تجزیه و تحلیل تاثیر مهاجرت های اعتباری در یک محیط پرتفولیو

Credit migration is an essential component of credit portfolio modeling. In this paper, we outline a framework for gauging the effects of credit migration on portfolio risk measurements. For a typical loan portfolio, we find credit migration can explain as much as 51% of volatility and 35% of economic capital. We compare through-the-cycle migration effects, implied by agency rating transitions, with point-in-time migration, implied by EDF™ (Expected Default Frequency) transitions, and find that migration of point-in-time credit quality accounts for a greater fraction of total portfolio risk when compared with through-the-cycle dynamics. In a stylized analytic setting, we show that, when controlling for PD term structure effects, higher likelihood of moving away from the current credit state does not necessarily imply greater risk. Finally, we review methods for generating high-frequency transition matrices, needed for analyzing instruments with cash flows or contingencies whose frequencies are asynchronous to an available transition matrix. We further demonstrate that the naïve application of such methods can result in material deviations to portfolio analytics.

Recent turmoil in the capital markets has led to a sharp rise in the number of negative rating actions taken by the leading rating agencies, signaling a deterioration in the credit quality of firms affected by adverse economic conditions. These credit quality dynamics highlight the importance of credit migration modeling as an integral part of modern credit risk solutions. Credit migration models play a vital role in pricing credit-risky instruments, as well as in assessing the risk of credit portfolios. Many credit instruments exhibit cash flows contingent on the credit quality of the reference entity, either implicitly, e.g., a prepayment option on a loan, which may be exercised in the event of improved credit quality of the borrower, or explicitly, e.g., a loan with grid pricing, which ties the contractual loan spread to the credit quality of the borrower. Pricing such credit-contingent cash flows requires a credit migration model. Further, to estimate the risk associated with a credit instrument or a portfolio, a credit migration model is needed to build a distribution of values at the analysis horizon. Most practical credit risk models typically employ a discretized representation of credit qualities and utilize a finite-state Markov model, which posits that the probability of migrating to another credit state in the future does not depend on the past.1 In a discrete-time framework, parameterizing a credit migration model amounts to estimating a Markov transition matrix, populated with migration probabilities over a specified time period. In a continuous-time framework, a generator matrix can be estimated, allowing for forecasts over any time horizon. Parameterization of a Markovian credit migration model with an appropriate transition matrix is an important step in the specification of a credit risk model. Within the context of a credit portfolio management (CPM) framework, common considerations affecting the choice of a transition matrix include: • Asset class/industry/domicile: A number of studies document the fact that rating transitions vary according to the industry or regional classification of the obligor, see, for example, Nickell et al., 2000 and Kadam and Lenk, 2008. Depending on the granularity of the model, portfolio managers may choose to employ a custom transition matrix that is industry- or region-specific or estimated for a particular asset class. • Through-the-cycle vs. point-in-time: Since agency ratings are considered through-the-cycle measures of credit quality ( Cantor and Mann, 2003 and Altman and Rijken, 2004), portfolio managers whom want to gain a long-term view of portfolio risk often employ a rating transition matrix as the driver of credit migration. Further, as Livingston et al., 2008 and Hill et al., 2010 document, different agencies’ credit ratings may imply different migration patterns, suggesting that managers take into consideration the particular agency data used. Alternatively, we can use point-in-time measures of credit quality, such as Moody’s Analytics EDF credit measure, to estimate a model that provides a more dynamic view of credit migration. Since credit migration is a fundamental component of a CPM model, it is important to understand to what extent different migration matrices impact measures of portfolio risk. In particular, if portfolio risk is sensitive to the choice of a specific migration model, we must take care to ensure that portfolio statistics are driven by economically meaningful data, rather than by numerical artifacts or estimation noise. While the literature on credit migration is plentiful, relatively little work addresses the parameterization of a credit migration model or the impact of credit migration on portfolio risk. Bangia et al. (2002) estimate transition matrices during economic expansion and contraction periods. They show that the loss distribution and economic capital of a synthetic bond portfolio vary significantly in different economic environments. Trück and Rachev (2005a) draw similar conclusions, finding that migration matrices estimated at different points of the business cycle produce significantly different VaR measures for a test portfolio. Further, they find a more dramatic impact on the confidence sets of estimated default probabilities. The experimental results documented in these papers suggest that the migration matrix choice can have a significant impact on the risk analysis of the portfolio. Realizing the need for a quantitative measure of difference between credit migration matrices, several authors propose distance measures that aim to capture economically significant differences. Jafry and Schuermann (2004) develop a metric for comparing migration matrices based on singular values (closely related to the induced matrix 2-norm) and use it to compare different methods of estimating migration probabilities. They demonstrate that under their proposed metric, statistically significant differences between transition matrices result in material differences in the economic capital of a fictitious bond portfolio. In related work, Trück and Rachev (2005b) suggest a class of so-called directed difference indices as a measure of distance between transition matrices. They demonstrate that their proposed measures exhibit high correlation with differences in credit VaR, indicating that these measures capture economically significant differences in migration dynamics. This paper complements existing work in three main areas. First, we construct a class of low-dimensional migration matrices parameterized by a varying diagonal element, and show that when the default probability term structure remains unchanged, increasing the probability mass assigned to the diagonal results in non-monotonic behavior of the volatility of the value distribution at the analysis horizon. This example supports observations made in Jafry and Schuermann, 2004 and Trück and Rachev, 2005b, that the diagonal dominance of a matrix, often a used as a visual indicator for the inherent “riskiness” of a migration matrix, is not a sufficient characteristic of migration risk. This example also highlights the interplay between default risk and migration risk implied by a transition matrix. Second, we propose a framework for measuring the portion of portfolio risk attributable to credit migration, which facilitates comparison of the impact of different transition matrices on portfolio risk. In contrast to the distance measures outlined in Jafry and Schuermann, 2004 and Trück and Rachev, 2005b, our framework yields a portfolio-referent distance measure. Further, our framework allows for decoupling of default risk and migration risk. We employ this framework to study the differences between two migration models, one based on agency ratings, and the other based on the EDF credit measure. We find that credit migration explains a significant portion of the risk attributed to a test portfolio, and that migration of point-in-time credit quality, measured by EDF transition rates, accounts for a greater fraction of total portfolio risk when compared with through-the-cycle dynamics reflected by agency rating migrations. The third contribution lies in studying the performance of numerical algorithms for generation of high-frequency transition matrices from an annual matrix. We demonstrate that in some cases, application of such methods results in distortions to the implied annual transition probabilities and, consequently, to the calculated risk measures as well. The remainder of this paper is organized as follows: Section 2 offers an overview of credit migration in the context of credit portfolio analysis. Section 3 describes a stylized analytic example that illustrates the behavior of an instrument’s volatility in value as a function of migration probabilities. Section 4 details a framework to measure migration impact on portfolio risk, which can be applied to compare the migration effects of two different migration models. We use this framework to contrast EDF-based migration with rating-based migration. Section 5 discusses the problem of generating high-frequency transition matrices from an annual transition matrix. We describe two numerical algorithms for this purpose and demonstrate their performance when applied to a rating transition matrix and an EDF credit measure transition matrix. Section 6 offers concluding remarks.

In this paper, we study the impact of credit migration on the risk of a credit portfolio and addressed some of the challenges a risk manager may face when specifying a transition matrix for portfolio analysis. The discussion highlights several important points to take into consideration in the parameterization process. First, we demonstrate that examination of a transition matrix’s diagonal weights is insufficient to assess the migration risk implied by it. In particular, we employ a stylized analytic setup to show that, when controlling for PD term structure effects, higher likelihood of moving away from the current credit state does not necessarily imply greater risk (as measured by the volatility of the value at horizon). In the context of a portfolio analysis, we outline a framework that allows for assessing the portion of risk attributed to migration. Applying this framework to a rating transition matrix and an EDF transition matrix, we demonstrate that for a synthetic test portfolio, a point-in-time view of credit migration accounts for a greater fraction of total portfolio risk, when compared with a through-the-cycle view of migration. Finally, we discuss the problem of estimating transition matrices to arbitrary time horizons from a given annual matrix. We review two methods for generating high-frequency transition matrices and document a case where the numerical artifacts caused by these methods results in material deviations of portfolio analytics. In conclusion, our findings suggest that migration explains a large portion of the risk of a credit portfolio, and that the choice of a migration matrix has a significant impact on analysis results. These findings shed new light on the importance of proper specification of migration matrices in addition to the identification of other risk inputs such as PD and LGD. Further, these findings suggest that care must be taken to ensure that the analysis results are not driven, in part, by numerical artifacts or estimation noise.