A Dependent Hidden Markov Model of Credit Quality

Abstract

We propose a dependent hidden Markov model of credit quality. We suppose that the "true" credit quality is not observed directly but only through noisy observations given by posted credit ratings. The model is formulated in discrete time with a Markov chain observed in martingale noise, where "noise" terms of the state and observation processes are possibly dependent. The model provides estimates for the state of the Markov chain governing the evolution of the credit rating process and the parameters of the model, where the latter are estimated using the EM algorithm. The dependent dynamics allow for the so-called "rating momentum" discussed in the credit literature and also provide a convenient test of independence between the state and observation dynamics.

1. Introduction

Credit ratings summarise a range of qualitative and quantitative information about the credit worthiness of debt issuers and are therefore a convenient signal for the credit quality of the debtor. The estimation of credit quality transition matrices is at the core of credit risk measures with applications to pricing and portfolio risk management. In view of pending regulations regarding the calculation of capital requirements for banks, there is renewed interest in efficiency of credit ratings as indicators of credit quality and models of their dynamics (Basel Committee on Banking Supervision [1]).

In the study of credit quality dynamics, it is convenient to assume that the credit rating process is a time-homogeneous Markov chain, with past changes in credit quality characterised by a transition matrix. The assumptions of time homogeneity and Markovian behaviour of the rating process have been challenged by some empirical studies; see, for example, Bangia et al. [2] or Lando and Skødeberg [3]. In particular, it has been proposed that ratings exhibit “rating momentum” or “drift,” where a rating change in response to a change in credit quality does not fully reflect that change in credit quality. As pointed out by Löffler in [4, 5], these violations of information efficiency could be the result of some of the agencies’ rating policies, namely, rating through the cycle and avoiding rating reversals.

In recent years, a number of modelling alternatives were suggested to address departures from the Markov assumption. In Frydman and Schuermann [6], a mixture of two independent continuous time homogeneous Markov chains is proposed for the ratings migration process, so that the future distribution of a firm’s ratings depends not only its current rating but also on the past history of ratings. Wendin and McNeil [7] suppose that credit ratings are subject to both observed and unobserved systematic risk. Rating transition patters (e.g., rating momentum) are captured within the context of a generalised linear mixed model (GLMM) that is estimated using Bayesian techniques. Stefanescu et al. [8] propose a Bayesian hierarchical framework, based on Markov Chain Monte Carlo (MCMC) techniques, to model non-Markovian dynamics in ratings migrations. In Wozabal and Hochreiter [9], a coupled Markov chain model is introduced to model dependency among rating migrations of issuers.

In this paper we follow the hidden Markov model (HMM) approach taken in Korolkiewicz and Elliott [10] and assume that the “true” credit quality evolution can be described by a Markov chain but we do not observe this Markov chain directly. Rather, it is hidden in “noisy” observations represented by posted credit ratings. The model is formulated in discrete time, with a Markov chain of “true” credit quality observed in martingale noise. However, we suppose that noise terms of the signal and observation processes are not independent, which allows for the presence of “rating momentum” in posted credit ratings. Application of such dependent hidden Markov model dynamics to modelling credit quality appears to be new. We employ hidden Markov filtering and estimation techniques described in Elliott et al. [11] and use the filter-based EM (Expectation Maximization) algorithm to estimate the parameters of the model. By construction parameters are revised as new information is obtained and so the resulting filters are adaptive and “self-tuning.”

The paper is organized as follows. In Section 2 we describe a hidden Markov model (HMM) of credit quality and in Section 3 the dependent dynamics. Recursive filters are given in Section 4 and the parameter estimation procedure is described in Section 5. Section 6 provides an implementation example.

2. Dynamics of the Markov Chain and Observations

Here we briefly describe a hidden Markov model as given in Chapter 2 of Elliott et al. [11]. Formally, a discrete-time, finite-state, time homogeneous Markov chain is a stochastic process {𝑋_𝑘} with the state space 𝑆={1,2,…,𝑁} and a transition matrix 𝐴=(𝑎_𝑗𝑖)_{1≤𝑖,𝑗≤𝑁}. Without loss of generality, we can assume that the elements of 𝑆 are identified with the standard unit vectors {𝑒₁,𝑒₂,…,𝑒_𝑁}, 𝑒_𝑖=(0,…,0,1,0,…,0)^′∈ℝ^𝑁.

Write ℱ_𝑘=𝜎{𝑋₀,𝑋₁,…,𝑋_𝑘} for a filtration {ℱ_𝑘} models all possible histories of 𝑋. The relationship between the state process at time 𝑘 and the state of the process at time 𝑘+1 is then given by 𝐸[𝑋_𝑘+1|𝑋_𝑘]=𝐴𝑋_𝑘.

Define 𝑉_𝑘+1=𝑋_𝑘+1−𝐴𝑋_𝑘∈ℝ^𝑁. Then, the semimartingale representation of the chain 𝑋 is 𝑋_𝑘+1=𝐴𝑋_𝑘+𝑉_𝑘+1,𝑘=0,1,…, (2.1) where 𝑉_𝑘+1 is a martingale increment with 𝐸[𝑉_k+1|ℱ_𝑘]=0∈ℝ^𝑁.

Suppose we do not observe 𝑋 directly. Rather, we observe a process 𝑌 such that 𝑌_𝑘+1=𝑐𝑋_𝑘,𝜔_𝑘+1,𝑘=0,1,…, (2.2) where 𝑐 is a function with values in a finite set and {𝜔_𝑘} is a sequence of i.i.d. random variables independent of 𝑋. Random variables {𝜔_𝑘} represent the noise present in the system. Suppose the range of 𝑐 consists of 𝑀 points which are identified with unit vectors {𝑓₁,𝑓₂,…,𝑓_𝑀},𝑓_𝑗=(0,…,0,1,0,…,0)^′∈ℝ^𝑀.

Write 𝒴_𝑘=𝜎𝑌₀,𝑌₁,…,𝑌_𝑘,𝒢_𝑘=𝜎𝑋₀,…,𝑋_𝑘,𝑌₀,…,𝑌_𝑘. (2.3) These increasing families of 𝜎-fields are filtrations representing possible histories of the state process 𝑋, the observation process 𝑌, and both processes (𝑋,𝑌). Write 𝑐_𝑗𝑖=𝑃(𝑌_𝑘+1=𝑓_𝑗∣𝑋_𝑘=𝑒_𝑖),1≤𝑖≤𝑁,1≤𝑗≤𝑀, for the probability of observing a state 𝑓_𝑗 when the signal process is in fact in state 𝑒_𝑖. Then, it can be shown that 𝐸[𝑌_𝑘+1|𝑋_𝑘]=𝐶𝑋_𝑘, where 𝐶=(𝑐_𝑗𝑖)_{1≤𝑖,𝑗≤𝑀} is a matrix with 𝑐_𝑗𝑖≥0 and ∑^𝑀_𝑗=1𝑐_𝑗𝑖=1.

Define 𝑊_𝑘+1=𝑌_𝑘+1−𝐶𝑋_𝑘. The semimartingale representation of the process 𝑌 is 𝑌_𝑘+1=𝐶𝑋_𝑘+𝑊_𝑘+1,𝑘=0,1,…, (2.4) where 𝑊 is a martingale increment with 𝐸[𝑊_𝑘+1|𝒢_𝑘]=0∈ℝ^𝑀. In our context, the process 𝑌 represents posted credit ratings and 𝑋 “true” credit quality. For reasons which will become apparent in the next section, we assume one-period delay between 𝑋 and 𝑌.

In summary, the model for the Markov chain 𝑋 hidden in martingale noise is as follows.

Hidden Markov Model (HMM)
Under a probability measure 𝑃, 𝑋_𝑘+1=𝐴𝑋_𝑘+𝑉_𝑘+1,𝑌signalequation,truecreditquality_𝑘+1=𝐶𝑋_𝑘+𝑊_𝑘+1observationequation,postedrating. (2.5)𝐴 and 𝐶 are matrices of transition probabilities whose entries satisfy ^𝑁_𝑗=1𝑎_𝑗𝑖=1,𝑎_𝑗𝑖≥0;^𝑀_𝑗=1𝑐_𝑗𝑖=1,𝑐_𝑗𝑖≥0. (2.6)𝑉_𝑘 and 𝑊_𝑘 are martingale increments satisfying 𝐸𝑉_𝑘+1∣ℱ_𝑘=0,𝐸𝑊_𝑘+1∣𝒢_𝑘=0. (2.7) Parameters of this model are (𝑎_𝑗𝑖),1≤𝑖,𝑗≤𝑁 and (𝑐_𝑗𝑖),1≤𝑗≤𝑀,1≤𝑖≤1.

3. Dependent Dynamics

The situation considered in this section is that of a hidden Markov model (HMM) for which the “noise” terms in the state and observation processes are possibly dependent.

The dynamics of the state process 𝑋 and the observation process 𝑌 are as given in Section 2. However, the noise terms 𝑉_𝑘 and 𝑊_𝑘 are not independent. Instead, we suppose that the joint distribution of 𝑌_𝑘 and 𝑋_𝑘 is given by 𝑌_𝑘+1𝑋^′_𝑘+1=𝑆𝑋_𝑘+Γ_𝑘+1,𝑘=0,1,…, (3.1) where 𝑆=(𝑠_𝑟𝑗𝑖) denotes a (𝑀×𝑁)×𝑁 matrix mapping ℝ^𝑁 into ℝ^𝑀×ℝ^𝑁 and 𝑠_𝑟𝑗𝑖=𝑃𝑌_𝑘+1=𝑓_𝑟,𝑋_𝑘+1=𝑒_𝑗∣𝑋_𝑘=𝑒_𝑖,1≤𝑟≤𝑀,1≤𝑖,𝑗≤𝑁. (3.2)Γ_𝑘+1 is a {𝒢_𝑘}-martingale increment with 𝐸[Γ_𝑘+1|𝒢_𝑘]=0. Write 𝟏=(1,1,…,1)^′ for the vector in ℝ^𝑁 or ℝ^𝑀 depending on the context. Then, for 𝟏∈ℝ^𝑀,⟨𝟏,𝑆𝑋_𝑘⟩=𝐴𝑋_𝑘 and for 𝟏∈ℝ^𝑁,⟨𝑆𝑋_𝑘,𝟏⟩=𝐶𝑋_𝑘, where ⟨,⟩ denotes the scalar product in ℝ^𝑀 and ℝ^𝑁, respectively.

Write 𝛾_𝑟𝑗𝑖=𝑃(𝑌_𝑘+1=𝑓_𝑟∣𝑋_𝑘+1=𝑒_𝑗,𝑋_𝑘=𝑒_𝑖)=𝑠_𝑟𝑗𝑖/𝑎_𝑗𝑖, and let 𝐶 be the 𝑀×(𝑁×𝑁) matrix (𝛾_𝑟𝑗𝑖),1≤𝑟≤𝑀,1≤𝑖,𝑗≤𝑁. Then it can be shown that 𝑌_𝑘+1=𝐶(𝑋_𝑘+1𝑋^′_𝑘)+𝑊_𝑘+1, where 𝐸[𝑊_𝑘+1|𝒢_𝑘]=0.

In summary, the model is now as follows.

Dependent Hidden Markov Model (Dependent HMM)
Under a probability measure 𝑃,𝑋_𝑘+1=𝐴𝑋_𝑘+𝑉_𝑘+1,𝑌_𝑘+1=𝐶𝑋_𝑘+1𝑋^′_𝑘+𝑊_𝑘+1. (3.3)𝐴 and 𝐶 are matrices of transition probabilities whose entries satisfy ^𝑁_𝑗=1𝑎_𝑗𝑖=1,^𝑀_𝑟=1𝛾_𝑟𝑗𝑖=1. (3.4)𝑉_𝑘 and 𝑊_𝑘 are martingale increments satisfying 𝐸𝑉_𝑘+1∣ℱ_𝑘=0,𝐸𝑊_𝑘+1∣𝒢_𝑘=0. (3.5) Parameters of this model are (𝑎_𝑗𝑖),1≤𝑖,𝑗≤𝑁,(𝑐_𝑗𝑖),1≤𝑗≤𝑀,1≤𝑖≤1, and (𝑠_𝑟𝑗𝑖),1≤𝑖,𝑗≤𝑁,1≤𝑟≤𝑀.

We are in a situation analogous to the dependent hidden Markov model case discussed in Chapter 2, Section  10 of Elliott et al. [11]. The difference is that we are assuming dynamics where the observation 𝑌_𝑘 depends on both 𝑋_𝑘 and 𝑋_𝑘−1. In other words, we suppose that the current credit rating contains information about both current and previous credit quality, thus allowing for the situation where a rating does not immediately reflect all available information about credit quality, as indicated by a number of empirical studies (see, e.g., Lando and Skødeberg [3]). Put differently, in this model 𝑋_𝑘 and observation 𝑌_𝑘 jointly depend on 𝑋_𝑘−1, which means that, in addition to previous period’s credit quality, knowledge of current credit rating carries information about current credit quality. Moreover, probabilities 𝛾_𝑟𝑗𝑖 provide the distribution of the next period’s credit rating given both current and next period’s credit quality, thus allowing us to capture “rating momentum” or “rating drift.”

In the following sections we will presents estimates for the state of the Markov chain 𝑋, the number of jumps from one state to another, the occupation time of 𝑋 in any state, the number of transitions of the observation process 𝑌 into a particular state of 𝑋, and the number of joint transitions of 𝑋 and 𝑌. We will then use the filter-based EM (expectation maximization) algorithm as described in Elliott et al. [11], to obtain optimal estimates of the model, making it adaptive or “self-tuning.”

Note that if the noise terms in the state 𝑋 and observation 𝑌 are independent, we have 𝑃𝑌_𝑘+1=𝑓_𝑟,𝑋_𝑘+1=𝑒_𝑗∣𝑋_𝑘=𝑒_𝑖=𝑃𝑌_𝑘+1=𝑓_𝑟∣𝑋_𝑘=𝑒_𝑖𝑃𝑋_𝑘+1=𝑒_𝑗∣𝑋_𝑘=𝑒_𝑖. (3.6) Hence if the noise terms are independent, 𝑠_𝑟𝑗𝑖=𝑐_𝑟𝑗𝑎_𝑗𝑖 (3.7) for 1≤𝑟≤𝑀,1≤𝑖,𝑗≤𝑁. Consequently, a test of independence is to check whether parameter estimates satisfy 𝑠̂_𝑟𝑗𝑖=𝑐̂_𝑟𝑗𝑎_𝑗𝑖. (3.8)

4. Recursive Filter

Following Elliott et al. [11], suppose that under some probability measure 𝑃 on (Ω,ℱ),{𝑌_𝑘} is a sequence of i.i.d. uniform variables, that is, 𝑃(𝑌_𝑘+1=𝑓_𝑗∣𝒢_𝑘)=𝑃(𝑌_𝑘+1=𝑓_𝑗)=1/𝑀. Further, under 𝑃,𝑋 is Markov chain independent of 𝑌, with state space 𝑆={𝑒₁,…,𝑒_𝑁} and transition matrix 𝐴=(𝑎_𝑗𝑖). That is, 𝑋_𝑘+1=𝐴𝑋_𝑘+𝑉_𝑘+1, where 𝐸[𝑉_𝑘+1∣𝒢_𝑘]=𝐸[𝑉_𝑘+1∣ℱ_𝑘]=0∈ℝ^𝑁. Suppose 𝐶=(𝛾_𝑟𝑗𝑖),1≤𝑟≤𝑀,1≤𝑖,𝑗≤𝑁, is a matrix with 𝛾_𝑗𝑖≥0, and ∑^𝑀_𝑗=1𝛾_𝑟𝑗𝑖=1.

Define 𝜆_𝑙=𝑀∑^𝑀_𝑗=1⟨𝐶(𝑋_𝑙𝑋^′_𝑙−1),𝑓_𝑗⟩⟨𝑌_𝑙,𝑓_𝑗⟩ and Λ_𝑘=∏^𝑘_𝑙=1𝜆_𝑙. Define a new probability measure 𝑃 by putting (𝑑𝑃/𝑑𝑃)∣_𝒢𝑘=Λ_𝑘. Then, under 𝑃,𝑋 remains a Markov chain with transition matrix 𝐴 and 𝑃(𝑌_𝑘+1=𝑓_𝑟∣𝑋_𝑘+1=𝑒_𝑗,𝑋_𝑘=𝑒_𝑖)=𝛾_𝑟𝑗𝑖. That is, under 𝑃,𝑋_𝑘+1=𝐴𝑋_𝑘+𝑉_𝑘+1 and 𝑌_𝑘+1=𝐶(𝑋_𝑘+1𝑋^′_𝑘)+𝑊_𝑘+1.

Suppose we observe 𝑌₀,…,𝑌_𝑘, and we wish to estimate 𝑋₀,…,𝑋_𝑘. The best (mean square) estimate of 𝑋_𝑘 given 𝒴_𝑘=𝜎{𝑌₀,…,𝑌_𝑘} is 𝐸[𝑋_𝑘∣𝒴_𝑘]∈ℝ^𝑁. However, 𝑃 is a much easier measure under which to work. Using Bayes’ Theorem as described in Elliott et al. [11], we have 𝐸𝑋_𝑘∣𝒴_𝑘=𝐸Λ_𝑘𝑋_𝑘∣𝒴_𝑘𝐸Λ_𝑘∣𝒴_𝑘. (4.1)

Write 𝑞_𝑘∶=𝐸[Λ_𝑘𝑋_𝑘∣𝒴_𝑘]∈ℝ^𝑁. 𝑞_𝑘 is then an unnormalized conditional expectation of 𝑋_𝑘 given the observations 𝒴_𝑘. Note that 𝐸[Λ_𝑘∣𝒴_𝑘]=⟨𝑞_𝑘,𝟏⟩, where 𝟏=(1,1,…,1)^′∈ℝ^𝑁. It then follows that 𝐸𝑋_𝑘∣𝒴_𝑘=𝑞_𝑘𝑞_𝑘.,𝟏 (4.2) Hence, to estimate 𝐸[𝑋_𝑘∣𝒴_𝑘] we need to know the dynamics of 𝑞. Using the methods of Elliott et al. [11], the following recursive formula for 𝑞_𝑘+1 is obtained: 𝑞_𝑘+1=𝑀𝑌^′_𝑘+1𝑆𝑞_𝑘. (4.3)

5. Parameter Estimates

To estimate parameters of the model, matrices 𝐴,𝐶, and 𝑆, we need estimates of the following processes: 𝒥_𝑘^𝑖𝑗=^𝑘_𝑛=1𝑋_𝑛−1,𝑒_𝑖𝑋_𝑛,𝑒_𝑗𝒪,1≤𝑖,𝑗≤𝑁,_𝑖𝑘=^𝑘_𝑛=1𝑋_𝑛−1,𝑒_𝑖𝒯,1≤𝑖≤𝑁,_𝑘𝑖𝑟=^𝑘_𝑛=1𝑋_𝑛−1,𝑒_𝑖𝑌_𝑛,𝑓_𝑟ℒ,1≤𝑖≤𝑁,1≤𝑟≤𝑀,_{𝑘𝑖𝑗𝑟}=^𝑘_𝑛=1𝑋_𝑛−1,𝑒_𝑖𝑋_𝑛,𝑒_𝑗𝑌_𝑛,𝑓_𝑟,1≤𝑟≤𝑀,1≤𝑖,𝑗≤𝑁. (5.1)

The above processes are interpreted as follows: 𝒥_𝑘^𝑖𝑗 is the number of jumps of 𝑋 from state 𝑒_𝑖 to state 𝑒_𝑗 up to time 𝑘. 𝒪^𝑖_𝑘 is the amount of time, up to time 𝑘−1,𝑋 has spent in state 𝑒_𝑖. 𝒯_𝑘^𝑖𝑟 is the number of transitions, up to time 𝑘, from state 𝑒_𝑖 to observation 𝑓_𝑟. ℒ_𝑘^𝑖𝑗𝑟 is the number of jumps of 𝑋 from state 𝑒_𝑖 to state 𝑒_𝑗 while 𝑌 was in state 𝑓_𝑟 up to time 𝑘.

Note that ∑^𝑁_𝑗=1𝒥_𝑘^𝑖𝑗=∑^𝑀_𝑗=1𝒯_𝑘^𝑖𝑗=∑^𝑀_𝑟=1∑^𝑁_𝑗=1ℒ_𝑘^𝑖𝑗𝑟=𝒪^𝑖_𝑘.

Consider first the jump process {𝒥_𝑘^𝑖𝑗}. We wish to estimate 𝒥_𝑘^𝑖𝑗 given the observations 𝑌₀,…,𝑌_𝑘. As in the case of a filter for the state 𝑋 described in Section 4, the best (mean-square) estimate is 𝐸𝒥_𝑘^𝑖𝑗∣𝒴_𝑘=𝐸Λ_𝑘𝒥_𝑘^𝑖𝑗∣𝒴_𝑘𝐸Λ_𝑘∣𝒴_𝑘𝜎𝒥∶=^𝑖𝑗𝑘𝑞_𝑘.,𝟏 (5.2) We wish to know how 𝜎(𝒥^𝑖𝑗)_𝑘 is updated as time passes and new information arrives. However, as noted in Elliott et al. [11], we work with 𝜎(𝐽^𝑖𝑗𝑋)_𝑘=𝐸[Λ_𝑘𝐽_𝑘^𝑖𝑗𝑋_𝑘∣𝒴_𝑘] rather than 𝜎(𝐽^𝑖𝑗)_𝑘=𝐸[Λ_𝑘𝐽_𝑘^𝑖𝑗∣𝒴_𝑘], in order to obtain closed-form recursions. The quantity of interest, namely, 𝜎(𝒥^𝑖𝑗)_𝑘, is then readily obtained as 𝜎(𝒥^𝑖𝑗)_𝑘=⟨𝜎(𝒥^𝑖𝑗𝑋)_𝑘,𝟏⟩. We have 𝜎𝒥^𝑖𝑗𝑋_𝑘+1=𝑀𝑌^′_𝑘+1𝑆𝜎𝒥^𝑖𝑗𝑋_𝑘+𝑞_𝑘,𝑒_𝑖𝑀^𝑀_𝑟=1𝑠_𝑟𝑗𝑖𝑌_𝑘+1,𝑓_𝑟𝑒_𝑗. (5.3) Similarly, we consider the best (mean square) estimates of 𝒪^𝑖_𝑘, 𝒯_𝑘^𝑗𝑟, and ℒ^𝑟𝑗𝑖 given 𝒴_𝑘: 𝐸𝒪^𝑖_𝑘∣𝒴_𝑘=𝐸Λ_𝑘𝒪^𝑖_𝑘∣𝒴_𝑘𝐸Λ_𝑘∣𝒴_𝑘𝜎𝒪∶=^𝑖𝑘𝑞_𝑘,𝐸𝒯,𝟏_{𝑘𝑗𝑖𝑟}∣𝒴_𝑘=𝐸Λ_𝑘𝒯_𝑘^𝑗𝑖𝑟∣𝒴_𝑘𝐸Λ_𝑘∣𝒴_𝑘𝜎𝒯∶=^𝑖𝑟𝑘𝑞_𝑘,𝐸ℒ,𝟏_{𝑘𝑖𝑗𝑟}∣𝒴_𝑘=𝐸Λ_𝑘ℒ_𝑘^𝑖𝑗𝑟∣𝒴_𝑘𝐸Λ_𝑘∣𝒴_𝑘𝜎ℒ∶=^{𝑖𝑗𝑟𝑘}𝑞_𝑘.,𝟏 (5.4) Recursive formulae for the processes 𝜎𝒪^𝑖𝑋_𝑘∶=𝐸Λ_𝑘𝒪^𝑖_𝑘𝑋_𝑘∣𝒴_𝑘,𝜎𝒯^𝑖𝑟𝑋_𝑘∶=𝐸Λ_𝑘𝒯_𝑘^𝑖𝑟𝑋_𝑘∣𝒴_𝑘,𝜎ℒ^𝑖𝑗𝑟𝑋_𝑘∶=𝐸Λ_𝑘ℒ_𝑘^𝑖𝑗𝑟𝑋_𝑘∣𝒴_𝑘 (5.5) are as follows: 𝜎𝒪^𝑖𝑋_𝑘+1=𝑀𝑌^′_𝑘+1𝑆𝜎𝒪^𝑖𝑋_𝑘+𝑞_𝑘,𝑒_𝑖^𝑀_𝑗=1𝑀^𝑀_𝑟=1𝑠_𝑟𝑗𝑖𝑌_𝑘+1,𝑓_𝑟𝑒_𝑗,𝜎𝒯^𝑖𝑟𝑋_𝑘+1=𝑀𝑌^′_𝑘+1𝑆𝜎𝒯^𝑖𝑟𝑋_𝑘+𝑀𝑞_𝑘,𝑒_𝑖^𝑁_𝑗=1𝑠_𝑟𝑗𝑖𝑒_𝑗𝑌_𝑘+1,𝑓_𝑟,𝜎ℒ^𝑖𝑗𝑟𝑋_𝑘+1=𝑀𝑌^′_𝑘+1𝑆𝜎ℒ^𝑖𝑗𝑟𝑋_𝑘+𝑞_𝑘,𝑒_𝑖𝑀𝑠_𝑟𝑗𝑖𝑌_𝑘+1,𝑓_𝑟𝑒_𝑗. (5.6) As in the case of the number of jumps of the state process 𝑋, quantities of interest 𝜎(𝒪^𝑖)_𝑘, 𝜎(𝒯^𝑖𝑟)_𝑘, and 𝜎(ℒ^𝑖𝑗𝑟)_𝑘 are obtained by taking inner products with 𝟏=(1,1,…,1)^′: 𝜎𝒪^𝑖_𝑘=𝜎𝒪^𝑖𝑋_𝑘,𝜎𝒯,𝟏_𝑖𝑟𝑘=𝜎𝒯^𝑖𝑟𝑋_𝑘,𝜎ℒ,𝟏_{𝑖𝑗𝑟𝑘}=𝜎ℒ^𝑖𝑗𝑟𝑋_𝑘,𝟏. (5.7) The model is determined by parameters 𝜃={𝑎_𝑗𝑖,1≤𝑖,𝑗≤𝑁;𝑐_𝑗𝑖,1≤𝑖≤𝑁,1≤𝑗≤𝑀;𝑠_𝑟𝑗𝑖,1≤𝑟≤𝑀,1≤𝑖,𝑗≤𝑁}. These satisfy 𝑎_𝑗𝑖≥0,^𝑁_𝑗=1𝑎_𝑗𝑖=1,𝑐_𝑗𝑖≥0,^𝑀_𝑗=1𝑐_𝑗𝑖=1,𝑠_𝑟𝑗𝑖≥0,^𝑀𝑁_𝑟=1𝑗=1𝑠_𝑟𝑗𝑖=1. (5.8) We want to determine a new set of parameters 𝜃={𝑎_𝑗𝑖,1≤𝑖,𝑗≤𝑁;𝑐̂_𝑗𝑖,1≤𝑖≤𝑁,1≤𝑗≤𝑀};  𝑠̂_𝑟𝑗𝑖,1≤𝑟≤𝑀,1≤𝑖,𝑗≤𝑁} given the arrival of new information embedded in the values of the observation process 𝑌. This requires maximum likelihood estimation. As in [11], we proceed by using the filter-based EM (Expectation Maximization) algorithm, which retains the well-established statistical properties of the EM algorithm while reducing memory costs and thus allowing for faster computation (see, e.g., Krishnamurthy and Chung [12]).

Consider first the parameter 𝑎_𝑗𝑖. Suppose that, under measure 𝑃_𝜃,𝑋 is a Markov chain with transition matrix 𝐴=(𝑎_𝑗𝑖). We define a new probability measure 𝑃𝜃 such that, under 𝑃𝜃,𝑋 is a Markov chain with transition matrix 𝐴=(𝑎_𝑗𝑖), that is, 𝑃𝜃𝑋_𝑘+1=𝑒_𝑗∣𝑋_𝑘=𝑒_𝑖=𝑎_𝑗𝑖, (5.9)𝑎_𝑗𝑖≥0,∑^𝑁_𝑗=1𝑎_𝑗𝑖=1. Define Λ₀Λ=1,_𝑘=^𝑘_𝑙=1^𝑁_𝑟,𝑠=1𝑎_𝑠𝑟𝑎_𝑠𝑟𝑋_𝑙,𝑒_𝑠𝑋_𝑙−1,𝑒_𝑟. (5.10) In case 𝑎_𝑗𝑖=0 take 𝑎_𝑗𝑖=0 and 𝑎_𝑠𝑟/𝑎_𝑠𝑟=1.

Define 𝑃𝜃 by setting 𝜃(𝑑𝑃/𝑑𝑃_𝜃)∣ℱ_𝑘=Λ_𝑘. It can then be shown that, under 𝑃𝜃,𝑋 is a Markov chain with transition matrix 𝐴=(𝑎_𝑗𝑖). Moreover, given the observations up to time 𝑘, {𝑌₀,𝑌₁,…,𝑌_𝑘}, and given the parameter set 𝜃={𝑎_𝑗𝑖,1≤𝑖,𝑗≤𝑁;𝑐_𝑗𝑖,1≤𝑖≤𝑁,1≤𝑗≤𝑀}, the EM estimates 𝑎_𝑗𝑖 are given by 𝑎_𝑗𝑖=𝜎𝒥^𝑖𝑗_𝑘𝜎𝒪^𝑖_𝑘. (5.11) Consider now the parameter 𝑐_𝑗𝑖. Suppose that, under measure 𝑃_𝜃, 𝑌_𝑘+1=𝐶𝑋_𝑘+𝑊_𝑘+1, (5.12) where 𝐶=(𝑐_𝑗𝑖). We define a new probability measure 𝑃𝜃 as follows. Put Λ₀Λ=1,_𝑘=^𝑘_𝑙=1^𝑁_𝑟,𝑠=1𝑐̂_𝑠𝑟𝑐_𝑠𝑟𝑋_𝑙−1,𝑒_𝑟𝑌_𝑙,𝑓_𝑠. (5.13) In case 𝑐_𝑗𝑖=0 take 𝑐̂_𝑗𝑖=0 and 𝑐̂_𝑠𝑟/𝑐_𝑠𝑟=1.

Define 𝑃𝜃 by setting 𝜃(𝑑𝑃/𝑑𝑃_𝜃)∣𝒢_𝑘=Λ_𝑘. Again it can be shown that, under 𝑃𝜃, 𝑌_𝑘+1=𝐶𝑋_𝑘+𝑊_𝑘+1, (5.14) that is, 𝑃𝜃(𝑌_𝑘+1=𝑓_𝑠∣𝑋_𝑘=𝑒_𝑟)=𝑐̂_𝑠𝑟. Moreover, given the observations up to time 𝑘,{𝑌₀,𝑌₁,…,𝑌_𝑘}, and given the parameter set 𝜃={𝑎_𝑗𝑖,1≤𝑖,𝑗≤𝑁;𝑐_𝑗𝑖,1≤𝑖≤𝑁,1≤𝑗≤𝑀}, the EM estimates 𝑐̂_𝑗𝑖 are given by 𝑐̂_𝑗𝑖=𝜎𝒯^𝑖𝑗_𝑘𝜎𝒪^𝑖_𝑘. (5.15) Finally, consider the parameter 𝑠_𝑟𝑗𝑖. A new probability measure 𝑃𝜃 is defined by putting Λ₀Λ=1,_𝑘=^𝑘_𝑙=1^𝑀𝑁_{𝑟=1𝑖,𝑗=1}𝑠̂_𝑟𝑗𝑖𝑠_𝑟𝑗𝑖𝑌_𝑙,𝑓_𝑟𝑋_𝑙,𝑒_𝑗𝑋_𝑙−1,𝑒_𝑖. (5.16) In case 𝑠_𝑟𝑗𝑖=0 take 𝑠̂_𝑟𝑗𝑖=0 and 𝑠̂_𝑟𝑗𝑖/𝑠_𝑟𝑗𝑖=1. Define 𝑃𝜃 by setting 𝜃(𝑑𝑃/𝑑𝑃_𝜃)∣𝒢_𝑘=Λ_𝑘. Then, under 𝑃𝜃, 𝑌_𝑘+1𝑋^′_𝑘+1=𝑆𝑋_𝑘+Γ_𝑘+1, that is, 𝑃𝜃𝑌_𝑘+1=𝑓_𝑟,𝑋_𝑘+1=𝑒_𝑗∣𝑋_𝑘=𝑒_𝑖=𝑠̂_𝑟𝑗𝑖. (5.17) Given the observations up to time 𝑘,{𝑌₀,𝑌₁,…,𝑌_𝑘}, and given the parameter set 𝜃={𝑎_𝑗𝑖,1≤𝑖,𝑗≤𝑁;𝑐_𝑗𝑖,1≤𝑖≤𝑁,1≤𝑗≤𝑀;𝑠_𝑟𝑗i,1≤𝑟≤𝑀,1≤𝑖,𝑗≤𝑁}, the EM estimates 𝑠̂_𝑟𝑗𝑖 are then given by 𝑠̂_𝑟𝑗𝑖=𝜎ℒ^𝑖𝑗𝑟_𝑘𝜎𝒪^𝑖_𝑘. (5.18)

6. Implementation Example

The dependent hidden Markov model (Dependent HMM) described in previous sections was applied to a dataset of Standard & Poor’s credit ratings. Description of the data and implementation results are given below.

6.1. Data Description

Our analysis takes advantage of the Standard & Poor’s COMPUSTAT database, which contains rating histories for 1,301 obligors over the period 1985–1999 (Standard & Poor’s [13]). The universe of obligors is mainly large US and Canadian corporate institutions. The obligors include industrials, utilities, insurance companies, banks and other financial institutions, and real-estate companies. The COMPUSTAT database provides annual ratings. Every year each of the rated obligors is assigned to one of the Standard and Poor’s 7 rating categories, ranging from 𝐴𝐴𝐴 (highest rating) to 𝐶𝐶𝐶 (lowest rating) as well as 𝐷 (payment in default) and the NR (not rated) state.

We have a total of 19,515 firm-years in our sample. However, only 34% of those observations correspond to one of the eight Standard & Poor’s rating labels in a given year. The remaining 66% of observations represent the so-called NR (not rated) status. As discussed in the literature, transitions to NR may be due to several reasons, such as expiration of the debt, calling of the debt, or the issuer deciding to bypass an agency rating (see, e.g., Bangia et al. [2]). Unfortunately, details of individual transitions to NR are not known.

Excluding NR, approximately 85% of the remaining ratings are in categories 𝐴 down to 𝐵. The median rating is 𝐵𝐵, the highest non-investment-grade rating. Approximately 1% of the observed ratings are 𝐴𝐴𝐴 and 2% are defaults. The most common rating is 𝐵, two rating categories above default, which accounts for 25.5% of the observations.

6.2. Implementation Results

Since individual firms generally experience few rating changes and changes that do occur are to neighbouring categories, we apply the Dependent HMM algorithm to an aggregate of firms in the dataset rather to allow for more observed transitions between rating categories and make inferences possible. Specifically, we follow the filter-based cohort approach adopted in Korolkiewicz and Elliott [10], and instead of estimating the distribution and parameters for the Markov chain 𝑋^𝑙_𝑘 for each firm 𝑙, we estimate the distribution and parameters for ∑^𝐿_𝑙=1𝑋^𝑙_𝑘 given the additivity of all stochastic processes discussed in Sections 4 and 5.

Given the fairly large number of parameters to be estimated compared to the number of rating transitions in the dataset, we have reclassified all firms in the sample as IG (investment grade), SG (speculative grade), 𝐷, or NR and then applied the Dependent HMM algorithm to the new dataset. This classification is motivated by the fact that a corporation which can issue higher rated debt usually receives better financing terms. Further, as a matter of policy or law, some institutional investors can only purchase investment-grade bonds. Hence it is often crucial for a borrower to maintain an investment-grade rating and so it is interesting to see if rating transition data reflects this.

Each modified credit rating category IG, SG, as well as default 𝐷 and NR, was identified with a unit vector in ℝ⁴. Given the relatively short time period, parameter estimates were updated with the arrival of every new observation for the 1,301 firms in the dataset. Repetition of the estimation procedures ensures that the model and estimates improve with each iteration. Estimated parameters of the model, namely, matrices 𝐴,𝐶, and 𝑆, are given in Table 1.

Table 1 

Estimates of matrices 𝐴, 𝐶, and 𝑆.

Considering the estimated transition matrix 𝐴, note that entries above the diagonal correspond to rating upgrades and those below the diagonal to rating downgrades. Nonzero transition probabilities are concentrated and highest on the diagonal and the second largest probability is in the last row, indicating that obligors generally either maintain their rating or enter the NR (not rated) category. Our results show that investment-grade firms generally hold on to their status. The probability of downgrade to speculative-grade status is estimated as 6.8%. However, for speculative-grade firms, the probability of upgrade to investment-grade status is lower (estimated probability of 1.8%). Speculative-grade firms tend to maintain their status or disappear from the dataset because of either default or withdrawn rating. The probability of transition to the NR status is higher for speculative-grade obligors (71.5%) than for investment-grade obligors (52.4%).

Recall from Section 3 that, given estimates of matrices 𝐴 and 𝐶, our Dependent HMM also provides the distribution of posted credit ratings at time 𝑘+1 given “true” credit quality at times 𝑘 and 𝑘+1, namely, estimates of conditional probabilities 𝛾_𝑟𝑗𝑖=𝑃(𝑌_𝑘+1=𝑓_𝑟∣𝑋_𝑘+1=𝑒_𝑗,𝑋_𝑘=𝑒_𝑖). To illustrate, consider a borrower with investment-grade “true” credit quality at times 𝑘 and 𝑘+1. The probability that this borrower is assigned to a speculative-grade rating class is 𝑃(𝑌_𝑘+1=SG∣𝑋_𝑘+1=IG,𝑋_𝑘=IG), which, given our model parameter estimates, is given by 𝑠̂_SG,IG,IG/𝑎_IG,IG=0.007/0.408=0.017. Similarly, for a borrower whose “true” credit quality improves from SG to IG, the probability of being assigned to an IG rating class is given by 𝑃(𝑌_𝑘+1=IG∣𝑋_𝑘+1=IG,𝑋_𝑘=SG), which we would estimate to be 0.007/0.018=0.389. These estimates again suggest that rating agencies may be somewhat reluctant to downgrade (upgrade) borrowers from (to) investment grade, thus introducing a degree of “rating momentum.”

6.3. Test of Independence

Recall that the Dependent HMM allows the “noise” terms in the state and observation processes to be possibly dependent. As indicated in Section 3, a convenient test of independence is to check whether the estimated parameters of the model satisfy 𝑠̂_𝑟𝑗𝑖=𝑐̂_𝑟𝑗𝑎_𝑗𝑖.

Given our estimates of matrices 𝐴 and 𝐶, products 𝑐̂_𝑟𝑗𝑎_𝑗𝑖 were calculated and then compared to corresponding entries of the estimated matrix 𝑆 using linear regression. The regression results are given in Table 2. As indicated by the high 𝐹-statistic (4728.10) and high 𝑅² value (98.71%), the fitted regression model is significant. The slope estimate is very close to one with low standard error and 𝑃 value of 0.000, while the intercept estimate is very close zero and not significant (𝑃 value of 0.91). These regression results suggest no major departures from independence, which seems to agree with findings in Kiefer and Larson [14] that indicate the Markov assumption, implicit in most credit risk models, does not seem to be “too wrong” for typical forecast horizons. However, longer rating histories may be necessary to verify these results.

Table 2 

Linear regression results.

7. Conclusion

We have proposed a Dependent Hidden Markov Model for the evolution of credit quality in discrete time with a Markov chain observed in martingale noise. We have applied the estimation techniques of hidden Markov models from Elliott et al. [11] to obtain the best estimate of the Markov chain representing “true” credit quality and estimates of the parameters. The estimation procedure was repeated to ensure that the model and estimates improved with each iteration. The model was applied to a dataset of Standard & Poor’s issuer ratings and our preliminary results agree with some qualitative observations made in the literature regarding credit rating systems but also indicate no significant dependence in the dynamics of the “state” (credit quality) and “observation” (credit rating) processes.