WP/05/219Estimating Markov Transition MatricesUsing Proportions Data:An Application to Credit RiskMatthew T. Jones
2005 International Monetary FundWP/05/219IMF Working PaperMonetary and Financial Systems DepartmentEstimating Markov Transition Matrices Using Proportions Data:An Application to Credit RiskPrepared by Matthew T. Jones1Authorized for distribution by S. Kal WajidNovember 2005AbstractThis Working Paper should not be reported as representing the views of the IMF.The views expressed in this Working Paper are those of the author(s) and do not necessarily representthose of the IMF or IMF policy. Working Papers describe research in progress by the author(s) and arepublished to elicit comments and to further debate.This paper outlines a way to estimate transition matrices for use in credit risk modeling witha decades-old methodology that uses aggregate proportions data. This methodology is idealfor credit-risk applications where there is a paucity of data on changes in credit quality,especially at an aggregate level. Using a generalized least squares variant of themethodology, this paper provides estimates of transition matrices for the United States usingboth nonperforming loan data and interest coverage data. The methodology can be employedto condition the matrices on economic fundamentals and provide separate transition matricesfor expansions and contractions, for example. The transition matrices can also be used as aninput into other credit-risk models that use transition matrices as a basic building block.JEL Classification Numbers: C13Keywords: Markov transition matrix; credit risk; nonperforming loans; interest coverageAuthor(s) E-Mail Address: mjones2 @imf.org1The author is grateful to colleagues in MFD for useful comments on earlier drafts of this paper, and to MarianneEl-Khoury and the FDIC for assistance with data.
-2ContentsPageI. Introduction .3II. Credit Quality Dynamics Using Transition Matrices.4A. Transition Matrices When Individual Transitions Known .4B. Transition Matrices When Individual Transitions Unknown.6III. Application to U.S. Data.8A. Nonperforming Loan Data .8B. Interest Coverage Data .12IV. Additional Applications.15V. Conclusions.16Technical Appendix I.17A. Estimating Transition Probabilities with Observed Transitions .17B. Markov Probability Model .18C. Data Descriptions .22References.24Tables1. Estimated Quarterly Transition Matrix for U.S. Commercial Bank Loans and Leases .102. Estimated Quarterly Transition Matrix, Split Sample .113. Estimated Quarterly Transition Matrix, Low and High Growth Estimates .124. Estimated Annual Transition Matrix Using Interest Coverage Ratio.14Figures1. U.S. Nonperforming Loan Ratios .102. Real GDP Growth and Loan Ratios.123. Using Interest Coverage Data to Estimate Transition Matrices.134. Interest Coverage Ratio for U.S. Companies.145. Interest Coverage Ratio, Performing Loans, and Real GDP.15Appendix Tables1. Illustrative Example of Using Count Proportions to Estimate Transition Probabilities.18
-3-I. INTRODUCTIONThe experience with banking crises in numerous countries has demonstrated the intricatelinks between deteriorations in creditor quality, macroeconomic conditions, and institutionalfailure.2 The costly lessons learned from recent banking crises have illustrated the importanceof proper credit-risk management to maintaining financial stability. Understanding theevolution of credit risk is thus an important step in preventing institutional failure andfinancial crises.In the past 10 years there has been a dramatic increase in the analysis and understanding ofthe evolution of market risk, but progress in understanding credit risk has been much slower.3Modeling credit risk is inherently more complex than modeling market risk, because thereturns on a credit portfolio tend to be asymmetric, causing the distribution of returns to behighly skewed with fat negative tails. In contrast, market returns tend to be distributed moresymmetrically and hence are more tractable analytically. Credit-risk events are also muchless frequent than changes in market returns, and tend to be monitored less effectively, givingrise to a paucity of data. Despite these difficulties, there have been significant advances inrecent years in the theory and application of credit-risk models.One strand of the credit-risk-modeling literature makes use of a matrix of transitionprobabilities to explain the migration of creditor quality, as measured by proxies such asbond ratings. These models of ratings migration show the evolution of creditor quality forbroad groups of creditors with the same approximate likelihood of default. This approachprovides matrices of transition probabilities that can be used as an input to models of creditevolution, because they summarize a broad range of possible creditor dynamics in a simpleand coherent fashion.This paper demonstrates how to use proportions data to estimate transition matrices incircumstances where individual transitions are not observed. The paper demonstrates theapplication of the technique using ratio data on nonperforming loans and corporate sectorinterest coverage to arrive at two independent estimates of transition matrices. Theseestimates provide a basis for comparing official sector estimates of credit quality (derivedfrom supervisory data) with corporate sector information on company earnings (derived frombalance sheet data). The transition matrices can then be conditioned on macroeconomicvariables to illustrate the impact of economic performance on creditor quality.Because of the minimal data requirements necessary to implement the techniques shown inthe paper, the approach is potentially applicable to a broad range of countries andcircumstances. The methodology demonstrated in this paper can be applied to individualcountries with sufficiently good data, or use average measures based on cross-country2See Lindgren, Garcia, and Saal (1996); and Caprio and Klingebiel (1996) for surveys.3See Altman and Saunders (1997) for an overview of credit risk, or www.defaultrisk.com for recent papers.
-4-experience for similar economies or financial systems. The framework can be appliedprospectively for stress testing of vulnerabilities to macroeconomic conditions, orretrospectively to understand the dynamics of linkages between loan portfolios andmacroeconomic outcomes.The paper is organized as follows: Section II discusses the ratings migration literature andpresents the analytic foundations of the use of proportions data to estimate transitionmatrices; Section III discusses the application of the methodology to nonperforming loan andcorporate sector data, using information from the United States to estimate transitionmatrices; Section IV discusses alternative applications of the estimated transition matrices;and Section V concludes.II. CREDIT QUALITY DYNAMICS USING TRANSITION MATRICESA. Transition Matrices When Individual Transitions KnownIn the credit-ratings literature, transition matrices are widely used to explain the dynamics ofchanges in credit quality. These matrices provide a succinct way of describing the evolutionof credit ratings, based on a Markov transition probability model. The Markov transitionprobability model begins with a set of discrete credit quality ranges (or states), into which allobservations (e.g., firms or institutions) can be classified. Suppose there are R discretecategories into which all observations can be ordered. We can define a transition matrix,P [pij], as a matrix of probabilities showing the likelihood of credit quality stayingunchanged or moving to any of the other R-1 categories over a given time horizon. Eachelement of the matrix, pij, shows the probability of credit quality being equal to i in period t-1and credit quality equal to j in period t: p11 pP 21 M p R1p12Lp 22LMOpR2 Lp1R p 2 R .M p RR (1)We impose a simple Markov structure on the transition probabilities, and restrict ourattention to first-order stationary Markov processes, for simplicity.4 The final state, R, whichcan be used to denote the loss category, can be defined as an absorbing state. This means thatonce an asset is classified as lost, it can never be reclassified as anything else.54A Markov process is stationary if pij(t) pij, i.e., if the individual probabilities do not change with time. SeeAppendix I for more details.5Thus the final row of the transition matrix [pR1 pR2 pRR] consists of zero entries everywhere, except for aone for pRR on the diagonal.
-5-Under this framework, the only relevant information for explaining the behavior of the seriesis its behavior in the previous period. This assumption of a first-order Markov process forcredit transitions may be somewhat restrictive if credit quality responds slowly to changes ineconomic fundamentals, for example. Under these circumstances, using a higher-orderMarkov process or a longer time horizon may be more appropriate. However, using higherorder processes or longer horizons increases the complexity and data requirements quitesubstantially, and may not be feasible with only a limited time series. It may also be the casethat credit quality itself responds quickly to changes in fundamentals, but observations oncredit quality are only made infrequently. Similarly, when using some sources of informationon credit quality such as supervisory data, the observed variable is not true credit quality butthe supervisor’s assessment of the data reported to it. Ideally, one could use hidden Markovchains to model the latent credit quality variable, using supervisory observations as theobserved (or emitted) model. However, the data requirements of this approach are immenseand thus are not practical for the applications considered in this paper.Estimating a transition matrix is a relatively straightforward process, if we can observe thesequence of states for each individual unit of observation, i.e., if the individual transitions areobserved. For example, if we observe the credit ratings of a group of firms at the beginningof a year and then again at the end of the year, then we can estimate the probability ofmoving from one credit rating to another.6 The probability of a firm having a particular creditrating at the end of the year, (e.g., A) given their rating at the beginning of the year (e.g., B)is given by the simple ratio of the number of firms that began the year with the same rating(B) and ended with an A rating to the total number of firms that began with a B rating.More generally, we can let nij denote the number of individuals who were in state i in periodt-1 and are in state j in period t. We can estimate the probability of an individual being instate j in period t given that they were in state i in period t-1, denoted by pij, using thefollowing formula:pij nij n.(2)ijjThus, the probability of transition from any given state i is equal to the proportion ofindividuals that started in state i and ended in state j as a proportion of all individuals in thatstarted in state i.Using the methods described above, it is possible to estimate a transition matrix using countdata. Anderson and Goodman (1957) show that the estimator given in equation (2) is amaximum-likelihood estimator that is consistent but biased, with the bias tending towardzero as the sample size increases. Thus, it is possible to estimate a consistent transitionmatrix with a large enough sample. Moody’s and Standard and Poor’s, for example, provide6Including no change in credit rating.
-6-estimates of transition matrices for different bond issuers, using observations on theindividual transitions of thousands of different entities issuing bonds.7B. Transition Matrices When Individual Transitions UnknownAs mentioned previously, the estimation of transition matrices is relatively simple whenindividual transitions are observed over time. Unfortunately, it is often the case that creditquality transitions are imperfectly observed, and the best information available is anaggregate ratio or proportion showing the percent of total observations in a particular ratingscategory at a point in time. It is not possible to obtain maximum-likelihood estimates usingthe count method shown in equation (2) using such a time series of aggregate proportionsdata. However, if the time