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DURATION MODELS FOR CREDIT RATING MIGRATION: EVIDENCE FROM THE FINANCIAL CRISIS.

I. INTRODUCTION

The evolution of ratings (commonly referred to as "rating migration"), assigned in particular by the three major credit rating agencies, plays a very important role within the financial sphere, especially in determining the credit spreads which influence the issue of bonds. The financial crisis has not fundamentally changed the deal. Financial ratings continue to play a key role in determining credit ratings given not only to the business world but also to sovereign countries. The credit rating's role is to analyze the economic and financial situation of the entity in question, anticipating its future prospects for development and yield. The reactions following a downgrade ("bad news") as well as those following an upgrade ("good news") show how much attention is paid by the financial world to the announcements made by the rating agencies. It is important to emphasize that a change in rating, particularly downwards, can have serious consequences on the market value and the financial cost for the concerned entity. When a business' rating drops, this directly affects the bond market as well as the stock market. Analysts subsequently revise their anticipated profits downwards not only for the firm but for the investors in these stocks. Additionally, as emphasized by Broto and Molina (2016) for sovereign ratings, the evolution of the ratings is not symmetric since downgrades seem to be deeper and quicker than upgrades. This suggests that amplitudes of rating changes and durations between two consecutive rating attributions are highly correlated.

Another question currently being asked is: What about the effect of the crisis on firms' ratings? The financial crisis of 2007-2008 did not spare many important business groups. Several sectors such as banks, services, construction, automotive, tourism, and property were hit hard, and more especially some specific geographical zones. For example, in France, according to the Financial Markets Authority, the rating agencies abruptly lowered their ratings during and after the crisis to 155 falls in 2008 against 56 falls in 2007. Given that downgrades doubled in 2008, many criticisms have been made against the rating agencies, especially from investors concerned with the actual performance of those financial assets with the best ratings. Indeed, numerous assets highly rated saw massive downgrades due to the methodology adopted by rating agencies. Other criticisms have mainly focused on the relationship between the rating agencies and the issuers. Since usually the agency is paid by the issuing business, the assigned ratings tend to be overvalued due to a relative dependence of the agencies with respect to their customers. We may therefore be tempted to question the quality of the agencies' expertise in assessing the matter of credit risk. Note, however, that as of the moment, no alternative has been implemented, despite some probably premature announcements. Compelled to review their ratings and methodologies, the agencies need to take better account of risks such as liquidity, counterparty, and contagion phenomena and also such as the economic context. Their ratings involve only the risk of default, but this is linked to the previous risks which turned out to be significant when the markets failed to operate properly during the crisis.

The aim of this paper is to propose a model to analyze the rating transition process and to estimate the basic properties of the migration process with respect to the latent economic context. Indeed, as emphasized by Frydman and Schuermann (2008), although credit migration matrices are usually assumed to be generated by Markovian models, empirical evidence shows that such assumption is too strong to fit actual financial data since future evolution of a firm's credit rating depends not only on its current rating but also on its past ratings. A number of studies have documented the fact that rating transition matrices vary according to the stage of the business cycle, the industry of the obligor, and the length of time that has elapsed since the issuance of the bond. These studies have either been conducted by the rating agencies themselves (e.g., Carty and Fons 1993 who have examined the autocorrelation of rating changes for Moody's) or by academic researchers, such as Altman and Kao (1992), who find that, over time, higher-rated bonds are more stable than lower-rated bonds as far as retaining their original ratings is concerned. Altman (1998) compares rating changes from the two major agencies, Moody's and Standard & Poor's, over the period 1970-1996, and highlights very significant differences between the various published reports on rating migration; Nickell, Perraudin, and Varotto (2000) show that the distribution of ratings changes plays a key role in many credit risk models. Nickell, Perraudin, and Varotto (2000) have also emphasized that probability transition matrices of bond ratings depend a lot on the business cycle; Bangia et al. (2002) examine the relationship between credit rating and business cycle and show the increase in default rates during recessions; Koopman and Lucas (2005) study the link between business and default cycles for credit risk. Lando and Sk0deberg (2002), Hamilton and Cantor (2004), and others have also demonstrated the existence of non-Markovian behavior, such as the effects of macroeconomic factors. In most of these studies, it is assumed that such economic conditions are observable and influence the rating changes for businesses. Kim and Sohn (2008) have also illustrated the impact of randomness on credit rating transitions. However, as emphasized by Gagliardini and Gourieroux (2005a, 2005b), the analysis of joint evolutions of individual risks based on observable macro-variables allows to better understand the reasons for joint rating movements (often related to the business cycle literature) but this type of model is difficult to implement for the prediction of future risk in a credit portfolio. Gagliardini and Gourieroux (2005a) examined instead both migration correlation and non-Markovian serial dependence by considering rating histories with stochastic transition matrices. They introduced an ordered probit model with unobservable dynamic factor illustrated on French data about corporate risk. Feng, Gourieroux, and Jasiak (2008) also proposed an ordered probit model to model the transition matrix viewed as a function of latent variables. They did not provide an exact specification for this factor and assumed it to be nonobservable, representing the economic conjunction of the rated businesses. The importance of this factor lies in anticipating the transition matrices.

This paper is concerned with modeling jointly the fluctuation of firms' ratings as well as the durations between variations of ratings. A fundamental application of our approach is to measure the effect of the crisis on this rating process. The main novelty of our model is that we simultaneously account for ratings transitions (possible category changes) as well as the durations between each rating (the period between two consecutive rating announcements). Indeed, contrary to Gagliardini and Gourieroux (2005a) who only consider stochastic transition matrices of ratings, we analyze jointly the distributions of durations between two rating changes. This approach allows a better analysis of the rating process in particular by linking the potential rating changes to the date of their announcement. Hence, this model is based on market events rather than on calendar time ("business time" vs "calendar time"). To this end, for each rating level, we consider an autoregressive model describing the conditional log durations. The accent is on estimating the hazard function parameters specific to the various possible adopted ratings so as to characterize the ratings episodes among risk classes. More precisely, each hazard function describes the change in instantaneous probability of migrating at time t from a given class j to a given class k (in the set of all possible ratings) as a function of in particular the durations which have already occurred before the episode being considered. In that sense, we propose to apply the autoregressive conditional duration (ACD) model introduced by Engle and Russell (1998) (also Engle 2000; Pacurar 2008) to financial rating so as to model the duration of assignment between two consecutive ratings. Then, this allows us to construct the appropriate transition matrices. To better analyze the impact of the financial crisis, we introduce an ACD model where the conditional mean of the duration between two ratings is modeled by means of a latent process so as to enable nonobservable variables to influence the durations between ratings. In our case, this factor allows us to explain the changes in firms' ratings and to examine the effects of the crisis on these changes. A dynamic-ordered probit model is also developed to describe the directions taken by the ratings in the presence of multiple states, and those before and during the crisis. The model parameters are estimated using the maximum likelihood method.

Indeed, probit models introducing a nonobservable latent factor to model the parameters (mean, thresholds, and variance) provide satisfactory results. The advantage of using a nonobservable latent factor is in calculating the a posteriori probability of group membership. In addition, this factor allows us to classify firms according to their risk class (via the correlation with changes in rating); this enables us to conclude that the rating agencies had a tendency to give excellent ratings to businesses but since the 2007-2008 crisis, these ratings were then suddenly and dramatically revised downwards, bringing into question the quality of the predictions by agencies having overvalued many firms. The conclusions of our study echo to those in the literature (see previous references) which emphasize the effect of a factor of heterogeneity (observable or not). (1)

The paper is organized as follows: Section II is devoted to the credit migration modeling. For this purpose, we introduce a general ACD model for the credit rating process and in particular Burr distributions with three parameters to describe durations with sufficient flexibility. We also propose an ACD model with latent process to highlight the effect of an unobservable factor on the process of firms' ratings during the financial crisis. Section III illustrates the implementation of our methodology, using a large sample of data on U.S., European, and Asian firms rated by Standard & Poor's during the period 1989-2009. Finally, Section IV concludes.

II. CREDIT RISK MIGRATION MODELING

In what follows, we model the dynamics of the rating process. For this purpose, first we introduce a dynamic-ordered probit model to describe rating changes. The main assumption is that the current risk class assigned to each firm is determined by evaluating a nonobservable continuous score. Second, we consider a LOG-ACD model to model the durations of ratings episodes. This crucially allows to take account of the time dependence between the ratings durations. Finally, we examine the likelihood function of the joint distribution of the rating changes and the associated durations.

A. The Dynamics of the Rating Process

When examining the movement of firms' ratings, we find that such a process corresponds to the joint observation of a series of arrival dates (dates on which ratings migrate or not to other risk classes) with the characteristics associated to them (types of rating characterizing the incoming and outgoing classes). This leads us to the general notion of marked point process (Last and Brandt 1995). Within this framework, the rating dynamics for each company can be reconstructed by retracing an individual path for each company marking the migration episodes (periods of time--number of months--between two consecutive changes of rating) which follow one another according to the ratings timetable. Figure 1 provides an example of the path illustrating, for a given company, the manner in which the rating process evolves from the date of its first rating to the date on which the period of observation ends.

During the period of observation, at each rating change is assigned one of the following 11 classes (we use here the Standard & Poor's classification since our data are based on Standard & Poor's rating process):

AAA - AA - A - BBB - BB - B - CCC - CC - C - SD - D.

Given the high number of risk classes (2) and in accordance with the aim of this paper which focuses on the effect of the crisis on the process of firms' ratings, we aggregate the rating changes so as to bring out the four possible directions taken by the rating on the issuance of the rating episode, namely "major downgrade," "minor downgrade," "minor upgrade," and "major upgrade." (3) For this purpose, we introduce a joint process to describe firms' ratings process as a function of the ratings durations [D.sup.f.sub.n] (separating the dates of two successive rating changes) and the directions [X.sup.f.sub.n] taken by the ratings resulting from the changes.

For each rating episode n (n = 1, 2, ..., [N.sup.f] - 1) and for a given firm f (f = 1, 2 ..., F), we write the joint distribution of the pair [mathematical expression not reproducible] as follows:

(1) [f.sub.Y] ([x.sup.f.sub.n], [d.sup.f.sub.n]/[z.sup.f.sub.n], v, [I.sup.f.sub.n-1]; [theta]) = [f.sub.X] ([x.sup.f.sub.n]/[d.sup.f.sub.n], [z.sup.f.sub.n], v, [I.sup.f.sub.n-1]; [theta]) x [f.sub.D] ([d.sup.f.sub.n]/[z.sup.f.sub.n], v, [I.sup.f.sub.n-1]; [theta])

where:

1. [f.sub.X] ([x.sup.f.sub.n], [d.sup.f.sub.n], [z.sup.f.sub.n], v, [I.sup.f.sub.n-1]; [theta]) is the Probability density function (pdf) of the different directions which could be taken by the rating at the end of the nth rating episode. These directions are modeled by a polytomic variable whose terms are defined as follows

[mathematical expression not reproducible]

This function is defined conditionally on a set of observable characteristics, [z.sup.f.sub.1,n], and nonobservable characteristics v, as well as on the duration of the rating episode giving rise to the nth change [d.sup.f.sub.n], announced for firm f.

2. The function [f.sub.D] ([d.sup.f.sub.n]/[z.sup.f.sub.n], v, [I.sup.f.sub.n-1]; [theta]) corresponds to the pdf of the duration conditional on [I.sup.f.sub.n-1] (all the information available up to that point on the dynamic situation describing the past history of the ratings paths for firm f). Here, again this function is expressed as a function of a set of observable characteristics, [z.sup.f.sub.n], and nonobservable characteristics v.

3. The term [theta] is the vector of parameters to be estimated.

A Dynamic-Ordered Probit Model for the Direction of Ratings. The use of a dynamic-ordered probit model to model rating directions is based on the idea that the current risk class assigned to a firm f at any given time is determined by evaluating a nonobservable continuous score. In our case, this score is considered to be a decreasing function of the estimated firms' probability of default. Let [S.sup.f.sub.n] be the value of the score registered by firm f at the end of the nth rating episode. We are interested in the variations of this score along time. Hence, we denote:

[mathematical expression not reproducible]

where:

(2) [DELTA] [S.sup.f.sub.n] = [S.sup.f.sub.n] - [S.sup.f.sub.n-1] = [[beta].sub.0] + [beta]' * [z.sup.f.sub.n] + [[beta].sub.v]v + u.

In Equation (2), [DELTA][S.sup.f.sub.n] indicates the difference between the scores at the start and at the end of the nth rating episode for firm f. This difference is expressed as a function of a set of observable variables [z.sup.f.sub.n] = ([z.sup.f.sub.1,n], [z.sup.f.sub.2,n]) (indicating on the one hand the duration and on the other hand the order the episode underway), a nonobservable factor of heterogeneity v, and a normally distributed error term u with mean zero and variance [[sigma].sup.2]. For identity reasons, the variance of the error term u is normalized to unity (i.e. [sigma] = 1). Consequently, three thresholds are adopted to bring out four scenarios determining the sense and magnitude of the rating variation (major downgrade, minor downgrade, minor upgrade, and major upgrade). The idea is therefore to estimate the parameter vector:

([c.sub.1], [c.sub.2], [c.sub.3], [beta]' [[beta.sub.v]),

where coefficients [c.sub.i] represent the thresholds to be estimated using the maximum likelihood method taking account of the [beta] = ([[beta].sub.1], [[beta].sub.2]) and [[beta].sub.v] parameters. The cumulative distribution function of the standard Gaussian distribution is denoted by [PHI](.). The probabilities associated to the four previous terms are defined as follows:

(3) [mathematical expression not reproducible]

A LOG-ACD Model for the Duration of Rating Episodes. To model the durations of ratings episodes, we introduce a LOG-ACD model. (4) This type of model allows to take account of the time dependence between the ratings durations for the same firm. This time dependence is expressed by an autoregressive structure which we adopt to model the conditional expectations of the durations. Hence, we assume that the duration [D.sup.f.sub.n] can be represented as the product of two independent components:

* A first component exp ([mathematical expression not reproducible]) where [mathematical expression not reproducible] is equal to the log of the conditional expectation of the duration of episode n, up to an additive constant term.

* A second random component corresponding to terms [[epsilon].sup.d.sub.n] which are identically and independently distributed, [I.sup.f.sub.n-1] being all the information available about firm f on date [[tau].sup.f.sub.n-1].

Formally, the rating duration process is expressed for each firm/ as follows:

(4) [mathematical expression not reproducible] for n = 2, ..., [N.sub.f] - 1,

(5) [mathematical expression not reproducible] with [absolute value of [beta]] < 1 for n = 1:

(6) [mathematical expression not reproducible] where E ([D.sup.f.sub.n]/[I.sup.f.sub.n-1]) = [[mu].sub.[epsilon]] exp ([mathematical expression not reproducible]) with E ([[epsilon].sub.n]/[I.sup.f.sub.n-1]) = [mu][epsilon] and Var ([[epsilon].sub.n]/[I.sup.f.sub.n-1]) = [[sigma].sup.2.sub.[epsilon]].

We assume that the terms [[epsilon].sup.d.sub.n] are Burr distributed according to Burr ([mu], [gamma], [[sigma].sup.2.sub.v]) probability distributions with the following pdf:

(7) [mathematical expression not reproducible]

with [gamma] > [[sigma].sup.2.sub.h] > 0 and mean given by (5) :

E ([[epsilon].sub.n]/[I.sup.f.sub.n-1]) = [mu][epsilon] = 1/[[mu].sup.1/[gamma]] x [GAMMA] [1 + 1/[gamma]] [GAMMA] [1/[[sigma].sup.2.sub.h] - 1/[gamma]]/[[sigma].sup.2(1+1/[gamma]).sub.h] [GAMMA] [1 + 1/[[sigma].sup.2.sub.h]]

Applying a simple change of variables, we can see that, conditionally on information [I.sup.f.sub.n-1] the distribution of [D.sup.f.sub.n] is also given by a Bundistribution with parameters:

[mathematical expression not reproducible]

with the following hazard function:

(8) [mathematical expression not reproducible]

Shaping the hazard function, the use of a Burrtype distribution provides higher flexibility than traditional distributions, such as the exponential, Weibull, and Log-logistic. In point of fact, these are special cases of the Burr distribution and consequently impose more restrictive assumptions when modeling the hazard functions. Indeed, when the parameter [[sigma].sup.2.sub.h] is close to zero, the Bundistribution reduces to a Weibull distribution. Unlike the Burr distribution, the conditional hazard function associated with a Weibull distribution is monotonie. It is strictly increasing as a function of the duration [gamma] > 1. strictly decreasing for [gamma] < 1, and constant for [gamma] = 1 * For the latter case, the Weibull distribution reduces to an exponential of parameter [mathematical expression not reproducible]. On the other hand, if the parameter [[sigma].sup.2.sub.h] is equal to 1, we obtain a Log-logistic distribution with [gamma] > 1. In this latter case, the hazard function initially increases before reaching a maximum and decreases thereafter. In another scheme (Lancaster 1990), the Burr distribution can be viewed as a Gamma mixture of Weibull distributions, allowing to take account of heterogeneity between firms. We know, of course, that the sources of this heterogeneity are unavoidable and that they bias the estimates as a consequence. This leads to introduce into the hazard functions a component which takes account of the unmeasured heterogeneity between firms. This heterogeneity is represented by a random term h assumed to follow a Gamma distribution of mean 1 and (unknown) variance [[sigma].sup.2.sub.h]. Estimating the parameter vector [theta] allows to calculate several functions useful for the understanding and analysis of firms' rating paths. In this sense, we can consider the probabilities of transition between types of rating. To avoid identification issues, we assume in what follows that parameter [mu] is equal to l. (6)

The Likelihood Function. We have expressed both pdfs [f.sub.X](.) and [f.sub.D](.) as functions of a nonobservable random term v so as to account for the heterogeneity of the observations in their own individual dimensions. This corresponds to nonobservable intertemporal factors of heterogeneity which influence the rating process. Statistically speaking, following Green (1990), we assume that v is distributed according to a discrete probability distribution defined over a set of m "mass points," denoted [v.sub.1],[v.sub.2], ..., [v.sub.m] with respective weights [p.sub.1] > 0, ..., [p.sub.m]> 0 satisfying [m.sup.m.sub.k=1] [p.sub.k] = 1. Note that this model is a hidden semi-Markov model due to the nonobservable factor of heterogeneity. Under previous conditions, the likelihood function for the set of firms in our sample is written as follows:

(9) [mathematical expression not reproducible]

The number of point masses to be used for y is determined by a scanning method which consists of maximizing the value of the likelihood function with respect to 9. To conclude about the model parameter estimation, we note that the values taken by the random term v are central to the creation of a set of classes enabling a typology of the ratings episodes. Allocation to the various classes is achieved by averaging the a posteriori probabilities of membership of a given group:

(10) [mathematical expression not reproducible]

This expression shows that the conditional probabilities of belonging to the different groups not only depend on all the parameters of the model but also on all the variables characterizing the joint process of durations and directions of the firms' ratings. This observation justifies the introduction of a latent class model since the process of group membership is governed by a nonobservable endogenous random process. Note also that, unlike the prior probability values that are common to all firms, those generated for the posterior probabilities are specific to each firm since they are obtained by adjusting the prior probabilities according to the evolution of its own rating process.

III. DATA AND SUMMARY STATISTICS

A. Data Description

The sample used in this section involves the ratings movements of 5,579 firms rated by Standard & Poor's 500 during the period 1989-2009. These firms are divided into three geographical zones as follows: 518 firms in the Asian region, 900 in the European zone, and 4,161 in the U.S. zone. During the period of observation, at each rating change business is assigned one of the following 11 classes:

AAA - AA - A - BBB - BB - B - CCC - CC - C - SD - D.

As in the previous section, the observation process of firms has served as the basis for constructing a sample of 18,261 ratings episodes distributed as follows: 1,320 episodes for firms in the Asian region, 2,816 episodes in Europe, and 14,125 episodes in the United States. Given the high number of risk classes, the latter ones were aggregated so as to bring out the four possible directions taken by the rating on the issuance of the rating episode, namely, "major downgrade," "minor downgrade," "minor upgrade," and "major upgrade."

In accordance with the aim of this paper, we focus on the effect of the crisis on the process of firms' ratings. Figure 2 shows how the sample is distributed at the end of the different episodes both before and during the crisis. These latter time periods correspond, respectively, to January 1989 to June 2007 and July 2007 to December 2009. (7) It shows that the set of those episodes penalized by a downgrade significantly increases during the crisis. It reveals that, before the crisis, 34.41% of ratings episodes led to a major downgrade while there has been 54.86% of rating episodes leading to a major downgrade during the crisis. In parallel, rating upgrades recorded a significant drop during the crisis: 18.37% before the crisis while 4.37% during the crisis.

These results show that the subprime crisis triggered a cascade of rating downgrades revealing the financial fragility of firms, especially their level of solvency. This instability led the rating agencies to lower their ratings for businesses often in an abrupt manner since they had overvalued the ratings, thereby amplifying the effects of the crisis. Figure 3 reveals the change in the mean duration of ratings episodes between periods before and during the crisis. It shows that the mean duration of a rating episode is 5 months; the mean duration before the crisis was 22 months (Broto and Molina 2016). Regardless of the type of event during the crisis, there is an acceleration in the changes in rating. Yet again, the crisis appears to influence the rating process via the episodes' durations.

To complete the analysis, we use the Student's t-test to compare the mean durations of ratings episodes (Table 2). At this significance level, the results of the test are unambiguous. The level of significance of the test statistic only reinforces the hypothesis that there is a difference in the means of the episodes' durations. This difference was of the order of 24 months, falling to 5 months during the crisis. Before the crisis, ratings for most of the businesses changed hardly at all. However, numerous indices showed that they were sensitive to changes in the economic cycle. The rating agencies maintained a constant watch on the business' situation, and did not react to minor alterations to their risk profiles. During the crisis, the rating agencies paid more attention to the slightest change which could affect the financial situation of a firm. Hence, they had a tendency to revise the rating very quickly once a particular event had occurred.

B. Estimation Results and Interpretation

In this subsection, we return to the problem of estimating the overall model, making use of the maximum likelihood method (Equation (9)). The contribution of each firm to this function is represented by the path taken by its rating. Within this framework, the dynamic-ordered probit model is combined with the ACD model to account for the direction of the rating and the duration between the ratings episodes, respectively. Four possible scenarios for the ordered probit model are adopted, and are classified from the least to the most favorable (i.e., "major downgrade" to "major upgrade") via intermediate situations, via "minor downgrade" and "minor upgrade":

[mathematical expression not reproducible]

Formally, the dynamic-ordered probit model is defined by the following equation:

[DELTA][S.sup.f.sub.n] = [S.sup.f.sub.n] - [S.sup.f.sub.n-1] = [[beta].sub.0] + [[beta].sub.1] 1n ([D.sup.f.sub.n]) + [[beta].sub.2]order + [[beta].sub.3] + [[beta].sub.4][DELTA][S.sup.f.sub.n-1] + u,

with u [right arrow] Normal (0, [[sigma].sup.2]).

The positive sign and the significance of the coefficients [[beta].sub.2], [[beta].sub.3], and [[beta].sub.4] indicate, respectively, a positive effect of the variables: "duration of ratings episode," the nonobservable heterogeneity factors for the probability of occupying a better position on the Standard & Poor's ratings scale and finally the "order of ratings episode." As regards the ACD model, it is defined by the following equation (8):

(11) [[PSI].sup.f.sub.n]/[I.sup.f.sub.n-1] = [[alpha].sub.0] + [[alpha].sub.1] log ([D.sup.f.sub.n-1]) + [[alpha].sub.2] order + [[alpha].sub.3] [[PSI].sup.f.sub.n-1]/[I.sup.f.sub.n-1] + v + [[epsilon].sub.n-1]

where the conditional distribution of [[epsilon].sub.n-1] knowing the latent variable v is given by a Burr (1, [[gamma].sub.v], [[sigma].sup.2.sub.v]) distribution with:

[[gamma].sub.v] = exp [[[gamma].sub.0] + [[gamma].sub.1] v] > 0

where [v.sub.1], [v.sub.2] = 0, [v.sub.3] and v4 have respective probabilities ([p.sub.1], 1-[[.sub.p1]+[p.sub.3]+[p.sub.4]],[p.sub.3],[p.sub.4]).

The results of the model parameters estimation using the maximum likelihood method are given in Table 3. The second and third columns in this table provide the results of the estimates for the Europe-Asia zone and the American zone, respectively. The fourth column contains the results for the estimates of the whole sample. Although these estimates are made over three different zones, the results appear to provide the same interpretation and at the same time confer a certain validity of the model. The first part of Table 3 provides the estimation results of the duration model. To document the interdependence of ratings episodes occurring for the same firm, we examine the parameters a,, a2, and a3. These parameters are all significant and, at the same time, corroborate the autoregressive specification used in Equation (11).

More precisely, the negative signs of the parameters [[alpha].sub.1], [[alpha].sub.2], and [[alpha].sub.3] show that the shorter the average duration of a given rating episode, the longer the respective durations of the preceding episode, and the higher the order of the current episode. Furthermore, the values recorded for the Gamma shape parameters highlight a positive dependence of the hazard function with regard to the latent variable v and lead us to state that the hazard function is generally strictly increasing with respect to it. In what follows, "economic context" refers to various economic regimes, which are not directly observable and modeled in particular by a latent process. In that case, roughly speaking, the wronger the context, the smaller the future rating duration.

The second part of Table 3 provides the parameters which are related to the qualitative part of the model (the ordered probit model). These results show that the recurrence of the rating, the lengthening of the average duration of the ratings episode, and the order of an episode are such as to predict an improvement in the rating.

The coefficients [c.sub.c1], [c.sub.c2], and [c.sub.3] indicate the values of the score (thresholds) from which one situation passes to another. These coefficients are, respectively, equal to:

* -2.4179 to indicate the transition of "major downgrade" to "minor downgrade";

* -0.9235 to indicate the transition of "minor downgrade" to "minor upgrade";

* 0.8118 to indicate the transition of "minor upgrade" to "major upgrade."

The coefficient [p.sub.0] refers to the constant in the score equation which displays a negative sign for the different samples.

The third part of Table 3 provides the parameters of the process which generates nonobservable heterogeneity in the sample. In our case, the latter is characterized by a support consisting of four point masses denoted by [v.sub.1], [v.sub.2], [v.sub.3], and [v.sub.4] whose respective values are equal to -2.1560, 0, 1.2370, and 3.2785. According to Equation (10), four observational classes are thus established; their values serve as a priori probabilities of membership to groups, 1, 2, 3, and 4, whose characteristics are given in Table 4. As previously indicated in the Discussion section, we have seen that the smaller the value of v (i.e., the more unfavorable the context), the higher the tendency for the rating duration to be reduced. This shows that the context is unfavorable. On the other hand, when v takes high values, it shows an improvement in the rating with longer durations, and in this case the context is favorable. In an unfavorable context, the rating agencies are rapidly forced to reassess the ratings.

In what follows, the groups resulting from the preceding stage are ordered in the following way to give rise to a polytomic variable with these terms for each rating:

[mathematical expression not reproducible]

Group allocation depends on the a posteriori probabilities mentioned in Equation (10). Hence, four probabilities are generated for each rating episode so that they can be allocated to the j groups for which the greatest probability [[pi].sup.f.sub.n] (j) is recorded. Formally, the group allocation rule is stated as follows:

if [[pi].sup.f.sub.n](j) = max {[[pi].sup.f.sub.n](1), (2), [[pi].sup.f.sub.n] (3), [[pi].sup.f.sub.n] (4)} then the nth episode for firm f will be classed in group j.

Table 4 provides the distribution of the overall sample according to the a posteriori probabilities of rating episodes.

According to Table 4, that part of the episodes associated with favorable contexts for firms' ratings is more dominant when a priori (nonconditional) probabilities are applied with respect to that part recorded with a posteriori probabilities, while for fairly favorable contexts, it is the converse. The a priori probabilities to have either a fairly favorable or a favorable context is higher for a priori probabilities (equal to about 82%) than for the a posteriori probabilities (equal to about 67%). Additionally, the a posteriori probability to have a fairly unfavorable context (equal to about 29%) is much higher than for the corresponding a priori probability (equal to about 9%). All the previous findings illustrate how the a posteriori probabilities, which are much more conditioned by the observed events, can better take account of the actual movements of firms' ratings.

Hence, the rating process does not seem to intervene fairly systematically in favor of rated businesses during the whole time period 1989-2009. (9) However, this result is mitigated when focusing around the financial crisis (Section III.C).

In what follows, we return to the analysis conducted initially for understanding the relationship between the variable v (rating context) and the two variables in the rating process (duration and direction of the rating). Based on an analysis of variance, Figure 4 shows that the more favorable the rating context, the higher the estimated mean duration of the episode, which is rather rational for the rating agency.

In parallel, this same positive relation is observed with the rating's direction variable. Table 5 is used to justify that the latent variable v reflects the context in which the notation events were awarded, specifically the relationship between this variable and the type of migration. The results of regression with optimal coding (10) display a significant positive coefficient for the direction variable of the rating (Table 5). In other words, for example, in the case of a rating upgrade, there is a tendency to end up in a favorable context, hence, in group 4.

These results naturally associate realizations of v with the rating context, a context going from the most unfavorable situations for the smallest values of v to the most favorable situations for the highest values of v. To summarize, we can clearly say that the smaller the value of v, the higher the tendency to cumulate a fall in ratings with a reduction in the mean duration of the ratings episodes. On the other hand, when v takes high values, we find ourselves in a "rating improvement" situation and a longer mean duration.

The hazard functions (Equation (8)) that model the duration probability distributions are displayed in Figures 5-7. They illustrate how rating duration depends on economic context. Visual examination of the curves shows that the hazard functions are increasing and concave except for the unfavorable context case for which they become decreasing thereafter. For this latter case, the hazard reaches a maximum then decreases. Note also that its values are much higher than for the other cases. It means that within unfavorable context, durations are much smaller. For the three other cases, the hazard function is concave and increasing, which implies that the default rate is positively dependent on duration (in other words, the longer the duration of an episode in progress, the higher the probability of default). The recurrence of ratings and the positive dependence of specific hazard rates with respect to past periods may be seen as a learning process or capitalizing on knowledge; in which case, the rating agency progressively learns from episodes to reduce the rating time scales.

C. Effect of the Crisis on the Ratings Process for Firms

As we might expect, the crisis has profoundly altered the conditions under which ratings were attributed. To better analyze the rating process and the effectiveness of rating agencies, we focus on the time period around the crisis. For this purpose, recall that we define "before crisis" as the time period corresponding to January 2005 to June 2007 and "during the crisis" as July 2007 to December 2009. According to Figure 8 which displays the a posteriori probabilities, the context degradation appears for the three major geographic zones (i.e., Europe, Asia, and North America). There is clearly excessive optimism exhibited by the agencies just before the crisis for the three zones. The crisis has increased the percentage of unfavorable situations for the European and U.S. regions. This is less pronounced for the Asian zone: approximately 23% overall for an unfavorable or fairly unfavorable context before and during the crisis but a significant increase of the percentage of unfavorable context.

In what follows, we analyze data in more detail to go further than the primary conclusions drawn from observation of Figure 8. The information we draw as regards the influence of the crisis on the process of rating firms is established from results of an ordered probit model expressing the context variable as a function of a binary variable denoted "crisis," which takes the value 1 when the rating occurs during the crisis date, and 0 otherwise. According to the likelihood ratio test, the model is significant. After estimating this model, we are able to determine three thresholds ranked in increasing order [c.sub.1], [c.sub.2], and [c.sub.3] to discriminate between the different contexts (we refer to Table 6).

The coefficients obtained show that overall the ratings context deteriorated after June 2007, that is the premise of the crisis. More explicitly, the value of -0.254 shows that, during the period before the crisis, the probability of being during the crisis is smaller than we could expect during the period of a favorable context. This latter situation is adopted as a reference situation and explains the value 0. These results reinforce the idea that the nonobservable factors of heterogeneity adopted in our model can indeed capture the context in which the ratings of the various firms apply, and hence in particular, the context before and during the crisis.

IV. CONCLUSION

We have introduced a specific nonobservable latent variable with the aim of pinning down the effect of the crisis on the firms' rating process. Our model can be considered as a joint model for the duration and direction of the rating, taking account of an unobservable latent factor corresponding to economic context. This factor enables us to estimate the a posteriori probabilities of belonging to reference groups distinguished by the mean duration and direction of the rating. The dynamic-ordered probit model shows that the rating episodes considered during the crisis tend to belong to the group characterized by a shorter duration and a rating downgrade. These facts lead us to consider this unobservable factor as a factor reflecting the natural state, and which might be influenced by the crisis. The effect of this factor was to bring about ratings downgrades rather than upward hikes. Looking at a posteriori probabilities just before and during the crisis, clearly credit rating agencies were too optimistic before the crisis. This empirical result also shows that the probability of a rating downgrade depends not only on the rating history of one specific firm but also on economic and financial conditions. Such statistical approach can potentially help rating agencies to take more account of their cumulated information, and, for example, to significantly reduce durations between two successive ratings for unfavorable economic contexts. Additionally, from the market participant's point of view, we can better gauge the ability of rating agencies to predict the credit risks. Finally, note that such marked point process can be useful to model various economic variables as soon as we want to take account of the dependency between changes of their values and the corresponding durations.

ABBREVIATIONS

ACD: Autoregressive Conditional Duration

pdf: Probability Density Function

REFERENCES

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Byrne, J. P., M.-E. Spaliara, and S. Tsoukas. "Firm Survival, Uncertainty, and Financial Frictions/ Is There a Financial Uncertainty Accelerator?" Economic Inquiry, 54, 2016, 375-90.

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Gagliardini, P., and C. Gourieroux. "Stochastic Migration Models with Application to Corporate Risk." Journal of Financial Econometrics, 3, 2005a, 188-226.

--. "Migration Correlation: Definition and Efficient Estimation." Journal of Banking and Finance, 29, 2005b, 865-94.

Grammig, J., and K. O. Maurer. "Non-monotonic Hazard Functions and the Autoregressive Conditional Duration Model." Econometrics Journal, 3, 2000, 16-38.

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Hamilton, D., and R. Cantor. "Rating Transition and Default Rates Conditioned on Outlooks." Journal of Fixed Income, 14, 2004,54-70.

Kim, Y., and S. Sohn. "Random Effects Model for Credit Rating Transitions." European Journal of Operational Research, 184, 2008, 561-73.

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Lancaster, T. The Econometric Analysis of Transition Data. Cambridge: Cambridge University Press, 1990.

Lando, D., and T. M. Skodeberg. "Analyzing Rating Transitions and Rating Drift with Continuous Observations." Journal of Banking and Finance, 26, 2002, 423-44.

Last, G., and A. Brandt. Marked Point Processes on the Real Line: The Dynamical Approach. Berlin, Heidelberg: Springer Verlag, 1995.

Nickell, P., W. Perraudin, and S. Varotto. "Stability of Rating Transitions." Journal of Banking and Finance, 24, 2000, 203-27.

Pacurar, M. "Autoregressive Conditional Duration Models in Finance: A Survey of the Theoretical and Empirical Literature." Journal of Economic Surveys, 22(4), 2008, 711-51.

* We thank participants at the 2017 IFC 9 conference for their helpful comments and suggestions on an earlier version of this paper.

Ben Ayed: University Sorbonne Abu Dhabi, Economy, Abu Dhabi, United Arab Emirates. E-mail benayedmyriam@gmail.com

Karaa: University of Tunis, ISG Tunis, Quantitative Methods, Le Bardo, 2000, Tunisia. E-mail adel.karaa@isg.rnu.tn

Prigent: THEMA and LabeX MME-DII, University of Cergy-Pontoise, Cergy, 95011, France. E-mail jeanluc.prigent@u-cergy.fr

doi: 10.1111/ecin.12561

(1.) Note that Byrne, Spaliara, and Tsoukas (2016) consider heterogeneity from the firm-level point of view, illustrating the impact of firm-level uncertainty in firms' hazard of failure, especially during the financial crisis.

(2.) As detailed in the empirical section, the process of observing firms is based on a sample of 18,261 rating episodes.

(3.) "Major downgrade" is understood as the transition of a rating from a given class to another class, for example, a transition, according to Standard & Poor's ratings scale, from category AA to category A. A "minor downgrade" corresponds to the transition of ratings with a "+" down to ratings with a "-," for example, a transition from AAA+ to AAA- or BBB+ to BBB-.

(4.) See Grammig and Maurer (2000) for all details about the Burr-ACD model that we use here to model the rating migration process.

(5.) Here, [GAMMA] denotes the function used to define the Gamma distribution, namely:

[GAMMA](x) = [[integral].sup.[infinity].sub.0] [t.sup.(x-1)][e.sup.-t]dt.

(6.) Writing [mu] = [e.sup.[??]], the hazard function expressed in Equation ((8)) becomes:

[mathematical expression not reproducible]

Now, keeping the parameter [mathematical expression not reproducible] in the expression of [mathematical expression not reproducible] (Equation (5)), we are not able to separate the parameters when estimating the model.

(7.) Taking July 2007 to December 2009 to define the crisis period allows considering a same crisis period for all regions.

(8.) We assume that the conditional distribution of [[PSI].sup.f.sub.n] with respect to information [I.sup.f.sub.n-1] (denoted by [[PSI].sup.f.sub.n]/[I.sup.f.sub.n-1]) is equal to [mathematical expression not reproducible] (i.e., the conditional distribution of [[PSI].sup.f.sub.n] with respect to information [I.sup.f.sub.n-1] only depends on [[PSI].sup.f.sub.n-1]).

(9.) Recall that our data are based on firms ratings and not, for example, on the rating of complex financial products, such as collateralized debt obligations.

(10.) This method of analysis is appropriate since it facilitates the parallel use of variables of all types, and secondly, the results are not too affected by potential multicollinearities between the independent variables (e.g., Breiman and Friedman 1985).

Caption: FIGURE 1 Example of Rating Path

Caption: FIGURE 4 Mean Duration as Function of Economic Context

Caption: FIGURE 5 Hazard Functions (Overall Case)

Caption: FIGURE 6 Hazard Functions (American Case)

Caption: FIGURE 7 Hazard Functions (Asian and European Cases)

Caption: FIGURE 8 Distribution of the A Posteriori Probabilities by Region
TABLE 1
Data Description

Note    Number   Frequency   Cumulative Frequency

AAA       50        0.3              0.3
AA       527        2.9              3.2
A        1909      10.5              13.6
BBB     3,177      17.4               31
BB      3,874      21.2              52.2
B       5,794      31.7               84
CCC     1,565       8.6              92.5
CC       446        2.4               95
C         7         0.0               95
D        711        3.9              98.9
SD       201        1.1              100
Total   18,261      100

TABLE 2
Student's Test for the Duration Variable

                             N       Mean     Median

Duration of the episode    11,468   24.4962     15
before the crisis

Duration of the episode    1,121    5.1392      3
during the crisis

                              Standard      Standard
                           Deviation Type    Error

Duration of the episode       24.61944       .22990
before the crisis

Duration of the episode        5.0536        .15094
during the crisis

Notes: [H.sub.0], equality of means; [H.sub.1] : different means.
Statistic of Student test: 136.33. Degree of freedom: 1. Level
of significance <0.01: We accept [H.sub.1].

TABLE 3
Parameters Estimation

Parameters                 Europe/Asia                USA

Parameters of the ACD model

[[alpha].sub.0]        2.8082 *** (0.0725)    2.9282 *** (0.0648)
[[alpha].sub.1]       -0.1645 *** (0.0191)    -0.1591 *** (0.0122)
[[alpha].sub.2]       -0.4066 *** (0.0457)    -0.5205 *** (0.0250)
[[alpha].sub.3]       -0.1258 *** (0.0229)    -0.1085 *** (0.0141)
[[gamma].sub.0]        1.6969 *** (0.0667)    1.3588 *** (0.0279)
[[gamma].sub.1]       -0.0711 *** (0.0298)    -0.0836 *** (0.0082)
[[sigma].sup.2]        0.0768 *** (0.0166)    0.0834 *** (0.0162)

Parameters of the dynamic probit model

[c.sub.1]               -2.4179 (0.0653)      -2.6069 *** (0.0531)
[c.sub.2]             -0.9235 *** (0.0290)    -0.9932 *** (0.0171)
[c.sub.3]              0.8118 *** (0.0327)    0.6232 *** (0.0155)
[[beta].sub.0]        -0.04623 *** (0.0373)   -0.5717 *** (0.0341)
[[beta].sub.1]         0.0404 *** (0.0167)    0.0285 *** (0.0094)
[[beta].sub.2]         0.0567 *** (0.0064)    0.0249 *** (0.0030)
[[beta].sub.3]         0.1838 *** (0.0219)    0.3616 *** (0.0262)
[[beta].sub.4]         0.8150 *** (0.0462)    0.7512 *** (0.0286)

Parameters associated with heterogeneity factors

[v.sub.1]             -2.1560 *** (0.0886)    -1.7960 *** (0.0706)
[p.sub.1]              0.0715 *** (0.0059)    0.0754 *** (0.0063)
[v.sub.3]              1.2370 *** (0.0560)    1.1190 *** (0.0436)
[p.sub.3]              0.4668 *** (0.0266)    0.4641 *** (0.0242)
[v.sub.4]              3.2785 *** (1.0999)    3.7719 *** (0.9966)
[p.sub.4]              0.1855 *** (0.0147)    0.1952 *** (0.0049)
Log-likelihood           -16,575.805584          -60,749.78875
No. of observations           4,136                  14,125

Parameters                  Overall

Parameters of the ACD model

[[alpha].sub.0]       2.8797 *** (0.0583)
[[alpha].sub.1]         -0.1592 (0.0106)
[[alpha].sub.2]       -0.4974 *** (0.0227)
[[alpha].sub.3]       -0.1113 *** (0.0123)
[[gamma].sub.0]       1.3962 *** (0.0264)
[[gamma].sub.1]       -0.0885 *** (0.0078)
[[sigma].sup.2]       0.09675 *** (0.0217)

Parameters of the dynamic probit model

[c.sub.1]             -2.5666 *** (0.0443)
[c.sub.2]             -0.9715 *** (0.0146)
[c.sub.3]             0.6634 *** (0.0139)
[[beta].sub.0]        -0.5424 *** (0.0282)
[[beta].sub.1]        0.0275 *** (0.0078)
[[beta].sub.2]        0.0294 *** (0.0027)
[[beta].sub.3]        0.3305 *** (0.0206)
[[beta].sub.4]        0.7635 *** (0.0232)

Parameters associated with heterogeneity factors

[v.sub.1]             -1.8198 *** (0.0662)
[p.sub.1]             0.0740 *** (0.0051)
[v.sub.3]             1.1428 *** (0.0403)
[p.sub.3]             0.4813 *** (0.0211)
[v.sub.4]             3.4862 *** (0.8535)
[p.sub.4]             0.1916 *** (0.0044)
Log-likelihood           -77,410.02249
No. of observations          18,261

Notes: Terms between parenthesis are estimated standard deviations.
Value of [v.sub.2] was chosen as being equal to 0 for reasons of
identification of the parameters. [p.sub.2] = 1 - ([p.sub.1] +
[p.sub.3] + [p.sub.4]).

*** Significant value.

TABLE 4
Characteristics of the Reference Groups

                A           A Posteriori
         Priori [p.sub.j]   Probabilities
          Probabilities      for Rating           Group
Groups         (%)            Episodes       Characteristics

1              8.10         8.7% (1,591)       Unfavorable
                                                 context
2              9.24         23.8% (4,349)   Fairly unfavorable
                                                 context
3             36.44         50.8% (9,272)    Fairly favorable
                                                 context
4             46.22         16.6% (3,028)   Favorable context

Notes: Probabilities of rating episodes. Terms between
parentheses denote sample numbers.

TABLE 5
Result of Regression with Optimal Coding

Spearman           Standard
Rho        Beta     Error     Dof   [chi square]    Sig

0.761      0.772    0.003      3      469.273      0.000

TABLE 6
Parameter Estimation by the Probit Method

Context       Estimate   Std.error   [[chi].sub.2]
                                     Wald (ddl = 1)

Threshold 1    -1.344      0.046       1,551.242
Threshold 2    -0.497      0.037        394.306
Threshold 3    2.615       0.070       1.155.121
Crisis = 1     -0.254     0.00080        56.901
Crisis = 0       0

                 Wald 95% Confidence
                      Intervals

Context       Lower bound   Upper bound

Threshold 1     -1.410        -1.277
Threshold 2     -0.546        -0.448
Threshold 3      2.464         2.766
Crisis = 1       0.188         0.320
Crisis = 0

Notes: Log-likelihood (under H0): 6,542.575;
Log-likelihood (under HI): 6,516.62; Test statistic
of the likelihood ratio: 25.94; Degree of freedom: 1;
Significance level < 0.01 ; Model significant overall.

FIGURE 2
Changes in Rating Before and During the Crisis

Asia

                      Before Crisis   During Crisis

Major Downngrade        19,5%           46,6%

Minor Downnggrade       25,0%           37,1%

Minor Upgrade           36,7%            9,7%

Major Upgrade           18,8%            6.5%

USA

                      Before Crisis   During Crisis

Major Downngrade        29,8%           56,3%

Minor Downnggrade       40,7%           29,6%

Minor Upgrade           17,3%            3,4%

Major Upgrade           12,3%           10,6%

Europe

                      Before Crisis   During Crisis

Major Downngrade        23,4%           50,5%

Minor Downnggrade       43,1%           36,8%

Minor Upgrade           22,3%            7,1%

Major Upgrade           11,2%            5,5%

Type

                      Before Crisis   During Crisis

Major Downngrade        34,41%           54,86%

Minor Downnggrade       31,93%           31,22%

Minor Upgrade           16,37%            4,37%

Major Upgrade           15,29%            9,55%

Note: Table made from bar graph.

FIGURE 3
Mean Duration of Rating Episodes Before and During the Crisis

Mean duration

                      Before Crisis   During Crisis

Major Downngrade        19,285           4,449

Minor Downnggrade       28,164           6,714

Minor Upgrade           27,689           8,102

Major Upgrade           24,727           2,598

Note: Table made from bar graph.
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Author:Ayed, Myriam Ben; Karaa, Adel; Prigent, Jean-luc
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Date:Jul 1, 2018
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