#  A Markov approach to credit rating migration
conditional on economic states

Michael Kalkbrener**footnotemark: *, Natalie Packham††footnotemark: †

###### Abstract

We develop a model for credit rating migration that accounts for the impact of
economic state fluctuations on default probabilities. The joint process for
the economic state and the rating is modelled as a time-homogeneous Markov
chain. While the rating process itself possesses the Markov property only
under restrictive conditions, methods from Markov theory can be used to derive
the rating process’ asymptotic behaviour. We use the mathematical framework to
formalise and analyse different rating philosophies, such as point-in-time
(PIT) and through-the-cycle (TTC) ratings. Furthermore, we introduce
stochastic orders on the bivariate process’ transition matrix to establish a
consistent notion of “better” and “worse” ratings. Finally, the construction
of PIT and TTC ratings is illustrated on a Merton-type firm-value process.

Keywords: Risk management, credit risk, credit ratings, Markov chains

MSC classification: 60J20, 91G40, 60J05, 60E1511footnotetext: Michael
Kalkbrener, Deutsche Bank AG, Otto-Suhr-Allee 16, 10585 Berlin, Germany.

22footnotetext:  Natalie Packham, Berlin School of Economics and Law,
Badensche Str. 52, 10825 Berlin, Germany. Email<EMAIL_ADDRESS>The
views expressed in this paper are those of the authors and do not necessarily
reflect the position of Deutsche Bank AG.

##  1 Introduction

It is generally accepted that corporate default rates depend on economic
conditions (Wilson, 1998; Nickell et al., 2000; Bangia et al., 2002;
Heitfield, 2005; Malik and Thomas, 2012). Rating systems, which are designed
to provide information on the credit quality of corporates, differ with
respect to the type of economic information reflected in the rating. The two
main rating philosophies are through-the-cycle (TTC) ratings, which are
insensitive to economic conditions and focus on company-specific criteria, and
point-in-time (PIT) ratings, which reflect the current economic situation as
well as expectations of future economic developments. However, no precise
definition of PIT and TTC characteristics of rating systems has been
established. One objective of this paper is to develop a mathematical
framework for rating migration processes, factoring in the economic situation,
that allows for a formal characterisation of PIT and TTC ratings.

The distinction between PIT and TTC rating concepts is particularly relevant
for banks who need to comply with accounting standards and capital regulation.
Assessing the credit risk of a company solely by its probability of default
(PD), incentivises procyclical adjustments of financial institutions’ capital
requirements, i.e., decreasing default rates in economic upturns would allow
financial institutions to reduce capital for their corporate credit
portfolios, while increasing default rates in economic downturns would
increase capital requirements, see e.g. Borio et al. (2001); Behn et al.
(2016) and references therein. As this procyclical capitalisation poses a
threat to financial stability, financial regulators prefer banks to assess
regulatory capital requirements for credit risk using TTC ratings and default
probabilities,333Capital requirements regulation and directive – CRR/CRD IV,
Article 502 which are designed to be stable through the business cycle. In
contrast, the International Financial Reporting Standard (IFRS) 9 as well as
the accounting standard Current Expected Credit Losses (CECL) require
financial institutions to calculate expected credit losses that reflect
current economic conditions and forecasts of future economic
conditions,444International Financial Reporting Standard (IFRS) 9, Paragraph
5.5.17. For CECL, see FASB Accounting Standards Update (ASU) 2016-13. leading
to the challenging problem of incorporating the impact of economic factors
into the estimation of default probabilities.

Models of rating migration processes specified as time-homogeneous Markov
chains, as is common in the credit risk literature (Duffie and Singleton,
2003; Jarrow et al., 1997; Lando, 2004; Bluhm et al., 2003), lack the
necessary flexibility to capture the complex link to the macroeconomic
environment. To resolve this problem, several extensions of the simple Markov
framework have been proposed in the literature: regime-switching models
specify different rating migration matrices for periods of economic
contraction and expansion (Bangia et al., 2002), mixtures of Markov chain
models define rating migration processes as a combination of several Markov
chains (Frydman and Schuermann, 2008; Fei et al., 2012).

The model presented in this paper extends the above-mentioned approaches by
incorporating an economic state process with finitely many states. Our
starting point is to model the evolution of economic states $\mathcal{A}$ as a
time-homogeneous Markov chain $(A_{t})_{t\in\mathbb{N}_{0}}$. Rating
migrations $(R_{t})_{t\in\mathbb{N}_{0}}$, specified on the state space of
rating classes $\mathcal{R}$, are not assumed to follow a Markov process but
are defined by transition matrices that depend on the state of the economy.
However, it is a key assumption that the rating process has the Markov
property if its state space is extended to the product space
$\mathcal{A}\times\mathcal{R}$ consisting of economic states and rating
classes. More formally, the combined process
$(X_{t})_{t\in\mathbb{N}_{0}}=(A_{t},R_{t})_{t\in\mathbb{N}_{0}}$ defined on
the product space $\mathcal{A}\times\mathcal{R}$ of economic states and
ratings is assumed to be a time-homogeneous Markov chain.

In the first part of the paper (Sections 2–4), we study conditions on the
conditional transition matrices that guarantee the Markov property for
$(R_{t})_{t\in\mathbb{N}_{0}}$ within the state space $\mathcal{R}$ itself.
Since $(R_{t})_{t\in\mathbb{N}_{0}}$ is a projection of the time-homogeneous
Markov chain $(X_{t})_{t\in\mathbb{N}_{0}}$, this problem is a special case of
the more general question of whether functions of Markov chains possess the
Markov property (Rosenblatt, 1971). We show that only a small subclass of TTC
rating migration processes are (time-inhomogeneous) Markov chains on the state
space $\mathcal{R}$, which provides theoretical support to the claim that
real-world rating systems do not have the Markov property, e.g. Altman (1998);
Lando and Skødeberg (2002); Güttler and Raupach (2010). However, it can be
shown that for any non-Markovian rating process $(R_{t})_{t\in\mathbb{N}_{0}}$
there exists a time-inhomogeneous Markov chain
$({\widetilde{R}}_{t})_{t\in\mathbb{N}_{0}}$on $\mathcal{R}$ with time-
dependent transition matrices $({\widetilde{P}}_{t})_{t\in{\mathbb{N}}}$ that
has identical default rates and future rating distributions as
$(R_{t})_{t\in\mathbb{N}_{0}}$, i.e.,

${\mathbb{P}}({\widetilde{R}}_{t}=r)={\mathbb{P}}(R_{t}=r),\quad
t\in\mathbb{N}_{0},\quad r\in{\mathcal{R}},$

where ${\mathbb{P}}(R_{t}=r)$ denotes the probability of $R_{t}$ being in
rating class $r$. Using the Perron-Frobenius Theorem, we prove that the
sequence of transition matrices $({\widetilde{P}}_{t})_{t\in{\mathbb{N}}}$
converges to a limit, denoted by ${\widetilde{P}}_{\infty}$. The corresponding
time-homogeneous Markov chain on $\mathcal{R}$ with transition matrix
${\widetilde{P}}_{\infty}$ replicates the rating distribution and default
behaviour of $(R_{t})_{t\in\mathbb{N}_{0}}$ in the asymptotic limit. In
particular, ${\widetilde{P}}_{\infty}$ specifies the vector of asymptotic
default probabilities of the different rating classes.

The explicit incorporation of an economic state process into the specification
of the rating process $(R_{t})_{t\in\mathbb{N}_{0}}$ allows for a unified
analysis of different rating philosophies. In the second part of the paper
(Section 5), we provide a formal definition of PIT and TTC rating migration
processes based on a decomposition of conditional transition matrices into
default and non-default components. In a nutshell, TTC ratings have a firm-
specific character, i.e., the non-default rating transitions are insensitive
to economic state fluctuations, whereas PIT non-default ratings are explicitly
assigned according to a firm’s PD (probability of default), whence they swing
with economic state fluctuations.

The third part of the paper (Section 6) is concerned with orderings on
distributions on the state space $\mathcal{A}\times\mathcal{R}$ of the
combined process $(X_{t})_{t\in\mathbb{N}_{0}}$, formalising the notions of
“better” or “worse” economic states and rating states. A natural approach to
order rating states is by first-order stochastic dominance, see Jarrow et al.
(1997), which, amongst other properties, preserves the order of long-term
default probabilities. Several extensions of first-order stochastic dominance
exist in a multivariate setting, amongst which stochastic dominance and the
upper orthant order turn out to be relevant for the combined process
$(X_{t})_{t\in\mathbb{N}_{0}}$. We analyse conditions for stochastic
monotonicity of transition matrices of $(X_{t})_{t\in\mathbb{N}_{0}}$ as well
as conditions for preserving the order of multi-year default probabilities
conditional on the initial economic state and rating.

In the fourth part (Section 7), we apply the theoretical framework to create
PIT and TTC ratings using a Merton-type firm-value process (Merton, 1974). The
economic state influences the firm-value by adding a systematic drift. PIT
ratings are determined by classifying firms according to their PDs. While the
PIT rating construction is relatively straightforward, there are various
methods to build TTC ratings, such as considering a firm’s idiosyncratic risk
or stressed PDs. In a worked example we select the TTC rating that most
closely aligns with the PIT rating by minimising the differences in multi-year
PDs conditional on various combinations of initial economic states and PIT
ratings. This provides valuable insights into how PIT and TTC ratings coexist,
as well as their similarities and differences.

We conclude by demonstrating how rating models within our framework
concurrently meet requirements of ratings and PDs in both accounting and
regulatory capital standards (Section 8).

##  2 Conditional rating transition matrices

###  2.1 Definition

We refer to Norris (1998) for basic definitions in the theory of Markov
chains. Let $(\Omega,\mathcal{F},{\mathbb{P}})$ be a probability space. On
this probability space there exists a stochastic process
$(A_{t})_{t\in\mathbb{N}_{0}}$ taking values in a set $\mathcal{A}$, which
consists of $K$ economic states. The economic state process is assumed to be a
time-homogeneous Markov chain with transition matrix
$M=(m_{ab})\in[0,1]^{K\times K}$, i.e.,
${\mathbb{P}}(A_{t+1}=b|A_{t}=a)=m_{ab}$ for $t\in\mathbb{N}_{0}$.

In addition, a stochastic process $(R_{t})_{t\in\mathbb{N}_{0}}$ denoting the
rating process of a firm exists, taking values in the set $\mathcal{R}$. The
set $\mathcal{R}$ has $J$ elements, which are interpreted as rating states
including an absorbing default state $\overline{r}$. The subset of non-default
ratings is denoted by $\widehat{\mathcal{R}}$, i.e.,
$\widehat{\mathcal{R}}={\mathcal{R}}\setminus\\{\overline{r}\\}$.

We do not require the rating process $(R_{t})_{t\in\mathbb{N}_{0}}$ to have
the Markov property. Instead we make the weaker assumption that the joint
process $(X_{t})_{t\in\mathbb{N}_{0}}=(A_{t},R_{t})_{t\in\mathbb{N}_{0}}$
taking values in the product space $\mathcal{X}=\mathcal{A}\times\mathcal{R}$
is a time-homogeneous Markov chain with transition matrix

$P=(p_{(a,r),(b,s)})\in[0,1]^{KJ\times KJ},$ (1)

where

$p_{(a,r),(b,s)}={\mathbb{P}}(A_{t+1}=b,R_{t+1}=s|A_{t}=a,R_{t}=r)$

denotes the probability of moving from the state $(a,r)$ to $(b,s)$. The
distribution of $X_{0}$ is denoted by $\lambda\in[0,1]^{KJ}$. W.l.o.g. we
assume a default probability of 0 at time 0, i.e.,
$\lambda_{(a,\overline{r})}=0$ for $a\in{\mathcal{A}}$.

It is a reasonable assumption that if the current economic state is known,
knowledge about the current rating does not add information about the future
economic state. The following Lemma establishes that, under this assumption,
the transition matrix $P$ is fully determined by

  1. 1.

the transition probabilities $m_{ab}$ of the economic state process and

  2. 2.

the rating transition probabilities conditional on economic state transitions,
i.e., the conditional probabilities
${\mathbb{P}}(R_{t+1}=s|R_{t}=r,A_{t+1}=b,A_{t}=a)$.

To formalize this statement, conditional rating migration matrices
$M^{(a,b)}=(m_{rs}^{(a,b)})\in[0,1]^{J\times J}$ are defined for each economic
state transition $(a,b)\in{\mathcal{A}}\times{\mathcal{A}}$, where

$m_{rs}^{(a,b)}={\mathbb{P}}(R_{t+1}=s|R_{t}=r,A_{t+1}=b,A_{t}=a).$

We assume that migrations from non-default ratings to any other rating class
have positive probability, i.e., $m_{rs}^{(a,b)}>0$ for all
$r\in\widehat{\mathcal{R}}$, $s\in{\mathcal{R}}$ and $a,b\in\mathcal{A}$.
Since $\overline{r}$ is absorbing, $m_{\overline{r}r}^{(a,b)}=0$ and
$m_{\overline{r}\,\overline{r}}^{(a,b)}=1$.

######  Lemma 2.1.

$p_{(a,r),(b,s)}=m_{ab}m_{rs}^{(a,b)}$ if and only if $A_{t+1}$ and $R_{t}$
are conditionally independent given $A_{t}$.

###### Proof.

“$\Leftarrow$”: Using that
${\mathbb{P}}(A_{t+1}=b|A_{t}=a)={\mathbb{P}}(A_{t+1}=b|A_{t}=a,R_{t}=r)$:

$\displaystyle m_{ab}m_{rs}^{(a,b)}$
$\displaystyle={\mathbb{P}}(A_{t+1}=b|A_{t}=a)\,{\mathbb{P}}(R_{t+1}=s|R_{t}=r,A_{t+1}=b,A_{t}=a)$
$\displaystyle={\mathbb{P}}(A_{t+1}=b|A_{t}=a,R_{t}=r)\,{\mathbb{P}}(R_{t+1}=s|R_{t}=r,A_{t+1}=b,A_{t}=a)$
$\displaystyle={\mathbb{P}}(A_{t+1}=b,R_{t+1}=s|R_{t}=r,A_{t}=a)$
$\displaystyle=p_{(a,r),(b,s)}.$

“$\Rightarrow$” follows in the same way. ∎

From now on we assume that $A_{t+1}$ and $R_{t}$ are conditionally independent
given $A_{t}$. It immediately follows from $P=(m_{ab}m_{rs}^{(a,b)})$ that the
process $X_{t}$ as well as the rating process $R_{t}$, which can be considered
as the projection of $X_{t}$ onto the state space ${\mathcal{R}}$, are
completely specified by the initial distribution $\lambda\in[0,1]^{KJ}$ of
$(A_{0},R_{0})$, the transition matrix $M\in[0,1]^{K\times K}$ of $A_{t}$ and
the $K^{2}$ conditional rating migration matrices $M^{(a,b)}\in[0,1]^{J\times
J}$. The probability of a path of economic states and ratings has the form

${\mathbb{P}}(X_{0}=(a_{0},r_{0}),\ldots,X_{T}=(a_{T},r_{T}))={\mathbb{P}}(X_{0}=(a_{0},r_{0}))\prod_{t=0}^{T-1}m_{a_{t}a_{t+1}}m_{r_{t}r_{t+1}}^{(a_{t},a_{t+1})}.$

###  2.2 Example of a rating system

The following example shows the economic state transition matrix $M$ and the
$M^{(a,b)}$ and $P$ matrices of a rating system. It is derived from a Merton
model (Merton (1974)), as will be explained in detail in Section 7.

Economic state transitions are assumed to follow a Markov process with
transition matrix

$M=\begin{pmatrix}0.8&0.175&0.025\\\ 0.1&0.8&0.1\\\
0.025&0.175&0.8\end{pmatrix}.$ (2)

Here, the first state is considered to be a good, the second state a neutral
and the third state a bad economic state.

Figure 1: $M^{(a,b)}$ matrices (left) and $P$ matrix (right). Economic states
are labelled 1 (good), 2 (neutral) and 3 (bad). Ratings are ordered from the
highest rating class to the worst (default). Darker shades indicate values
closer to one, whereas white indicates a value of zero or close to zero.

There are four rating classes, of which the first three – ordered from best to
worst – are non-default ratings and the fourth is the (absorbing) default
state. The left hand side of Figure 1 shows the $M^{(a,b)}$ matrices for
economic states $a,b\in\\{1,2,3\\}$. In this example, the economic state
transition $(a,b)$ has a strong impact on rating migration: if $b$ is the best
economic state 1, upgrades clearly dominate downgrades (as illustrated by the
three matrices in the first column), whereas a transition to the worst state
$b=3$ increases the probability of downgrades. Two examples of $M^{(a,b)}$
matrices are:

$M^{(2,1)}=\begin{pmatrix}0.9808&0.0192&0.0001&4\cdot 10^{-7}\\\
0.8371&0.1598&0.0030&0.0001\\\ 0.6137&0.3660&0.0195&0.0008\\\
0&0&0&1\end{pmatrix},\ \ \
M^{(2,3)}=\begin{pmatrix}0.1737&0.5058&0.3188&0.0017\\\
0.0213&0.2460&0.7004&0.0323\\\ 0.0033&0.0911&0.7814&0.1242\\\
0&0&0&1\end{pmatrix}.$

The right-hand side of Figure 1 shows the $P$ matrix resulting from
$p_{(a,r),(b,s)}=m_{ab}m^{(a,b)}_{rs}$, for all $a,b\in\mathcal{A}$ and
$r,s\in\mathcal{R}$.

##  3 Analysis of the Markov property of rating processes

###  3.1 A class of Markovian rating processes

As a consequence of the dependence on the underlying economic states, rating
processes $(R_{t})_{t\in\mathbb{N}_{0}}$ typically do not have the Markov
property if their state space is restricted to the set of rating classes. In
this section, we establish conditions on the transition matrices $M^{(a,b)}$
that ensure that $(R_{t})_{t\in\mathbb{N}_{0}}$ is a Markov process on
$\mathcal{R}$.

It is a key model assumption in this paper that rating processes have the
Markov property on the extended state space
$\mathcal{X}=\mathcal{A}\times\mathcal{R}$, which consists of all combinations
of economic states and rating classes. More formally,
$(R_{t})_{t\in\mathbb{N}_{0}}$ is the projection of the time-homogeneous
Markov chain $(X_{t})_{t\in\mathbb{N}_{0}}$ with state space $\mathcal{X}$
onto the state space $\mathcal{R}$. Hence, the question whether
$(R_{t})_{t\in\mathbb{N}_{0}}$ is a Markov process can be addressed in the
more general context of studying the Markov property of functions of Markov
processes, see Rosenblatt (1971).

The general results in Rosenblatt (1971) can be significantly improved in our
setup by utilizing the structure of the transition matrix
$P=(p_{(a,r),(b,s)})\in[0,1]^{KJ\times KJ}$ given in Lemma 2.1, i.e.,
representing $p_{(a,r),(b,s)}$ in the form
$p_{(a,r),(b,s)}=m_{ab}m_{rs}^{(a,b)}$. Not surprisingly, severe restrictions
on the conditional transition matrices $M^{(a,b)}$ are required to ensure the
Markov property of the rating process $(R_{t})_{t\in\mathbb{N}_{0}}$. It turns
out that the following definition formalizes the key property of conditional
transition matrices $M^{(a,b)}$ of Markovian rating processes
$(R_{t})_{t\in\mathbb{N}_{0}}$.

######  Definition 3.1.

The conditional transition matrices $M^{(a,b)}$ have identical ratios of non-
default migration probabilities if for all
$r,r^{\prime},s,s^{\prime}\in\widehat{\mathcal{R}}$ and
$a,a^{\prime},b,b^{\prime}\in\mathcal{A}$

$\frac{m_{rs}^{(a,b)}}{m_{r^{\prime}s^{\prime}}^{(a,b)}}=\frac{m_{rs}^{(a^{\prime},b^{\prime})}}{m_{r^{\prime}s^{\prime}}^{(a^{\prime},b^{\prime})}}.$
(3)

It is easily verified that the two matrices provided in the example in Section
2.2 do not fulfil this: $m_{1,1}^{(2,1)}/m_{1,2}^{(2,1)}>1$, while
$m_{1,1}^{(2,3)}/m_{1,2}^{(2,3)}<1$.

As an example of a simple class of rating migration processes that have
identical ratios of non-default migration probabilities, consider processes
based on just two ratings $r$ and $\overline{r}$. Since $r$ is the only
element of $\widehat{\mathcal{R}}$ condition (3) is always satisfied,
regardless of the specification of the conditional transition matrices.

The above condition significantly reduces the free parameters that are
required for specifying conditional transition matrices. Choose
$a,b\in{\mathcal{A}}$ and $r,s\in\widehat{\mathcal{R}}$ and specify
$M^{(a,b)}$ and $m_{rs}^{(a^{\prime},b^{\prime})}$ for all
$(a^{\prime},b^{\prime})\not=(a,b)$. All other parameters
$m_{r^{\prime}s^{\prime}}^{(a^{\prime},b^{\prime})}$ are defined by (3), i.e.,
for $(r^{\prime},s^{\prime})\not=(r,s)$

$m_{r^{\prime}s^{\prime}}^{(a^{\prime},b^{\prime})}=\frac{m_{r^{\prime}s^{\prime}}^{(a,b)}}{m_{rs}^{(a,b)}}m_{rs}^{(a^{\prime},b^{\prime})}.$

Since $M^{(a,b)}$ is a stochastic matrix with one absorbing state, only
$(J-1)^{2}+K^{2}-1$ free parameters are required for fully specifying
conditional transition matrices with identical ratios of non-default migration
probabilities. In the case $K=1$, just one rating transition matrix is
defined, which is equivalent to the standard specification of a time-
homogeneous Markov chain. For $K>1$, the number of conditional rating
migration matrices $M^{(a,b)}$ as well as the number of free parameters
increases by $K^{2}$.

The set of Equations (3) is a sufficient condition for the Markov property,
see Corollary 3.3. In fact, Theorem 3.2 shows that the Equations (3) are
equivalent to the economic state variable $A_{t}$ being independent of the
rating history up to time $t$ (conditional on no default), which is a slightly
stronger condition than the Markov property of $(R_{t})_{t\in\mathbb{N}_{0}}$.

More formally, let $t\in\mathbb{N}_{0}$. Conditional on no default, the
economic state variable $A_{t}$ is independent of the rating history up to
time $t$ if for all $a\in\mathcal{A}$ and
$r_{0},s_{0},\ldots,r_{t},s_{t}\in\widehat{\mathcal{R}}$ with
${\mathbb{P}}(R_{0}=r_{0},\ldots,R_{t}=r_{t})>0$ and
${\mathbb{P}}(R_{0}=s_{0},\ldots,R_{t}=s_{t})>0$,

${\mathbb{P}}(A_{t}=a|R_{0}=r_{0},\ldots,R_{t}=r_{t})={\mathbb{P}}(A_{t}=a|R_{0}=s_{0},\ldots,R_{t}=s_{t})={\mathbb{P}}(A_{t}=a|R_{t}\not=\overline{r}).$
(4)

######  Theorem 3.2.

The following conditions are equivalent.

  1. (i)

The conditional transition matrices $M^{(a,b)}$ have identical ratios of non-
default migration probabilities.

  2. (ii)

For every joint distribution $\lambda\in[0,1]^{KJ}$ of independent variables
$A_{0}$ and $R_{0}$ and every stochastic matrix $M\in[0,1]^{K\times K}$ the
rating process $R_{t}$ specified by the distribution $\lambda$ and transition
matrices $M$ and $M^{(a,b)}$ satisfies (4) for every $t\in\mathbb{N}_{0}$.

###### Proof.

In this proof we abbreviate a rating path
$\\{R_{0}=r_{0},\ldots,R_{t}=r_{t}\\}$ and a path
$\\{A_{0}=a_{0},\ldots,A_{t}=a_{t}\\}$ of economic states by $r_{0..t}$ and
$a_{0..t}$ respectively.

“$\Rightarrow$”: For $t\in\mathbb{N}_{0}$ let
$a_{0},b_{0},\ldots,a_{t},b_{t}\in\mathcal{A}$,
$r_{0},s_{0},\ldots,r_{t},s_{t}\in\widehat{\mathcal{R}}$ with
${\mathbb{P}}(a_{0..t})>0$ and ${\mathbb{P}}(b_{0..t})>0$. We assume that
$A_{0}$ and $R_{0}$ are independent variables and
${\mathbb{P}}(R_{0}=r_{0})>0$ and ${\mathbb{P}}(R_{0}=s_{0})>0$. We first
prove by induction on $t$ that

$\frac{{\mathbb{P}}(r_{0..t}|a_{0..t})}{{\mathbb{P}}(s_{0..t}|a_{0..t})}=\frac{{\mathbb{P}}(r_{0..t}|b_{0..t})}{{\mathbb{P}}(s_{0..t}|b_{0..t})}.$
(5)

Since $A_{0}$ and $R_{0}$ are independent, Equation (5) holds for $t=0$. For
$t>0$, we assume that (5) holds for $t-1$. Note that

${\mathbb{P}}(r_{0..t}|a_{0..t})=\frac{{\mathbb{P}}(r_{0..t-1},a_{0..t-1})m_{a_{t-1}a_{t}}m_{r_{t-1}r_{t}}^{(a_{t-1},a_{t})}}{{\mathbb{P}}(a_{0..t-1})m_{a_{t-1}a_{t}}}={\mathbb{P}}(r_{0..t-1}|a_{0..t-1})m_{r_{t-1}r_{t}}^{(a_{t-1},a_{t})}.$
(6)

Hence, by induction and (3),

$\frac{{\mathbb{P}}(r_{0..t}|a_{0..t})}{{\mathbb{P}}(s_{0..t}|a_{0..t})}=\frac{{\mathbb{P}}(r_{0..t-1}|a_{0..t-1})m_{r_{t-1}r_{t}}^{(a_{t-1},a_{t})}}{{\mathbb{P}}(s_{0..t-1}|a_{0..t-1})m_{s_{t-1}s_{t}}^{(a_{t-1},a_{t})}}=\\\
\frac{{\mathbb{P}}(r_{0..t-1}|b_{0..t-1})m_{r_{t-1}r_{t}}^{(b_{t-1},b_{t})}}{{\mathbb{P}}(s_{0..t-1}|b_{0..t-1})m_{s_{t-1}s_{t}}^{(b_{t-1},b_{t})}}=\frac{{\mathbb{P}}(r_{0..t}|b_{0..t})}{{\mathbb{P}}(s_{0..t}|b_{0..t})}.$

For proving (4) we apply Bayes formula. For $t=0$, Equation (4) is implied by
the independence of $A_{0}$ and $R_{0}$. Otherwise it follows from (6) that
Bayes formula can be expressed as

${\mathbb{P}}(a_{0..t}|r_{0..t})=\frac{{\mathbb{P}}(a_{0..t}){\mathbb{P}}(r_{0..t-1}|a_{0..t-1})m_{r_{t-1}r_{t}}^{(a_{t-1},a_{t})}}{\sum_{(b_{0},\ldots,b_{t})\in{\mathcal{A}}^{t+1}}{\mathbb{P}}(b_{0..t}){\mathbb{P}}(r_{0..t-1}|b_{0..t-1})m_{r_{t-1}r_{t}}^{(b_{t-1},b_{t})}}.$
(7)

Hence, ${\mathbb{P}}(a_{0..t}|r_{0..t})={\mathbb{P}}(a_{0..t}|s_{0..t})$ is
equivalent to

$\sum_{(b_{0},\ldots,b_{t})\in{\mathcal{A}}^{t+1}}{\mathbb{P}}(b_{0..t}){\mathbb{P}}(r_{0..t-1}|a_{0..t-1}){\mathbb{P}}(s_{0..t-1}|b_{0..t-1})m_{r_{t-1}r_{t}}^{(a_{t-1},a_{t})}m_{s_{t-1}s_{t}}^{(b_{t-1},b_{t})}=\\\
\sum_{(b_{0},\ldots,b_{t})\in{\mathcal{A}}^{t+1}}{\mathbb{P}}(b_{0..t}){\mathbb{P}}(r_{0..t-1}|b_{0..t-1}){\mathbb{P}}(s_{0..t-1}|a_{0..t-1})m_{s_{t-1}s_{t}}^{(a_{t-1},a_{t})}m_{r_{t-1}r_{t}}^{(b_{t-1},b_{t})}.$
(8)

By (5), Equation (8) holds if and only if

$\sum_{(b_{0},\ldots,b_{t})\in{\mathcal{A}}^{t+1}}{\mathbb{P}}(b_{0..t}){\mathbb{P}}(r_{0..t-1}|a_{0..t-1}){\mathbb{P}}(s_{0..t-1}|b_{0..t-1})(m_{r_{t-1}r_{t}}^{(a_{t-1},a_{t})}m_{s_{t-1}s_{t}}^{(b_{t-1},b_{t})}-m_{s_{t-1}s_{t}}^{(a_{t-1},a_{t})}m_{r_{t-1}r_{t}}^{(b_{t-1},b_{t})})=0.$

Since the conditional transition matrices $M^{(a,b)}$ have identical ratios of
non-default migration probabilities it follows that
$m_{r_{t-1}r_{t}}^{(a_{t-1},a_{t})}m_{s_{t-1}s_{t}}^{(b_{t-1},b_{t})}-m_{s_{t-1}s_{t}}^{(a_{t-1},a_{t})}m_{r_{t-1}r_{t}}^{(b_{t-1},b_{t})}=0$.
Hence, we obtain
${\mathbb{P}}(a_{0..t}|r_{0..t})={\mathbb{P}}(a_{0..t}|s_{0..t})$, which
implies (4).

“$\Leftarrow$”: Let $a,a^{\prime},b,b^{\prime}\in\mathcal{A}$ and
$r,r^{\prime},s,s^{\prime}\in\widehat{\mathcal{R}}$ such that
$m_{rs}^{(a,b)}m_{r^{\prime}s^{\prime}}^{(a^{\prime},b^{\prime})}\not=m_{r^{\prime}s^{\prime}}^{(a,b)}m_{rs}^{(a^{\prime},b^{\prime})}$.
If $b=b^{\prime}$ we choose $c\in\mathcal{A}$ with $c\not=b$. Obviously, at
least one of the two inequalities

$\frac{m_{rs}^{(a,b)}}{m_{r^{\prime}s^{\prime}}^{(a,b)}}\not=\frac{m_{rs}^{(a,c)}}{m_{r^{\prime}s^{\prime}}^{(a,c)}},\
\ \ \ \ \
\frac{m_{rs}^{(a^{\prime},b^{\prime})}}{m_{r^{\prime}s^{\prime}}^{(a^{\prime},b^{\prime})}}\not=\frac{m_{rs}^{(a,c)}}{m_{r^{\prime}s^{\prime}}^{(a,c)}}$

holds. Hence, we can assume that $b\not=b^{\prime}$. We now define $A_{0}$ and
$R_{0}$ as independent variables specified by

${\mathbb{P}}(A_{0}=a)={\mathbb{P}}(A_{0}=a^{\prime})=\frac{1}{|\\{a,a^{\prime}\\}|},\
\ \
{\mathbb{P}}(R_{0}=r)={\mathbb{P}}(R_{0}=r^{\prime})=\frac{1}{|\\{r,r^{\prime}\\}|}.$

Furthermore, it is assumed that the transition matrix $M$ satisfies
$m_{ab}=m_{a^{\prime}b^{\prime}}=1/2$ if $a=a^{\prime}$ and
$m_{ab}=m_{a^{\prime}b^{\prime}}=1$ if $a\not=a^{\prime}$.

In both cases,
${\mathbb{P}}(A_{0}=a,A_{1}=b)={\mathbb{P}}(A_{0}=a^{\prime},A_{1}=b^{\prime})=1/2$
and Bayes formula (7) specializes to

${\mathbb{P}}(A_{0}=a,A_{1}=b|R_{0}=r,R_{1}=s)=\frac{m_{rs}^{(a,b)}}{m_{rs}^{(a,b)}+m_{rs}^{(a^{\prime},b^{\prime})}}.$

Since
$m_{rs}^{(a,b)}(m_{r^{\prime}s^{\prime}}^{(a,b)}+m_{r^{\prime}s^{\prime}}^{(a^{\prime},b^{\prime})})\not=m_{r^{\prime}s^{\prime}}^{(a,b)}(m_{rs}^{(a,b)}+m_{rs}^{(a^{\prime},b^{\prime})})$
we obtain

$\displaystyle{\mathbb{P}}(A_{0}=a,A_{1}=b|R_{0}=r,R_{1}=s)$ $\displaystyle=$
$\displaystyle\frac{m_{rs}^{(a,b)}}{m_{rs}^{(a,b)}+m_{rs}^{(a^{\prime},b^{\prime})}}$
$\displaystyle\not=$
$\displaystyle\frac{m_{r^{\prime}s^{\prime}}^{(a,b)}}{m_{r^{\prime}s^{\prime}}^{(a,b)}+m_{r^{\prime}s^{\prime}}^{(a^{\prime},b^{\prime})}}$
$\displaystyle=$
$\displaystyle{\mathbb{P}}(A_{0}=a,A_{1}=b|R_{0}=r^{\prime},R_{1}=s^{\prime}).$

It follows from $b\not=b^{\prime}$ that
${\mathbb{P}}(A_{0}=a,A_{1}=b)={\mathbb{P}}(A_{1}=b)$ and therefore

${\mathbb{P}}(A_{1}=b|R_{0}=r,R_{1}=s)\not={\mathbb{P}}(A_{1}=b|R_{0}=r^{\prime},R_{1}=s^{\prime}).$

Hence, (4) does not hold. ∎

######  Corollary 3.3.

The rating process $R_{t}$ has the Markov property if

  1. (i)

the variables $A_{0}$ and $R_{0}$, which are specified by their joint
distribution $\lambda\in[0,1]^{KJ}$, are independent and

  2. (ii)

the conditional transition matrices $M^{(a,b)}$ have identical ratios of non-
default migration probabilities.

###### Proof.

By Theorem 3.2, Equation (4) holds for $t\in\mathbb{N}_{0}$, which implies

${\mathbb{P}}(A_{t}=a|R_{0}=r_{0},\ldots,R_{t}=r_{t})={\mathbb{P}}(A_{t}=a|R_{t}=r_{t})$

for $a\in\mathcal{A}$ and $r_{0},\ldots,r_{t}\in\widehat{\mathcal{R}}$ with
${\mathbb{P}}(R_{0}=r_{0},\ldots,R_{t}=r_{t})>0$. Hence, in case of no default
at time $t$, i.e., $R_{t}\not=\overline{r}$, we obtain

${\mathbb{P}}(R_{t+1}=r_{t+1}|R_{0}=r_{0},\ldots,R_{t}=r_{t})\\\
\begin{aligned}
&=\sum_{a,b\in\mathcal{A}}{\mathbb{P}}(R_{t+1}=r_{t+1},A_{t}=a,A_{t+1}=b|R_{0}=r_{0},\ldots,R_{t}=r_{t})\\\
&=\sum_{a,b\in\mathcal{A}}{\mathbb{P}}(R_{t+1}=r_{t+1},A_{t+1}=b|A_{t}=a,R_{0}=r_{0},\ldots,R_{t}=r_{t})\,{\mathbb{P}}(A_{t}=a|R_{0}=r_{0},\ldots,R_{t}=r_{t})\\\
&=\sum_{a,b\in\mathcal{A}}{\mathbb{P}}(R_{t+1}=r_{t+1},A_{t+1}=b|R_{t}=r_{t},A_{t}=a){\mathbb{P}}(A_{t}=a|R_{t}=r_{t})\\\
&={\mathbb{P}}(R_{t+1}=r_{t+1}|R_{t}=r_{t}).\end{aligned}$

If $R_{t}=\overline{r}$ then the Markov property holds because $\overline{r}$
is absorbing. ∎

###  3.2 Time-inhomogeneous Markov chains on ratings

As shown above, rating processes typically do not have the Markov property.
However, it is easy to prove that for each rating process
$(R_{t})_{t\in\mathbb{N}_{0}}$, a time-inhomogeneous Markov process can be
constructed that replicates the unconditional distribution of
$(R_{t})_{t\in\mathbb{N}_{0}}$ for each $t\in\mathbb{N}_{0}$.555A Markov chain
is time-inhomogeneous if the transition matrices are time-dependent, see e.g.
Chapter 1 of Brémaud (1999). Using mathematical results from Markov chain
theory, this process will then be used to determine the long-term and
asymptotic rating transition behaviour of $(R_{t})_{t\in\mathbb{N}_{0}}$, see
Section 4.

More precisely, we define a time-inhomogeneous Markov chain on the state space
$\mathcal{R}$ that replicates the rating distribution

$({\mathbb{P}}(R_{t}=r))_{r\in{\mathcal{R}}},\quad t\in\mathbb{N}_{0}.$

######  Proposition 3.4.

There exists a time-inhomogeneous Markov process
$({\widetilde{R}}_{t})_{t\in\mathbb{N}_{0}}$ with

  1. (i)

state space $\mathcal{R}$,

  2. (ii)

initial distribution ${\tilde{\lambda}}$ on $[0,1]^{J}$ defined by
${\tilde{\lambda}}=({\mathbb{P}}(R_{0}=r))_{r\in{\mathcal{R}}}$,

  3. (iii)

transition matrices $({\widetilde{P}}_{t})_{t\in\mathbb{N}_{0}}$ that can be
determined from $(\lambda,P)$, such that

${\mathbb{P}}({\widetilde{R}}_{t}=r)={\mathbb{P}}(R_{t}=r),\quad
t\in\mathbb{N}_{0},\quad r\in{\mathcal{R}}.$ (9)

In particular, long-term default probabilities and distributions of non-
default ratings can be immediately derived via

$({\mathbb{P}}(R_{T}=r))_{r\in{\mathcal{R}}}={\tilde{\lambda}}\cdot\prod_{t=1}^{T}{\widetilde{P}}_{t},\quad
T\in\mathbb{N}_{0}.$ (10)

Essentially, the distributions of the process ${\widetilde{R}}$ are averaged
with respect to the economic state transitions. Depending on the initial
distribution $\lambda$ of
$(X_{t})_{t\in\mathbb{N}_{0}}=(A_{t},R_{t})_{t\in\mathbb{N}_{0}}$, information
about the initial economic state may enter the rating distributions, which
creates the time-inhomogeneity in the transitions.

###### Proof.

First, we construct transition matrices
$({\widetilde{P}}_{t})_{t\in\mathbb{N}_{0}}$. For $t\in\mathbb{N}_{0}$, let
$\lambda_{t}\in[0,1]^{KJ}$ be the distribution of $X_{t}$, i.e.,
$\lambda_{t,(a,r)}={\mathbb{P}}(X_{t}=(a,r))$ for $(a,r)\in\mathcal{X}$, and
define the matrix ${\widetilde{P}}_{t}=({\tilde{p}}_{t,rs})\in[0,1]^{J\times
J}$ for $t\geq 1$: if ${\mathbb{P}}(R_{t-1}=r)>0$, then

$\displaystyle{\tilde{p}}_{t,rs}$
$\displaystyle={\mathbb{P}}(R_{t}=s|R_{t-1}=r)=\sum_{a\in\mathcal{A}}{\mathbb{P}}(R_{t}=s|R_{t-1}=r,A_{t-1}=a)\cdot{\mathbb{P}}(A_{t-1}=a|R_{t-1}=r)$
$\displaystyle=\sum_{a\in\mathcal{A}}{\mathbb{P}}(R_{t}=s|R_{t-1}=r,A_{t-1}=a)\cdot\frac{{\mathbb{P}}(A_{t-1}=a,R_{t-1}=r)}{{\mathbb{P}}(R_{t-1}=r)}$
$\displaystyle=\frac{\sum_{a,b\in\mathcal{A}}p_{(a,r)(b,s)}\cdot\lambda_{t-1,(a,r)}}{\sum_{a^{\prime}\in\mathcal{A}}\lambda_{t-1,(a^{\prime},r)}}.$

Otherwise, ${\tilde{p}}_{t,rs}=0$ for $s\in\widehat{\mathcal{R}}$ and
${\tilde{p}}_{t,r\overline{r}}=1$. By definition, the matrices
${\widetilde{P}}_{t}$, $t\geq 1$, are stochastic and can be calculated
iteratively from $(\lambda,P)$. Hence, the process
$({\widetilde{R}}_{t})_{t\geq\mathbb{N}_{0}}$ can be defined as a time-
inhomogeneous Markov chain. It is easily verified that
${\mathbb{P}}({\widetilde{R}}_{t}=r)={\mathbb{P}}(R_{t}=r)$, $t>0$: From

${\mathbb{P}}(R_{t}=s)=\sum_{r\in{\mathcal{R}}}{\mathbb{P}}(R_{t}=s|R_{t-1}=r)\cdot{\mathbb{P}}(R_{t-1}=r),$

the claim follows by definition of ${\tilde{\lambda}}$ and by induction. ∎

Bluhm and Overbeck (2007) propose a parametric class of time-inhomogeneous
Markov chains to calibrate observed multi-year default rates. Proposition 3.4
provides theoretical support to the claim that a convincing fit can be
achieved even if the underlying process is not Markovian.

##  4 Asymptotic properties

In this section, we use standard techniques from the theory of Markov chains
to prove that the sequence of transition matrices
$({\widetilde{P}}_{t})_{t\in\mathbb{N}_{0}}$ converges to a limit, denoted by
${\widetilde{P}}_{\infty}$. The proof is based on the Perron-Frobenius Theorem
(Perron (1907); Frobenius (1912)). In order to apply the theorem we assume in
this section that the economic state process $(A_{t})_{t\in\mathbb{N}_{0}}$ is
irreducible and aperiodic. Hence, the transition matrix $M$ is primitive,
i.e., there exists an integer $k$ such that $M^{k}>0$. Furthermore, recall
that $m^{(a,b)}_{rs}>0$ for all $r\in\widehat{\mathcal{R}}$,
$s\in{\mathcal{R}}$ and $a,b\in\mathcal{A}$. This assumption implies that
$\widehat{P}$ is a primitive matrix, where
$\widehat{\mathcal{X}}=\mathcal{A}\times\widehat{\mathcal{R}}$ and
$\widehat{P}$ is the restriction of $P$ to $\widehat{\mathcal{X}}$, i.e.,
$\widehat{P}=(\hat{p}_{(a,r),(b,s)})\in[0,1]^{K(J-1)\times K(J-1)}$ with
$\hat{p}_{(a,r),(b,s)}=p_{(a,r),(b,s)}$ for
$(a,r),(b,s)\in\widehat{\mathcal{X}}$. Likewise, we denote the restriction of
$\lambda$ to $\widehat{\mathcal{X}}$ by $\hat{\lambda}\in[0,1]^{K(J-1)}$. Note
that the assumption ${\mathbb{P}}(X_{0}\in\widehat{\mathcal{X}})=1$ implies
that $\hat{\lambda}$ is a distribution on $\widehat{\mathcal{X}}$.

The matrix $\widehat{P}$ is not stochastic but it is primitive. Hence, the
following properties are a consequence of the Perron-Frobenius Theorem (see
e.g. Theorems 1.1 and 1.2 of Seneta (1973), Chapter 6 of Brémaud (1999)):

  1. (P1)

The primitive matrix $\widehat{P}$ has a real eigenvalue $\rho$ with algebraic
as well as geometric multiplicity one such that $\rho>0$ and
$\rho>|\rho^{\prime}|$ for any other eigenvalue $\rho^{\prime}$. Moreover, the
left eigenvector $\mu$ and the right eigenvector $\nu$ associated with $\rho$
can be chosen positive and such that $\mu\nu=1$.666Note that the left
eigenvector $\mu$ is a row-vector and the right eigenvector $\nu$ is a column-
vector, both of dimension $K(J-1)$. Hence, $\mu\nu$ is a real number whereas
$\nu\mu$ is a $K(J-1)\times K(J-1)$ matrix.

  2. (P2)

The matrix $((1/\rho)\widehat{P})^{n}$ converges to the matrix $\nu\mu$.

We assume that the components of $\mu$ add up to 1, i.e., $\mu$ is a
distribution on $\widehat{\mathcal{X}}$. Let ${\tilde{\mu}}$ be the
corresponding distribution on $\mathcal{R}$ specified by

${\tilde{\mu}}_{r}=\sum_{a\in\mathcal{A}}\mu_{(a,r)}\text{ for
}r\in\widehat{\mathcal{R}}\quad\text{ and
}\quad{\tilde{\mu}}_{\overline{r}}=0,$ (11)

and define the stochastic matrix
${\widetilde{P}}_{\infty}=({\tilde{p}}_{\infty,rs})\in[0,1]^{J\times J}$ by

${\tilde{p}}_{\infty,rs}=\sum_{a\in\mathcal{A}}\mu_{(a,r)}\sum_{b\in\mathcal{A}}p_{(a,r),(b,s)}/{\tilde{\mu}}_{r},\quad
r\in\widehat{\mathcal{R}},s\in{\mathcal{R}},$ (12)

${\tilde{p}}_{\infty,\overline{r}r}=0$ for $r\in\widehat{\mathcal{R}}$ and
${\tilde{p}}_{\infty,\overline{r}\,\overline{r}}=1$.

We will now study the asymptotic properties of the Markov chain
$(X_{t})_{t\in\mathbb{N}_{0}}$. Since ${\mathcal{A}}\times\\{\overline{r}\\}$
is a closed subset of the state space of $(X_{t})_{t\in\mathbb{N}_{0}}$ it
immediately follows from the assumptions on the transition matrices $M$ and
$M^{(a,b)}$ that

$\lim_{t\rightarrow\infty}{\mathbb{P}}(X_{t}\in{\mathcal{A}}\times\\{\overline{r}\\})=1.$

The more relevant question is the asymptotic behaviour of
$(X_{t})_{t\in\mathbb{N}_{0}}$ conditionally on no default, which is
summarized in the following theorem. In particular, it is shown that the
conditional distribution of non-default rating states of $X$ converges to
$\mu$, which implies convergence of $({\widetilde{P}}_{t})_{t\in{\mathbb{N}}}$
to ${\widetilde{P}}_{\infty}$. Furthermore, $1-\rho$ is the limit of the
marginal default rates.777Marginal default rates refer to probabilities of
default conditional on no prior default, as opposed to cumulative default
rates.

In the following, note that
$\\{R_{t}=\overline{r}\\}=\\{X_{t}\in(\cdot,\overline{r})\\}$ and
$\\{R_{t}\not=\overline{r}\\}=\\{X_{t}\not\in(\cdot,\overline{r})\\}$. For
ease of exposition, we use both notations.

######  Theorem 4.1.

  1. (i)

For all $(a,r)\in\widehat{\mathcal{X}}$,

$\lim_{t\rightarrow\infty}{\mathbb{P}}(X_{t}=(a,r)|X_{t}\not\in(\cdot,\overline{r}))=\mu_{(a,r)}.$

Furthermore,

$\lim_{t\rightarrow\infty}{\widetilde{P}}_{t}={\widetilde{P}}_{\infty}\quad\text{
and
}\quad\lim_{t\rightarrow\infty}{\mathbb{P}}(R_{t}=\overline{r}|R_{t-1}\not=\overline{r})=1-\rho.$
(13)

  2. (ii)

If the limit distribution $\mu=(\mu_{(a,r)})\in[0,1]^{K(J-1)}$ is the
distribution of $X_{0}$ on the set of non-default states
$\widehat{\mathcal{X}}$, i.e., $\hat{\lambda}=\mu$, then for all
$t\in{\mathbb{N}}$

${\mathbb{P}}(X_{t}=(a,r)|X_{t}\notin(\cdot,\overline{r}))=\mu_{(a,r)}\quad\text{
and }\quad{\widetilde{P}}_{t}={\widetilde{P}}_{\infty}.$

###### Proof.

(i) Since $\nu$ is a column-vector with positive entries and
$\hat{\lambda}\not=0$ has only non-negative entries, the vector product
$c=\hat{\lambda}\cdot\nu$ is a positive number. By (P2),

$\lim_{t\rightarrow\infty}\hat{\lambda}\cdot((1/\rho)\widehat{P})^{t}=\hat{\lambda}\cdot\nu\cdot\mu=c\cdot\mu.$
(14)

Let $t\in{\mathbb{N}}$ and define $\alpha_{t}\in[0,1]^{K(J-1)}$ as the
distribution of $X_{t}$ conditional on $R_{t}\not=\overline{r}$, i.e.,

$\alpha_{t}={\mathbb{P}}(X_{t}=(a,r)|X_{t}\not\in(\cdot,\overline{r}))_{(a,r)\in\widehat{\mathcal{X}}}.$

Since the set of default states ${\mathcal{A}}\times\\{\overline{r}\\}$ is
closed,

$\alpha_{t}=\frac{{\mathbb{P}}(X_{t-1}\notin(\cdot,\overline{r}))}{{\mathbb{P}}(X_{t}\notin(\cdot,\overline{r}))}\cdot\alpha_{t-1}\cdot\widehat{P}=\frac{1}{{\mathbb{P}}(X_{t}\notin(\cdot,\overline{r}))}\cdot\hat{\lambda}\cdot\widehat{P}^{t}$
(15)

and therefore

$\alpha_{t}=\frac{\rho^{t}}{{\mathbb{P}}(X_{t}\notin(\cdot,\overline{r}))}\cdot\hat{\lambda}\cdot((1/\rho)\widehat{P})^{t}.$

Using (14) and the fact that the $L_{1}$-norm of $\mu$ and $\alpha_{t}$ is 1
for each $t\in{\mathbb{N}}$ we obtain

$\lim_{t\rightarrow\infty}\alpha_{t}=\mu.$ (16)

Since $\widehat{P}$ is primitive, ${\mathbb{P}}(X_{t}=(a,r))>0$ for all
$(a,r)\in\widehat{\mathcal{X}}$ and sufficiently large $t\in{\mathbb{N}}$. By
the definition of the matrix
${\widetilde{P}}_{t}=({\tilde{p}}_{t,rs})\in[0,1]^{J\times J}$, for all
$r\in\widehat{\mathcal{R}}$ and $s\in{\mathcal{R}}$

$\displaystyle{\tilde{p}}_{t+1,rs}$ $\displaystyle=\left(\sum_{a\in
A}{\mathbb{P}}(X_{t}=(a,r))\sum_{b\in
A}p_{(a,r),(b,s)}\right)/{\mathbb{P}}(X_{t}\in(\cdot,r))$
$\displaystyle=\left(\sum_{a\in
A}({\mathbb{P}}(X_{t}=(a,r))/{\mathbb{P}}(X_{t}\notin(\cdot,\overline{r})))\sum_{b\in
A}p_{(a,r),(b,s)}\right)/({\mathbb{P}}(X_{t}\in(\cdot,r))/{\mathbb{P}}(X_{t}\notin(\cdot,\overline{r})))$
$\displaystyle=\left(\sum_{a\in A}\alpha_{t,(a,r)}\sum_{b\in
A}p_{(a,r),(b,s)}\right)/\sum_{a\in A}\alpha_{t,(a,r)}$ (17)

and ${\tilde{p}}_{t+1,\overline{r}r}=0$ for $r\in\widehat{\mathcal{R}}$ and
${\tilde{p}}_{t+1,\overline{r}\,\overline{r}}=1$. Hence, by (16),

$\lim_{t\rightarrow\infty}{\widetilde{P}}_{t}={\widetilde{P}}_{\infty}.$

From (16) and the first equation in (15) it follows that

$\mu=\left(\lim_{t\rightarrow\infty}\frac{{\mathbb{P}}(X_{t-1}\notin(\cdot,\overline{r}))}{{\mathbb{P}}(X_{t}\notin(\cdot,\overline{r}))}\right)\cdot\mu\cdot\widehat{P}.$

Since $\mu$ is an eigenvector with eigenvalue $\rho$,

$\lim_{t\rightarrow\infty}\frac{{\mathbb{P}}(X_{t-1}\notin(\cdot,\overline{r}))}{{\mathbb{P}}(X_{t}\notin(\cdot,\overline{r}))}=\frac{1}{\rho}\quad\text{
and
}\quad\lim_{t\rightarrow\infty}{\mathbb{P}}(X_{t}\in(\cdot,\overline{r})|X_{t-1}\notin(\cdot,\overline{r}))=1-\rho.$

(ii) We now assume that $\hat{\lambda}=\mu$. It follows from (15) and
$\mu\cdot\widehat{P}=\rho\cdot\mu$ that

$\alpha_{t}=\frac{1}{{\mathbb{P}}(X_{t}\notin(\cdot,\overline{r}))}\cdot\mu\cdot\widehat{P}^{t}=\frac{1}{{\mathbb{P}}(X_{t}\notin(\cdot,\overline{r}))}\cdot\rho^{t}\cdot\mu=\mu$

for every $t\in{\mathbb{N}}$ and therefore, by (17),
${\widetilde{P}}_{t}={\widetilde{P}}_{\infty}$. ∎

######  Definition 4.2.

Let $({S}_{t})_{t\in\mathbb{N}_{0}}$ be the time-homogeneous Markov chain with
state space $\mathcal{R}$, transition matrix
${\widetilde{P}}_{\infty}=({\tilde{p}}_{rs})\in[0,1]^{J\times J}$ and initial
distribution ${\tilde{\lambda}}$. Then $({S}_{t})_{t\in\mathbb{N}_{0}}$ is
called the asymptotic approximation of $(R_{t})_{t\in\mathbb{N}_{0}}$.

The following corollary summarises the properties of the asymptotic
approximation.

######  Corollary 4.3.

  1. (i)

For any $r\in\widehat{\mathcal{R}}$,

$\lim_{t\rightarrow\infty}{\mathbb{P}}({S}_{t}=r|{S}_{t}\not=\overline{r})=\lim_{t\rightarrow\infty}{\mathbb{P}}(R_{t}=r|R_{t}\not=\overline{r}).$
(18)

  2. (ii)

If the limit distribution $\mu=(\mu_{(a,r)})\in[0,1]^{K(J-1)}$ is the
distribution of $X_{0}$ on $\widehat{\mathcal{X}}$ then

${\mathbb{P}}({S}_{t}=r)={\mathbb{P}}(R_{t}=r)={\mathbb{P}}(X_{t}\in(\cdot,r))$
(19)

for $t\in\mathbb{N}_{0}$ and $r\in{\mathcal{R}}$. Furthermore, for
$t\in{\mathbb{N}}$,

${\mathbb{P}}({S}_{t}=\overline{r}|{\mathbb{P}}({S}_{t-1}\not=\overline{r})={\mathbb{P}}(R_{t}=\overline{r}|{\mathbb{P}}({R}_{t-1}\not=\overline{r})=(1-\rho)\rho^{t-1}.$
(20)

###### Proof.

We first assume that the limit distribution
$\mu=(\mu_{(a,r)})\in[0,1]^{K(J-1)}$ is the distribution of $X_{0}$ on
$\widehat{\mathcal{X}}$. It follows from Theorem 4.1 that
${\widetilde{R}}_{t\in\mathbb{N}_{0}}$ is a time-homogeneous Markov process,
which equals the asymptotic approximation $({S}_{t})_{t\in\mathbb{N}_{0}}$,
and

${\mathbb{P}}(R_{t}\not=\overline{r}|R_{t-1}\not=\overline{r})=\rho\ \ \ \
\hbox{ and }\ \ \ \
{\mathbb{P}}(R_{t}=\overline{r}|R_{t-1}\not=\overline{r})=1-\rho$

for all $t\in{\mathbb{N}}$. Hence, Equations (19) and (20) follow from
Proposition 3.4. Furthermore, we obtain from Theorem 4.1 that
$({\tilde{\mu}}_{r})_{r\in\widehat{\mathcal{R}}}$ is a positive left
eigenvector of the restriction of ${\widetilde{P}}_{\infty}$ to
$\widehat{\mathcal{R}}$. Analogously to the proof of Theorem 4.1, it follows
from the Perron-Frobenius Theorem that

$\lim_{t\rightarrow\infty}{\mathbb{P}}({S}_{t}=r|{S}_{t}\not=\overline{r})={\tilde{\mu}}_{r},\quad
r\in\widehat{\mathcal{R}},$ (21)

holds for any initial distribution of ${S}_{0}$. Hence, (18) follows from
Theorem 4.1. ∎

##  5 Rating philosophies: Point-in-time and through-the-cycle

###  5.1 Definition of TTC and PIT ratings

This section deals with the characterisation of rating systems. Ratings are
designed to provide information on the credit quality of an obligor. However,
rating systems may differ with respect to the type of information reflected in
the rating. There exist two main rating philosophies:

  1. (i)

Point-in-time (PIT) ratings reflect the likelihood of default over a future
period, e.g. one year, and are therefore dependent on firm-specific as well as
systematic factors. The PIT rating of an obligor reflects the current state of
the economy as well as all available information about the current and future
economic development. As such, PIT ratings fluctuate over economic cycles
(e.g. economic downturns lead to an increased number of downgrades), whereas
expected one-period default probabilities of PIT rating classes are stable.

  2. (ii)

In contrast, through-the-cycle (TTC) ratings have a longer rating horizon
covering multiple periods, i.e., the rating is performed through the cycle. As
a consequence, economic effects cancel out and TTC ratings are therefore
independent of the state of the economy and, as such, are insensitive to
economic or credit cycles. However, since actual corporate default rates
depend on the economic environment the expected one-period default
probabilities of TTC rating classes inevitably fluctuate with the economic
cycle.888The characteristics of expected one-period default probabilities of
TTC rating classes is different to TTC PDs defined in the literature as long-
run averages of one-year default probabilities (e.g. Vallés (2006); Gordy and
Howells (2004); Altman and Rijken (2004)) or stressed default probability
(e.g. Heitfield (2005); Crouhy et al. (2001)). A comparison of these concepts
in our mathematical framework is provided in Sections 7.3 and 8 of this paper.

Figure 2: Example paths of economic states (top), PIT (middle) and TTC
(bottom) rating distributions over 30 time periods for a firm starting in the
best PIT rating. Economic states are ordered from best (1) to worst (3).

Figure 2 shows an example of an economic path (top) together with rating
distributions of PIT and TTC ratings for a firm starting in the best PIT
rating. This shows how PIT ratings fluctuate with the economy while TTC non-
default ratings are insensitive to the economic environment. The rating models
used in this illustration will be discussed in Section 7.

Both rating philosophies are interesting theoretical concepts. For practical
applications, however, it is almost impossible to implement TTC or PIT ratings
in a strict sense. First of all, it is very difficult to completely separate
firm-specific components from macroeconomic developments. It is equally
challenging to implement a rating system that comprehensively reflects the
current state of the economy as well as all available information about the
future economic development in a timely manner. As a consequence, real-world
rating systems are typically hybrids in the following sense: Individual
ratings are impacted by macroeconomic developments, e.g. upgrades are more
frequent in economic upturns, but not all the available economic information
is reflected leading to fluctuations of default rates of rating classes across
credit cycles. Rating systems mainly differ with respect to the weights of TTC
and PIT components, i.e., whether they are predominantly through-the-cycle or
point-in-time, see Morone and Cornaglia (2009); Rubtsov and Petrov (2016) for
estimating the degree of PIT-ness of a rating system. The consistent
calculation of TTC and PIT default probabilities is intensively discussed in
the literature, e.g. in Aguais et al. (2008); Carlehed and Petrov (2012);
Rubtsov and Petrov (2016); Miu and Ozdemir (2017); Rubtsov (2021).

In order to formally define point-in-time rating migration processes we
consider expected (one-period) default probabilities for each combination of
economic state $a$ and rating $r$:

${\rm
PD}(a,r)=\sum_{b\in\mathcal{A}}p_{(a,r)(b,\overline{r})}={\mathbb{P}}(R_{t+1}=\overline{r}\mid
A_{t}=a,R_{t}=r)$

for any $t\in{\mathbb{N}}_{0}$. Since the PIT rating $r$ of a company already
reflects the impact of the state of the economy on its creditworthiness the
corresponding default probability is completely determined by $r$.

######  Definition 5.1.

A rating migration process is point-in-time (PIT) if ${\rm PD}(a,r)={\rm
PD}(b,r)$ for all ratings $r\in{\mathcal{R}}$ and economic states
$a,b\in{\mathcal{A}}$.

Essentially, this captures that ratings are assigned such that expected
default and survival probabilities of a rating class do not depend on the
economic state. However, ex-post default frequencies
${\mathbb{P}}(R_{t+1}=\overline{r}|R_{t}=r,A_{t}=a,A_{t+1}=b)$ still depend on
the economic state transition, e.g. if the economy transitions into a worse
state, more defaults will be observed compared to a neutral or positive
transition.

In contrast, through-the-cycle ratings are insensitive to economic or credit
cycles. More formally, let $Q^{(a,b)}\in[0,1]^{J\times J}$ be the matrix of
transition probabilities from rating $r$ to rating $s$ conditionally on
economic states $a,b\in{\mathcal{A}}$ and no default:

$q_{rs}^{(a,b)}=\begin{cases}{\mathbb{P}}(R_{t+1}=s|R_{t}=r,A_{t}=a,A_{t+1}=b,R_{t+1}\not=\overline{r})=\displaystyle\frac{m_{rs}^{(a,b)}}{1-m_{r\overline{r}}^{(a,b)}},&\text{
if }r,s\not=\overline{r},\\\ 1,&\text{ if }r=s=\overline{r},\\\ 0,&\text{
else.}\end{cases}$ (22)

######  Definition 5.2.

A rating migration process is through-the-cycle (TTC) if there exists a
stochastic matrix $Q\in[0,1]^{J\times J}$ such that $Q^{(a,b)}=Q$ for all
$a,b\in\mathcal{A}$.

The definition captures that, conditional on no-default, rating transitions
are independent of the economic state.

###  5.2 Decomposition into rating and default components

For all $a,b\in\mathcal{A}$ the rating migration matrix $M^{(a,b)}$ can be
decomposed into a matrix product

$M^{(a,b)}=D^{(a,b)}\cdot Q^{(a,b)},$ (23)

where the non-default component $Q^{(a,b)}$ has already been specified in (22)
and the default component $D^{(a,b)}\in[0,1]^{J\times J}$ is defined as a
stochastic matrix with entries

$\displaystyle d_{r\overline{r}}^{(a,b)}$
$\displaystyle={\mathbb{P}}(R_{t+1}=\overline{r}|R_{t}=r,A_{t}=a,A_{t+1}=b)=m_{r\overline{r}}^{(a,b)},$
$\displaystyle d_{rr}^{(a,b)}$
$\displaystyle={\mathbb{P}}(R_{t+1}\not=\overline{r}|R_{t}=r,A_{t}=a,A_{t+1}=b)=1-m_{r\overline{r}}^{(a,b)},\quad\text{
if }r\not=\overline{r},$

and all other entries taking value zero. The default component $D^{(a,b)}$ can
be further decomposed as

$D^{(a,b)}=D^{a}\cdot C^{(a,b)},$

where $D^{a}\in[0,1]^{J\times J}$ is fully specified by the default
probabilities ${\rm PD}(a,r)$, i.e., $D_{a}$ depends on the current economic
state $a$ of the economy but not on futures states, whereas
$C^{(a,b)}\in[0,1]^{J\times J}$ specifies the ex-post deviation from the
expected default probability ${\rm PD}(a,r)$ if the economy transitions from
$a$ to $b$. More precisely,

  * •

for each state $a\in\mathcal{A}$ of the economy,
$D^{a}=(d_{rs}^{a})\in[0,1]^{J\times J}$ is a stochastic matrix containing
default and survival probabilities unconditional on the future state of the
economy; its entries are, for $r\not=\overline{r}$,

$d_{r\overline{r}}^{a}={\rm
PD}(a,r)={\mathbb{P}}(R_{t+1}=\overline{r}|A_{t}=a,R_{t}=r),\ \ \ \ \ \
d_{rr}^{a}=1-d_{r\overline{r}}^{a},\ \ \ \ \ \
d_{\overline{r}\,\overline{r}}^{a}=1;$

all other entries take value zero;

  * •

$C^{(a,b)}\in[0,1]^{J\times J}$ has entries, for $r\not=\overline{r}$,

$c_{rr}^{(a,b)}=\frac{d_{rr}^{(a,b)}}{d_{rr}^{a}},\ \ \ \ \ \
c_{r\overline{r}}^{(a,b)}=1-c_{rr}^{(a,b)},\ \ \ \ \ \
c_{\overline{r}\,\overline{r}}^{(a,b)}=1;$

all other entries in $C^{(a,b)}$ are zero.

######  Proposition 5.3.

For $a,b\in\mathcal{A}$,

$M^{(a,b)}=D^{(a,b)}\cdot Q^{(a,b)}\ \ \ \hbox{and}\ \ \ D^{(a,b)}=D^{a}\cdot
C^{(a,b)}.$

The matrices $C^{(a,b)}$ satisfy

$\sum_{b\in\mathcal{A}}m_{ab}C^{(a,b)}=I$

for all $a\in\mathcal{A}$ and identity matrix $I\in[0,1]^{J\times J}$.

###### Proof.

The equation $M^{(a,b)}=D^{(a,b)}\cdot Q^{(a,b)}$ immediately follows from the
definition of $D^{(a,b)}$ and $Q^{(a,b)}$.

All matrix elements of $D^{a}\cdot C^{(a,b)}$ and $D^{(a,b)}$ are 0 except the
diagonal elements and the last columns. By definition,

$d_{rr}^{a}c_{rr}^{(a,b)}=d_{rr}^{(a,b)}$ (24)

and therefore the matrices $D^{a}\cdot C^{(a,b)}$ and $D^{(a,b)}$ have the
same diagonal elements. The matrix elements in the last column satisfy the
following equations:

$d_{rr}^{a}c_{r\overline{r}}^{(a,b)}+d_{r\overline{r}}^{a}=d_{rr}^{a}(1-c_{rr}^{(a,b)})+d_{r\overline{r}}^{a}=d_{rr}^{a}-d_{rr}^{(a,b)}+d_{r\overline{r}}^{a}=d_{r\overline{r}}^{(a,b)}$

and therefore $D^{(a,b)}=D^{a}\cdot C^{(a,b)}$.

It follows from (24) and

$d_{rr}^{a}=\frac{{\mathbb{P}}(A_{t}=a,R_{t}=r,R_{t+1}\not=\overline{r})}{{\mathbb{P}}(A_{t}=a,R_{t}=r)},\
\ \ \ \ \
d_{rr}^{(a,b)}=\frac{{\mathbb{P}}(A_{t}=a,A_{t+1}=b,R_{t}=r,R_{t+1}\not=\overline{r})}{{\mathbb{P}}(A_{t}=a,A_{t+1}=b,R_{t}=r)}$

that

$c_{rr}^{(a,b)}=\frac{{\mathbb{P}}(A_{t+1}=b|A_{t}=a,R_{t}=r,R_{t+1}\not=\overline{r})}{{\mathbb{P}}(A_{t+1}=b|A_{t}=a,R_{t}=r)}.$

By Lemma 2.1,

$\sum_{b\in\mathcal{A}}m_{ab}c_{rr}^{(a,b)}=\sum_{b\in\mathcal{A}}{\mathbb{P}}(A_{t+1}=b|A_{t}=a,R_{t}=r)c_{rr}^{(a,b)}=1$

for all $a\in\mathcal{A}$. Since all matrix elements of $C^{(a,b)}$ are 0
except the diagonal and the last column and each row sum equals 1 we obtain

$\sum_{b\in\mathcal{A}}m_{ab}C^{(a,b)}=I.$

∎

PIT and TTC rating migration processes have a rather intuitive interpretation
in the context of the decomposition

$M^{(a,b)}=D^{a}\cdot C^{(a,b)}\cdot Q^{(a,b)},\quad a,b\in\mathcal{A},$

given in the proposition above:

  1. 1.

A rating migration process is point-in-time (PIT) iff there exists a
stochastic matrix $D\in[0,1]^{J\times J}$ such that $D^{a}=D$ for all
$a\in\mathcal{A}$, i.e.,

$M^{(a,b)}=D\cdot C^{(a,b)}\cdot Q^{(a,b)},\quad a,b\in\mathcal{A}.$

Note that the matrix $D$, which does not depend on economic states, specifies
the expected default and survival probabilities for each rating class whereas
the economic-state dependent matrix $C^{(a,b)}$ reflects that ex-post default
frequencies ${\mathbb{P}}(R_{t+1}=\overline{r}|R_{t}=r,A_{t}=a,A_{t+1}=b)$
still depend on the economic state transition.

  2. 2.

A rating migration process is through-the-cycle (TTC) if there exists a
stochastic matrix $Q\in[0,1]^{J\times J}$ such that $Q^{(a,b)}=Q$ for all
$a,b\in\mathcal{A}$, i.e.,

$M^{(a,b)}=D^{(a,b)}\cdot Q=D^{a}\cdot C^{(a,b)}\cdot Q,\quad
a,b\in\mathcal{A}.$ (25)

In contrast to rating migrations specified by $Q$ for all economic states,
expected default probabilities are economic-state dependent for TTC processes,
which is reflected by the specification of the default component $D^{a}$.

We will now provide an alternative definition of TTC and PIT rating migration
processes in terms of the asymptotic rating migration matrix
${\widetilde{P}}_{\infty}$. Analogously to the decomposition of the
conditional rating migration matrices $M^{(a,b)}$ (see Equation (23)),
${\widetilde{P}}_{\infty}$ can be decomposed into a default component
${\widetilde{D}}_{\infty}$ and a non-default component
${\widetilde{Q}}_{\infty}$ with

${\widetilde{P}}_{\infty}={\widetilde{D}}_{\infty}\cdot{\widetilde{Q}}_{\infty}.$
(26)

######  Theorem 5.4.

  1. (i)

$(R_{t})_{t\in\mathbb{N}_{0}}$ is a TTC rating migration process if and only
if ${\widetilde{Q}}_{\infty}=Q^{(a,b)}$ for each $a,b\in{\mathcal{A}}$.

  2. (ii)

$(R_{t})_{t\in\mathbb{N}_{0}}$ is a PIT rating migration process if and only
if ${\widetilde{D}}_{\infty}=D^{a}$ for each $a\in{\mathcal{A}}$.

Note that the condition in Theorem 5.4(ii) can also be expressed as follows.

######  Corollary 5.5.

$(R_{t})_{t\in\mathbb{N}_{0}}$ is a PIT rating migration process if and only
if the default vector of ${\widetilde{P}}_{\infty}$ equals the PD vector
$({{\rm PD}(a,r))}_{r\in{\mathcal{R}}}$ for all economic states
$a\in\mathcal{A}$.

The proof of Theorem 5.4 is an immediate consequence of the following lemma,
which provides a characterisation of the rating migration matrices
${\widetilde{P}}_{t}$ of the time-inhomogeneous Markov process
$({\widetilde{R}}_{t})_{t\in\mathbb{N}_{0}}$ if $(R_{t})_{t\in\mathbb{N}_{0}}$
is PIT or TTC.

######  Lemma 5.6.

  1. (i)

Let $(R_{t})_{t\in\mathbb{N}_{0}}$ be a TTC rating migration process and let
$Q$ denote the uniquely defined non-default component of the rating transition
matrices $M^{(a,b)}$. Then $Q$ is the non-default component of each of the
rating transition matrices ${\widetilde{P}}_{t}$ for $t>0$.

  2. (ii)

Let $(R_{t})_{t\in\mathbb{N}_{0}}$ be a PIT rating migration process and write
each rating transition matrix $M^{(a,b)}$ in the form

$M^{(a,b)}=D\cdot C^{(a,b)}\cdot Q^{(a,b)}.$

Then $D$ is the default component of each of the rating transition matrices
${\widetilde{P}}_{t}$ for $t>0$.

###### Proof.

(i) Let $r,s\in\widehat{\mathcal{R}}$ and denote the distribution of $X_{t}$
by $\lambda_{t}\in[0,1]^{KJ}$, i.e.,
$\lambda_{t,(a,r)}={\mathbb{P}}(X_{t}=(a,r))$ for $(a,r)\in\mathcal{X}$ and
$t\in\mathbb{N}_{0}$. Using the expression of $\tilde{p}_{t,rs}$ from the
proof of Proposition 3.4, we obtain

$\displaystyle\tilde{p}_{t,rs}$
$\displaystyle=\frac{\sum_{a,b\in\mathcal{A}}p_{(a,r)(b,s)}\cdot\lambda_{t-1,(a,r)}}{\sum_{a\in\mathcal{A}}\lambda_{t-1,(a,r)}}=\frac{\sum_{a,b\in\mathcal{A}}m_{a,b}m_{rs}^{(a,b)}\lambda_{t-1,(a,r)}}{\sum_{a\in\mathcal{A}}\lambda_{t-1,(a,r)}}$
$\displaystyle=\frac{\sum_{a,b\in\mathcal{A}}m_{a,b}q_{rs}(1-m_{r\overline{r}}^{(a,b)})\lambda_{t-1,(a,r)}}{\sum_{a\in\mathcal{A}}\lambda_{t-1,(a,r)}}=q_{rs}c_{r},$

where $c_{r}$ is defined by

$c_{r}=\frac{\sum_{a,b\in\mathcal{A}}m_{a,b}(1-m_{r\overline{r}}^{(a,b)})\lambda_{t-1,(a,r)}}{\sum_{a\in\mathcal{A}}\lambda_{t-1,(a,r)}}.$

Since $\sum_{s\in\hat{R}}q_{rs}c_{r}+\tilde{p}_{t,r\overline{r}}=1$, it
follows, together with $\sum_{s\in\hat{R}}q_{rs}=1$, that
$c_{r}=1-\tilde{p}_{t,r\overline{r}}$. As a consequence,
$\tilde{p}_{t,rs}=q_{rs}\left(1-\tilde{p}_{t,r\overline{r}}\right)$ and
therefore $q_{rs}=\tilde{p}_{t,rs}/(1-\tilde{p}_{t,r\overline{r}})$.

(ii) It follows from Proposition 5.3 that for every $a\in\mathcal{A}$

$\sum_{b\in\mathcal{A}}m_{ab}D^{(a,b)}=D\sum_{b\in\mathcal{A}}m_{ab}C^{(a,b)}=D.$

Hence, for $r\in{\mathcal{R}}$,

${\tilde{p}}_{t,r\overline{r}}=\frac{\sum_{a,b\in\mathcal{A}}p_{(a,r)(b,\overline{r})}\lambda_{t-1}(a,r)}{\sum_{a\in\mathcal{A}}\lambda_{t-1}(a,r)}=\frac{\sum_{a\in\mathcal{A}}\lambda_{t-1}(a,r)\sum_{b\in\mathcal{A}}m_{ab}d_{r\overline{r}}^{(a,b)}}{\sum_{a\in\mathcal{A}}\lambda_{t-1}(a,r)}=d_{r\overline{r}}.$
(27)

The default component ${\widetilde{D}}_{t}$ of ${\widetilde{P}}_{t}$ is a
stochastic matrix and all matrix element are zero except the diagonal elements
and the last column, which equals
$({\tilde{p}}_{t,r\overline{r}})_{r\in{\mathcal{R}}}$. Hence, by (27),
${\widetilde{D}}_{t}=D$ for every $t>0$. ∎

We finish this subsection by showing an interesting property of TTC rating
migration processes. In Section 3, a class of Markovian rating processes has
been identified. These processes are characterized by the property that their
rating migration matrices $M^{(a,b)}$ have identical ratios of non-default
migration probabilities. It is easy to prove that this class of rating
processes is a subclass of TTC rating migration processes.

######  Definition 5.7.

The conditional transition matrices $M^{(a,b)}$ have identical survival ratios
if for all $r,s\in\widehat{\mathcal{R}}$ and
$a,a^{\prime},b,b^{\prime}\in\mathcal{A}$

$\frac{d_{rr}^{(a,b)}}{d_{ss}^{(a,b)}}=\frac{d_{rr}^{(a^{\prime},b^{\prime})}}{d_{ss}^{(a^{\prime},b^{\prime})}},$
(28)

where $D^{(a,b)}=(d_{rs}^{(a,b)})$ is the default component of $M^{(a,b)}$,
see Equation (23).

######  Proposition 5.8.

The following conditions are equivalent for a rating migration process
$(R_{t})_{t\in\mathbb{N}_{0}}$.

  1. (i)

The matrices $M^{(a,b)}$ have identical ratios of non-default migration
probabilities (3).

  2. (ii)

$(R_{t})_{t\in\mathbb{N}_{0}}$ is TTC and the matrices $M^{(a,b)}$ have
identical survival ratios (28).

###### Proof.

“$\Rightarrow$”: Let $r,r^{\prime},s\in\widehat{\mathcal{R}}$ and
$a,a^{\prime},b,b^{\prime}\in\mathcal{A}$. It follows from (3) that

$m_{rs}^{(a,b)}d_{r^{\prime}r^{\prime}}^{(a^{\prime},b^{\prime})}=\sum_{s^{\prime}\in\widehat{\mathcal{R}}}m_{rs}^{(a,b)}m_{r^{\prime}s^{\prime}}^{(a^{\prime},b^{\prime})}=\sum_{s^{\prime}\in\widehat{\mathcal{R}}}m_{rs}^{(a^{\prime},b^{\prime})}m_{r^{\prime}s^{\prime}}^{(a,b)}=m_{rs}^{(a^{\prime},b^{\prime})}d_{r^{\prime}r^{\prime}}^{(a,b)}.$

By setting $r^{\prime}$ to $r$ and to $s$ respectively we obtain

$q_{rs}^{(a,b)}=\frac{m_{rs}^{(a,b)}}{d_{rr}^{(a,b)}}=\frac{m_{rs}^{(a^{\prime},b^{\prime})}}{d_{rr}^{(a^{\prime},b^{\prime})}}=q_{rs}^{(a^{\prime},b^{\prime})}~{}\hbox{
and
}~{}\frac{d_{rr}^{(a,b)}}{d_{rr}^{(a^{\prime},b^{\prime})}}=\frac{m_{rs}^{(a,b)}}{m_{rs}^{(a^{\prime},b^{\prime})}}=\frac{d_{ss}^{(a,b)}}{d_{ss}^{(a^{\prime},b^{\prime})}}.$

“$\Leftarrow$”: Let $r,r^{\prime},s,s^{\prime}\in\widehat{\mathcal{R}}$ and
$a,a^{\prime},b,b^{\prime}\in\mathcal{A}$. Since
$(R_{t})_{t\in\mathbb{N}_{0}}$ is TTC there exists a matrix
$Q=(q_{rs})\in[0,1]^{J\times J}$ such that
$m_{rs}^{(a,b)}=q_{rs}d_{rr}^{(a,b)}$. Together with (28),

$\frac{m_{rs}^{(a,b)}}{m_{r^{\prime}s^{\prime}}^{(a,b)}}=\frac{q_{rs}d_{rr}^{(a,b)}}{q_{r^{\prime}s^{\prime}}d_{r^{\prime}r^{\prime}}^{(a,b)}}=\frac{q_{rs}d_{rr}^{(a^{\prime},b^{\prime})}}{q_{r^{\prime}s^{\prime}}d_{r^{\prime}r^{\prime}}^{(a^{\prime},b^{\prime})}}=\frac{m_{rs}^{(a^{\prime},b^{\prime})}}{m_{r^{\prime}s^{\prime}}^{(a^{\prime},b^{\prime})}}.$

∎

##  6 Ordered rating processes

###  6.1 Ordered rating distributions

In this section, we formalize the notions of “better” or “worse” economic and
rating states. A rating $r$ is typically considered better than a rating $s$
if the multi-year PDs of $r$ are consistently smaller than the multi-year PDs
of $s$. In order to satisfy this condition, not only the 1y PDs of $r$ and $s$
have to be ordered accordingly but the rating migration matrices also have to
meet certain requirements. Informally, the probability that an $r$-rated
company is downgraded to a bad rating has to be lower than the corresponding
migration probability for an $s$-rated company. It is well-know that these
conditions on migration probabilities can be formalised by monotonicity
properties of rating migration matrices, where a migration matrix is called
monotone with respect to a given (partial) order on the set of rating
distributions if the order is preserved under multiplication with the
respective migration matrix. The main objective of the section is to study
conditions on the transition matrices $M$, $M^{(a,b)}$ and $P$ to ensure
monotonicity of the matrices with respect to a given order.

Our starting point is the formal specification of a total order on the set
${\mathcal{R}}$ of ratings. We assume that the vector of asymptotic PDs
${({\tilde{p}}_{r\overline{r}})}_{r\in{\mathcal{R}}}$ consists of $J$ distinct
values and introduce the total order $\leq$ on ${\mathcal{R}}$ by $r\leq
r^{\prime}$ if
${\tilde{p}}_{r\overline{r}}\leq{\tilde{p}}_{r^{\prime}\overline{r}}$ for
$r,r^{\prime}\in{\mathcal{R}}$. Note that the default rating $\overline{r}$ is
the biggest element, i.e., $r\leq\overline{r}$ for all $r\in{\mathcal{R}}$.

Consider rating distributions $\lambda$ and $\mu$ represented by their density
functions, i.e., $\lambda={(\lambda_{r})}_{r\in{\mathcal{R}}}$ and
$\mu={(\mu_{r})}_{r\in{\mathcal{R}}}$ are elements of $[0,1]^{J}$ with
$\sum_{r\in{\mathcal{R}}}\lambda_{r}=\sum_{r\in{\mathcal{R}}}\mu_{r}=1$. The
distribution $\mu$ is typically considered to be more risky than $\lambda$ if
for all $r\in{\mathcal{R}}$

$\sum_{r\leq s}\lambda_{s}\leq\sum_{r\leq s}\mu_{s},$ (29)

see Jarrow et al. (1997). The partial order on rating distributions defined by
(29) is called stochastic dominance or usual stochastic order, written
$\lambda\leq_{st}\mu$. It is obvious that (29) is equivalent to
$\bar{F}_{\lambda}(r)\leq\bar{F}_{\mu}(r)$ for all $r\in{\mathcal{R}}$, where
$\bar{F}_{\lambda}(r)=\sum_{r\leq s}\lambda_{s}$ denotes the survival function
of $\lambda$. In the context of a rating transition matrix, where each row
represents a rating distribution, tail probabilities thus induce the notion of
“better” and “worse” ratings. Another condition equivalent to (29) is

$\lambda x^{T}\leq\mu x^{T}\hbox{ for each increasing real-valued sequence
}x=(x_{1},\ldots,x_{J})\in{\mathbb{R}}^{J},$ (30)

see (Müller and Stoyan, 2002; Föllmer and Schied, 2002; Shaked and
Shanthikumar, 2007; Rolski et al., 2009).

A stochastic matrix $L\in[0,1]^{J\times J}$ is called monotone with respect to
a partial order $\leq$ on the set of distributions if $\lambda L\leq\mu L$ for
all distributions with $\lambda\leq\mu$, see e.g. (Rolski et al., 2009). In
particular, a stochastic matrix $L$ is called stochastically monotone if it is
monotone with respect to $\leq_{st}$. It is easy to show (Theorem 7.4.1 of
Rolski et al. (2009)) that $L$ is stochastically monotone iff

$r\leq s\hbox{ implies }l_{r}\leq_{st}l_{s}\hbox{ for all
}r,s\in{\mathcal{R}},$ (31)

where the distribution $l_{r}$ denotes the $r$-th row of matrix $L$.

We now apply the concept of stochastic monotonicity to the (conditional)
rating migration matrices $M^{(a,b)}\in[0,1]^{J\times J}$. It follows
immediately from the definition that the $M^{(a,b)}$ are stochastically
monotone if and only if the order relation $\leq_{st}$ is preserved on each
economic path.

######  Theorem 6.1.

The following two conditions are equivalent:

  1. (i)

For each $a,b\in{\mathcal{A}}$, the (conditional) rating migration matrix
$M^{(a,b)}\in[0,1]^{J\times J}$ is stochastically monotone.

  2. (ii)

For all rating distributions $\lambda,\mu\in[0,1]^{J}$ and each path
$a_{0},\ldots,a_{T}\in{\mathcal{A}}$ of economic states,

$\lambda\leq_{st}\mu\hbox{ implies
}\lambda\prod_{t=0}^{T-1}M^{(a_{t},a_{t+1})}\leq_{st}\mu\prod_{t=0}^{T-1}M^{(a_{t},a_{t+1})}.$

###  6.2 Ordered distributions of economic states and ratings

We will now order the set of distributions on the product space
$\mathcal{X}=\mathcal{A}\times\mathcal{R}$ of economic states and ratings.
First we fix a total order $\leq$ on the set of economic states
${\mathcal{A}}$. The total orders on ${\mathcal{A}}$ and ${\mathcal{R}}$ are
now extended to the (partial) product order $\leq$ on ${\mathcal{X}}$:
$(a,r)\leq(b,s)$ in ${\mathcal{X}}$ if $a\leq b$ in ${\mathcal{A}}$ and $r\leq
s$ in ${\mathcal{R}}$.

Let $\lambda=(\lambda_{a,r})_{(a,r)\in{\mathcal{X}}}$ and
$\mu=(\mu_{a,r})_{(a,r)\in{\mathcal{X}}}$ be distributions on ${\mathcal{X}}$,
i.e., elements of $[0,1]^{KJ}$ with
$\sum_{(a,r)\in{\mathcal{X}}}\lambda_{a,r}=1$. The following definition
generalizes the concept of stochastic dominance (or the usual stochastic
order) to distributions on ${\mathcal{X}}$: the distribution $\mu$
stochastically dominates $\lambda$, written $\lambda\leq_{st}\mu$, if

$\sum_{(a,r)\in U}\lambda_{a,r}\leq\sum_{(a,r)\in U}\mu_{a,r},\ \ \ \ \
\hbox{for all upper sets }U\subseteq{\mathcal{X}},$ (32)

where $U\subseteq{\mathcal{X}}$ is called an upper set if $(a,r)\in U$ and
$(a,r)\leq(b,s)$ implies $(b,s)\in U$.999 Upper sets of the ordered set
$\\{1,2\\}\times\\{1,2,3\\}$ with Hasse diagram
$(1,1)$$(1,2)$$(1,3)$$(2,1)$$(2,2)$$(2,3)$:
$\\{\\},\left\\{\begin{array}[]{c}(2,3)\\\
\end{array}\right\\},\left\\{\begin{array}[]{c}(2,3)\\\ (1,3)\\\
\end{array}\right\\},\left\\{\begin{array}[]{c}(2,3)\\\ (2,2)\\\
\end{array}\right\\},\left\\{\begin{array}[]{c}(2,3)\\\ (2,2)\\\ (1,3)\\\
\end{array}\right\\},\left\\{\begin{array}[]{c}(2,3)\\\ (2,2)\\\ (2,1)\\\
\end{array}\right\\},\left\\{\begin{array}[]{c}(2,3)\\\ (2,2)\\\ (1,3)\\\
(1,2)\\\ \end{array}\right\\},\left\\{\begin{array}[]{c}(2,3)\\\ (2,2)\\\
(2,1)\\\ (1,3)\\\ \end{array}\right\\},\left\\{\begin{array}[]{c}(2,3)\\\
(2,2)\\\ (2,1)\\\ (1,3)\\\ (1,2)\\\
\end{array}\right\\},\left\\{\begin{array}[]{c}(2,3)\\\ (2,2)\\\ (2,1)\\\
(1,3)\\\ (1,2)\\\ (1,1)\\\ \end{array}\right\\}$. As for rating distributions,
stochastic dominance on ${\mathcal{X}}$ can be characterized via increasing
real-valued sequences, see e.g. Theorem 3.3.4 of Müller and Stoyan (2002).
More precisely, the relation $\lambda\leq_{st}\mu$ holds for distributions on
${\mathcal{X}}$ iff $\lambda x^{T}\leq\mu x^{T}$ for each increasing real-
valued sequence $x=(x_{a,r})_{(a,r)\in{\mathcal{X}}}$ in ${\mathbb{R}}^{KJ}$,
where $x$ is called increasing if $(a,r)\leq(b,s)$ implies $x_{a,r}\leq
x_{b,s}$.

Another potential generalisation of the partial order on rating distributions
defined in (29) is formalised by the concept of upper orthant orders. In our
setup, the upper orthant order on ${\mathcal{X}}$ can be defined as follows.
For $(a,r)\in{\mathcal{X}}$ let $U_{(a,r)}$ denote the upper set generated by
$(a,r)$, i.e., $U_{(a,r)}=\\{(b,s)\in{\mathcal{X}}\mid(a,r)\leq(b,s)\\}$ and
let $\lambda=(\lambda_{a,r})_{(a,r)\in{\mathcal{X}}}$ and
$\mu=(\mu_{a,r})_{(a,r)\in{\mathcal{X}}}$ be distributions on ${\mathcal{X}}$.
The upper orthant order $\leq_{uo}$ is defined by $\lambda\leq_{uo}\mu$ if

$\sum_{(b,s)\in U_{(a,r)}}\lambda_{b,s}\leq\sum_{(b,s)\in U_{(a,r)}}\mu_{b,s}\
\ \ \ \ \hbox{for all }(a,r)\in{\mathcal{X}}.$ (33)

Note that the definition of $\leq_{uo}$ is obtained by restricting the upper
sets in the definition (32) of $\leq_{st}$ to upper sets that are generated by
single elements $(a,r)\in{\mathcal{X}}$.101010From the list of upper sets in
the previous footnote, this excludes the first, fifth, eigth and ninth sets.
It is an immediate consequence that $\lambda\leq_{st}\mu$ implies
$\lambda\leq_{uo}\mu$. Again, it is obvious that (33) is equivalent to
$\bar{F}_{\lambda}(a,r)\leq\bar{F}_{\mu}(a,r)$ for all
$(a,r)\in{\mathcal{X}}$, where
$\bar{F}_{\lambda}(a,r)=\sum_{(a,r)\leq(b,s)}\lambda_{b,s}$ denotes the
survival function of $\lambda$.

In analogy to the ordering of ratings in a rating transition matrix, a minimum
requirement on the transition matrix $P=(p_{(a,r),(b,s)})$ is

$(a,r)\leq(a^{\prime},r^{\prime})\text{ implies
}p_{(a,r)}\leq_{uo}p_{(a^{\prime},r^{\prime})}\hbox{ for every
}(a,r),(a^{\prime},r^{\prime})\in{\mathcal{X}},$ (34)

where the row vectors of $P$ are denoted by $p_{(a,r)}$. In other words,
$(a,r)\leq(a^{\prime},r^{\prime})$ implies ${\mathbb{P}}(A_{t+1}\geq
b,R_{t+1}\geq s|A_{t}=a,R_{t}=r)\leq{\mathbb{P}}(A_{t+1}\geq b,R_{t+1}\geq
s|A_{t}=a^{\prime},R_{t}=r^{\prime})$, for all $(b,s)\in\mathcal{X}$. It turns
out, however, this is not sufficient to ensure that the order of multi-year
PDs is preserved.

To this end, we will now analyse the monotonicity of the transition matrix $P$
with respect to the partial orders $\leq_{st}$ and $\leq_{uo}$. First of all,
it can be shown (e.g. Corollary 5.2.4 of Müller and Stoyan (2002)) that $P$ is
stochastically monotone, i.e., $\lambda\leq_{st}\mu$ implies $\lambda
P\leq_{st}\mu P$, if and only if the following generalisation of (31) holds:

$(a,r)\leq(a^{\prime},r^{\prime})\hbox{ implies
}p_{(a,r)}\leq_{st}p_{(a^{\prime},r^{\prime})}\hbox{ for every
}(a,r),(a^{\prime},r^{\prime})\in{\mathcal{X}}.$ (35)

If $\leq_{st}$ is replaced by $\leq_{uo}$, i.e., (34) is considered instead of
(35), a condition for monotonicity with respect to $\leq_{uo}$ is obtained,
which is necessary but not sufficient. In order to ensure the
$\leq_{uo}$-monotonicity of $P$, further conditions on the row vectors of $P$
are required, which can be derived from results in Müller (2013).

For $(a,r)\in{\mathcal{X}}$ define the distribution
$1^{(a,r)}=(1^{(a,r)}_{b,s})_{(b,s)\in{\mathcal{X}}}$ by $1^{(a,r)}_{a,r}=1$
and $1^{(a,r)}_{b,s}=0$ for $(b,s)\not=(a,r)$. Note that if $P$ is monotone
with respect to $\leq_{st}$ or $\leq_{uo}$ the matrix products $P^{t}$ satisfy

$(a,r)\leq(a^{\prime},r^{\prime})\hbox{ implies
}1^{(a,r)}P^{t}\leq_{uo}1^{(a^{\prime},r^{\prime})}P^{t},$ (36)

for $(a,r),(a^{\prime},r^{\prime})\in{\mathcal{X}}$ and $t\in{\mathbb{N}}$. An
immediate consequence of condition (36) is that multi-year default
probabilities are ordered with respect to the initial economic state and
rating. More formally, we define the multi-year default probabilities

${\rm PD}_{t}(a,r)={\mathbb{P}}(R_{t}=\overline{r}\mid A_{0}=a,R_{0}=r),\ \ \
t\in{\mathbb{N}},$

for initial economic state and rating $(a,r)\in{\mathcal{X}}$. If the
transition matrix $P$ satisfies (36) then

$(a,r)\leq(a^{\prime},r^{\prime})\hbox{ implies }{\rm PD}_{t}(a,r)\leq{\rm
PD}_{t}(a^{\prime},r^{\prime})\hbox{ for }t\in{\mathbb{N}}.$ (37)

For the rest of this section we will study monotonicity properties of the
transition matrix $M\in[0,1]^{K\times K}$ and the conditional rating migration
matrices $M^{(a,b)}\in[0,1]^{J\times J}$ that imply stochastic monotonicity of
the transition matrix $P\in[0,1]^{KJ\times KJ}$. It turns out that $M$ and the
$M^{(a,b)}$ have to be stochastic monotone and, in addition, the $M^{(a,b)}$
have to satisfy a condition formulated in terms of the following order
relation: $M^{(a,b)}\leq_{st}M^{(a^{\prime},b^{\prime})}$ if
$m_{r}^{(a,b)}\leq_{st}m_{r}^{(a^{\prime},b^{\prime})}$ for all
$r\in{\mathcal{R}}$, where $m_{r}^{(a,b)}$ and
$m_{r}^{(a^{\prime},b^{\prime})}$ denote the rows of $M^{(a,b)}$ and
$M^{(a^{\prime},b^{\prime})}$ respectively.

######  Theorem 6.2.

Assume that the transition matrix $M\in[0,1]^{K\times K}$ and the rating
migration matrices $M^{(a,b)}\in[0,1]^{J\times J}$ have the following
properties:

  1. (i)

$M$ is stochastically monotone.

  2. (ii)

$M^{(a,b)}$ is stochastically monotone for all $a,b\in{\mathcal{A}}$.

  3. (iii)

$a\leq a^{\prime}$ and $b\leq b^{\prime}$ imply
$M^{(a,b)}\leq_{st}M^{(a^{\prime},b^{\prime})}$ for all
$a,a^{\prime},b,b^{\prime}\in{\mathcal{A}}$.

Then the transition matrix $P\in[0,1]^{KJ\times KJ}$ is stochastically
monotone.

###### Proof.

Let $a,a^{\prime}\in{\mathcal{A}}$ and $r,r^{\prime}\in{\mathcal{R}}$ such
that $a\leq a^{\prime}$ and $r\leq r^{\prime}$. In order to prove (35) it
suffices to show that

$\sum_{(b,s)\in U}p_{(a,r),(b,s)}\leq\sum_{(b,s)\in
U}p_{(a,r^{\prime}),(b,s)}\ \ \ \hbox{ and }\ \ \ \sum_{(b,s)\in
U}p_{(a,r),(b,s)}\leq\sum_{(b,s)\in U}p_{(a^{\prime},r),(b,s)},$ (38)

where $U\subseteq{\mathcal{X}}$ is an arbitrary upper set with respect to the
order $\leq$ on ${\mathcal{X}}$.

We define the partition

$U=\bigcup_{b\in{\mathcal{A}}}\\{(b,r)\mid r\in U_{b}\\},\ \ \
U_{b}=\\{r\in{\mathcal{R}}\mid(b,r)\in U\\},$

where each $U_{b}$ is an upper set in $\mathcal{R}$. By Lemma 2.1,
$p_{(a,r),(b,s)}=m_{ab}m_{rs}^{(a,b)}$. Hence, the left inequality in (38)
follows from the assumption that the rating migration matrices $M^{(a,b)}$ are
stochastically monotone for all $a,b\in{\mathcal{A}}$:

$\sum_{(b,s)\in U}p_{(a,r),(b,s)}=\sum_{b\in{\mathcal{A}}}m_{ab}\sum_{s\in
U_{b}}m_{rs}^{(a,b)}\leq\sum_{b\in{\mathcal{A}}}m_{ab}\sum_{s\in
U_{b}}m_{r^{\prime}s}^{(a,b)}=\sum_{(b,s)\in U}p_{(a,r^{\prime}),(b,s)}.$

It remains to prove the right inequality in (38). By definition of the product
order $\leq$ on ${\mathcal{X}}$, $b\leq c$ implies $U_{b}\subseteq U_{c}$ for
all $b,c\in{\mathcal{A}}$. Together with $M^{(a,b)}\leq_{st}M^{(a,c)}$ it
follows that

$\sum_{s\in U_{b}}m_{rs}^{(a,b)}\leq\sum_{s\in
U_{b}}m_{rs}^{(a,c)}\leq\sum_{s\in U_{c}}m_{rs}^{(a,c)}.$

Hence, by defining $x_{b}=\sum_{s\in U_{b}}m_{rs}^{(a,b)}$ we obtain a real-
valued increasing sequence $(x_{b})_{b\in{\mathcal{A}}}$, i.e., $b\leq c$
implies $x_{b}\leq x_{c}$. Since $M$ is stochastically monotone it follows
from (30) that

$\sum_{(b,s)\in U}p_{(a,r),(b,s)}=\sum_{b\in{\mathcal{A}}}m_{ab}\sum_{s\in
U_{b}}m_{rs}^{(a,b)}\leq\sum_{b\in{\mathcal{A}}}m_{a^{\prime}b}\sum_{s\in
U_{b}}m_{rs}^{(a,b)}.$ (39)

Since $M^{(a,b)}\leq_{st}M^{(a^{\prime},b)}$,

$\sum_{b\in{\mathcal{A}}}m_{a^{\prime}b}\sum_{s\in
U_{b}}m_{rs}^{(a,b)}\leq\sum_{b\in{\mathcal{A}}}m_{a^{\prime}b}\sum_{s\in
U_{b}}m_{rs}^{(a^{\prime},b)}=\sum_{(b,s)\in U}p_{(a^{\prime},r),(b,s)}.$

Together with (39), we obtain the right inequality in (38). ∎

##  7 Ratings in the Merton model

To demonstrate the above and further properties of PIT and TTC ratings, we
consider a Merton firm-value model (Merton, 1974), where an additional drift
component, reflecting the economic state, affects the firm’s
profitability.111111For an application to real data using historical S&P
rating data and GDP growth rates, see (Gera, 2020). More specifically,
consider a firm’s asset value process (credit quality process, ability-to-pay
process) $(V_{t})_{t\geq 0}$, with

$V_{t+1}=V_{t}{\bf e}^{(\mu+\mu_{t+1}^{A}-1/2\sigma^{2})+\sigma\Delta
W_{t+1}},$ (40)

where, in the standard Merton-model, $\mu,\sigma>0$ are constants and
$W=(W_{t})_{t\geq 0}$ is a Brownian motion (without loss of generality, we set
$\Delta t=1$). The economic state process $A_{t}$, independent of $W$
introduces an additional drift in the credit quality process with magnitude
given by the one-to-one mapping $v:\mathcal{A}\rightarrow{\mathbb{R}}$, i.e.,
the drift is given by $\mu_{t}^{A}=v(A_{t})$.

The firm is in default at time $t+1$ if $\\{V_{t+1}<D_{t+1}\\}$, with
$D_{t+1}$ the $\mathcal{F}_{t}$-measurable default threshold. In a Markov
setting, $D_{t+1}$ will be $\sigma(V_{t},\mu_{t}^{A})$-measurable. If one
associates with $V$ the firm’s asset value, then $D$ corresponds to the firm’s
debt value, with default taking place if the asset value drops below the debt
value. We give details on how the firm updates its debt level $D$ in
Assumption 1 below.

To ease notation, we write the log asset-to-debt ratio process
$Z_{t}=\ln(V_{t}/D_{t})$. Because default takes place if $Z_{t}<0$, this
process has an interpretation as the “distance to default”. Because the firm
updates $Z_{t}$ (e.g. by updating $D_{t}$), we also write
$Z_{(t+1)-}=\ln(V_{t+1}/D_{t})$ to denote the random variable, which initially
at time $t$ reflects the “distance to default”, and in response to which the
firm updates $D_{t}$ to $D_{t+1}$.

###  7.1 Point-in-time rating

At any time, a firm’s PIT rating is determined by its probability of default.
The non-default rating classes $r_{1},\ldots,r_{J-1}$ are specified via PD
buckets $(d_{j-1},d_{j}]$, $j=1,\ldots J-1$, with PD boundaries
$\\{d_{0}=0,d_{1},\ldots,d_{J-1}=1\\}$. In other words, rating $r_{j}$ is
assigned at time $t$ if the firm’s probability of default at time $t$ is in
$(d_{j-1},d_{j}]$. To ensure that the overall Markov process is time-
homogeneous, we assume that the firm makes adjustments to its capital
structure over time. This assumption is justified e.g. by (Lemmon et al.,
2008) who observe that capital structures and leverage ratios tend to remain
stable over time. In the following, we assume that the firm is not in default
at time $t$.

######  Assumption 1.

At time $t$, the initial debt level is $D_{t}$. Denote
$Z_{(t+1)-}=\ln(V_{t+1}/D_{t})$. The rating class $r_{j}$ at time $t$ is
determined from
${\mathbb{P}}(Z_{(t+1)-}<0|Z_{t},\mu_{t}^{A})\in(d_{j-1},d_{j}]$. Associated
with each rating class is a PD $p_{j}\in(d_{j-1},d_{j}]$, $j=1,\ldots,J$.
Based on the rating class $r_{j}$, the firm chooses its debt level $D_{t+1}$
such that ${\mathbb{P}}(Z_{t+1}<0|Z_{t},\mu_{t}^{A})=p_{j}$.

By construction, a PIT rating is fully determined by a firm’s PD, so a firm
with rating $r_{j}$ has a probability of default of $p_{j}$, regardless of the
state of the economy. Formally,

$R_{t}=r_{j}\quad\iff\quad{\mathbb{P}}(Z_{(t+1)-}<0|Z_{t},\mu_{t}^{A})\in(d_{j-1},d_{j}]\quad\iff\quad{\mathbb{P}}(Z_{t+1}<0|Z_{t},\mu_{t}^{A})=p_{j}.$

This is how PIT ratings are assigned. The following proposition gives details
about the rating process obtained by this construction.

######  Proposition 7.1.

  1. (i)

At time $t$, conditional on no-default, i.e., $R_{t}\in\widehat{\mathcal{R}}$,
which is equivalent to $\\{Z_{t}\geq 0\\}$, the firm’s initial probability of
default is given by

${\mathbb{P}}(Z_{(t+1)-}<0|Z_{t},\mu_{t}^{A})=\sum_{a\in\mathcal{A}}{\rm
N}\left(-\frac{Z_{t}+\mu+v(a)-\frac{1}{2}\sigma^{2}}{\sigma}\right){\mathbb{P}}(\mu_{t+1}^{A}=v(a)|\mu_{t}^{A}),$

where ${\rm N}$ denotes the standard normal distribution function.

  2. (ii)

For each economic state, there exist asset-to-debt ratios
$\alpha_{1},\ldots,\alpha_{J-1}$, determined by solving, for each rating class
$r_{j}$,

${\mathbb{P}}(R_{t+1}=\overline{r}|R_{t}=r_{j},A_{t})=p_{j}=\sum_{a\in\mathcal{A}}{\rm
N}\left(-\frac{\ln(\alpha_{j})+\mu+v(a)-\frac{1}{2}\sigma^{2}}{\sigma}\right){\mathbb{P}}(\mu_{t+1}^{A}=v(a)|\mu_{t}^{A}),$
(41)

where $p_{j}$ is the default probability associated with rating $r_{j}$. The
asset-to-debt ratio correspond to $\alpha_{j}=V_{t}/D_{t+1}$,
$j=1,\ldots,J-1$. In particular, the left equation implies that the rating is
a PIT rating, see Definition 5.1.

  3. (iii)

There exist rating boundaries
$c_{J+1}=-\infty<c_{J}=0<c_{J-1}<\cdots<c_{1}=\infty$ depending on $A_{t}$,
$A_{t+1}$ and $R_{t}$ such that the conditional rating transition
probabilities $m_{r\,r_{j}}^{(a,b)}$ are given by

$m_{r\,r_{j}}^{(a,b)}={\mathbb{P}}(R_{t+1}=r_{j}|R_{t}=r,A_{t}=a,A_{t+1}=b)={\rm
N}(c_{j})-{\rm N}(c_{j+1}),\quad j=1,\ldots,J.$

Using that the asset-to-debt ratio is $\alpha$ given $R_{t}=r$ and $A_{t}=a$,
the boundaries are

$\displaystyle c_{j+1}$
$\displaystyle=\frac{\min(\tilde{c}_{j+1},0)-\ln(\alpha)-\mu-v(b)+1/2\sigma^{2}}{\sigma}$
(42) $\displaystyle c_{j}$
$\displaystyle=\frac{\tilde{c}_{j}-\ln(\alpha)-\mu-v(b)+1/2\sigma^{2}}{\sigma},$
(43)

where $\tilde{c}_{j},\tilde{c}_{j+1}$, depending on $A_{t}=a$, are such that

${\mathbb{P}}(Z_{(t+1)-}<0|Z_{t},\mu_{t}^{A}=v(a))\in(d_{j-1},d_{j}]\quad\iff\quad
Z_{t}\in[\tilde{c}_{j+1},\tilde{c}_{j}).$

  4. (iv)

The process $(R_{t},A_{t})$ is a time-homogeneous Markov process.

###### Proof.

(i) The claim is obtained from
$Z_{(t+1)-}=Z_{t}+\mu+\mu_{t+1}^{A}-\frac{1}{2}\sigma^{2}+\sigma\Delta
W_{t+1}$ and taking expectation with respect to $\mu_{t+1}^{A}$.

(ii) By construction $R_{t}=r_{j}$ if and only if
${\mathbb{P}}(Z_{t+1}<0|Z_{t},\mu_{t}^{A})=p_{j}$. Since the asset-to-debt
ratio $V_{t}/D_{t+1}$ is $(Z_{t},\mu_{t}^{A})$-measurable, $V_{t}/D_{t+1}$ may
be replaced by $\alpha_{j}$, so that

$\displaystyle Z_{t+1}$
$\displaystyle=\ln(V_{t}/D_{t+1})+\mu+\mu_{t+1}^{A}-1/2\sigma^{2}+\sigma\Delta
W_{t+1}$
$\displaystyle=\ln(\alpha_{j})+\mu+\mu_{t+1}^{A}-1/2\sigma^{2}+\sigma\Delta
W_{t+1}.$

The left equation of (41) holds by definition of the rating classes, while the
right equation holds by construction of the PIT rating system on the firm-
value process. It remains to observe that the sum in (41) is a mixture of
functions strictly monotone in $\alpha_{j}$; hence, $\alpha_{j}$ is uniquely
determined given $p_{j}$. (iii) If $j=J$, then

${\mathbb{P}}(R_{t+1}=\overline{r}|R_{t},\mu_{t}^{A},\mu_{t+1}^{A})={\mathbb{P}}(Z_{(t+1)-}<0|R_{t},\mu_{t}^{A},\mu_{t+1}^{A}),$

so $c_{J+1}=-\infty$ and $c_{J}=0$. In the case $j\not=J$, we have

${\mathbb{P}}(R_{t+1}=r_{j}|R_{t},A_{t},A_{t+1})\\\ \begin{aligned}
&={\mathbb{P}}\left({\mathbb{P}}(Z_{t+2}<0|Z_{t+1},\mu_{t+1}^{A})=p_{j},Z_{t+1}>0\big{|}Z_{t},\mu_{t}^{A},\mu_{t+1}^{A}\right)\\\
&={\mathbb{P}}\left({\mathbb{P}}(Z_{(t+2)-}<0|Z_{t+1},\mu_{t+1}^{A})\in(d_{j-1},d_{j}],Z_{t+1}>0\big{|}Z_{t},\mu_{t}^{A},\mu_{t+1}^{A}\right).\end{aligned}$
(44)

On $\\{Z_{t+1}\geq 0\\}$, holding $\mu_{t+1}^{A}$ fixed, the PD
${\mathbb{P}}(Z_{(t+2)-}<0|Z_{t+1},\mu_{t+1}^{A})$ is invertible in $Z_{t+1}$,
since from (i):

${\mathbb{P}}(Z_{(t+2)-}<0|Z_{t+1},\mu_{t+1}^{A})=\sum_{k=1}^{K}{\rm
N}\left(-\frac{Z_{t+1}+\mu+\mu_{t+2}^{A}-\frac{1}{2}\sigma^{2}}{\sigma}\right)\,{\mathbb{P}}(\mu_{t+2}^{A}=v(a_{k})|\mu_{t+1}^{A}),$
(45)

which is a mixture of functions strictly monotone in $Z_{t+1}$. Hence, there
exist boundaries $\tilde{c}_{j+1},\tilde{c}_{j}$ dependent on $\mu_{t+1}^{A}$
such that,

${\mathbb{P}}(Z_{(t+2)-}<0|Z_{t+1},\mu_{t+1}^{A})\in(d_{j-1},d_{j}]\quad\iff\quad
Z_{t+1}\in[\tilde{c}_{j+1},\tilde{c}_{j}).$

Inserting in (45) and using that, if $R_{t}=r$,

$Z_{t+1}=\ln(\alpha)+\mu+\mu_{t+1}^{A}-1/2\sigma^{2}+\sigma\Delta W_{t+1},$

gives

${\mathbb{P}}(R_{t+1}=r_{j}|R_{t}=r,A_{t}=a,A_{t+1}=b)\\\ \begin{aligned}
&={\mathbb{P}}(Z_{t+1}\in[\min(\tilde{c}_{j+1},0),\tilde{c}_{j})|V_{t}/D_{t+1}=\alpha,\mu_{t}^{A}=v(a),\mu_{t+1}^{A}=v(b))\\\
&={\mathbb{P}}(\Delta W_{t+1}\in[c_{j+1},c_{j})),\end{aligned}$

where $c_{j+1}$ and $c_{j}$ are given by (42) and (43), respectively. (iv) It
is a consequence of Assumption 1 and parts (ii), (iii) that the rating
transition probabilities do not depend on time or on the history of the asset-
to-debt ratio. ∎

###  7.2 Through-the-cycle rating

The PIT rating construction implies that, when observing the economic state,
it is equivalent to observe a firm’s credit quality or rating process. The
Markov property is secured via Assumption 1, which links the credit quality
(via the asset-to-debt ratio) to the firm’s rating class.

Conditional TTC rating changes are determined independent of the economic
state, for example due to credit quality changes idiosyncratic to the firm. As
a consequence, building a TTC rating on the process (40) following Assumption
1 will, in general, be non-Markovian, as observing the history of economic
state transitions will reveal additional information on a firm’s probability
of default. Hence, a similar construction for a TTC rating requires different
assumptions on the firm’s activities: achieving the Markov property requires
the firm to eliminate credit quality information from systematic changes
(i.e., the economic state) at each point in time. This stronger assumption
will, in general, lead to a time-inhomogeneous Markovian PIT rating. Depending
on the economic state, one would have to adjust rating boundaries to yield
probabilities of default that are independent of the economic state.

Not only is the stronger assumption suggested above economically implausible,
but from a practical point of view, it is not desirable to have two
disconnected rating systems yielding entirely different rating classes.
Rather, it is reasonable to build on the PIT property, i.e., the connection
between rating class and probability of default, while enforcing the TTC
property.

As a consequence, we shall stick to the construction of the PIT rating, in
particular Assumption 1, and establish the closest TTC rating by minimising
the discrepancy between TTC and PIT default term structures, conditional on
economic states and PIT ratings. This provides a feasible construction of a
TTC rating from a given PIT rating, even without the explicit PIT rating
construction from the previous section.

In the following, we assume throughout that the number of rating classes are
the same for both PIT and TTC ratings.

######  Definition 7.2.

A firm’s default term structure at time $0$ is defined as

$t\mapsto
p_{t}:={\mathbb{P}}(R_{t}=\overline{r}|R_{t-1}\not=\overline{r}),\quad t>0.$

PDs entering the default term structure are calculated as follows: with
$\lambda_{0}$ the initial distribution of rating and economic state,
$\lambda_{t}=\lambda_{0}\cdot P^{t}$ is the time-$t$ distribution, where $P$
is the rating transition matrix given by (1). The time-$t$ default probability
is given by
${\mathbb{P}}(R_{t}=\overline{r})=\sum_{a\in\mathcal{A}}\lambda_{t,(a,\overline{r})}$,
where $\lambda_{t,(a,r)}$, $a\in\mathcal{A}$, $r\in\mathcal{R}$, denote the
entries of $\lambda_{t}$.

The construction of the TTC rating could be done in one step, but due to the
computational burden, we break it down into two steps:

  1. (I)

Define the TTC $Q$ matrix (see Definition 5.2), which embodies the non-default
rating transitions and is independent of the economic state.

  2. (II)

Determine the TTC $P$ matrix and TTC initial distributions conditional on
economic state and PIT rating.

The two steps are now described in detail:

(I). There are different plausible ways to build a TTC’s rating $Q$ matrix:
one can average the PIT rating’s $Q^{(a,b)}$ matrices, which depend on the
economic state transitions. The averaging may be weighed according to the
probabilities of economic state transitions:

${\mathbb{P}}(A_{t+1}=b,A_{t}=a)={\mathbb{P}}(A_{t+1}=b|A_{t}=a)\,{\mathbb{P}}(A_{t}=a)=m_{ab}\cdot\mu_{a}^{A},\quad
a,b\in\mathcal{A},$

where $\mu^{A}$ denotes the stationary distribution of the economic states.

Another method – the one used in the numerical example below – would use the
PIT ratings’ asymptotic $Q$ matrix, which is denoted by $\tilde{Q}$ and
derived from $\tilde{P}_{\infty}$, see Equations (12) and (26).

(II). From Equation (25), a TTC rating transition matrix is characterised by

$M^{(a,b)}=D^{(a,b)}\cdot Q,\quad a,b\in\mathcal{A}.$ (46)

It remains to specify the default probabilities
$m_{r_{j}\overline{r}}^{(a,b)}$, $j=1,\ldots,J-1$, of the default component
matrix $D^{(a,b)}$. The goal is to determine the default probabilities by
calibrating default-term structures for each combination of PIT rating and
economic state. This requires calibrating probabilities of the initial TTC
rating for each of these combinations as well. More specifically, for each
initial state $(a,r)$, the initial probabilities under the TTC rating,
$\lambda_{(a,r)}^{\text{TTC}}\in[0,1]^{KJ}$, are determined by calibrating the
default term structure conditional on the economic state $a$ and the PIT
rating $r$. In addition, default-term structures are fitted where only the
economic state is given and the initial PIT rating is specified by the
stationary distribution of the non-default states. The corresponding TTC
default-term structures (for each economic state) are also determined from the
TTC rating’s stationary distributions. Formally, the calibration problem is
specified as

$\min_{\\{m_{r\overline{r}}^{(a,b)},\lambda_{(a,r)}^{\text{TTC}}|r\in\widehat{\mathcal{R}},a,b\in\mathcal{A}\\}}\left(\sum_{r\in\widehat{\mathcal{R}}}\sum_{a\in\mathcal{A}}\sum_{t\in\mathcal{T}}\left(p_{t,(a,r)}^{\text{TTC}}-p_{t,(a,r)}^{\text{PIT}}\right)^{2}+\sum_{a\in\mathcal{A}}\sum_{t\in\mathcal{T}}(p_{t,a}^{\text{TTC}}-p_{t,a}^{\text{PIT}})^{2}\right)^{1/2}.$
(47)

As outline above, $(p_{t,(a,r)}^{\text{PIT}})_{t\in\mathcal{T}}$ and
$(p_{t,(a,r)}^{\text{TTC}})_{t\in\mathcal{T}}$ as well as
$\lambda_{(a,r)}^{\text{TTC}}$ refer to default term structures, resp. an
initial distribution, conditional on economic state $a$ and PIT rating $r$,
whereas $(p_{t,a}^{\text{TTC}})_{t\in\mathcal{T}}$ and
$(p_{t,a}^{\text{PIT}})_{t\in\mathcal{T}}$ refer to default term structures
where the initial rating distribution is determined from the respective
stationary distribution conditional on economic state $a\in\mathcal{A}$.
Constraints of the optimisation problem ensure that the initial distribution
is a probability measure, that the matrix $P^{\text{TTC}}$ is a stochastic
matrix and that PDs of default term structures with varying initial
distributions are well-ordered.

###  7.3 Example

Parameter | Value
---|---
Merton model
$\mu$ | 0.003
$\sigma$ | 0.006
$\Delta t$ | 1
Economic states
# economic states | 3
Economic state transition matrix $M$ | $\begin{pmatrix}0.8&0.175&0.025\\\ 0.1&0.8&0.1\\\ 0.025&0.175&0.8\end{pmatrix}$
Economic state drift $\mu^{A}$ | $\\{\sigma,0,-\sigma\\}$
Rating classes
# non-default rating classes | 3
PIT default probabilities | $p\in\\{0.0002,0.005,0.025\\}$
PIT rating boundaries | $d\in\\{0,0.0003,0.02,1\\}$
Table 1: Parameters used in the example.

This section illustrates PIT and TTC ratings determined from a Merton model as
outlined in the previous sections. The parameters used are shown in Table 1.

Figure 3: $M^{(a,b)}$ matrices (left) and $P$ matrix (right) of TTC rating.
Economic states are labelled 1 (good), 2 (neutral) and 3 (bad). Darker shades
indicate values closer to one, whereas white indicates a value of zero or
close to zero.

Figure 1 in Section 2.2 shows the PIT $M^{(a,b)}$ matrices and $P$ matrix.
Figure 3 shows the corresponding matrices of the TTC rating. The $M^{(a,b)}$
matrices reveal that, as expected, rating transitions in the PIT rating depend
more strongly on the economic state transition, while transitions to non-
default ratings of the TTC system vary little across economic state
transitions. The PDs of TTC rating classes (last column in each matrix) show
variation across the economic states. It is easily verified that the $P$
matrices are monotone with respect to the upper orthant order. They are even
stochastically monotone, so powers are monotone with respect to the upper
orthant order as well.

Figure 4: Default curves generated by PIT (left) and TTC (right) ratings.
Left, from top to bottom: initial PIT ratings are ordered from best to worst
non-default rating; each curve denotes the economic state in which the curve
starts (blue: good, red: neutral, green: bad). Right: Corresponding TTC
ratings, where initial distributions
$\lambda^{\text{TTC}}_{(a,r)},r\in\mathcal{\hat{R}},a\in\mathcal{A}$, are
calibrated to respective initial PIT rating, see Equation (47).

In the following, we compare various default curves generated by PIT and TTC
ratings. The left-hand side of Figure 4 shows default curves for all
combinations of initial PIT non-default ratings and economic states. The
right-hand side shows the corresponding calibrated TTC ratings. Regardless of
the economic state, the PIT one-period PD depends only on the initial PIT
rating. This is not the case for TTC ratings, although default probabilities
are close, given that each TTC default curve represents firms in one PIT
rating. The default curves converge at the long end across ratings and
economic states. Because TTC rating classes are more stable, due to the
constant $Q$ matrix, convergence tends to occur faster.

Figure 5: Default curves generated by TTC ratings. Colour-coded curves denote
the initial economic state (blue: good, red: neutral, green: bad). The grey
curve denotes the default probabilities of the asymptotic approximation (12),
which corresponds to a TTC rating that has no economic-state information at
all.

Figure 5 shows default curves for every combination of initial TTC rating and
initial economic state. In each graph, the curves are again ordered according
to the initial economic state (blue: good, red: neutral, green: bad). As the
one-period PDs for the best and medium TTC ratings are practically zero,
regardless of the economic state, a firm’s rating path will nearly certainly
pass through the worst TTC rating before it defaults.

Also shown in Figure 5 (in grey) are default curves derived from the
asymptotic approximation (12) of the TTC rating. As before, (conditional)
rating migrations are insensitive to the economic state, but in addition, this
is now also the case for PDs. These PDs meet regulatory capital requirements
by being based on the observed historical average one-year default rate, Basel
Committee on Banking Supervision (2016). It can be seen that these PDs over-
or underestimate real PDs depending on the economic state.

Figure 6: Default curves generated by PIT and TTC ratings where only the
initial economic state is known. The initial distributions of the PIT and TTC
ratings are determined from the stationary distribution of $P$. Colour-coded
curves denote the initial economic state (blue: good, red: neutral, green:
bad). The grey curve denotes the default probabilities of the asymptotic
approximation (12), which corresponds to a TTC rating that has no economic-
state information at all.

Finally, Figure 6 shows default curves of PIT and TTC ratings, where only the
initial economic state is known. The initial distributions are chosen as the
stationary distributions, which are determined from the $P$ matrices with
default states removed, and then conditioned on each economic state. The
curves can be interpreted as representing – conditional on the economic state
– the default probabilities of a randomly picked firm. As such, ideally, the
curves would be equal, as the default probabilities of a randomly picked firm
should not depend on a particular rating philosophy. However, as can be seen,
with the TTC rating being “more stable”, it is harder to differentiate the
neutral and bad economic states than in the case of the PIT ratings where the
initial default probabilities are given.

The grey curve in Figure 6 shows the default curve of the TTC rating’s
asymptotic approximation (12). Given that there is no initial information
about the firm and the economic state, these PDs remain constant through time.

##  8 Conclusion

To conclude, we point out how the rating framework introduced in this paper
complies with requirements of ratings and PDs in both accounting and
regulatory capital standards.

The International Financial Reporting Standard (IFRS) 9 as well as the
accounting standard Current Expected Credit Losses (CECL) require financial
institutions to calculate expected credit losses that reflect current economic
conditions and forecasts of future economic conditions121212International
Financial Reporting Standard (IFRS) 9, Paragraph 5.5.17. Hence, default
probabilities that are used for calculating expected credit losses for
accounting purposes have to be realistic estimates of the actual probability
of default taking the current economic state as well as the future
(stochastic) development of economic conditions into account. As the
accounting standards do not contain any requirements on rating systems, even
default probabilities of a rating process $(R_{t})_{t\in\mathbb{N}_{0}}$ that
is predominantly TTC satisfies the accounting requirements as long as the PD
curve ${{\mathbb{P}}(R_{t}=\overline{r})}_{t\in\mathbb{N}_{0}}$ specified by
$(R_{t})_{t\in\mathbb{N}_{0}}$ provides a realistic estimate of the actual
probabilities of default. This requirement typically implies that default
probabilities ${\mathbb{P}}(R_{t+1}=\overline{r}|R_{t}=r)$ show a strong
dependence on economic states to make up for the low sensitivity of non-
default ratings to economic conditions if $(R_{t})_{t\in\mathbb{N}_{0}}$ is
predominantly TTC.

The PD curve ${{\mathbb{P}}(R_{t}=\overline{r})}_{t\in\mathbb{N}_{0}}$ can be
derived from the time-inhomogeneous transition matrices ${\widetilde{P}}_{t}$
via Equation (10). In our setup, the transition matrices ${\widetilde{P}}_{t}$
depend on the probabilities of the different economic paths
$(a_{0},a_{1},\ldots,a_{t})\in{\mathcal{A}}^{t+1}$ specified by the Markov
process $(A_{t})_{t\in\mathbb{N}_{0}}$. In practice, the specification of
economic scenarios $(a_{0},a_{1},\ldots,a_{T})$ is based on actual macro-
economic forecasts, which provide the economic information required for the
transition matrices ${\widetilde{P}}_{1},\ldots,{\widetilde{P}}_{T}$ up to a
forecast horizon $T$. For extending the default curve beyond $T$, the
asymptotic transition matrix ${\widetilde{P}}_{\infty}$ is typically used,
which formalizes the long-term average migrations of the rating process, see
Equation (12).

In contrast to accounting requirements, financial regulators wish to avoid
procyclical capitalisation of financial institutions, as this would
incentivise firms to reduce their capital when the economy thrives, which in
turn would require firms to increase their capital in economic downturns, when
this is particularly difficult to achieve. As this procyclical capitalisation
poses a threat to financial stability, financial regulators prefer banks to
assess regulatory capital requirements for credit risk using through-the-cycle
(TTC) ratings,131313Capital requirements regulation and directive – CRR/CRD
IV, Article 502 which are designed to be stable through the business cycle. In
addition, default probabilities in regulatory capital requirement calculations
should be based on the observed historical average one-year default rate, see
Basel Committee on Banking Supervision (2016). In order to satisfy these
requirements in our setup, consider a rating migration process
$(R_{t})_{t\in\mathbb{N}_{0}}$ that is predominantly TTC. The asymptotic PDs
of $(R_{t})_{t\in\mathbb{N}_{0}}$ correspond to the historical average one-
year default rate of this rating migration process. Hence, the asymptotic
approximation $(S_{t})_{t\in\mathbb{N}_{0}}$ based on the transition matrix
${\widetilde{P}}_{\infty}$ (see Definition 4.2) is a rating migration process
that is consistent with the regulatory requirements listed above. However, as
the example in Section 7.3 demonstrates, $(S_{t})_{t\in\mathbb{N}_{0}}$ does
not provide realistic PD estimates. In particular, these PDs – shown in
Figures 5 and 6 – tend to underestimate the actual default risk in a negative
economic environment. Alternatively, the asymptotic default probabilities
specified by ${\widetilde{P}}_{\infty}$ could be replaced by default
probabilities taken from a conditional transition matrix $M^{(a,b)}$, where
the choice of the economic states $a,b\in{\mathcal{A}}$ reflects the
underlying economic assumptions. For example, if the transition from $a$ to
$b$ represents a negative economic scenario the default probabilities in
$M^{(a,b)}$ could be interpreted as stressed PDs. If default probabilities are
specified in this way the underestimation of the actual default risk could be
avoided, even in a negative economic environment, but at the expense of overly
conservative PD estimates during benign economic times.

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