Cuadernos de economía
versão On-line ISSN 0717-6821
Cuad. econ. v.46 n.133 Santiago maio 2009
Cuadernos de Economía, Vol. 46 (Mayo), pp. 33-50, 2009
EVIDENCE OF NON-MARKOVIAN BEHAVIOR IN THE PROCESS OF BANK RATING MIGRATIONS*
JOSÉ E. GÓMEZ-GONZÁLEZ1, NlCHOLAS M. KIEFER2
This paper estimates transition matrices for the ratings on financial institutions, using an unusually informative data set. We show that the process of rating migration exhibits significant non-Markovian behavior, in the sense that the transition intensities are affected by macroeconomic and bank-specific variables. We illustrate how the use of a continuous time framework may improve the estimation of the transition probabilities. However, the time homogeneity assumption, frequently done in economic applications, does not hold, even for short time intervals. Thus, the information provided by migrations alone is not enough to forecast the future behavior of ratings. The stage of the business cycle should be taken into account, and individual characteristics of banks must be considered as well.
JEL: C4, E44, G21, G23, G38
Keywords: Financial Institutions, Macroeconomic Variables, Capitalization, Supervision, Transition Intensities
Este documento estima matrices de transición para ratings de instituciones financieras, utilizando un conjunto de datos excepcionales. Se muestra que el proceso de migración en el rating exhibe un comportamiento no Markoviano, en el sentido que la transición se ve afectada por variables macroeconómicas y por variables bancarias específicas. Mostramos que el uso de un marco de tiempo continuo puede mejorar la estimación de las probabilidades de transición. Sin embargo, el supuesto de la homogeneidad en el tiempo, frecuentemente utilizado en aplicaciones económicas, no se sostiene, incluso por breves intervalos de tiempo. Por lo tanto, la información proporcionada por las migraciones no es suficiente para predecir, por sí sola, el futuro comportamiento de los ratings, sino que debe tenerse en cuenta la fase del ciclo económico, así como las características individuales de los bancos.
External ratings provide important information for managers, investors and supervisors about a firm's default risk. They summarize the firm's overall financial health by placing it into a specific category according to the perception of the risk of default, and fherefore complement the information available fhrough financial markets. In emerging market economies, where financial markets are not well developed, external ratings constitute a fundamental piece of information in the process of investment allocation. It is a regular practice for firms to pay a fee to rating companies in order to receive a grading, which will be used by investors (and also by supervisors) to make decisions that will affect the firm's future.
In the Basel Accord (Basel Committee on Banking Supervision, 2004), external ratings promote market discipline in the financial intermediation industry, in the sense that by signaling a bank's default probability to other economic agents, ratings give the bank incentives to adopt more conservative risk taking policies. If ratings are accurate, and therefore reflect closely the default probability of an institution, a bank taking higher risks is more likely to be downgraded, because higher risks imply a greater default probability. Therefore, the rating will provide a signal to investors and supervisors, and banks will be more inclined toward sound financial practices.
In order to be accurate, rating agencies need to have an adequate knowledge of the firm and the environment in which it operates. When issuing a credit rating, rating agencies use qualitative and quantitative information obtained both from public and private sources (see, for instance, Grey et al, 2006). Several studies have argued that the methodologies used by international rating agencies such as Standard & Poor's and Moody's to evalúate the risk of default of firms on emerging market economies are not completely adequate, in the sense that in order to provide "uniformity" in the rating policies across countries they sacrifice precision, because they do not take into account idiosyncratic effects properly (see, for instance, Rojas-Suarez,2001 and Ferri and Liu, 2003). Those studies argue that there exist a high positive correlation between the grade given to sovereign debt of a developing country and the grading received by firms in that country, which does not appear in the case of developed economies. Therefore, in the case of emerging economies, although ratings from international agencies are important because they provide foreign investors information about domestic firms, alternative external ratings by domestic agencies provides important complementary information for decision makers. These complementary external ratings are particularly important for financial firms' supervisors, who benefit from tools that provide a signal about possible threats to the stability of the financial system.
Rating transition matrices are at the core of risk modeling and are a standard starting point for risk dynamics. In their application to banks, migration matrices are particularly attractive for supervisors in the sense that they are in the set of available early warning tools. The main objective is to use current ratings and the past history of rating migrations to predict future downgrades and defaults. Usually, transition dynamics are analyzed using Markov chains. In many important economic applications (e.g. J.R Morgan's Credit Metrics), transition matrices are estimated in a discrete-time setting using a cohort method under the assumption of time-homogeneity; in a discrete and finite space setting, the probability of migrating from state i to state j is estimated by dividing the number of observed migrations from i to j in a given time period by the total number of firms in state i at the beginning of the period. One implication of this cohort method is that if no firm migrates from state i to j during the observation period, the estimate of the corresponding probability is zero. This is a not desirable feature, especially when dealing with the estimation of rare event probabilities which, in case of occurring, may have a deep impact.
Various studies have proposed using continuous time methodologies as an alternative to the cohort approach, which not only overcomes the problem of the zero estimates for rare event probabilities, but also offer additional advantages such as allowing simple tests for non-Markovian behavior. Lando and Skodeberg (2002) present the way of estimating transition probabilities in a continuous time framework, both with and without the assumption of time homogeneity. With a data set covering several years of rating history of Standard and Poor's, and using survival analysis techniques, they study two deviations from the Markov assumption: the dependence on previous rating, and waiting time effects, and find evidence that supports the hypothesis of non-Markovian behavior of migration dynamics. Other studies have reported different types of non-Markovian behavior. For instance, Kavvathas (2000) finds dependence of rating migrations on macro-economic variables, while Jonker (2002), using a data set of ratings of banks in Europe, USA and Japan, finds that the country of origin of the bank matters in the downgrading process.
The question is not whether the ratings are in fact Markovian. With an absorbing state of default the Markovian assumption essentially implies all assets will eventually default. The question is rather whether the Markovian specification, which provides simplicity, is adequate, and if so on what time scale.
This study contributes to the literature on rating transition dynamics by presenting evidence of non-Markovian behavior in the process of rating transition, using a rich data set on ratings of financial institutions in Colombia. Using monthly data1 covering the period December 1996 to November 2005, we find that macroeconomic variables, as well as bank specific variables (summarized in the capitalization ratio) affect significantly the probability of migrating from one rating category to another. The paper shows how moving from a discrete time to a continuous time framework improves the estimation of transition probabilities in the sense that the number of zero estimates is reduced, but still non-homogeneities remain. By introducing macroeconomic variables using survival analysis techniques we show that upgrades are procyclical while downgradings are countercyclical. This fact, together with evidence on the influence of bank specific factors on the migration process indicates that a simple Markov chain is not adequate for explaining the bank rating migration process in Colombia.
The dataset used in this paper is unique, in the sense that in contrast with traditional datasets from external rating agencies in which the frequency of the data is annual, the frequency of the data used here for estimation is monthly. This allows identifying with more precision the moment in which a transition occurs, and also increases the number of observed transitions, which allows a finer estimation of migration probabilities. We expect that our qualitative results hold more generally. Certainly this is worth investigating as other data become available.
Section 2 describes the data. Section 3 presents the estimation of a Markov chain using the data, both under a discrete time and a continuous time framework. It shows how the results differ whether estimation is done assuming time homo-geneity or without that assumption. Section 4 presents the results of tests for the dependence of rating migration on macroeconomic and bank specific variables, and Section 5 presents conclusions.
2. DESCRIPTION OF THE DATA
In 1994, the Department of Financial Stability (DFS) of the Banco de la República (the Central Bank of Colombia) began grading fmancial institutions in Colombia. Based on fmancial indicators derived from their balance sheets and on expert opinión, each institution is rated into one of four non-default catego-ries, denoted I, II, III and IV. Category I corresponds to the highest rating, while category IV corresponds to the lowest one. All institutions that are in operation at the moment in which the rating is done are rated. During the first two years, ratings were computed only once a year. However, since December 1996 the DFS decided to produce monthly ratings in order to have a tool to evalúate frequently potential risks to the soundness of the financial system in Colombia. Several differ-ent financial indicators are taken into account in the grading process. Taking into account these indicators and also considering expert opinión, a number is given to each bank. Then, the number is compared to predetermined threshold valúes, and each bank is assigned to one of the four categories.
Because few institutions were ever rated I, categories I and II were combined. The new categories are denoted A, B, C and the default category is D.
In this study, we consider all ratings of commercial banks and financial companies2, from December 1996 to November 2005. Table 1 presents a summary of the data, showing the number of financial institutions at the beginning and the end of the observad on period, as well as the number of transitions observed among the different categories.
The reduction in the number of institutions during the observation period obeys to consequences of the fmancial crisis that took place during the late 1990s, leading to several bank failures, and to mergers and acquisitions (for details on the effects of the crisis on bank failure see Gomez-Gonzalez and Kiefer, 2006). Regarding the fraction of average annual transitions, Table 1 shows that better rated institutions are more likely to remain in the same category. For instance, on average 72% of institutions rated A at the beginning of a year are rated A at the end of the year, while only 56% of those rated B at the beginning of a year are in the same rating at the end of the year. Migrations outside of a given category concéntrate on neighbor categories. For example, migrations from A to B are more frequent than migrations from A to C or D. It is important to keep in mind that Table 1 considers annual migrations only, Le. changes in rating comparing December of a given year with December of the next year. Therefore, it does not take into account migrations occurring wifhin the year3. For example, a bank rated C in December 1996 that was rated B in June 1997 but went back to category C in December 1997 will be considered as a transition from C to C. Therefore, the diagonal elements of this matrix tend to be higher than those of a transition matrix that considers transitions within the year.
3. MARKOV CHAIN ESTIMATION
Markov chains are widely used to estimate migration probabilities. This section shows the results of estimating the probabilities of bank rating transitions assuming that the stochastic process underlying the observed migration dynamics can be represented adequately by a Markov chain. We present the results of estimations in a discrete time setting and in a continuous time setting. Within each of these two settings, we present results when time-homogeneity is assumed, and when such assumption is not made.
3.1 Estimation of discrete time Markov chains
Suppose we have a sample of N banks, which are observed during T+1(discrete) periods of time. At every moment of time, each bank is given a particular rating. The number of ratings is finite, and transitions from one rating to another, which are assumed to be independent across banks, are observed. Let ni(t) denote the number of banks in category i at time t, and nij(t) the number of banks migrating from category i to; between dates t-1 and t. The total number of banks exposed to migration from category i, during the whole period of observation, is given by , while the total number of transitions from rating i to j is given by , The rating of the bank in the first period of time observed (t = 0) is given.
If time-homogeneity is assumed, then for all t, and the log-likelihood function is given by
Maximizing the log-likelihood function above, subject to the constraint, which indicates that every bank is rated in exactly one of the S possible categories at every date, we get the maximum likelihood estimator for the probability of migrating from one category to another at time t is given by
If the time-homogeneity assumption is removed, the maximum likelihood estimator is given by
Although time-homogeneity has the inconvenience that it is hard to justify for long periods of time, it is a very convenient assumption, especially for forecast purposes, and therefore many Markov chain applications rely on this assumption. In credit rating applications, it is frequently assumed that the process can be represented by a discrete time-homogeneous Markov chain for a one year period. Using our dataset, we performed a likelihood-ratio test, to check whether the hypothesis of time-homogeneity in a discrete time setup is adequate. Suppose we are interested in testing whether the i-th rows of the transition matrix for different periods are statistically equal. We can test that using a likelihood-ratio test where
is a random variable with (S-1)x(T'-t') degrees of freedom (see Thomas, Edelman and Crook, 2002)4.
We performed tests of the time-homogeneity assumption for different time periods, using a roll over technique. To avoid problems with zeros in the division or log(O), we bounded each transition probability below by 10-7. This appears to be a sufficiently low bound to avoid changing the results of the test, and trying different bounds did not change the results significantly. We performed this test for different periodicity, ranging from four months to two years, using a roll over technique. We calculated the statistic, together with the corresponding p-value, and found that the hypothesis of time-homogeneity can be rejected at very low significance levéis (even for four months, in most cases). In all cases the nuil hypothesis can be rejected at the 5 percent level for eleven months or more.
These results provide strong evidence that the rating migration process underlying our data set is not time-homogeneous. In fact, misleading conclusions can be derived from imposing this assumption on the data.
We now turn to continuous-time estimation, avoiding the awkward question of the definition of the period5.
3.2 Estimation of continuous time Markov chains
External rating systems may have trouble when estimating continuous time Markov chains using migrations data, if they do not have sufficiently frequent information to update the rating of a firm as soon as a change occurs that takes the firm to a different risk proñle. Internal rating systems do not have this problem, because they have the required information at every moment of time; particularly, with internal information the exact moment at which a transition occurs can be recorded. Our data is somewhere in between. We do not have information about the exact moment in which the migration occurred, but we have a good approxi-mation to it due to the relatively high frequency of the data; the fact that we have ratings available on a monthly basis allows us to estimate continuous time Markov chains for different time periods.
Estimation under time-homogeneity assumption
A starting point for estimating continuous time Markov chains is the assumption of time-homogeneity. Above we showed that this assumption does not seem adequate in the discrete-time specification and therefore is unlikely to hold in continuous time; however, it provides a good starting point, a benchmark to compare the results obtained when this assumption is removed, and when covariates are included in the estimation.
Suppose we observe the ratings of N banks between time 0 and time T. Assume that the state space is finite, being one the highest category and S the lowest one. For a given time period, let P(t) be the transition matrix. This matrix can be expressed in terms of transition intensities, which appears to be a more natural way to formúlate statistical hypotheses (Lando, 2004), by noting that
where Λ represents the generator matrix. Given that for any t, the transition matrix is a function of the generator matrix, we can obtain maximum likelihood estimates of the elements of the transition matrix by obtaining first maximum likelihood estimators of the elements of the generator matrix, and then applying the exponential matrix function to this estimates, after scaling appropriately by t. The elements of the generator matrix are the transition intensities, whose maximum likelihood estimator (Kuchler and Sorensen, 1997) is given by
whereYi(s) is the number of banks rated i at time s. The diagonal elements are . The key point here is that the denominator takes into account every bank that has been rated i during some time during the observation period. Therefore, this method uses information differently than the cohort method.
The advantage of this method is that it takes into account not only direct transitions from one rating class to another, but also "indirect" transitions. In particular, the estimation of a transition will be strictly positive if during the observation period there was a sequence of migrations between intermedíate rating classes, even if there is no single bank that experienced all those migrations. For example, if we are interested in estimating the probability of a rare event, say the one year transition from category A to default, but no bank experienced this transition directly, we can still obtain a positive estimate if there was at least one bank which migrated from A to B, at least one which migrated from B to C, and at least one which migrated from C to default, even if the migrating banks are different, during the observation period. Using our dataset we still had some zero estimates for some probabilities in some time intervals, because some periods of time presented very few transitions.
For illustration purposes only, we present the average one year transition matrix of the data set:
Note that all probabilities are strictly positive, except for transitions out of default, which is assumed to be an absorbing state.
One may be tempted to assume that rating dynamics can be modeled adequately by using a continuous time homogeneous Markov chain. This would indeed be very conveniently, as using current data one could calculate the aggregate number of transitions between any two categories; this would in turn be very useful for supervisors. However, using a rollover estimation technique, it is clear that non homogeneities appear. Figures 1, 2 and 3 show time series for one year transition intensities away from categories A, B and C, respectively, estimated under the time homogeneity assumption. From these figures it can be seen clearly that transition intensities vary a lot in time. This holds true when the estimation period of the Markov chains is modiñed. Therefore, even though it would be useful to assume time homogeneity in rating migrations estimation, this assumption does not seem to be adequate.
Estimation without imposing the time homogeneity assumption
An alternative non-parametric method exists to estimate continuous time Markov chains without assuming time homogeneity. The method is based in the Aalen-Johansen estimator (for a discussion see Lando and Skodeberg (2002)). Suppose m transitions are observed during a period of time s. The transition matrix, P(s), is consistently estimated by
where I is the identity matrix and Ti is a jump time occurring in the observation period; is a matrix in which the non-diagonal entry ij is given by the ratio of the number of transitions observed from state i to state; at date Ti and the total number of banks in state i at the instant right before the time of the jump. The diagonal enfries are given by the negative of the summation of the non-diagonal entries of the row, so each row in the matrix adds up to zero. The last row of this matrix is a zero vector, as there are no transitions out of default. As it can be seen, this method also allows censoring properly.
The average one year continuous time transition matrix estimated without using the time homogeneity assumption is given by:
Note that the average one year continuous time transition matrix estimated without using the time homogeneity assumption looks similar to the one estimated under the time homogeneity assumption. Therefore, if the rollover method were not used, one may be tempted to conclude that the homogeneity assumption seems appropriate. However, as it was discussed above, the huge variations over time of the transition matrix estimated under the time homogeneity assumption show clearly that this assumption is not adequate in this context.
4. INTRODUCING COVARIATES TO EXPLAIN MIGRATION DYNAMICS
Above we showed that rating dynamics vary over time. It is not clear why. Different studies have shown that different covariates influence significantly the transition probabilities. Jonker (2002), using a data set of ratings of banks in Europe, USA and Japan, finds that the country of origin of the bank matters in the downgrading process. Bangia et al. (2002), using data from the Standard & Poor's CreditPro 3.0 datábase, show that the business cycle influences significantly credit migration matrices, by separating the economy into two states (contraction and expansión) and computing transition matrices for these states separately. Lando and Skodeberg (2002), and Kavvathas (2000) use survival analysis techniques to show the influence of migration matrices on previous rating and waiting time effects, and on macroeconomic variables, respectively.
This study introduces macroeconomic variables and bank specific variables (summarized by the capitalization ratio) to explain bank rating dynamics. Covariates are introduced using survival analysis techniques, which appears to be a very convenient way of doing so -for an introduction to these mefhods in general see Klein and Moeschberger (2003), and for an introduction to the apphcation of these methods in economics see Kiefer (1988)-, because censoring is handled, and the time a bank spends in a given category provides useful information for estimating the transition probabilities.
Given the frequency of the data, the set of macroeconomic variables that can be used effectively is limited6. Two different macroeconomic variables were used: the monthly average interest rate on loans (RIR), computed by the Banco de la República, and the real production index (RPI) provided by the Department of National Statistics of Colombia (DANE). Monthly information for these two variables was collected from November 1996 to November 2005. Both macroeconomic variables are included in the regressions with one lag. Additionally, the capitalization ratio (CAP), given by the ratio of equity and assets, was used as a proxy for the financial institutions' overall financial health. Although other financial variables are also important bank specific indicators, CAP is a special indicator determining the probability of bank failure in Colombia (see Gómez-González and Kiefer, 2006), and therefore it seems to be a variable which summarizes compactly the overall financial performance of a bank7.
Let denote the transition intensity from category i to category j of bank n. Then,
where is an indicator function which takes the valué one if the firm is rated in category i at time t and cero otherwise; is a function both of time and of a vector of covariates of bank n at time t, denoted . In this study, we use time varying covariates; however, if time varying covariates are not available or if the covariates to be included do not vary during the observation period, a vector of fixed covariates can be used. It is assumed that the function has the multiplicative (proportional hazards) form, as in Cox (1972):
where represents the baseline intensity, common to all banks, which captures the direct effect of time on the transition intensity. For estimation purposes, a functional form is specified for , while the baseline intensity is let unspecified (the only restriction is that it is non-negative). A functional form which is frequently chosen for is an exponential form, ,which has the advantage of guaranteeing non-negativity without imposing any restrictions on the valúes of the parameters of interest . The model is estimated by the method of partial likelihood estimation, developed by Cox (1972).
Tables 2 to 4 present the results of the estimation when only macroeconomic variables are included as covariates. Note that RIR affects significantly all transition intensities, except for that from category C to default. The sign of the coefficient corresponding to this covariate is the expected one in all regressions in which it is significant: when the transition implies a downgrading, the sign of RIR is positive, indicating that increases in the real interest rate lead to increases in the probability of a downgrading. When the transition implies an upgrading, the sign of RIR is negative, indicating that increases in the real interest rate lead to decreases in the probability of an upgrading. Taking into account that the interest rate is counter-cyclical, this implies that migrations depend on the business cycle.
Meanwhile, the impact of the RPI on the transition intensities is non-significantly different from zero in most of the cases (at a 5 percent level of significance). However, the sign of the coefficient corresponding to this variable is always the expected one: positive when the transition implies an upgrading and negative when the transition implies a downgrading. Additionally, RPI and RIR are jointly significant at the 5 percent level in all regressions except on those from category A to category B (they are significant at the 10 percent level in this case) and from category C to default.
It is interesting to note that, contrary to what occurs with all other transitions, no macroeconomic variable is significant in explaining migrations from category C to default. A possible reason is that few transitions from C to default are observed (relative to the number of banks exposed in category C).
Note RIR is a better explanatory variable than RPI in terms of fit and when both are included RPI is typically insignificant.
Tables 5 to 7 present the results of the estimation when the capitalization ratio is included as a covariate. Two different models are presented for each rating migration: one in which CAP is the only covariate included, and another in which CAP and RIR are included. It is interesting to note that, similar to the case in which only macro variables were included, neither CAP ñor RIR appear to affect significantly the transition intensity from category C to default. This can probably be explained by the low proportion of defaults out of bank exposures in category C, together with the fact that banks that spend a long time in category C are already in bad financia! health, independently on whether they default or not.
Another important feature is that in all other regressions the two covari-ates included result jointly significant at the 5 percent level. The signs of the coefficients of these two variables are the expected in all cases (the coefficient of CAP is positive when the transition implies an upgrading and negative when the transition implies a downgrading, while the coefficient of RIR is negative when the transition implies an upgrading and negative when the transition implies a downgrading), except for the sign of CAP in the migration from A to B, which is the opposite to the expected one.
Altogether, the results of the regressions indícate that the process of rating dynamics depends on external covariates related to the business cycle and on bank specific ratios. This provides evidence that supports the idea that a simple Markov model is not adequate to represent this process. The evidence reported here complements evidence of non-Markovian behavior on rating migrations reported in other studies that use dataseis with different time scales, and test for dependence in different sets of covariates.
This paper estimates transition matrices for the ratings on financial in-stitutions in Colombia. Using an unusually informative data set, we show that the process of rating migration exhibits significant non-Markovian behavior, in the sense that the transition intensities are affected by macroeconomic and bank specific variables. The monthly real interest rate influences significantly all transition migrations to neighboring ratings, except for the migration from category C to default. The same conclusión holds when the capitalization ratio in included in the regression.
The use of a continuous time framework may improve the estimation of the transition probabilities, in the sense that problems related to zero probability estimates of rare events can be avoided, but, as well as in other studies, this study finds that the time homogeneity assumption, commonly assumed in important economic applications, does not hold, not even for short periods of time. Therefore, the information provided by migrations alone is not enough to forecast the future behavior of ratings. The stage of the business cycle should be taken into account, and individual characteristics of banks must be considered as well.
* The views herein are those of the authors and do not necessarily represent the views of the Banco de la República or the Office of the ComptroUer of the Currency. We thank seminar participants at Cornell University, and useful comments from anonymous referees. Email: firstname.lastname@example.org; email@example.com.
1 Asa robustness test of our results, we performed estimations using quarterly data. The results did not change significantly. In this paper we present results using monthly data, which is the frequency with which financial institutions are graded at the Central Bank of Colombia.
2 Financial companies specialized in commercial leasing are not included, because they are quite different, in the sense that they have different purposes than the other intermediaries mentioned before, and their activities and portfolio composition are also very different. Therefore, for the purpose of this paper, data are collected only from commercial banks and financial companies.
4 A likelihood ratio test for the assumption of time homogeneity when data are not observed in regular intervals is developed in Kiefer and Larson (2008).
5 One important advantage of continuous time estimation is that it avoids the problem of defining the period: should data be observed monthly, quarterly, annually? Does the frequency with which datáis observed correspond to the frequency with which transitions occur?
6 We expect that one or two variables would be enough to capture the major macroeconomic effects. Recall that the theoretical model underlying the Basel II regulation is a single-factor model.
7 Following a useful comment of an anonymous referee, we also ran regressions using the Basel capital adequacy ratio replacing CAP. The results we obtained were essentially the same. We only report results using CAP as the proxy for overall financial health in this study.
Bangia, A.; EX. Diebold; A. Kronimus; C. Schagen, and T. Schuermann (2002), "Rating migration and the business cycle, with application to credit portfolio stress testing". Journal of Banking and Finance, 26, 445-474. [ Links ]
Basel Committee on Banking Supervision (2004), International Convergence of Capital Measurement and Capital Standards: a revised Framework. Bank for International Settlements. [ Links ]
Cox, D.R. (1972), "Regression Models and Life-Tables". Journal of the Royal Statistical Society, B 34, 187-220. [ Links ]
Ferri, G., and L. Liu (2003), "How do global credit-rating agencies rate firms from developing countries?", Asian Economic Papers, 2, 30-56. [ Links ]
Gomez-Gonzalez, J.E., and N.M. Kiefer (2009), "Bank failure: evidence from the Colombia financial crisis", The International Journal of Business and Finance Research, forthcoming. [ Links ]
Grey, S.; A. Mirkovic, and V. Ragunathan (2006), "The determinants of credit ratings: Australian evidence", Australian Journal of Management, 31, 333-354. [ Links ]
Jonker, N. (2002), "Credit ratings of the banking sector", De Nederlandsche Bank Research, Research Memorándum N° 714/0236. [ Links ]
Kavvathas, D. (2000), "Estimating credit rating transition probabilities for corporate bonds", AFA 2001 New Orleans Meetings. [ Links ]
Kiefer, N.M. (1988), "Economic duration data and hazard functions". Journal of Economic Literature, XXVI, 646-679. [ Links ]
Kiefer, N.M., and E. Larson (2008), "A simulation estimator for testing the time homogeneity of credit rating transitions ", Journal of Empirical Finance, 15, 818-835. [ Links ]
Klein, J.P., and M.L. Moeschberger (2003), Survival Analysis. Techniques for Censored and Truncated Data, Second Edition, Springer. [ Links ]
Küchler, U. and M. Søvensen (1997), "Exponential families of stochastic processes". Springer Series in Statistic. Springer-Verlag, New York. [ Links ]
Lando, D. (2004), Credit Risk Modeling, Princeton University Press. [ Links ]
Lando, D., and T. Skodeberg (2002), "Analyzing rating transitions and rating drift with continuous observations", Journal of Banking and Finance, 26, 423-444. [ Links ]
Mahlmann, T. (2006), "Estimation of rating class transition probabilities with incomplete data", Journal of Banking and Finance, 30, 3235-3256. [ Links ]
Rojas-Suarez, L. (2001), "Rating banks in emerging markets: what credit rating agencies should learn from financial indicators" Institute for International Economics, Working Paper N° 01-06. [ Links ]
Thomas, L.; D. Edelman, and J. Crook (2002), Credit Scoring and Its Applications, SIAM, Philadelphia. [ Links ]