that had influence on credit risk In

general, banks set a recovery rate according to the kind, liquidity, and value of collateral prior to

lending Altman, Resti and Sironi (2004) present a detailed review of default probability, recovery rate

and their relationship They found that most credit risk models treated recovery rate as an exogenous

variable either as structural-form models or reduced-form models For structure-form models, recovery

rate is exogenous and independent from the firm’s asset value (Kim, Ramaswamy and Sundaresan

4 The motive of this work is also due to a limit in the sample size

1989; Hull and White, 1995; Longstaff and Schwartz, 1995) Reduced-form models also assume an exogenous recovery rate that is either a constant or a stochastic variable independent from default probability (Litterman and Iben, 1991; Madan and Unal, 1995; Jarrow and Turnbull, 1995; Jarrow, Lando and Turnbull, 1997; Lando, 1998; Duffee, 1999) According to previous studies, there is no clear definition of the recovery rate Fons (1987), Longstaff and Schwartz (1995), Briys and de Varenne (1997) and assumed a constant recovery rate according to the historic level Therefore, for secured loans, recovery rates were taken from 0.1-0.9 (Lu and Kuo, 2005, 2006) For unsecured loans, the recovery rate was zero (Copeland and Jones, 2001)

Finally, the default risk for at least a one-year horizon was analyzed and therefore excluded observations for short-term loans and incomplete data Loans that had an overly low rate were also excluded because they were likely to have resulted from aggressive accounting politics and would have biased the results Consequently, the credit risk of mid- and long-term loans, including secured and unsecured loans, were analyzed for 28 domestic banks in Taiwan

5 Empirical Results

5.1 Summary statistics

In this paper, the credit risk of 28 domestic banks in Taiwan was estimated 5 Since a key feature in any lending and loan-pricing decision is the degree of collateral of the loan, we consider secured (collaterialized) and unsecured (uncollaterialized) loans in this paper Table 1 presents the summary statistics of the sample From Panel A, average collateral loan rates and their corresponding government bond yields (risk-free rates) were 6.392% and 5.344%, respectively From Panel B, the average rates of unsecured loans and their corresponding government bond yields were 5.451% and 4.625%, respectively Generally, the risky rates were higher than the risk-free rates as can be seen from the results of both Panel A and Panel B Furthermore, loan rates had greater volatility than risk-free rates The average lending periods for secured and unsecured loans were 5.603 and 4.226 years, respectively This phenomenon may be due to unsecured loans give a bank a more risky claims to this debt In general, a loan with collateral had longer lending periods than an unsecured loan Finally, the kurtosis is excess implying that loan rates and risk-free rates were not normal

5

The 28 domestic banks include: (1) Agricultural Bank of Taiwan; (2) Bank of Taiwan; (3) Bank of Overseas Chinese; (4) Bank of Sinopac Company Ltd.; (5) Bowa Bank; (6) Cathay United Bank; (7) Chang Hwa Commercial Bank; (8) Chaio Tung Bank; (9) China Development Industrial Bank Inc.; (10) Chinatrust Commercial Bank; (11) Chinfon Commercial Bank; (12) Cosmos Bank, Taiwan; (13) EnTie Commercial Bank; (14) E Sun Commercial Bank; (15) Far Eastern International Bank; (16) First Commercial Bank; (17) Fuhwa Commercial Bank; (18) Hua Nan Commercial Bank; (19) Jih Sun International Bank; (20) Land Bank of Taiwan; (21) International Bank of Taipei; (22) Ta Chong Bank Ltd.; (23) Taiwan Cooperative Bank; (24) Taipei Fubon Commercial Bank; (25) Taishin International Bank; (26) The Chinese Bank; (27) The International Commercial Bank of China; (28) Taiwan Business Bank

Table 1: Summary statistics

This table was summary statistics for bank loans and government bonds in Taiwan The government

bond yield was taken as a proxy for the risk-free rate In general, the loan rates (or risky rates) were

higher than risk-free rates For example, the average loan rates of secured loans and their

corresponding government bonds’ rates were 6.392% and 5.344%, respectively In Taiwan, the

average lending periods of secured loans always longer than unsecured loans

Panel A Secured loans

5.2 The transition matrix

First, the transition matrices estimated from discrete- and continuous-time Markov chain models in

Table 2 were compared Both transition matrices were diagonally dominant, meaning that they had a

heavy concentration around the diagonal Interestingly, the default probabilities of unsecured loans

estimated by discrete-time method were not observed in Panel B of Table 2 However, the

continuous-time method could provide an estimate of such loans Regarding the discrete-continuous-time method, there were

many default probabilities close to zero On the other hand, most default probabilities in discrete-time

estimator were lower than those in the continuous-time estimator This is because the dynamic rating

process is ignored in the discrete-time estimator result in lower estimated default probabilities

Table 2: Average transition matrix, 1997-2005

This table shows the average 9 one-year transition matrices in the period 1997-2005 Panel A and B present average transition matrices based on discrete-time Markov chain model that are estimated by cohort method as equation (4) On the other hand, Panel C and D show average transition matrices using continuous-time Markov chain model, which are estimated by equation (5) and (6), including all movements within the sample period However, whether incorporating the dynamic rating process results in different estimated transition matrices between discrete- and continuous-time estimators Rather than extracting default rates from the discrete-time transition matrix, the continuous-time transition matrix analyze default probabilities by incorporating all information of migration within the sample periods, which is more elaborate

Rating at the end of year