International Research Journal of Finance and Economics ISSN 1450-2887 Issue 30 (2009) © EuroJournals Publishing, Inc. 2009 http://www.eurojournals.com/finance.htm Measurement and Comparison of Credit Risk by a Markov Chain: An Empirical Investigation of Bank Loans in Taiwan Su-Lien Lu Assistant Professor,Department of Finance,National United University No. 1, Lien-Da, Kung-Ching Li, Miao-Li, 360 Taiwan R.O.C E-mail: 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 E-mail: kjlee@npic.edu.tw Abstract Transition matrices are at the center of modern credit risk management. In this paper, the estimation of transition matrices based on discrete- and continuous-time Markov chain models is presented. These different models were applied to bank loans, including secured and unsecured loans, for twenty-eight banks in Taiwan. Furthermore, the differences between discrete- and continuous-time methods are compared with a statistics, mobility estimator. Substantial differences between the credit risks of the two methods were found. The continuous-time Markov chain model can hood up nicely with rating- based term structure modeling. The empirical results indicate that the discrete-time Markov chain model may underestimate default probabilities when the dynamic rating process is not taken into consideration. Consequently, the conclusion is made that care has to be taken when discrete- and continuous-time Markov chain models are employed for dynamic credit risk management. Keywords: Credit risk, discrete-time Markov chain model, continuous-time Markov chain model, bank loans JEL Classification Codes: G10, G21 1. Introduction In the past ten years, major developments in financial markets have led to a more sophisticated approach to credit risk management. The origination of credit is still based on the relationship between the banker and his client. In the banking industry, the classic risk is credit risk that may cause a financial institution to become insolvent or result in a significant drain on capital and net worth that may adversely affect its growth prospects and ability to compete with other financial institutions. Therefore, credit risk management has become a major concern for the banking industry and other financial intermediaries. This is also stated by the Basel Committee on Banking Supervision (“the Committee”) that formalizes a universal approach to credit risk in financial institutions. In fact, the intention of the Committee is to assure the safety and soundness of the financial system. To achieve this goal, the Committee issued the “International Convergence of Capital International Research Journal of Finance and Economics - Issue 30 (2009) 109 Measurement and Capital Standards” document, published in July 1988. Furthermore, the treatment of market and operational risk were incorporated in 1996 and 2001, respectively. In June 2004 the Committee published the final draft of the revised framework for capital measurement and capital standards. Traditional credit analysis is an expert system that relies on the subjective judgment of trained professionals, implying that credit decisions are the reflection of personal judgment about a borrower’s ability to repay. However, traditional credit analysis has often lulled banks into a false sense of security, failing to protect them against the many risks embedded in their business. In contrast with the traditional approach, the other approach is primarily based on statistical methods, such as presented by Jarrow and Turnbull (1995) and Jarrow, Lando and Turnbull (1997). Jarrow and Trunbull (1995) used matrices of historical transition probabilities from original ratings and recovery values at each terminal state. The Jarrow, Lando and Turnbull (1997) method is based on the risk-neutral probability valuation model for pricing securities by transition matrices. Consequently, a crucial element in such models is the transition matrix. Theoretically, transition matrices can be estimated for any desired transition horizon. Generally, transition matrices are estimated by a yearly time horizon, such as Carty and Lieberman (1996), Wei (2003) and Lu and Kuo (2006). The computation of such transition matrices estimated from yearly data implies the assumption that the underlying process is a discrete-time Markov chain model. However, large movements are often achieved via some intermediary steps implying that there are no transitions from AAA to default, but there are transitions from AAA to AA and from AA to default. Thus, transitions of the intermediate state ( defaultAAAAA →→ ) contribute to the estimation of the transition matrices. In other words, the shorter the measurement interval, the fewer rating changes are omitted. Therefore, transition matrices estimated over short time periods best reflect the rating process and the underlying process is a continuous-time Markov chain model. The information is gained using the full information of exact transitions, which the discrete-time model ignores, but the continuous- time model does not. The purpose of this paper is to assess the credit risk of bank loans using two different Markov chain models, the discrete- and continuous-time models. The different Markov chain models depend on the generation of transition matrices. The discrete-time Markov chain model uses the discrete multinomial (or cohort) method and the continuous-time Markov chain model is estimated with continuous hazard rate (or duration) methods. Therefore, we also compare the estimated results of different methods. Since continuous-time methods incorporate full information of rating transitions, it seems that continuous-time approaches bring more efficient results than discrete-time methods. There are four contributions in this paper. First, the credit risk of bank loans is discussed; including secured and unsecured loans, both from analytical and empirical perspectives. To our knowledge, not much research has been done on the estimation of loans’ transition matrices considering both discrete- and continuous-time approaches. This paper adopts discrete- and continuous-time Markov chain models for measuring the credit risk of bank loans from a more comprehensive perspective than previous studies. Second, although risk premium plays a crucial role in gauging the credit risk of bank loans, previous research has handled the risk premium as a time-invariant (Jarrow, Lando and Turnbull, 1997; Wei, 2003). In fact, the risk premium is actually always a time-variant parameter (Kijima and Komoribayashi, 1998; Lu and Kuo, 2006). Therefore, the assumptions made in previous research were relaxed by incorporating the time-variant risk premium into the transition matrices making it more elaborate. Third, a comparison of the estimated results of discrete- and continuous-time Markov chain models is given. It was found that the continuous-time approach has a more reliable default probability than the discrete-time approach. The discrete-time estimator may underestimate the default probabilities due to the neglect of some rating transitions whereas the continuous-time estimator incorporates all information on the exact timing of rating transitions. 110 International Research Journal of Finance and Economics - Issue 30 (2009) Fourth, a statistics mobility estimator was extended for investigating the migration size of discrete- and continuous-time transition matrices. The mobility estimator was designed to give a measure of the transition matrices propensity, which had been developed by Jafry and Schuermann (2004). It was found that the discrete-time method had a higher mobility estimator than the continuous- time method, implying that a higher off-diagonal probability is concentrated in a discrete-time transition matrix rather than diluted in a continuous-time transition matrix. On the whole, credit risk modeling is crucial for bank regulators in providing an effective credit risk review, not only in helping to detect borrowers in difficulty, but also in facilitating to the Basel Capital Accord. We expect that this study can provide a suitable model to gauge the credit risk for financial institutions. This paper is organized as follows: Section 1 provides the motivation for this study. Section 2 reviews literature concerning models of credit risk. Section 3 presents the formal methodology, and Section 4 describes the sample data used in this paper. Section 5 shows empirical results and robustness tests. Finally, Section 6 includes a discussion of our findings with a conclusion. 2. Literature Review Credit risk research has gained considerable momentum over the last decade. Many different classes of models have been put forward to measure, manage, and price credit risk. In general, these models can be divided into two main categories: (a) structural-form models and (b) reduced-form models. Both categories have advantages and limitations in valuing credit risk. One important difference between these two categories of models is the implicit assumption they make about managerial decisions regarding capital structure. The structural-form approach imposes assumptions on the evolution of the value of the firm’s underlying assets. The reduced form approach does not make this implicit assumption. The structural-form models include the original work of Black and Scholes (1973) and Merton (1974). In such a framework, the securities issued by a firm as contingent claims on its own value and, therefore, the credit risk is driven by the value of the company’s assets. The basic intuition behind the Merton model is that default occurs when the value of a firm’s asset is lower than that of its liabilities. Furthermore, the basic Merton model has subsequently been extended by removing one or more of Merton’s assumptions. Black and Cox (1976) suggest that bondholders can force the reorganization or the bankruptcy of the firm if its value falls to a specific value. Kim, Ramaswamy and Sundaresan (1989) and Collin-Dufresne and Goldstein (2001) propose a model similar to the Black and Cox (1976) model, suggesting that capital structure is explicitly considered and default occurs if the value of total assets is low enough to reach a trigger value, which is assumed to be exogenous. Leland (1994) endogenizes the bankruptcy while accounting for taxes and bankruptcy costs. Leland and Toft (1996) propose a Barrier option model, suggesting that expected default probabilities depend on the endogenously defined bankruptcy threshold. In spite of these improvements on Merton’s original framework, structural-form models still suffer some drawbacks, which are the main reasons behind their relatively poor empirical performance (Altman, Resti and Sironi, 2004; Emo, Helwege and Huang, 2004). First, since the firm’s value is not a tradable asset, the parameters of the structural-form model are difficult to estimate consistently. In other words, unlike the stock price in Black and Scholes model for valuing equality options, the current market value of a firm is not easily observable. Second, the inclusion of some frictions like tax shields and liquidation costs would break the last rule. Third, corporate bonds undergo credit downgrades before they actually default, but structural-form models cannot incorporate these credit-rating changes. Finally, most structural-form models assume that the value of the firm is continuous in time and, consequently, the time of default can be predicted just before it happens. Reduced-form models attempt to overcome the above mentioned shortcomings of structural- form models. These include Jarrow and Turnbull (1995), Jarrow, Lando and Trunbull (1997), Lando (1998), Duffie (1998) and Duffie and Singleton (1999). Unlike structural-form models, reduced-form models do not default on the firm’s value, and parameters related to the firm’s value need not be International Research Journal of Finance and Economics - Issue 30 (2009) 111 estimated to implement them. These variables related to default risk are modeled independently from the structural features of the firm, its asset value and leverage. The calibration of the credit risk for reduced-form models is made with respect to rating agencies’ data. Therefore, rating systems have become increasingly important for reduced-form models. Their key purpose is to provide a simple qualitative classification of the solidity, solvency and prospects of a debt issuer. The importance of credit ratings has increased significantly with the introduction of the Basel II. It is obvious that the present rating of an obligor is a strong predictor of his rating in the nearest future. A cardinal feature of any credit rating is the past and present rating influencing the evolution. Therefore, the Markov chain is a stochastic process, in which the transition probabilities, given all past ratings, depend only on the present state. It allows all transition probabilities for a specific time-horizon to be collected in a so-called transition matrix, such as presented by both Jarrow, Lando and Turnbull (1997) and Lando (1998) using transition matrices to determine credit risk. In most applications, transition matrices are estimated by discrete-time observations with a yearly time horizon. For example, Lu and Kuo (2006) have applied the discrete-time Markov chain model to assess the credit risk of bank loans by yearly transition matrices. However, if borrowers seldom change their rating, then transition matrices typically concentrate along the main diagonal. That is, the most probability mass resides along the diagonal and most of the time there is no migration. The low occurrence of certain transitions may be a problem when estimating default probabilities. For the lowest risk grade, such as AAA in Standard and Poor’s rating, defaulting in a given period is a rare event. Although there are no transitions from AAA to default, there are transitions from AAA to AA and from AA to default. As a result, the estimator for transitions from AAA to default should be non- zero and these rare events are ignored by the discrete-time Markov chain model. In order to avoid the embedding problem for discrete-time observations, the continuous-time Markov chain model has been adopted to estimate meaningful default probabilities. On the other hand, differences between discrete- and continuous-time Markov chain models were compared to determine the credit risk of bank loans. Bangia et al. (2002) found that only the diagonal elements were estimated with high precision, since transition matrices are dominated diagonally. They found that if it was one transition away from the diagonal, then the degree of estimated precision decreases. However, Jafry and Schuermann (2004) also suggest a criterion, distribution discriminatory, which is particularly relevant for transition matrices that are sensitive the distribution of off-diagonal probability mass. They also propose statistics to compare the differences between the discrete- and continuous-time methods. Consequently, Jafry and Schuermann’s (2004) estimation was used to compare the differences of transition matrices generated by two distinct Markov chain models. 3. Model Specification 3.1. The discrete-time Markov chain model Let t x represent the credit rating of a bank’s borrower at time t. Assume that , }2,1,0t,x{x t == is a Markov chain on the finite state space S={1, 2,…, C, C+1}, where state 1 represents the highest credit class; and state 2 the second highest, …, state C the lowest credit class; and state C+1 designates the default. It is usually assumed for the sake of simplicity that the state C+1 is the absorbing state. Furthermore, let P(s, t) denote the )1C()1C( + × + transition matrix generated by a Markov chain model with transition probability as () ixjxP)t,s(p stij === , Sj,i ∈ , ts < , t=0, 1, 2,… (1) Equation (1) is the probability that a borrower rated i at time s migrates to rating j at time t. Let P ~ andP denote transition matrices for the discrete-time estimator and continuous-time estimator. 112 International Research Journal of Finance and Economics - Issue 30 (2009) Hereafter, let “P” and “ ij p ” be generally termed the transition matrices and transition probabilities, respectively. The first Markov chain model applied to the transition matrix is the discrete-time Markov chain model based on annual migration frequencies. Generally, estimation in a discrete-time Markov chain can be viewed as a multinomial experiment since it is based on the migration away from a given state over a one-year horizon. Let )t(N i denote the number of firms in state i at the beginning of the year and )t(N ij represent the number of firms with rating i at date t migrated to state j at time t+1. Thus, the one-year transition probability is estimated as )t(N )t(N )t(p i ij ij = (2) If the rating process is assumed to be a time-homogeneous Markov chain, i.e., time- independent, then the transitions for different borrowers away from a state can be viewed as independent multinomial experiments. Therefore, the maximum likelihood estimator (MLE) for time- independent probability is defined as ∑ ∑ = = = T 1t i T 1t ij ij )t(N )t(N p (3) where T is the number of sample years. For a special case, the number of firms are the same over the sample period, ii N)t(N = , the estimator for the transition probabilities is the average of the year-on- year transition matrices, such as Bangia et al. (2002) and Hu, Kiesel and Perraudin (2002). However, the special case is implausible. Accordingly, the estimator of transition probabilities is always modified by the number of firms during the sample years. If P denotes transition matrix for a Markov chain over a year horizon, then the discrete-time transition matrix is as ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ = + + + 1000 pppp pppp pppp P 1C,CCC2C1C 1C,2C22221 1C,1C11211 L L MMOMM L L (4) where ,j,i,0p ij ∀≥ and ∑ + = ∀= 1C 1j ij i,1p . Since the information concerning within-year rating transition is ignored in the discrete-time Markov chain model, the continuous-time Markov chain model had to be used to determine the additional migration within the year. According to Christensen, Hansen and Lando (2004), the advantages of the continuous-time Markov chain model can be summarized as: (i) The duration method can obtain non-zero estimates for probabilities of rare events whereas the cohort method estimates to zero. (ii)The duration method uses all available information in the data including information of a firm even when it enters a new state. In the discrete-time estimator, the exact date within the year that a firm changed its rating cannot be distinguished. Therefore, the continuous-time Markov chain models were also adopted for valuing the credit risk of bank loans. 3.2. The continuous-time Markov chain model As for continuous-time models, the non-parametric method of Aalen and Johansen (1978) was adopted to replace the cohort methods. The Aalen-Johansen estimator imposes fewer assumptions on the data generating process by allowing for time heterogeneity while fully accounting for all movements within International Research Journal of Finance and Economics - Issue 30 (2009) 113 the sample period. In other words, the Aalen-Johansen estimator can be applied to an extremely short time interval and observe a borrower’s rating movement during the sample period. Let )t,s(P ~ be the transition matrix over the horizon [s, t] and take the Aalen-Johansen estimator (or product-limit estimator) for the transition matrix. The estimator for the transition matrix, )t,s(P ~ , from time s to time t is given by [] ∏ = Δ+= m 1i i )T(A ˆ I)t,s(P ~ (5) where i T represents the transition point over the sample period [s, t] and m is the total number of transitions over the sample period from s to t. The estimator is clearly a duration approach, which allows for time non-homogeneous while fully accounting for all movements with the sample period (estimated horizon). The matrix )T(A ˆ i Δ is given by ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ Δ Δ − ΔΔ ΔΔ Δ − Δ ΔΔ ΔΔ − =Δ + 000 )T(Y )T(N )T(Y )T(N )T(Y )T(N )T(Y )T(N )T(Y )T(N )T(Y )T(N )T(Y )T(N )T(Y )T(N )T(Y )T(N )T(Y )T(N )T(Y )T(N )T(Y )T(N )T (A ˆ iC i1C,C iC iC iC i2,C iC i1,C i2 iC2 i2 i23 i2 i2 i2 i21 i1 iC1 i1 i13 i1 i12 i1 i1 i LL L MLOMM L L (6) where )T(N ihj Δ denotes the number of transitions observed from state h to j at date i T 1 . The diagonal element )T(N iK Δ counts the total number of transitions away from state k at date i T and )T(Y ik is the number of firms in state k prior to date i T . Hence, the off-diagonal elements { } hj i )T(A ˆ Δ ,jh ≠ denote the fraction of the firms at state h just before date i T that migrate to state j at date i T . The bottom row is zero since firms leaving the default state, the absorbing state, were not taken into consideration. Note that the sum of each row of )T(A ˆ i Δ is zero and the rows of ))T(A ˆ I( i Δ+ automatically sum to one. In summary, the Aalen-Johansen estimator is equal to the cohort method for short time intervals. For a short time horizon, one could neglect the differences between the discrete- and continuous-time estimators. However, as the time horizon extends, differences between the two estimators increase, because of the higher migration potential for longer time horizons. 3.3. Risk premium Consider the corresponding stochastic process },2,1,0t,x ~ {x ~ t L = = of credit rating under the risk- neutral probability measure. For valuation purposes, the transition matrices, P, need to be transformed into a risk-neutral transition matrix under the equivalent martingale measure. Therefore, let M ~ denote the risk-neutral transition matrix. 2 Thus, the transition matrix under the risk-neutral probability measure is given by 1 )1t,t(N hj + counts the total number of transitions from state h to state j from date t to t+1, and )T(N ihj Δ is an increment of this process 2 The assumption is also made in Copeland and Jones (2001) and Lu and Kuo (2005, 2006). 114 International Research Journal of Finance and Economics - Issue 30 (2009) () ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ Ο ++ = ⎥ ⎥ ⎥ ⎥ ⎥ ⎥ ⎦ ⎤ ⎢ ⎢ ⎢ ⎢ ⎢ ⎢ ⎣ ⎡ +++ +++ +++ =+ × × ×× + + + )11( )C1( )1C()CC( 1C,CCC1C 1C,2C221 1C,1C111 1 ~ )1t,t(D ~ )1t,t(A ~ 100 )1t,t(m ~ )1t,t(m ~ )1t,t(m ~ )1t,t(m ~ )1t,t(m ~ )1t,t(m ~ )1t,t(m ~ )1t,t(m ~ )1t,t(m ~ 1t,tM ~ L L MMOM L L (7) where 0m ~ ij > the risk-neutral transition probability and ∑ + = = 1C 1j ij 1m ~ , i ∀ . The submatrix )CC( A ~ × is defined on non-absorbing states }C ,,2,1{S ˆ = The components of submatrix A ~ denote the regime-switching of credit classes for the bank’s borrower. However, it excludes default state C+1. )1C( D ~ × is the column vector with components 1C,i m ~ + , which represent the transition probability of banks’ borrowers in any rating class, i.e., i=1, 2, …,C, transiting to default, i.e., j=C+1. Assume for the sake of simplicity that bankruptcy (state C+1) is an absorbing state, so that )C1( ~ × Ο is the zero row vector giving a transition probability from the default state at initial time until the final time. Once the process enters the default state, it does not return to the credit class state, so that 1m ~ 1C,1C = ++ .In such a case, it can be said that default state C+1 is an absorbing state. If transition matrix, P, is multiplied by the corresponding risk premium, then the transition matrix will be a risk-neutral transition matrix as equation (7). Therefore, risk premium is the risk adjustment that transforms the actual probability into the risk-neutral probability. First, let )T,t(V 0 be the time-t price of a risk-free bond maturing at time T, and let )T,t(V i be its higher risk, i.e., riskier counterpart for the rating class, i. Since a loan does not lose all interest and principal if the borrower defaults, one has to realistically consider that a bank will receive some partial repayment even if the borrower goes into bankruptcy. Let δ be the proportions of the loan’s principle and interest, which is collectable on default, 10 ≤δ< , where in general δ will be referred to as the recovery rate. If there is no collateral or asset backing, then δ =0. As shown by Jarrow, Lando and Turnbull (1997), it can be assumed that ijijj,i p)t()1t,t(m ~ ⋅μ=+ , Sj,i ∈ , and )t()t( iij μ = μ , for ij ≠ and their procedure for risk premium is 1C,i0 i0 i p)1,0(V)1( )1,0(V)1,0(V )0( + δ− − =μ (8) In equation (8), it is apparent that a zero or near-zero default probability, 0p 1C,i ≈ + , would cause the risk premium estimate to explode and it is also implied that the credit rating process (including default state) of every borrower is independent, which is inappropriate and irrational for bank loans. If the borrower defaults, the default probability for the future is not to be estimated. Consequently, the assumption that every borrower’s credit rating class is independent only before entering the default state has to be modified. Redefine the risk premium as )t,0(V)1( )t,0(V)t,0(V )t,0(m ~ p1 1 )t( 0 0i C 1j 1 ij 1C,i i δ− δ− − = ∑ = − + l , i=1, 2,…,C and t=1,…,T (9) )1t,t(A ~ )t,0(A ~ )1t,0(A ~ +=+ (10) International Research Journal of Finance and Economics - Issue 30 (2009) 115 where )t,0(m ~ 1 ij − are the components of the inverse matrix )t,0(A ~ 1− and )t,0(A ~ will be invertible. The denominator of equation (9) is not that 1C,i p + , but that )p1( 1C,i + − , the estimation problem in equation (8) is avoided this way. For equation (10), A)t()1t,t(A ~ ⋅Ω=+ and )t( Ω is the )CC( × diagonal matrix with diagonal components being the risk premium, which is adjusted to S ˆ j),t( j ∈l . In particular, the risk premium of t=0 is )1,0(V)1( )1,0(V)1,0(V m1 1 )0( 0 i0 1C,i i δ− δ− − = + l , for i=1,2,…,C (11) Therefore, estimate risk premium using a recursive method for all loan periods, t=0, 1,…, T. On the whole, it is found that the risk-neutral transition matrix varies over time to accompany the changes in the risk premium by equation (9) and (11). Then, assume the indicator function to be {}( ) {}( ) ⎩ ⎨ ⎧ ≤∈ >∈ = TtimebeforedefaultTIif,0 TtimebeforedefaultnotTIif,1 1 }I{ τ τ (12) Since the Markov processes and the interest rate are independent under the equivalent martingale measure, the value of the loan is equal to {} [] { } () () [] {} ()( ) {} TQ ~ 1)T,t(V TQ ~ 1TQ ~ )T,t(V 11E ~ )T,t(V)T,t(V i t0 i t i t0 }T{Tt0i >−+= >−+>= += ≤> τδδ τδτ δ ττ (13) where () TQ ~ i t >τ is the probability under the risk-neutral probability measure that the loan with rating class i will not be in default before time T. It is clear that )T,t(m ~ 1)T,t(m ~ )T,t(V)1( )T,t(V)T,t(V )T(Q ~ 1C,i C 1j ij 0 0i i t + = −= ∑ = δ− δ− =>τ (14) which holds for time Tt ≤ , including the current time, t=0. Similarly, the default probability occurs before time T as )T,t(V)1( )T,t(V)T,t(V )T(Q ~ 0 i0 i t δ− − =≤τ , for i=1,….,C and T=1,2, (15) Consequently, the default probability of bank loans under a risk-neutral probability measure can be estimated by incorporating time-varying risk premium. Furthermore, for three different Markov chain models to generate transition matrices, their risk premiums also need to be estimated to construct transition matrices under risk-neutral probability measurement. 3.4. Mobility Jafry and Schuermann (2004) propose a statistics, mobility estimator, to compare the differences in the estimated models. Let P be the transition matrix and the dynamic part be measured by mobility matrix asP ~ ~ 3 IPP ~ ~ −= (16) where I is an identity matrix, i.e., the static (no migration) matrix. That is, the state vector of the matrix is unchanged from one period to the next. Thus, subtract the identity matrix, I, leaving only the 3 Decompose the transition matrix into a static and dynamic component, whereas Geweke, Marshall and Zarkin (1986) use the original transition matrix, P. 116 International Research Journal of Finance and Economics - Issue 30 (2009) dynamic part of the original matrix, which reflects the “magnitude” of the matrix in terms of the implied mobility. Therefore, the mobility estimator as 1C )P ~ ~ P ~ ~ ( )P(m 1C 1i i + ′ = ∑ + = λ (17) where i λ denotes the i-th eigenvalue of P ~ ~ P ~ ~ ′ . An interpretation can be contributed to equation (17), )P(m , in term of “average migration rate”, as it would yield exactly the average probability of transition if such probability were constant across all possible states. Let two matrices dis P ~ ~ and con P ~ ~ are mobility matrices of discrete- and continuous-time observations, respectively. Then, we use the difference, )P ~ ~ ,P ~ ~ (m condis Δ , to take into account estimation uncertainty and measurement errors in the transition matrices. )P ~ ~ (m)P ~ ~ (m)P ~ ~ ,P ~ ~ (m condiscondis −=Δ (18) The continuous-time transition matrix spreads the transition probability mass more off-diagonal which implies a considerable decrease in the )P ~ ~ (m metric. In the absence of any theory on the asymptotic properties of equation (18), a resampling technique of bootstrapping is a reasonable and feasible alternative. Therefore, the mobility estimator is adopted to measure the dispersion in transition matrices by a bootstrapping. 4. Data The sample data come from two databases of the Taiwan Economic Journal (TEJ), namely the Taiwan Corporate Risk Index (TCRI) and long and short-term bank loans. The sample period is from Quarter 1, 1997 to Quarter 4, 2005. The TCRI is a complete credit rating record for Taiwan’s corporations. TEJ applies a numerical class from 1 to 9 and D for each rating classification. The categories are defined in terms of default risk and the likelihood of payment for each individual borrower. Obligation rated number 1 is generally considered as the lowest in terms of default risk, which is similar to the investment grade for Standard & Poor’s and Moody. Obligation number 9 is the most risky and rating class D denotes the default borrower. The definitions of the rating categories of TCRI for long-term credit are similar to Standard & Poor’s and Moody. The TEJ also define rating classes 1-4 as investment grade and 7-9 as speculative grade. Therefore, we group credit ratings 1 through 4 into * 1 4 . Similarly, numbers 5-6 and 7-9 are grouped into * 2 and * 3 , respectively. Thus, there are four rating classes * 1, * 2, * 3 and D. The long- and short-term bank loan database records all debts of corporations in Taiwan, including lender names, borrower names, rate of debt, and debt issuance dates. The credit risk of bank loans was investigated according to every borrower’s lending structure. The government bond yield was taken as a proxy for the risk-free rate which is published by the Central Bank in Taiwan. Since the maturity of bank loans and government bonds differ, the yields of government bonds had to be adjusted by interpolating the yield of the government bond whose maturity was the closest and used as the risk-free rate. The recovery rate served as a security for bank loans that had influence on credit risk. In general, banks set a recovery rate according to the kind, liquidity, and value of collateral prior to lending. Altman, Resti and Sironi (2004) present a detailed review of default probability, recovery rate and their relationship. They found that most credit risk models treated recovery rate as an exogenous variable either as structural-form models or reduced-form models. For structure-form models, recovery rate is exogenous and independent from the firm’s asset value (Kim, Ramaswamy and Sundaresan 4 The motive of this work is also due to a limit in the sample size. International Research Journal of Finance and Economics - Issue 30 (2009) 117 1989; Hull and White, 1995; Longstaff and Schwartz, 1995). Reduced-form models also assume an exogenous recovery rate that is either a constant or a stochastic variable independent from default probability (Litterman and Iben, 1991; Madan and Unal, 1995; Jarrow and Turnbull, 1995; Jarrow, Lando and Turnbull, 1997; Lando, 1998; Duffee, 1999). According to previous studies, there is no clear definition of the recovery rate. Fons (1987), Longstaff and Schwartz (1995), Briys and de Varenne (1997) and assumed a constant recovery rate according to the historic level. Therefore, for secured loans, recovery rates were taken from 0.1-0.9 (Lu and Kuo, 2005, 2006). For unsecured loans, the recovery rate was zero (Copeland and Jones, 2001). Finally, the default risk for at least a one-year horizon was analyzed and therefore excluded observations for short-term loans and incomplete data. Loans that had an overly low rate were also excluded because they were likely to have resulted from aggressive accounting politics and would have biased the results. Consequently, the credit risk of mid- and long-term loans, including secured and unsecured loans, were analyzed for 28 domestic banks in Taiwan. 5. Empirical Results 5.1. Summary statistics In this paper, the credit risk of 28 domestic banks in Taiwan was estimated. 5 Since a key feature in any lending and loan-pricing decision is the degree of collateral of the loan, we consider secured (collaterialized) and unsecured (uncollaterialized) loans in this paper. Table 1 presents the summary statistics of the sample. From Panel A, average collateral loan rates and their corresponding government bond yields (risk-free rates) were 6.392% and 5.344%, respectively. From Panel B, the average rates of unsecured loans and their corresponding government bond yields were 5.451% and 4.625%, respectively. Generally, the risky rates were higher than the risk-free rates as can be seen from the results of both Panel A and Panel B. Furthermore, loan rates had greater volatility than risk-free rates. The average lending periods for secured and unsecured loans were 5.603 and 4.226 years, respectively. This phenomenon may be due to unsecured loans give a bank a more risky claims to this debt. In general, a loan with collateral had longer lending periods than an unsecured loan. Finally, the kurtosis is excess implying that loan rates and risk-free rates were not normal. 5 The 28 domestic banks include: (1) Agricultural Bank of Taiwan; (2) Bank of Taiwan; (3) Bank of Overseas Chinese; (4) Bank of Sinopac Company Ltd.; (5) Bowa Bank; (6) Cathay United Bank; (7) Chang Hwa Commercial Bank; (8) Chaio Tung Bank; (9) China Development Industrial Bank Inc.; (10) Chinatrust Commercial Bank; (11) Chinfon Commercial Bank; (12) Cosmos Bank, Taiwan; (13) EnTie Commercial Bank; (14) E. Sun Commercial Bank; (15) Far Eastern International Bank; (16) First Commercial Bank; (17) Fuhwa Commercial Bank; (18) Hua Nan Commercial Bank; (19) Jih Sun International Bank; (20) Land Bank of Taiwan; (21) International Bank of Taipei; (22) Ta Chong Bank Ltd.; (23) Taiwan Cooperative Bank; (24) Taipei Fubon Commercial Bank; (25) Taishin International Bank; (26) The Chinese Bank; (27) The International Commercial Bank of China; (28) Taiwan Business Bank. [...]... effective credit review process to measure the credit risk of bank loans, including secured and unsecured loans, for 28 banks in Taiwan The credit risk of bank loans was estimated and compared using discrete- and continuous-time Markov chain models Four conclusions are presented in this paper First, investigating the credit risk of bank loans using discreteand continuous-time Markov chain models was found... “Corporate bond valuation and the term structure of credit spreads”, Financial Analysis Journal, Spring, pp.52-64 Longstaff, F A and E S Schwartz, 1995 A simple approach to valuing risky fixed and floating rate debt”, Journal of Finance, 50, pp.789-819 Lu, S L and C J Kuo, 2005 “How to gauge the credit risk of guarantee issues in Taiwanese bills finance company: an empirical investigation using a market-based... transform the transition matrix into a risk- neutral transition matrix was estimated The average risk premium estimated using equation (9) and (11) is shown in Table 4 123 International Research Journal of Finance and Economics - Issue 30 (2009) Table 4: Average risk premium This table was average risk premium estimated by equations (9) and (10) Panel A and C represent bank loans with collateral and. .. market-based approach”, Applied Financial Economics, 15, pp.1153-1164 Lu, S L and C J Kuo, 2006 “The default probability of bank loans in Taiwan: an empirical investigation by Markov chain model”, Asia Pacific Management Review, 11(2), pp.405-413 Madan, D and H Unal, 1995 “Pricing the risks of default”, University of Maryland working paper Merton, R C., 1974 “On the pricing of corporate debt: the risk. ..118 Table 1: International Research Journal of Finance and Economics - Issue 30 (2009) 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... time Markov chains”, Econometrica, 54, pp.1407-1423 Hull, J and A White, 1995 “The impact of default risk on the prices of options and other derivative securities”, Journal of Banking and Finance, 19, pp.299-322 Hu, Y-T, R Kiesel and W Perraudin, 2002 “The estimation of transition matrices for sovereign credit ratings”, Journal of Banking and Finance, 26, pp.1383-1406 Jafry, Y and T Schuermann, 2004 Measurement, ... Schuermann, 2004 Measurement, estimation and comparison of credit migration matrices”, Journal of Banking and Finance, 28, pp.2603-2639 Jarrow, R A and S M Turnbull, 1995 “Pricing derivatives on financial securities subject to credit risk , Journal of Finance, 50, pp.53-86 Jarrow, R A. , D Lando and S M Turnbull, 1997 A Markov model for the term structure of credit risk spreads”, Review of Financial Studies,... 119 International Research Journal of Finance and Economics - Issue 30 (2009) 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... International Research Journal of Finance and Economics - Issue 30 (2009) Risk- neutral transition matrix, 1997-2005 This table was risk- neutral discrete- and continuous-time transition matrices that incorporate timevarying risk premium by averaging 9 one-year risk- neutral transition matrices in the period 19972005 Panel A and C represent bank loans with collateral and therefore, we set the recovery rate is... rating duration and may underestimate the default probability Since transition matrices are sensitive to offdiagonal probability, the process for generating transition matrices has a significant effect on measuring the credit risk In general, credit matrices are cardinal inputs into many risk management applications; therefore, the accurate estimation of the transition matrix is vital 129 International . 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; . 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. Comparison of Credit Risk by a Markov Chain: An Empirical Investigation of Bank Loans in Taiwan Su-Lien Lu Assistant Professor,Department of Finance,National United University No. 1, Lien-Da,