[subsection]section *[subsubsection]subsection

# An introduction to rating triggers for collateral-inclusive XVA in an ICTMC
framework

Kevin Kamm Dipartimento di Matematica, Università di Bologna, Bologna, Italy.
e-mail<EMAIL_ADDRESS>
###### Abstract

In this paper, we model the rating process of an entity as a piecewise
homogeneous continuous time Markov chain. We focus specifically on calibrating
the model to both historical data (rating transition matrices) and market data
(CDS quotes), relying on a simple change of measure to switch from the
historical probability to the risk-neutral one. We overcome some of the
imperfections of the data by proposing a novel calibration procedure, which
leads to an improvement of the entire scheme. We apply our model to compute
bilateral credit and debit valuation adjustments of a netting set under a CSA
with thresholds depending on ratings of the two parties.

Keywords: Ratings, Rating Triggers, Default Probability, First to default,
Inhomogeneous Markov processes, ICTMC, Exponential Change of Measure, XVA.
Acknowledgements: This project has received funding from the European Union’s
Horizon 2020 research and innovation programme under the Marie Sklodowska-
Curie grant agreement No 813261 and is part of the ABC-EU-XVA project.

Special thanks goes to the whole team of Banco Santander’s XVA desk in Madrid,
in particular to my supervisor Luca Caputo, Michael Belk and Jerome Vincent
Maetz for great discussions and help in implementing this approach.

##  1 Introduction

In this paper, we study bilateral credit and debit valuation adjustments
(hereafter referred to as $\mathrm{CVA}$ and $\mathrm{DVA}$) of a portfolio of
trades between two parties having signed a collateral agreement dependent on
ratings. Recall that $\mathrm{CVA}$ and $\mathrm{DVA}$ are adjustment to the
fair price of a financial portfolio accounting for the potential loss in case
of default of the counterparty and the owner, respectively. These are usually
defined as an average of the exposure (positive and negative, respectively)
weighted by the probability of default. The two parties usually sign a so-
called netting agreement, so as to consider the exposure at portfolio level
(as opposed to trade-wise). Attached to the netting agreement, one often has a
Credit Support Annex (CSA) by which each of the two parties further agrees to
interchange securities (referred to as collateral) to reduce the exposure of
the other party. In the case of bankruptcy the collateral account can be used
to mitigate the losses of the non-defaulting party, although collateral is
often non-segregated and therefore also at risk. Since posting collateral is
another expense for an entity, it is desirable to keep the postings as small
as possible while simultaneously keeping the losses due to a default event
small as well. To this goal, more and more CSAs specify thresholds of
permitted unsecured exposure in terms of the credit quality of the parties:
the higher the credit quality of a party, the smaller the amount collateral it
has to post (and the larger the unsecured exposure of the other party).

A customary way to measure the credit quality of an entity is to use credit
ratings. A high rating means that the entity is very likely to fulfill its
financial obligations towards its contracting party, while a low rating
associates an increasing risk for meeting the aforementioned obligations. In
this line of thought, the default can be viewed as the worst possible rating.
CSAs dependent on ratings are said to have rating triggers: a change of rating
of one of the parties triggers a change of threshold of that party.

Since the exposure depends on the amount of collateral posted or received, to
compute $\mathrm{CVA}$ and $\mathrm{DVA}$ in presence of a CSA with rating
triggers it is necessary to model the rating processes of the contracting
parties. As always in modeling, there are trade-offs between describing the
real-world behavior and numerical or analytical tractability. We decided in
favor of numerical tractability to use Markov processes. The memoryless
property of such processes make them somewhat unrealistic as models for the
rating process, since an entity with a history of successive downgrades is
more likely to be considered risky than a competitor with long-time constant
rating. Another aspect to take into account is whether the rating process can
be assumed to be time homogeneous. It is empirically evident that a time
homogeneity assumption is often violated. However, the data published by
rating agencies is scarce, for example for short times, e.g. one year, only
one, three, six and twelve month rating matrices are usually available. This
is precluding the possibility to fit a totally inhomogeneous process to the
data in a meaningful way. In most of the literature, this motivates the usage
of pure time-homogeneous models. However, we decided to use piecewise
homogeneous continuous time Markov chains (PHCTMC) to model the rating
evolutions to capture at least some time-dependent features of the rating
evolution. Both assumption are discussed by [22] in greater details.

We calibrate the rating processes simultaneously to historical rating matrices
published by rating agencies (such as Moody’s, S&P and Fitch) and default
probabilities, derived from Credit Default Swaps (CDS) market quotes. These
are not only two different sources of data inputs but also have a different
underlying probability space. Rating transitions are considered under the
historical measure, while CDS market quotes are under a risk-neutral
probability measure. Hence, it is necessary to have an explicit change of
measure formula to ensure a fast calibration procedure.

In the literature, this problem is overcome by extracting the generator of the
underlying Markov chain from the rating transition data and calibrating a
parametrized change of measure formula to the CDS quotes afterwards. Note,
that the risk-neutral measure is an output of the calibration procedure. We
will comment on this later on in Remark 3.3 in more detail.

We found that this split procedure does not perform very well and use instead
a penalized calibration procedure allowing for some error with respect to the
previously obtained the market generators. This means that the output of the
calibration procedure is simultaneously a generator under the historical
measure and the parameters for the change of measure. This type of calibration
procedure seems to be novel compared to the results in the literature and
performs very well.

To assess the performance of this model we will compare three different
collateral agreements: an agreement with perfect collateralization, no
collateralization and with rating triggers. As expected the model with rating
triggers will reduce the losses due to default events but its performance is
dependent on the immediate jumps from high ratings to default. We will call
the distribution of jumps prior to default the pre-default distribution and
study it in the end of the paper.

###  1.1 Review of the literature and comparison

Rating models have been studied for several years. As described in [5, p. 76
Section 4.12.1 Standing Assumptions] one can identify two different approaches
to rating modeling.

On the one hand, one can model ratings in a HJM-framework, independently
proposed by [7] and [30], and, on the other hand, there are intensity-based
models, introduced by the pioneering work of [18]. As this paper also belongs
to the intensity-based models, we will briefly summarize the available
literature on the intensity-based approach, focusing particularly on those
results which provide context to our results.

The characteristic of this approach is that the historical and risk-neutral
measure are given and the rating model itself is defined under a third
measure, which is calibrated to the risk-neutral measure. Usually, Markov
processes are used in this framework, in particular CTMCs, whose generator
takes the role of the intensities known from default modeling. However,
contrary to default modeling, one is not only interested in the first jump
time of the process but in all transitions over time from one state to the
other.

In [18] the authors propose a continuous-time Markov chain model for the
rating process. To tackle the problem of historical versus risk-neutral data
they start with a CTMC under the historical measure and assume that there
exists a risk-neutral generator given by
$A_{t}=\mathrm{diag}\left(\mu_{1}(t),\dots,\mu_{K-1}(t),1\right)A$, where $A$
denotes the generator under the historical measure and $\mu_{i}(t)$ are
positive integrable functions. In the main application the $\mu_{i}$ are
assumed to be constants which amounts to assuming that the rating process is a
time-homogeneous CTMC. The coefficients $\mu_{i}$, which are calibrated to
credit risky bonds, can be thought as risk-premia relating the historical
measure to the risk-neutral measure, although the underlying change of measure
is not described explicitly in the paper (a rigorous proof that such a change
of measure exists can be found in [8, p. 12 Example 2.9.]).

In [6] the authors are interested in bilateral CVA under rating triggers, as
well, but are focusing on so-called _Close-outs_ , namely clauses dictating
the termination of the portfolio whenever a given credit rating is reached by
one of the parties. To model the rating evolution they use a Markov Copulae
for multivariate time-homogeneous Markov chains to include the possibility of
calibrating the rating processes of different sectors in a consistent way. We
will discuss this issue further in Section 4. For the necessary change of
measure they apply, in contrast to [18], an exponential change of measure
technique proposed by [26], which we will also use in this paper.

In more recent works of Bielecki et al. ([9] and [10]) conditional Markov
chains and Markov copulae approaches are proposed and analysed in a more
general mathematical framework.

A detailed overview of CTMC approaches can be found in [8] in which they also
discuss the time-inhomogeneous case briefly. Additionally, in [11, pp. 351 ff.
Chapter 12 Markovian Models for Credit Migrations] is a detailed overview of
further models for credit migrations.

We will compare our work mainly to [18] and [6]. Also we will elaborate how
the different change of measure techniques therein relate to each other.

Like [6] we are using CTMCs but for simplicity we are not using a copulae
approach, which would be more realistic and is subject to future research.
However, we are using a simple inhomogeneous extension instead of time-
homogeneous CTMCs. For the calibration they are restricting the space of
possible parameters, which will not be necessary in our case and allows for
better calibration results.

Thus, the main contribution of this paper is to propose a model to price
$\mathrm{CVA}$ and $\mathrm{DVA}$ in presence of collateralization with rating
triggers in the context of piecewise-homogeneous continuous time Markov
chains. Moreover, we propose a simple calibration procedure with major
improvements to straightforward techniques. Last but not least, we highlight
the importance of studying the pre-default rating distribution in the context
of collateralization with rating triggers.

The paper is organized as follows. In Section 2 we introduce the mathematical
framework of inhomogeneous continuous-time Markov chains and discuss the
change of measure technique. This is followed by a brief discussion about
extracting a generator from real data, called _embedding problem_ , in Section
2.1. Then, we start with the numerical tests in Section 3. We will first
explain the data we use and its issues in Section 3.1 and explain how to
calibrate the PHCTMC model to the data in Section 3.2. Afterwards, we will use
the Gillespie Stochastic Simulation Algorithm to simulate our PHCTMC under
both measures and use the simulated processes as our rating processes in the
context of rating triggers in Section 4. In Section 4.1 we will first
calculate $\mathrm{CVA}$, $\mathrm{DVA}$ and $\mathrm{BVA}$ and dicuss the
pre-default rating distribution and its impact on $\mathrm{XVA}$ in Section
4.2. Last but not least, we will conclude the paper in Section 5 and discuss
future extensions and research.

##  2 ICTMCs and an exponential change of measure formula

We begin this section by describing the features making CTMC a good choice for
a rating model. First, it is a natural choice to consider a discrete state-
space consisting of all ratings and a continuous-time framework. Moreover,
rating transition probabilities have to be time and state-dependent to reflect
the different intensities associated to the different rating-changes over
time. About this dependency, two aspects must be considered (see [30, Section
8.2] for more details): time homogeneity and the Markov property. Both
assumptions are not completely realistic although time homogeneity can be seen
as a stronger restriction than the Markovianity assumption.

As far as calibration is concerned, it is desirable for the model to be
consistent with all the available data, namely both the historical rating
transition matrices and the market quoted credit spreads (either coming from
bonds or from credit default swaps). For this to be feasible, one needs a
change of measure formula, preferably preserving some of the feature of the
model (for instance, Markovianity).

Finally, for numerical and/or analytical tractability of the model is
certainly a fundamental need. This led us to consider Markov processes for the
rating process. CTMCs are then a natural choice in view of the above
(continuous-time, a discrete state-space and Markovianity). As far as time
dependence is concerned, we propose a simple non-homogeneous CTMC model,
namely a piecewise-homogeneous CTMC (PHCTMC).

Let us briefly recall the relevant terminology of ICTMCs (cf. [11, pp. 326 ff.
Chapter 11 Markov Chains]). An ICTMC is a Markov process
$\left(X_{t}\right)_{t\in[0,T]}$ with a discrete state space $S$. We will only
consider the case of a finite state space, i.e.
$S=\left\\{1,\dots,K\right\\}$. The functions
$p_{ij}(s,t)\coloneqq\mathbb{P}\left(\left.X_{t}=j\right|X_{s}=i\right)$ are
called _transition functions_ and their time-derivative from above is called
_generator_ of the ICTMC, i.e. $A_{ij}(t)\coloneqq\lim_{h\downarrow
0}\frac{p_{ij}(t,t+h)-\delta_{ij}}{h}$, $i,j=1,\dots,K$ and $\delta_{ij}$ is
the Dirac-delta. We have the following immediate properties of generator:
$A_{ij}(t)\geq 0$ for all $i\neq j$ and $A_{ii}(t)=-\sum_{j\neq
i}{A_{ij}(t)}$.

For the change of measure we have the following result from [14, pp. 13–15
Section 3 The Equivalent Martingale Measure] or [11, pp. 334 ff. Chapter
11.2.5 Change of Probability Measure]:

######  Theorem 2.1.

Let $\left(X_{t}\right)_{0\leq t\leq T}$ be an ICTMC on the probability space
$\left(\Omega,\mathcal{F},\left(\mathcal{F}_{t}\right),\mathbb{P}\right)$
taking values in $S\coloneqq\left\\{1,\dots,K\right\\}$ with generator
$A^{\mathbb{P}}_{t}\coloneqq\left(A_{i,j}^{\mathbb{P}}(t)\right)_{i,j=1,\dots,K}$.
Define

$\displaystyle L_{t}^{\kappa}\coloneqq\exp\left(-M_{t}\right)\prod_{0<u\leq
t}{\left(1+\sum_{i,j=1}^{K}{\kappa_{ij}(u)\left(H_{ij}(u)-H_{ij}(u-)\right)}\right)},$

where

  1. (i)

the stochastic processes $\kappa_{i,j}(t)$ are bounded, real-valued and
$\mathbb{F}$-predictable, such that $\kappa_{ij}(t)>-1$ and
$\kappa_{ii}(t)=0$;

  2. (ii)

the number of jumps from rating $i$ to $j$ are $H_{ij}(t)\coloneqq\sum_{0\leq
u\leq t}{\mathbbm{1}_{X_{u-}=i}\mathbbm{1}_{X_{u}=j}}$;

  3. (iii)

$M_{t}\coloneqq\int_{0}^{t}{\sum_{i,j=1}^{K}{\kappa_{ij}(u)A^{\mathbb{P}}_{i,j}(u)\mathbbm{1}_{X_{u}=i}}du}.$

Then $L_{t}^{\kappa}$ is a strictly positive martingale under $\mathbb{P}$
satisfying $\mathbb{E}^{\mathbb{P}}\left[L_{T}\right]=1$ and the equivalent
probability measure $\mathbb{Q}^{\kappa}$ given by

$\displaystyle\left.\frac{d\mathbb{Q}^{\kappa}}{d\mathbb{P}}\right|_{\mathcal{F}_{t}}=L^{\kappa}_{t}$

$\mathbb{P}$-almost surely, is well-defined.

Furthermore, $X_{t}$ is an ICTMC under $\mathbb{Q}^{\kappa}$ as well with
generator

$\displaystyle A_{i,i}^{\kappa}(t)$ $\displaystyle=-\sum_{i\neq
j}{A_{i,j}^{\kappa}(t)}$ $\displaystyle A_{i,j}^{\kappa}(t)$
$\displaystyle=\left(1+\kappa_{i,j}(t)\right)A_{i,j}^{\mathbb{P}}(t).$

We can see that for any admissible family
$\kappa\coloneqq\left(\kappa_{ij}\right)_{i,j=1,\dots,K}$ we get a valid
change of measure and know the generator of the ICTMC after the change of
measure as well.

As mentioned in the introduction, there are two major examples of this change
of measure in the rating community.

######  Example 2.2.

On the one hand, if we set $\kappa_{ij}(t)\coloneqq h_{i}(t)-1$ for strictly
positive deterministic functions $h_{i}(t)>0$, $i\neq j$, we recover the
change of measure by [18], which we will call JLT-change of measure from now
on.

On the other hand, if we set
$\kappa_{ij}(t)\coloneqq\frac{h_{i}(t)}{h_{j}(t)}-1$ we get a time-
inhomogeneous version of the exponential change of measure technique by [26],
which we will call exponential change of measure throughout the entire paper.

In both cases, we will denote the resulting measure by $\mathbb{Q}^{h}$ and
the generators under this measure by $A^{h}_{t}$.

The reason for choosing only $K$ different functions $h_{i}$ is due to the
fact that after the change of measure we know only data with respect to the
default column, and there seems to be no meaningful way to use an entire
matrix $\kappa_{ij}$.

With this change of measure formula, we can calibrate the model (see Section
3.2) under the historical measure $\mathbb{P}$ and risk-neutral measure
$\mathbb{Q}$ by finding the appropriate functions $h$, such that
$\mathbb{Q}^{h}$ and $\mathbb{Q}$ are close with respect to the default
probabilities.

But before we can start with the calibration procedure we need to discuss how
one extracts the generator $A^{\mathbb{P}}$ from the historical data and
associated problems, which is the subject of the following subsection.

###  2.1 Embedding problem

Another important aspect, which has to be studied for rating models in the
context of ICTMCs, is how to extract the generator of an ICTMC from the
historical data, because we will need it for the change of measure formula and
the simulation of the process.

In the homogeneous case, the question boils down to asking if the given
transition matrix has a matrix logarithm, such that all off-diagonal entries
are positive and all row sums are zero.

In general, this is still an open problem and in the literature referred to as
_embedding problem_ , which is deeply connected to root finding techniques.
The paper by [23] provides a nice overview of the known conditions on the
existence of a generator for both the homogeneous and inhomogeneous case.

As aforementioned, in general the embedding problem is not fully studied at
this point in time and has only been solved for $2\times 2$ and $3\times 3$
matrices in general. A condition for $4\times 4$ can be found in [13] with a
good overview of the existing literature.

For our approach the paper by [16] is of particular interest, since it deals
with the embedding problem in a rating context and provides us with a
numerical justification to approximate a generator from given historical data.
They also note, that in general the data does not yield a valid generator,
therefore an adjustment to either the rating matrices or the extracted
generator is necessary. Another numerical approach to approximate a generator
from given historical data can be found in [20]. They use a best approximation
technique in a suitable space for transition matrices.

In Section 3.2 we will apply the simple justification by [16] and mitigate the
effects of this approximation in our calibration procedure. We use the simple
approach because this part is not our main focus in this paper. However, we
would like to caution the reader that this choice might have a significant
impact on the outcome. Further studies would be needed to estimate the
sensitivity of this choice but for this more data is required. We will briefly
discuss this issue in Section 3.1: note that the published data does not even
yield directly a valid transition matrix due to _withdrawals_ , namely those
cases where companies stop being rated (for different reasons) during the
period of calculation of a rating matrix.

##  3 Numerical implementation of ICTMCs for rating triggers

After establishing the change of measure in Theorem 2.1 for the general case
of ICTMCs, we will now consider piecewise homogeneous CTMCs (PHCTMC)
$X_{\cdot}$ in this section. To be more precise, let $T_{0}\in[0,T]$ be the
initial time and $T_{k}$, $k=1,\dots,n$ be the points in time when historical
rating matrices—denoted by $R^{M}_{t}$—are available in an increasing order
with $T_{n}=T$, then $X_{\cdot}$ is assumed to be homogeneous on each
$[T_{k},T_{k+1})$ $k=0,\dots,n-1$.

We used for the calculations Matlab 2021a with the (Global) Optimization
Toolbox running on Windows 10 Pro, on a machine with the following
specifications: processor Intel(R) Core(TM) i7-8750H CPU @ 2.20 GHz and 2x32
GB (Dual Channel) Samsung SODIMM DDR4 RAM @ 2667 MHz.

The section is organized as follows: First, we have a look at the available
historical data and the problems intertwined with the data in Section 3.1.
Then, we will calibrate a PHCTMC to the market data in Section 3.2 and we
compare the exponential change of measure to the JLT change of measure in
Section 3.3. Last but not least, we simulate the trajectories of the PHCTMC
using Gillespie’s stochastic simulation algorithm in Section 3.4.

###  3.1 Historical and Market Data

Appendix B contanins the collection of the rating matrices under the
historical measure and default probabilities under the risk neutral measure,
which we use in this paper.

The result presented in the paper are obtained using the historical data
published by Fitch Ratings from 2014 containing one, three, six and twelve
month transition matrices. We verified that our method also works with the
matrices published by S&P from 2020 but we decided not to include the results,
since they mirror the ones obtained from the first data set. Note that,
regardless of the source of data, the rows of historical transition matrices
do not in general sum up to one due to companies, which stop being rated
during the matrix calculation period (see for instance Table 18).
Unfortunately, it is not known publicly when and with what rating a company
drops out of the rating procedure, making it very difficult to reconstruct the
data. We will come back to this problem in Remark 3.4.

The default probabilities for the different initial ratings are obtained from
market CDS spreads published at the beginning of 2022. Specifically we
considered the spreads for the the financial sector, without distinguishing by
geography. To derive default probability from spreads we used a standard
bootstrapping procedure.

###  3.2 Calibration and rating matrix adjustments

The calibration procedure has the following three steps: First, we need to
adjust the historical data to remove the withdrawal component, then we have to
extract an approximation of the generator and finally calibrate the ICTMC to
the previously obtained generators with the exponential change of measure
technique.

Let us first of all fix some notations throughout this section: As mentioned
above, $R^{M}_{t}$ will denote the historical rating matrix at time $t$ with
possible row-sum less than one, $R^{A}_{t}$ will be the adjusted rating matrix
with row-sum equal to one and $A^{M}_{t}$ will denote the approximation of the
generator of $R^{A}_{t}$.

#### Historical rating matrix adjustment

As mentioned in Section 3.1, this step is performed without further knowledge
on the time of withdrawal and rating of the company at the time of withdrawal.
When performing the adjustment, one would like to have some general properties
the adjusted matrix should satisfy. However, there seems to be no consensus on
what properties a rating matrix should have, e.g. monotone increasing rows
till the diagonal element and monotone decreasing afterwards. The only
property, which is mentioned in the literature is [18, p. 15 Lemma 2 (Credit
Ratings Versus Risk)], ensuring that lower credit ratings should be riskier.
Nevertheless, this does not seem to be enough to describe the usual observed
shape of a rating matrix and is up to future research to find a suitable
mathematical and economical characterization. We noticed that there are three
prominent components: first, the default column and row have different
properties compared to the rest of the matrix, second the neighboring region
to the default column (usually one or two ratings), which seems to stretch
over time, will be influenced by the default column over time and last but not
least, the rest of the matrix has usually a skewed Gaussian shape around the
diagonal, making it more likely to decrease in rating.

We decided to use the approach described in Algorithm 1 for the initial data
reconstruction. The effects of this adjustment will be mitigated in the
calibration step but this choice might have a significant impact on the
outcome of the entire model. First, we iterate over the rows of the input
historical rating matrix $R^{M}_{t}$. Then we calculate the withdrawal. If the
withdrawal is greater than zero, we first decide, that a transition
probability should never be zero and set it to a small number. Afterwards we
calculate the weights of each rating in the current row by dividing each entry
by the total mass of the row. Now, we calculate the adjustment for each entry
by multiplying the weights with the withdrawal. The adjusted rating matrix
will then be the sum of the adjustment and the current row of the historical
rating matrix. If the withdrawal was zero, we set the current row of the
adjusted rating matrix to the row of the historical rating matrix.

Input : $R_{t}^{M}\in\mathbb{R}^{K,K}$ with row-sums less or equal to $1$

Output : $R_{t}^{A}\in\mathbb{R}^{K,K}$ with row-sums equal to $1$

for  _$i\leftarrow 1$ to K_ do

wd $\leftarrow\sum_{j=1}^{\textnormal{{K}}}{\left(R_{t}^{M}\right)_{ij}}$ ;

if  _wd $>0$_ then

y $\leftarrow\left(R_{t}^{M}\right)_{i,j=1,\dots,\textnormal{{K}}}$;

y $\left(\textnormal{{y}}==0\right)\leftarrow 1e^{-10}$;

b
$\leftarrow\frac{\textnormal{{y}}}{\sum_{j=1}^{K}{\textnormal{{y}}_{j}}}\cdot\left(1-\textnormal{{wd}}\right)$;

$\left(R_{t}^{A}\right)_{i,j=1,\dots,\textnormal{{K}}}\leftarrow\left(R_{t}^{M}\right)_{i,j=1,\dots,\textnormal{{K}}}+\textnormal{{b}}$

else

$\left(R_{t}^{A}\right)_{i,j=1,\dots,\textnormal{{K}}}\leftarrow\left(R_{t}^{M}\right)_{i,j=1,\dots,\textnormal{{K}}}$

end if

end for

Algorithm 1 Adjustment of the historical rating matrices.

#### Market generator extraction

As discussed in Section 2.1 not all given datasets yield a valid generator.
Therefore, an approximation is needed, which we perform in two steps and is
described in Algorithm 2. First, we apply the matrix logarithm (cf. [2] and
[1]) to the adjusted rating matrix. Then, we set the diagonal and any negative
entries of the matrix logarithm to zero. For a justification of setting the
negative entries to zero we refer to [16, p. 6 Section 3 The Non-Negativity
Condition]. Now, we sum up the rows and set the new diagonal to the negative
value of the sums.

A more sophisticated approach is presented in [20] by using a best-
approximation approach in a suitable space.

Input : $R_{t}^{A}\in\mathbb{R}^{K,K}$

Output : $A_{t}^{M}\in\mathbb{R}^{K,K}$ approximated generator of $R_{t}^{A}$

A $\leftarrow\textnormal{{logm}}\left(R_{t}^{A}\right)$;

$\textnormal{{A}}\left(\textnormal{{A}}<0\right)\leftarrow 0$;

for  _$i\leftarrow 1$ to K_ do

$\left(\textnormal{{A}}\right)_{ii}\leftarrow-\sum_{j\neq
i}{\left(\textnormal{{A}}\right)_{ij}}$;

end for

$A_{t}^{M}\leftarrow\textnormal{{A}}$;

Algorithm 2 Approximation of the historical generators.

#### Calibrating the PHCTMC

As aforementioned, we need to calibrate the rating model to the historical
data and market data with their respective measures. On the $\mathbb{P}$-side
we have the rating matrices $R^{M}_{t}\in\mathbb{R}^{K,K}$, discussed above,
and on the $\mathbb{Q}$-side we have the default probabilities
$\mathrm{PD}(t)\in\mathbb{R}^{K,1}$ for each initial rating.

Let us start under the historical measure $\mathbb{P}$.

As aforementioned, let $T_{0}\in[0,T]$ be the initial point and $T_{k}$,
$k=1,\dots,n$ be the points in time, when rating matrices are available in an
increasing order with $T_{n}=T$, then $X_{\cdot}$ is assumed to be homogeneous
on each $[T_{k},T_{k+1})$ $k=0,\dots,n-1$.

Now, by the Chapman-Kolmogorov equation we get

$\displaystyle U^{\mathbb{P}}_{T_{0},T}=U^{\mathbb{P}}_{T_{0},T_{1}}\cdot
U^{\mathbb{P}}_{T_{1},T_{2}}\cdots
U^{\mathbb{P}}_{T_{n-1},T_{n}}=\prod_{k=1}^{n}{U^{\mathbb{P}}_{T_{k-1},T_{k}}}.$
(3.1)

By homogeneity on each sub-interval we know that the evolution system will
reduce to a semigroup and its generator will be time-constant with explicit
formula

$\displaystyle
U^{\mathbb{P}}_{T_{k-1},t}=U^{\mathbb{P}}_{t-T_{k-1}}=\exp\left(A_{k}^{\mathbb{P}}\left(t-T_{k-1}\right)\right),\quad
t\in[T_{k-1},T_{k}).$ (3.2)

Hence, to extract these generators from the historical data, we denote by
$R^{M}_{k}$ the rating matrices at times $T_{k}$, $k=1,\dots,n$ and solve

$\displaystyle U^{\mathbb{P}}_{T_{0},T_{k}}\overset{!}{=}R^{M}_{k},$

which is by (3.2) under the assumption that $U^{\mathbb{P}}_{T_{0},T_{k-1}}$
is invertible and $\left(U^{\mathbb{P}}_{T_{0},T_{k-1}}\right)^{-1}\cdot
R^{M}_{k}$ has a matrix logarithm equivalent to

$\displaystyle
A_{k}^{\mathbb{P}}=\frac{\log\left(\left(U^{\mathbb{P}}_{T_{0},T_{k-1}}\right)^{-1}\cdot
R^{M}_{k}\right)}{T_{k}-T_{k-1}},\qquad
U^{\mathbb{P}}_{T_{0},T_{k-1}}=\prod_{l=1}^{k-1}{U^{\mathbb{P}}_{T_{l-1},T_{l}}}.$

This yields an iterative scheme on the $\mathbb{P}$-side and we will use the
same procedure described in Algorithm 2 to retrieve valid generators at each
time.

On the risk-neutral side we proceed similarly. The generator on the
$\mathbb{Q}$-side is assumed to have the same properties as the one on the
$\mathbb{P}$-side, hence the change of measure formula in Example 2.2 will be
piecewise constant on each sub-interval, as well, such that we solve a
minimization problem at each time $T_{k}$ by finding the appropriate vector
$h_{k}\in\mathbb{R}^{K}_{>0}$.

Additionally, let us assume that the default state is absorbing, i.e. we have
$e_{K}^{T}R^{M}_{t}e_{i}=\delta_{Ki}$, $i=1,\dots,K$, $t\geq 0$, and therefore
$Ae_{i}=0$. Hence, we can choose the last entry in the vector
$\left(h_{k}\right)_{K}$ freely in each interval and will assume
$\left(h_{k}\right)_{K}\equiv 1$ without altering the following minimization
problem for finding the right change of measure:

$\displaystyle\min_{\begin{subarray}{c}h_{k}\in\mathbb{R}^{K,1}_{>0},\left(h_{k}\right)_{K}=1,\\\
A_{k}\in\mathcal{A}\end{subarray}}\left\|U_{T_{0},T_{k-1}}^{\mathbb{Q}^{L}}\cdot\exp\left(A_{k}^{h}\left(T_{k}-T_{k-1}\right)\right)\cdot
e_{K}-\mathrm{PD}\left(T_{k}\right)\right\|_{\mu_{\mathbb{Q}}}+\left\|A_{k}-A_{k}^{\mathbb{P}}\right\|_{\mu_{\mathbb{P}}},$
(3.3) $\displaystyle\mathcal{A}\coloneqq\left\\{A\in\mathbb{R}^{K,K}:\text{
for all $i,j=1,\dots,K$ }A_{K,j}=0,\ A_{i,j}\geq 0\text{ for $i\neq j$ and
}A_{i,i}\leq 0\right\\}.$ $\displaystyle
A_{k}^{h}\coloneqq\begin{cases}A_{k}\frac{\left(h_{k}\right)_{j}}{\left(h_{k}\right)_{i}},&i\neq
j,\\\ -\sum_{k\neq
i}{A_{k}\frac{\left(h_{k}\right)_{j}}{\left(h_{k}\right)_{i}}},&i=j,\\\
\end{cases}$ $\displaystyle
U_{T_{0},T_{k-1}}^{\mathbb{Q}^{L}}=\prod_{l=1}^{k-1}{U^{\mathbb{Q}^{L}}_{T_{l-1},T_{l}}}.$

The norms
$\left\|\cdot\right\|_{\mu_{\mathbb{Q}}},\left\|\cdot\right\|_{\mu_{\mathbb{P}}}$
are weighted norms and defined as follows: Let $X\in\mathbb{R}^{K,K}$,
$x\in\mathbb{R}^{K,1}$ and $M^{\mathbb{P}}\in\mathbb{R}^{K,K}_{\geq 0}$, as
well as $m^{\mathbb{Q}}\in\mathbb{R}^{K,1}_{\geq 0}$, we set

$\displaystyle\left\|X\right\|_{\mu_{P}}$
$\displaystyle\coloneqq\left\|M^{\mathbb{P}}\odot X\right\|_{F},$
$\displaystyle\left\|x\right\|_{\mu_{Q}}$
$\displaystyle\coloneqq\left\|m^{\mathbb{Q}}\odot x\right\|_{2},$

where $\odot$ denotes the elementwise or Hadamard product. The idea of using
weighted sums in the calibration procedure is explained in Remark 3.1 in more
detail.

Moreover, after this piecewise calibration, the whole right-generator of the
ICTMC is then defined as
$A^{P}_{t}=\sum_{k=1}^{n}{A^{P}_{k}\mathbbm{1}_{[T_{k-1},T_{k})}(t)}$,
$P=\mathbb{P},\mathbb{Q}$ and to evaluate the evolution system one has either
to use (3.1) or solve the Kolmogorov forward equation with the inhomogeneous
generator $A^{P}_{t}$.

Table 1: One year rating matrix under measure $\mathbb{Q}$ after calibration with $M^{\mathbb{P}}\equiv+\infty$ using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ |  $0.000$ % |  $98.047$ % |  $0.000$ % |  $0.000$ % |  $0.838$ % |  $0.351$ % |  $0.764$ %
F1 |  $0.000$ % |  $97.033$ % |  $0.000$ % |  $0.000$ % |  $1.468$ % |  $0.696$ % |  $0.802$ %
F2 |  $0.000$ % |  $89.720$ % |  $0.000$ % |  $0.000$ % |  $5.570$ % |  $2.860$ % |  $1.850$ %
F3 |  $0.000$ % |  $85.545$ % |  $0.000$ % |  $0.000$ % |  $7.724$ % |  $4.049$ % |  $2.683$ %
B |  $0.000$ % |  $1.868$ % |  $0.000$ % |  $0.000$ % |  $56.863$ % |  $32.511$ % |  $8.758$ %
C |  $0.000$ % |  $0.044$ % |  $0.000$ % |  $0.000$ % |  $13.455$ % |  $71.167$ % |  $15.334$ %
D |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $100.000$ %

######  Remark 3.1.

Let us explain why we decided to include the generator under the measure
$\mathbb{P}$ in the calibration procedure. First of all, we know that our
initial adjustment has been done in an arbitrary way and impacts the final
results. In the case, where $M^{\mathbb{P}}\equiv+\infty$, i.e. the
calibration is without the generator under $\mathbb{P}$, we get results that
look like Table 1. We see columns consisting of zero entries. This is
obviously nothing like we expect a rating matrix to look like. For example,
the first column would suggest that it is impossible to transition to or even
stay in rating F1+, which is does not make sense, since companies with the
best rating usually have the highest probability to stay in their rating.
Therefore, we added the generator under $\mathbb{P}$ to the calibration
procedure to mitigate the effects of our initial choice of data reconstruction
leading to a significant improvement and making the model feasible if one
compares Table 1 to Table 14. In Table 14 we chose $M^{\mathbb{P}}\equiv 1$
and $m^{\mathbb{Q}}\equiv 1$.

As a side-note the choice of restricting the parameter space in the case
$M^{\mathbb{P}}\equiv+\infty$ to $h_{i}\coloneqq e^{\alpha i}$ as in [6] did
not help to remove the columns with zero entries and decreased the performance
for $M^{\mathbb{P}}\equiv 1$ and $m^{\mathbb{Q}}\equiv 1$.

Having now two different components in the objective function, we added the
possibility to add weights corresponding to the user’s trust in the data, e.g.
they may depend on liquidity and the size of the withdrawal. If the data is
assumed to be trustworthy, one can choose the individual weight parameter
large, which ensures a better fit in this particular entry.

######  Remark 3.2.

We tried different versions of (3.3). In particular, we adjusted $A_{k}$ by
Algorithm 2 in the objective function but this led to a decrease in accuracy
and increase in computational time. Moreover, we added the condition that the
sum of each row in the generator is supposed to be zero to the constraints,
which led to badly conditioned matrices and unpredictable behavior in the
calibration algorithm.

Therefore, we decided to leave it unconstrained but keep the bounds on the
matrix entries and adjust the outcome of the calibration for the generator
under $\mathbb{P}$ by Algorithm 2, which leads to the presented results. To
make this more precise, we did not add the constraint to $\mathcal{A}$ that
all rows must sum to zero. Therefore, the matrix $A_{k}$ as an output of the
calibration procedure is not necessary a valid generator but we use Algorithm
2 after the calibration procedure to repair it. Let us denote this adjustment
$\widetilde{A}_{k}$ for the moment.

The average error of this final adjustment
($\frac{1}{K^{2}}\sum_{k,i,j}\left|\widetilde{A}_{k}-A_{k}\right|$) after the
calibration procedure was of magnitude $10^{-4}$ for both Fitch and S&P’s
datasets, when summing the absolute values of the differences of all entries
and dividing by the number of entries, i.e. the error of this adjustment is
negligible and makes the calibration algorithm more robust and justifies why
we did not include the “sum to zero”-constraint in $\mathcal{A}$.

Moreover, we added the constraint that lower credit ratings should be riskier
introduced by [18, p. 15 Lemma 2 (Credit Ratings Versus Risk)] to the
minimization problem as well, but it did not change the results significantly.
Conclusively, we decided not to include it in the paper.

In particular, these choices make it possible to implement the calibration
procedure as a weighted non-linear least squares problem.

######  Remark 3.3.

Notice, that in our calibration procedure, we are in fact introducing an error
in measure. This is because the default probabilities coming from CDS quotes
were derived in a measure $\mathbb{Q}$ and our change of measure is
parametrized by parameters $h$.

We are not aware of any literature discussing this issue and its impact is
unknown. However this procedure seems to be common practice, since a-priori
the Radon-Nikodym derivative in our rating setting is not known.

We used Matlab 2021a’s function fmincon to solve the minimization problem with
bounds (3.3) and display the corresponding errors in Table 4. For the errors
we used the Frobenius norm to compare the resulting rating matrices under
$\mathbb{P}$ to the adjusted rating matrices and the Euclidean norm to compare
the resulting probabilities of default to the market data under $\mathbb{Q}$.
We divided both norms by the number of entries, i.e. $K^{2}$ for the Frobenius
norm and $K$ for the Euclidean norm. The total computational time for the
calibration was 4.77 seconds with Fitch’s data and 15 seconds with S&P’s data.

Table 2: Calibration errors using Fitch’s data. First row is the error of fmincon, second the mean error of the model rating transitions and adjusted market rating transitions and the third row contains the errors under the risk neutral measure of the model and market probabilities of default. Error Time | $t=\frac{1}{12}$ | $t=\frac{3}{12}$ | $t=\frac{6}{12}$ | $t=1$
---|---|---|---|---
fmincon | 1.52e-07 | 8.99e-08 | 6.91e-08 | 6.37e-08
$\frac{1}{K^{2}}\left\|R_{t}^{\mathbb{P}}-R_{t}^{A}\right\|_{\mathbb{R}^{K,K}}$ | 2.69e-06 | 2.35e-05 | 0.000101 | 0.000464
$\frac{1}{K}\left\|R_{t}^{\mathbb{Q}}e_{K}-\mathrm{PD}(t)\right\|_{\mathbb{R}^{K}}$ | 1.85e-06 | 4.93e-09 | 1.31e-08 | 6.51e-09

All results using Fitch’s data can be found in Appendix A starting with the
generators after calibration in Appendix LABEL:sec:generatorsP.

######  Remark 3.4.

We were also interested in the case of long term historical calibrations, i.e.
using three or five year rating matrices. However, we noticed that the
withdrawal column was the dominant column. Therefore, it is questionable if it
makes sense to reconstruct a row of a rating matrix with a withdrawal of e.g.
50 percent without further knowledge.

Another aspect is that the property we hinted at in Section 3.2 of a Gaussian
on the diagonal is not justifiable for long-terms. This might be connected to
the aging-effects discussed in [3]: older bonds might have a tendency to be
up- or downgraded in a smaller period of time than newer bonds. Whether this
effect is taken into account by rating agencies nowadays is not clear and
should be addressed in future research.

Additionally, it is worth pointing out that rating agencies could use the so-
called Aalen-Johansen estimator (cf. [21]) instead of a cohort method
(sometimes called static pool method) to compute the rating matrices. This
would naturally remove the withdrawal rates. Moreover, let us mention that
rating agencies are required by “Rule 17g-7 of the Securities Exchange Act of
1934” to publish certain rating changes of individual companies. We
implemented a cohort method using this data. However, the results were not
close to the published rating matrices. Possible reasons for this could be
unpublished sensitive rating data and an initial step to remove correlation
structures within the data by using additional knowledge about the companies.

We confirmed that S&P includes such data taking these two points into
consideration in their usual subscription for the interested reader.

Conclusively, for the time being we restrict ourselves to historical
transition matrices up to one year published by the rating agencies in this
paper.

###  3.3 Comparison to [18] change of measure

To make this comparison, let us first of all recall their change of measure
formula. They assume that the risk-neutral generator is given by

$\displaystyle
A^{\mathbb{Q}}\coloneqq\mathrm{diag}\left(\widetilde{h}_{1},\dots,\widetilde{h}_{K-1},1\right)A^{\mathbb{P}},$

where $\widetilde{h}_{i}\geq 0$ for $i=1,\dots,K-1$.

Furthermore, they suppose that the generator under the historical measure can
be extracted from the data: here we will use Algorithm 1 and 2 to extract it.

Contrary to [18] we will compare the methods using CDS quotes instead of risky
bonds to calibrate the parameters $\widetilde{h}_{i}$ but this is no major
change.

Since this is a homogeneous CTMC, we will first calibrate it to the one year
CDS data and compare it to the one year transitions of the PHCTMC.

The calibration works excellent: Since, we use the adjusted rating matrix,
$R_{t}^{\mathbb{P}}$ is the same as $R_{t}^{A}$ and for the default
probabilities we obtain
$\frac{1}{K}\left\|R_{t}^{\mathbb{Q}}e_{K}-\mathrm{PD}(t)\right\|_{\mathbb{R}^{K}}=1.79e-10$.
It took only $0.2$ seconds to calibrate the values $\widetilde{h}$.

However, the rating transitions apart from the default probabilities look
rather peculiar under the risk-neutral measure, which are printed in Table 3.
In particular, the rating transitions far from the diagonal entries seem too
high and consequently the probability to stay in a rating seems too low.

Table 3: One year rating matrix under measure $\mathbb{Q}$ after calibration with [18] change of measure using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ | 57.20 % | 26.53 % | 12.77 % | 1.56 % | 1.19 % | 0.21 % | 0.50 %
F1 | 14.37 % | 39.27 % | 36.17 % | 4.72 % | 3.87 % | 0.84 % | 0.74 %
F2 | 2.47 % | 9.88 % | 66.04 % | 10.00 % | 8.37 % | 2.10 % | 1.11 %
F3 | 1.73 % | 5.07 % | 39.91 % | 18.13 % | 23.89 % | 7.53 % | 3.70 %
B | 0.42 % | 1.29 % | 12.81 % | 13.05 % | 42.86 % | 20.85 % | 8.68 %
C | 0.085 % | 0.257 % | 3.040 % | 4.98 % | 26.15 % | 50.14 % | 15.33 %
D | 0 % | 0 % | 0 % | 0 % | 0 % | 0 % | 100 %

###  3.4 Simulation of the ICTMC

There are several techniques in the literature concerning the efficient
simulation of ICTMCs and its forward equation. For a detailed discussion we
refer to [24] and [4] among many others. However, for our PHCTMC described in
the previous Section 3.2 we will use the Gillespie Stochastic Simulation
Algorithm (SSA) (cf. [15]), which is also called _Kinetic Monte Carlo (KME)_ ,
on each sub-interval, where the PHCTMC is homogeneous. It turns out that this
approach is sufficiently fast, because our state space has few states. The
algorithm is displayed in Algorithm 3 and works as follows:

Remember that our PHCTMC is homogeneous on each interval $[T_{k-1},T_{k}]$,
$k=1,\dots,n$, and now iterate over those intervals, i.e. assume that we are
already at $t=T_{k-1}$ with current rating $i$. On each sub-interval we
proceed as follows:

  1. (i)

If $t\leq T_{k}$ and $\left(A_{k}\right)_{ii}\neq 0$ draw two uniform random
numbers $r_{1}$, $r_{2}$, otherwise end and set $R^{i_{0}}_{t}=i$ on
$[t,T_{k}]$;

  2. (ii)

Retrieve the exponentially distributed transition waiting time with parameter
$-\left(A_{k}\right)_{ii}$ as

$\displaystyle\tau=\frac{-\log\left(r_{1}\right)}{-\left(A_{k}\right)_{ii}}=\frac{\log(r_{1})}{\left(A_{k}\right)_{ii}}.$

If $t+\tau\geq T_{k}$ set $R^{i_{0}}=i$ and go to the next interval, starting
with step (i), else continue to calculate the next state;

  3. (iii)

Now, sample from the discrete state transition distribution
$\left[\frac{\left(A_{k}\right)_{ij}}{-\left(A_{k}\right)_{ii}}\right]_{j\neq
i}$. This can be done by choosing the first integer $j$, such that
$\sum_{k=1,k\neq
i}^{j}{\frac{\left(A_{k}\right)_{ij}}{-\left(A_{k}\right)_{ii}}}>r_{2}$, which
is equivalent to

$\displaystyle\min_{j}\sum_{l=1,l\neq
i}^{j}{\left(A_{k}\right)_{il}}>-\left(A_{k}\right)_{ii}r_{2}$

Now, go back to (i) with $R^{i_{0}}_{t+\tau}=j$.

Input : $A\in\mathbb{R}^{n,K,K}$ generator, $i_{0}\in\\{1,\dots,K\\}$ initial
state, $M\in\mathbb{N}$ number of trajectories

Output : $R_{t}^{i_{0}}\in[0,T]\times\mathbb{R}^{1,K,M}$ trajectories of the
simulated PHCTMC starting in $i_{0}$

for  _$m\leftarrow 1$ to M_ do

$i\leftarrow i_{0}$;

for  _$k\leftarrow 1$ to n_ do

while  _$t <T_{k}$_ do

if  _$A_{k,i,i}==0$_ then

\\\Absorbing state

$t\leftarrow T_{k}$;

else

\\\Calculate waiting time

$r_{1}\leftarrow\textnormal{{Uniform}}\left(0,1\right)$;

$\tau\leftarrow\frac{\log\left(r_{1}\right)}{A_{k,i,i}}$;

\\\Update time

if  _$t+\tau\geq T_{k}$_ then

$t\leftarrow T_{k}$;

break ;

else

$t\leftarrow t+\tau$;

end if

\\\Calculate state transition;

$\textnormal{{temp}}\leftarrow\textnormal{{cumsum}}\left(\frac{A_{k,i,j=1,\dots,i-1,i+1,\dots,K}}{-A(k,i,i)}\right)$;

$r_{2}\leftarrow\textnormal{{Uniform}}\left(0,1\right)$;

$j\leftarrow\textnormal{{findFirst}}\left(\textnormal{{temp}}\geq
r_{2}\right)$;

if  _$j <i$_ then

$i\leftarrow j$;

else

$i\leftarrow j+1$;

end if

end if

$\textnormal{{update}}\left(R_{t}^{i_{0}}\right)$;

end while

$\textnormal{{update}}\left(R_{t}^{i_{0}}\right)$;

end for

end for

Algorithm 3 Iterative SSA for the PHCMTC starting in
$\left\\{1,\dots,K\right\\}$.

On the other hand, we could also implement a time-dependent version of the
SSA, which is discussed in e.g. [29, pp. 5 ff. Section 4 The Stochastic
Simulation Algorithm], [28] and [17] among many others. In particular, we will
use the method described in [27, pp. 7 ff.]. The principal of the algorithm is
essentially the same as before, but now the waiting time is a general
exponential random variable depending on an integral. This leads to an
integral equation
$\int_{t}^{t+\tau}{\left(A_{s}\right)_{i,i}ds}=\log\left(r_{1}\right)$, where
$r_{1}$ is a uniform random number, in each iteration, which has to be solved
for $\tau$ and the performance of the algorithm is highly dependent on the
ability of solving this integral equation very fast.

Table 4: Calibration errors using Fitch’s data. First row is the mean error of the model rating transitions and simulated rating transitions under the historical measure. The second row contains the errors under the risk neutral measure of the model and simulated rating transitions. Error Time | $t=\frac{1}{12}$ | $t=\frac{3}{12}$ | $t=\frac{6}{12}$ | $t=1$
---|---|---|---|---
$\frac{1}{K^{2}}\left\|R_{t}^{\mathbb{P}}-R_{t}^{\text{Sim},\mathbb{P}}\right\|_{\mathbb{R}^{K,K}}$ | 0.000179 | 0.000469 | 0.000632 | 0.000806
$\frac{1}{K^{2}}\left\|R_{t}^{\mathbb{Q}}-R_{t}^{\text{Sim},\mathbb{Q}}\right\|_{\mathbb{R}^{K,K}}$ | 0.000332 | 0.00077 | 0.00121 | 0.00172

It turned out that both algorithms performed the same regarding simulation
errors, which are given in the last two rows of Table 4. The errors were
computed by first calculating the transition matrices from the simulated
rating processes $R^{i,P}_{t}$, $P=\mathbb{P},\mathbb{Q}$, by counting how
many trajectories are at each state and dividing by the total amount of
trajectories. The result of this is denoted by $R^{\text{Sim},P}_{t}$ and we
used the Frobenius norm divided by the squared number of ratings to evaluate
the error. However, the iterative version was significantly faster than a
naive implementation of the general scheme, i.e. roughly 1.69 seconds using
Fitch’s data seconds compared 500 seconds. For S&P’s data Algorithm 3 took
7.66 seconds. We calculated the SSAs for each initial rating in parallel on a
CPU. Therefore, we used Algorithm 3 for all further calculations.

Figure 1: Simulated trajectories of PHCTMC $X_{\cdot}$ calibrated to Fitch’s
data set starting in rating F2. The top picture is under measure $\mathbb{P}$
and the bottom picture is under measure $\mathbb{Q}$.

##  4 Application to rating triggers for collateral-inclusive bilateral
valuation adjustments

In this section, we discuss collateral-inclusive bilateral valuation
adjustments with rating-dependent CSA thresholds. The main references for the
general theory of XVA are [25, pp. 375 ff. Chapter 12.3 Credit Valuation
Adjustment and Risk Management] and [12, pp. 305 ff. Chapter 13 Collateral,
Netting, Close-Out and Re-Hypothecation]. Throughout this section, we are
taking the point of view of a bank having a portfolio of deals with a
counterparty. The two parties have signed a netting set agreement with a CSA
having rating-dependent thresholds. The stochastic process representing the
future mark-to-market of the portfolio is denoted by $V_{t}$. We will assume
that both contracting parties are subject to default and the default time will
be denoted by $\tau_{B}$ and $\tau_{C}$ for the bank and the counterparty,
respectively. Additionally, we will suppose that the same rating matrices
apply to both, i.e. they are in the same industrial sector and the
consideration of two different sectors (cf. [6] for a copula approach) is left
for future research. For our illustration, we will assume that the bank has
the highest rating today, whereas the counterparty has a mid-range rating
today, since we expect a bank to default less likely than the majority of
companies. The evolution of their ratings over time will be denoted by
$\mathcal{R}_{t}^{B}$ and $\mathcal{R}_{t}^{C}$, respectively, and we will set
$\mathcal{R}_{t}\coloneqq\left(\mathcal{R}_{t}^{B},\mathcal{R}_{t}^{C}\right)$
to shorten notation.

Let $C_{t}$ be the stochastic process representing the value of the collateral
account. In particular $C_{t}>0$ if the collateral is received by the bank. We
will assume for simplicity that $C_{t}$ depends on $V$ only through the value
$V_{t}$: more precisely we will suppose that $C_{t}\coloneqq
f\left(V_{t},\mathcal{R}_{t}\right)$. In particular to avoid path dependencies
we assume there are no minimum transfer amounts, the impact of this assumption
being not material for our purposes.

We will discuss the following three scenarios of collateral agreements:

  1. (i)

_uncollateralized_ , i.e. no collateral is interchanged and $f\equiv 0$;

  2. (ii)

_perfectly collateralized_ , i.e. collateral is posted instantaneously at a
discrete set of times, e.g. daily, and is equal to the mark-to-market
($V_{t}=C_{t}$), $f(v,r)=v$;

  3. (iii)

_rating-trigger dependent_ , more precisely we will focus on the case of
thresholds depending on rating (see below for a description of $f$ in this
case).

The relation of the bank to the counterparty is illustrated in Figure 2 and
reads as follows:111 We will use the same conventions as in [12, pp. 310 ff.
Chapter 13.2 Bilateral CVA Formula under Collateralization], in particular
$X^{+}=\max\left(X,0\right)$ and $X^{-}=\min\left(X,0\right)$. :

To illustrate the impact of collateral in risk mitigation, let us assume
instantaneous posting and no rehypothecation of collateral for simplicity.
Assume the counterparty (but not the bank) defaults at time $\tau$. Then on
the one hand we have the value of the portfolio $V_{\tau}$ and on the other
hand we have the value of the collateral account just before default
$C_{\tau-}$. We distinguish four cases:

  * •

$V_{\tau}\geq 0,C_{\tau^{-}}\geq 0$: the portfolio generates a positive
exposure for the bank but this is mitigated by the collateral (which can be
fully retrieved by the bank because of no rehypotecation). Therefore the
outstanding claim is $V_{t}\tau-C_{\tau-}$.

  * •

$V_{\tau}\geq 0,C_{\tau-}\leq 0$: although the portfolio generates a positive
exposure for the bank, the bank had posted collateral just before default.
Because of no rehypothecation, the bank can fully get back its collateral and
the outstanding claim is therefore $V_{\tau}$.

  * •

$V_{\tau}\leq 0,C_{\tau-}\geq 0$: the counterparty gets back the collateral
posted to the bank and also gets the value of the portfolio
$\left|V_{\tau}\right|$.

  * •

$V_{\tau}\leq 0,C_{\tau-}\leq 0$: the counterparty keeps the collateral posted
by the bank and also gets the remaining value of the portfolio
$\left|V_{\tau}-C_{\tau-}\right|$.

The behavior in case of default of the bank is symmetrical.

Additionally, the individual collateral postings depending on the collateral
agreement are depicted by $f\left(V_{t},\mathcal{R}_{t}\right)^{-}$, meaning
that the bank has to post collateral if this value is greater than zero and
its analogue for the counterparty is given by
$f\left(V_{t},\mathcal{R}_{t}\right)^{+}$.

A comprehensive explanation of all default events in this bilateral setup can
be found in [12, pp. 311–312 Chapter 13.2.1 Collection of CVA Contributions].

Bank Counterparty Portfolio $V_{t}$ Collateral Account $C_{t}$ Potential Loss
$V_{t}-C_{t}$ $\left(V_{t}-C_{t}\right)^{+}$$\left(V_{t}-C_{t}\right)^{-}$
$f\left(V_{t},\mathcal{R}_{t}\right)^{-}<0$: post collateral $C_{t}>0$ in
favor for $f\left(V_{t},\mathcal{R}_{t}\right)^{+}>0$: post collateral
$C_{t}<0$ in favor for$V_{t}>0$ creditor on default$V_{t}<0$ creditor on
default

Figure 2: Illustration of bank and counterparty relations in terms of exposure
and collateral agreements.

Next, we will discuss the impact of rating triggers compared to the
aforementioned scenarios of collateral agreements on collateral-inclusive CVA,
DVA and BVA, followed by a discussion on the pre-default distribution of the
rating processes in Section 4.2.

###  4.1 XVA with different collateral agreements

We are interested in the impact of rating triggers on $\mathrm{BVA}$,
$\mathrm{CVA}$ and $\mathrm{DVA}$ (Collateralized Bilateral, Credit, Debit
Valuation Adjustments) without the possibility of re-hypothecation and zero
interest rate at mid-market to simplify the investigation.

Before we dive into this topic, let us first of all discuss our benchmark
portfolio. Since our main purpose is to analyse the general behavior of the
rating model, we are not interested in setting up an accurate model for the
computation of $V_{t}$. In particular the $V_{t}$ that we consider doesn’t
represent the value of a portfolio of real deals. Rather we decided to
simulate $V_{t}$ using a number of independent Brownian motions with different
volatilities and life-times to account for the cash-flows of the portfolio. To
be more precise

$\displaystyle V_{t}\coloneqq
V_{0}+\sigma_{0}W^{0}_{t}+\sum_{i=1}^{n}{\sigma_{i}W^{i}_{t}\mathbbm{1}_{t\leq\tau^{i}}},$

where $V_{0}\in\mathbb{R}_{\geq 0}$ is the initial value, $W^{i}$,
$i=0,\dots,n$, $n\in\mathbb{N}$, are independent Brownian motions,
$\sigma_{i}\in\mathbb{R}$ are volatilities and $\tau^{i}\in[0,T]$ are
uniformly distributed random variables describing the different life-times of
the cash-flows. In the experiment we use $V_{0}=0$, $n=24$ and $\sigma_{i}$
are the standard normal random variables multiplied by $10$ for scaling and
its sign indicates a positive or negative cash-flow (from the bank
perspective). Also notice that we designed the portfolio in such a way that at
least one cash-flow survives till $T$ by not adding a finite life-time to
$W_{t}^{0}$.

Now, let us briefly recall the relevant definitions of $\mathrm{XVA}$ (cf.
[12, p. 314 Equation 13.4], [12, p. 316 Equation 13.10]) without re-
hypothecation

$\displaystyle\mathrm{BVA}\left(t,T,\mathrm{C}\right)$
$\displaystyle\coloneqq\mathrm{DVA}\left(t,T,\mathrm{C}\right)-\mathrm{CVA}\left(t,T,\mathrm{C}\right),$
(4.1) $\displaystyle\mathrm{DVA}\left(t,T,\mathrm{C}\right)$
$\displaystyle\coloneqq-\mathbb{E}^{\mathbb{Q}}\left[\left.\mathbbm{1}_{\tau=\tau_{B}<T}\mathrm{LGD}_{B}\left(V_{\tau}^{-}-C_{\tau}^{-}\right)^{-}\right|\mathcal{G}_{t}\right],$
(4.2) $\displaystyle\mathrm{CVA}\left(t,T,\mathrm{C}\right)$
$\displaystyle\coloneqq\mathbb{E}^{\mathbb{Q}}\left[\left.\mathbbm{1}_{\tau=\tau_{C}<T}\mathrm{LGD}_{C}\left(V_{\tau}^{+}-C_{\tau}^{+}\right)^{+}\right|\mathcal{G}_{t}\right],$
(4.3)

where $\mathcal{G}_{t}$ is the filtration containing all the default-free
market information plus default monitoring. These values are calculated under
a risk-neutral measure $\mathbb{Q}$, which explains why we were interested in
changing the measure of our rating model from the historical probabilities
$\mathbb{P}$ to the risk-neutral measure in the first place.

The evaluation of the collateral account at the exact time of the default
event, i.e. $C_{\tau}$, might seem confusing. We could imagine a scenario in
which bonds or stocks could be used as collateral, making it necessary to
evaluate the collateral account at the default event. In our case, we will
assume that the collateral account will be a pure cash account, meaning that
upon a default event the value will not be updated from its previous value
$C_{\tau-}$. Therefore, it is very important to study the distribution of
ratings prior to default, which is subject to Section 4.2.

We now describe the function $f\left(\mathrm{V}_{t},\mathcal{R}_{t}\right)$ in
the case of rating-triggers dependent agreements, following [12, pp. 316 ff.
Chapter 13.5.2 Collateralization Through Margining]. Let $r_{i}^{x}\geq 0$,
$x\in\left\\{B,C\right\\}$, $i=1,\dots,K$ denote the threshold for the party
$x$ in case $x$ has rating $i$: this means that the maximum unsecured exposure
of the other party will be at most $r_{i}^{x}$.

Now, we introduce the rating triggers $\rho^{x}$ with corresponding thresholds
$r_{i}^{x}$ as

$\displaystyle\rho^{x}(i)\coloneqq\sum_{j=1}^{K}{r_{j}^{x}\mathbbm{1}_{j}(i)}.$

As a small example, setting for all $i=1,\dots,K$ and $x=\left\\{B,C\right\\}$
the thresholds $r_{i}^{x}=+\infty$ leads to the uncollateralized scenario and
$r_{i}^{x}=0$ to the perfectly collateralized scenario.

The amount of collateral to be posted by the bank at time $t_{j}$ is then

$\displaystyle\left(\mathrm{V}_{t_{j}}+\rho^{B}(\mathcal{R}^{B}_{t_{j}})\right)^{-}-C_{t_{j}-}^{-}.$

For the counterparty we have analogously

$\displaystyle\left(\mathrm{V}_{t_{j}}-\rho^{C}(\mathcal{R}^{C}_{t_{j}})\right)^{+}-C_{t_{j}-}^{+}.$

As aforementioned, we assume for simplicity that the value $C_{t_{j}}$ of the
collateral account at time $t_{j}$ is equal to $C_{\beta(t_{j})}$ where
$\beta(u)$ is the last collateral posting date before $u$. In particular we
assume there is no remuneration on the collateral account. We then have

$\displaystyle C_{t_{0}}\coloneqq 0,\quad C_{t_{n}}\coloneqq 0,\quad
C_{u-}\coloneqq C_{\beta(u)}.$ $\displaystyle C_{t_{j}}\coloneqq
C_{t_{j}-}+\left(\left(\mathrm{V}_{t_{j}}+\rho^{B}(\mathcal{R}^{B}_{t_{j}})\right)^{-}-C_{t_{j}-}^{-}\right)+\left(\left(\mathrm{V}_{t_{j}}-\rho^{C}(\mathcal{R}^{C}_{t_{j}})\right)^{+}-C_{t_{j}-}^{+}\right).$

This can be rewritten as

$\displaystyle
C_{t_{j}}=\left(\mathrm{V}_{t_{j}}+\rho^{B}(\mathcal{R}^{B}_{t_{j}})\right)^{-}+\left(\mathrm{V}_{t_{j}}-\rho^{C}(\mathcal{R}^{C}_{t_{j}})\right)^{+}\eqqcolon
f\left(\mathrm{V}_{t_{j}},\mathcal{R}_{t_{j}}\right).$

We will use in all experiments 365 posting dates per year. In Figure 3 one can
see one trajectory of the portfolio, collateral account and individual
postings by both counterparties in the top picture. The picture in the middle
indicates the ratings of both counterparties over time for this particular
trajectory and the bottom picture shows the corresponding threshold for each
point in time. The orange boxes are magnifications of the indicated sections
in the figure.

One can see that for this choice of trajectory the bank has no rating
transition and the counterparty has many, ranging through all thresholds
defined in Table 5.

At the section “Zoom A” one can see that only the bank has to post collateral
(green dots), since the portfolio is negative and the allowed threshold is
exceeded. In section “Zoom B” we can see the first impact due to the change of
the countparty’s threshold, which now has to post collateral (red dots). We
can see that the distance between the blue dashed line (the collateral
account) and the black line (the portfolio) is closer than e.g. in “Zoom A”,
because the threshold is now five million euros. After time $t=0.35$ the
rating of the counterparty reaches C. Therefore, the threshold is set to zero
and we can see that the collateral stays close to the value of the portfolio,
whenever its positive, which is also the content of “Zoom C”.

Table 5: Rating thresholds for both counterparties using Fitch’s data. Values in Euro. F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---
Bank
$1e+07$ | $1e+07$ | $1e+07$ | $5e+06$ | $5e+06$ | $0$ | $0$
Counterparty
$1e+07$ | $1e+07$ | $1e+07$ | $5e+06$ | $5e+06$ | $0$ | $0$
Zoom AZoom BZoom CZoom AZoom BZoom CZoom AZoom BZoom CZoom AZoom BZoom C
Figure 3: One trajectory of a collateral agreement with rating triggers using
Fitch’s data. Top picture shows the collateral account and portfolio over
time, middle one the rating evolution and bottom one the corresponding rating
thresholds.

In Table 6 are the values of $\mathrm{DVA}$ (4.2), $\mathrm{CVA}$ (4.3) and
$\mathrm{BVA}$ (4.1) using the Loss-Given-Default $\mathrm{LGD}_{B}=0.6$,
$\mathrm{LGD}_{C}=0.6$ and $M=10000$ simulations for the three collateral
agreements: no collateralization, perfect collateralization and
collateralization with rating triggers.

One can see, that the collateralization with rating triggers lies in between
the values of the uncollateralized case and the perfectly collateralized case,
which is the expected behaviour, because as illustrated in Figure 3 one has a
possible transition from unsecured money to the perfectly collaterlized
scenario, where rating thresholds are zero. The difference to the perfectly
collateralized case is that there can be transitions from high ratings to
default in one instant, which will be subject of the next subsection.

Table 6: $\mathrm{XVA}$ with the different collateral agreements (no, perfectly and rating triggers) using Fitch’s data and $\mathrm{LGD}_{B}=0.6$, as well as $\mathrm{LGD}_{C}=0.6$ with $M=10000$ simulations and thresholds defined in Table 5. CXVA | Uncollateralized | Rating Triggers | Perfectly collateralized
---|---|---|---
CDVA | $9.13e+04$ | $5.05e+04$ | $2.39e+04$
CCVA | $9.96e+04$ | $5.67e+04$ | $3.09e+04$
CBVA | $-8.3e+03$ | $-6.23e+03$ | $-7.07e+03$

###  4.2 Pre-default rating distribution

As it is apparent from the definition of the rating thresholds, it is
important to study the distribution of the rating process one time-instant
prior to default, because this will determine the unsecured amount of money at
the default event. We will call this henceforth pre-default distribution and
will also compare the distribution under $\mathbb{P}$ to the one under
$\mathbb{Q}$ with the help of Figure 4, which were obtained by Monte-Carlo
simulation.

Now, let us have a closer look at Figure 4. First of all, one can see the pre-
default distribution under the measure $\mathbb{P}$ in the top picture and
under the measure $\mathbb{Q}$ in the bottom picture. Disregarding the
individual colors, the probability of being in a certain rating prior to
default is given by the total height of the column. The composition of the
individual colors of each column indicate the contribution of each starting
rating, e.g. in the column prior to default we can see that the most prominent
contributions are resulting from the two ratings prior to default, e.g. in
Fitch’s data rating B and C, but there are contributions of the other ratings
as well.

In the market, it can be observed that the default probabilities in the risk
neutral world are usually higher than the default probabilities quoted under
the historical measure in the rating matrices. This phenomenon has an impact
in our model on all other ratings as well, which can already be seen in Figure
1 by the spread of the grey lines indicating all simulated trajectories. In
the risk-neutral world there seem to be more transitions than in the
historical world causing this spread of grey lines. The reason for this is
that the calibration of this model has essentially one parameter for each
rating because $h_{i}\in\mathbb{R}^{K}$ for each time interval where the
process is homogeneous. Therefore, the high probability of default under the
measure $\mathbb{Q}$ compared to the one under $\mathbb{P}$ has a significant
impact on the other ratings as well.

We can see that under the measure $\mathbb{P}$, the top picture, almost all
the defaults had a prior rating of C, while under the measure $\mathbb{Q}$,
this is still the most prominent case but significantly smaller. It is more
likely under the measure $\mathbb{Q}$ that a company starting with a high
rating defaults and this without transitioning to the rating prior to default
first, which is indicated by the different heights of the each individual
color for each rating.

It is yet an open question and needs thorough economical investigation whether
this behavior makes sense or not, because it has a significant impact on the
performance of collateralization with rating triggers. To be more precise, the
more likely it is, that a company starting in a good rating defaults without
first transitioning to a rating, where a low threshold is defined, the more
unsecured money we have at a default event.

Figure 4: Pre-default distribution using Fitch’s data. The upper picture is
under the historical measure and the lower plot under the risk-neutral
measure.

##  5 Conclusion and future research

In this paper, we have seen how to use piecewise homogeneous continuous-time
Markov chains to model rating transitions. We have demonstrated how the most
popular change of measure techniques perform on our dataset and found an
improvement by introducing a penalized calibration procedure. After that, we
computed $\mathrm{CVA}$ and $\mathrm{DVA}$ by simulating rating processes from
the rating transitions by means of the stochastic simulation algorithm. We
showed that rating triggers lowered both $\mathrm{CVA}$ and $\mathrm{DVA}$ as
expected and discussed the pre-default distribution in the end.

Allowing for small errors of the generator in the calibration under the
historical measure revealed a high sensitivity of the model to its generator.
In this line of thought, the impact of the initial adjustment of the defect
rating matrices published by the agencies might be very severe. In future
research, we would like to consider a stochastic model for rating transitions,
maybe including a stochastic rating matrix adjustment to reduce the
sensitivity of the model. A first step towards this direction has been taken
in [19] by means of Machine Learning and SDEs on Lie-Groups.

Moreover, for the application to rating triggers it will be important that the
entities can be in different sectors. Including the possibility of copulae as
in [6] will be left for future research as well.

##  Appendix A Model data

In this appendix we list the rating transitions under the historical measure
and under the risk-neutral measure after calibration with
$M^{\mathbb{P}}\equiv 1\equiv m^{\mathbb{Q}}$.

Table 7: Rating matrix under measure $\mathbb{P}$ for one month using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ |  $99.366$ % |  $0.613$ % |  $0.020$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ %
F1 |  $0.261$ % |  $99.086$ % |  $0.622$ % |  $0.030$ % |  $0.000$ % |  $0.000$ % |  $0.000$ %
F2 |  $0.030$ % |  $0.281$ % |  $99.085$ % |  $0.512$ % |  $0.090$ % |  $0.000$ % |  $0.000$ %
F3 |  $0.010$ % |  $0.040$ % |  $0.725$ % |  $98.388$ % |  $0.806$ % |  $0.020$ % |  $0.010$ %
B |  $0.000$ % |  $0.000$ % |  $0.040$ % |  $0.343$ % |  $99.223$ % |  $0.363$ % |  $0.030$ %
C |  $0.000$ % |  $0.000$ % |  $0.001$ % |  $0.005$ % |  $2.886$ % |  $95.201$ % |  $1.907$ %
D |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $100.000$ %
Table 8: Rating matrix under measure $\mathbb{P}$ for three months using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ |  $98.079$ % |  $1.818$ % |  $0.081$ % |  $0.020$ % |  $0.001$ % |  $0.000$ % |  $0.000$ %
F1 |  $0.758$ % |  $97.300$ % |  $1.810$ % |  $0.091$ % |  $0.030$ % |  $0.000$ % |  $0.010$ %
F2 |  $0.091$ % |  $0.842$ % |  $97.241$ % |  $1.491$ % |  $0.304$ % |  $0.010$ % |  $0.020$ %
F3 |  $0.041$ % |  $0.123$ % |  $2.219$ % |  $95.225$ % |  $2.331$ % |  $0.041$ % |  $0.020$ %
B |  $0.000$ % |  $0.010$ % |  $0.133$ % |  $1.057$ % |  $97.630$ % |  $1.025$ % |  $0.144$ %
C |  $0.000$ % |  $0.001$ % |  $0.006$ % |  $0.049$ % |  $8.991$ % |  $85.931$ % |  $5.023$ %
D |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $100.000$ %
Table 9: Rating matrix under measure $\mathbb{P}$ for six months using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ |  $96.151$ % |  $3.570$ % |  $0.206$ % |  $0.041$ % |  $0.010$ % |  $0.000$ % |  $0.021$ %
F1 |  $1.503$ % |  $94.682$ % |  $3.486$ % |  $0.215$ % |  $0.092$ % |  $0.002$ % |  $0.020$ %
F2 |  $0.165$ % |  $1.699$ % |  $94.615$ % |  $2.811$ % |  $0.638$ % |  $0.041$ % |  $0.031$ %
F3 |  $0.084$ % |  $0.230$ % |  $4.406$ % |  $90.801$ % |  $4.333$ % |  $0.084$ % |  $0.063$ %
B |  $0.001$ % |  $0.021$ % |  $0.264$ % |  $2.174$ % |  $95.310$ % |  $1.829$ % |  $0.400$ %
C |  $0.000$ % |  $0.002$ % |  $0.023$ % |  $0.216$ % |  $18.610$ % |  $72.787$ % |  $8.362$ %
D |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $100.000$ %
Table 10: Rating matrix under measure $\mathbb{P}$ for twelve months using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ |  $92.470$ % |  $6.753$ % |  $0.585$ % |  $0.096$ % |  $0.043$ % |  $0.001$ % |  $0.053$ %
F1 |  $2.845$ % |  $89.782$ % |  $6.485$ % |  $0.523$ % |  $0.305$ % |  $0.008$ % |  $0.053$ %
F2 |  $0.276$ % |  $3.380$ % |  $89.732$ % |  $4.954$ % |  $1.456$ % |  $0.105$ % |  $0.096$ %
F3 |  $0.209$ % |  $0.439$ % |  $9.059$ % |  $82.653$ % |  $7.181$ % |  $0.206$ % |  $0.252$ %
B |  $0.007$ % |  $0.034$ % |  $0.513$ % |  $4.522$ % |  $90.826$ % |  $3.046$ % |  $1.053$ %
C |  $0.003$ % |  $0.004$ % |  $0.076$ % |  $0.989$ % |  $37.477$ % |  $49.160$ % |  $12.292$ %
D |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $100.000$ %
Table 11: Rating matrix under measure $\mathbb{Q}$ for one month using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ |  $99.469$ % |  $0.418$ % |  $0.010$ % |  $0.002$ % |  $0.002$ % |  $0.059$ % |  $0.042$ %
F1 |  $0.383$ % |  $98.942$ % |  $0.453$ % |  $0.070$ % |  $0.003$ % |  $0.086$ % |  $0.062$ %
F2 |  $0.060$ % |  $0.379$ % |  $97.394$ % |  $1.617$ % |  $0.314$ % |  $0.144$ % |  $0.092$ %
F3 |  $0.006$ % |  $0.017$ % |  $0.224$ % |  $97.662$ % |  $0.879$ % |  $0.897$ % |  $0.314$ %
B |  $0.000$ % |  $0.000$ % |  $0.010$ % |  $0.271$ % |  $86.058$ % |  $12.906$ % |  $0.754$ %
C |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.073$ % |  $98.549$ % |  $1.378$ %
D |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $100.000$ %
Table 12: Rating matrix under measure $\mathbb{Q}$ for three months using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ |  $94.201$ % |  $5.182$ % |  $0.317$ % |  $0.012$ % |  $0.008$ % |  $0.153$ % |  $0.127$ %
F1 |  $0.485$ % |  $96.977$ % |  $2.146$ % |  $0.063$ % |  $0.032$ % |  $0.112$ % |  $0.186$ %
F2 |  $0.068$ % |  $0.779$ % |  $96.702$ % |  $1.361$ % |  $0.511$ % |  $0.300$ % |  $0.280$ %
F3 |  $0.048$ % |  $0.461$ % |  $12.141$ % |  $76.121$ % |  $7.866$ % |  $2.425$ % |  $0.939$ %
B |  $0.001$ % |  $0.011$ % |  $0.159$ % |  $0.307$ % |  $74.814$ % |  $22.464$ % |  $2.245$ %
C |  $0.000$ % |  $0.000$ % |  $0.001$ % |  $0.000$ % |  $0.431$ % |  $95.492$ % |  $4.077$ %
D |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $100.000$ %
Table 13: Rating matrix under measure $\mathbb{Q}$ for six months using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ |  $93.630$ % |  $5.530$ % |  $0.374$ % |  $0.024$ % |  $0.025$ % |  $0.164$ % |  $0.253$ %
F1 |  $2.868$ % |  $93.168$ % |  $3.081$ % |  $0.162$ % |  $0.173$ % |  $0.178$ % |  $0.371$ %
F2 |  $0.427$ % |  $2.115$ % |  $91.390$ % |  $2.759$ % |  $1.575$ % |  $1.176$ % |  $0.559$ %
F3 |  $0.241$ % |  $0.710$ % |  $12.960$ % |  $68.795$ % |  $11.911$ % |  $3.514$ % |  $1.870$ %
B |  $0.005$ % |  $0.016$ % |  $0.178$ % |  $0.553$ % |  $67.670$ % |  $27.138$ % |  $4.440$ %
C |  $0.001$ % |  $0.000$ % |  $0.001$ % |  $0.005$ % |  $1.788$ % |  $90.218$ % |  $7.987$ %
D |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $100.000$ %
Table 14: Rating matrix under measure $\mathbb{Q}$ for twelve months using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ |  $91.273$ % |  $6.951$ % |  $0.938$ % |  $0.073$ % |  $0.092$ % |  $0.169$ % |  $0.505$ %
F1 |  $5.070$ % |  $84.704$ % |  $8.207$ % |  $0.458$ % |  $0.582$ % |  $0.238$ % |  $0.741$ %
F2 |  $0.558$ % |  $2.910$ % |  $87.492$ % |  $3.919$ % |  $2.533$ % |  $1.474$ % |  $1.115$ %
F3 |  $0.389$ % |  $0.871$ % |  $17.517$ % |  $58.447$ % |  $14.750$ % |  $4.322$ % |  $3.704$ %
B |  $0.010$ % |  $0.019$ % |  $0.290$ % |  $1.390$ % |  $61.375$ % |  $28.233$ % |  $8.682$ %
C |  $0.002$ % |  $0.001$ % |  $0.008$ % |  $0.081$ % |  $8.393$ % |  $76.179$ % |  $15.336$ %
D |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $100.000$ %

##  Appendix B Historical and market data

In this appendix we listed the rating matrices used in this paper. They
contain the transition probabilities at a certain point for a particular
period of time. For example in Table 15 the probability to stay at rating
`F1+`, if one started in `F1+`, over the period of one month is $98.84\,\%$.

Table 15: Rating matrix under measure $\mathbb{P}$ for one month using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ |  $98.840$ % |  $0.610$ % |  $0.020$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ %
F1 |  $0.260$ % |  $98.740$ % |  $0.620$ % |  $0.030$ % |  $0.000$ % |  $0.000$ % |  $0.000$ %
F2 |  $0.030$ % |  $0.280$ % |  $98.610$ % |  $0.510$ % |  $0.090$ % |  $0.000$ % |  $0.000$ %
F3 |  $0.010$ % |  $0.040$ % |  $0.720$ % |  $97.680$ % |  $0.800$ % |  $0.020$ % |  $0.010$ %
B |  $0.000$ % |  $0.000$ % |  $0.040$ % |  $0.340$ % |  $98.380$ % |  $0.360$ % |  $0.030$ %
C |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $2.830$ % |  $93.350$ % |  $1.870$ %
D |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $100.000$ %
Table 16: Rating matrix under measure $\mathbb{P}$ for three months using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ |  $96.550$ % |  $1.790$ % |  $0.080$ % |  $0.020$ % |  $0.000$ % |  $0.000$ % |  $0.000$ %
F1 |  $0.750$ % |  $96.230$ % |  $1.790$ % |  $0.090$ % |  $0.030$ % |  $0.000$ % |  $0.010$ %
F2 |  $0.090$ % |  $0.830$ % |  $95.870$ % |  $1.470$ % |  $0.300$ % |  $0.010$ % |  $0.020$ %
F3 |  $0.040$ % |  $0.120$ % |  $2.170$ % |  $93.140$ % |  $2.280$ % |  $0.040$ % |  $0.020$ %
B |  $0.000$ % |  $0.010$ % |  $0.130$ % |  $1.030$ % |  $95.170$ % |  $1.000$ % |  $0.140$ %
C |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $8.520$ % |  $81.470$ % |  $4.760$ %
D |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $100.000$ %
Table 17: Rating matrix under measure $\mathbb{P}$ for six months using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ |  $93.190$ % |  $3.460$ % |  $0.200$ % |  $0.040$ % |  $0.010$ % |  $0.000$ % |  $0.020$ %
F1 |  $1.470$ % |  $92.610$ % |  $3.410$ % |  $0.210$ % |  $0.090$ % |  $0.000$ % |  $0.020$ %
F2 |  $0.160$ % |  $1.650$ % |  $91.900$ % |  $2.730$ % |  $0.620$ % |  $0.040$ % |  $0.030$ %
F3 |  $0.080$ % |  $0.220$ % |  $4.210$ % |  $86.760$ % |  $4.140$ % |  $0.080$ % |  $0.060$ %
B |  $0.000$ % |  $0.020$ % |  $0.250$ % |  $2.060$ % |  $90.460$ % |  $1.740$ % |  $0.380$ %
C |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $16.870$ % |  $66.140$ % |  $7.580$ %
D |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $100.000$ %
Table 18: Rating matrix under measure $\mathbb{P}$ for twelve months using Fitch’s data. From To | F1+ | F1 | F2 | F3 | B | C | D
---|---|---|---|---|---|---|---
F1+ |  $86.950$ % |  $6.350$ % |  $0.550$ % |  $0.090$ % |  $0.040$ % |  $0.000$ % |  $0.050$ %
F1 |  $2.720$ % |  $85.850$ % |  $6.200$ % |  $0.500$ % |  $0.290$ % |  $0.000$ % |  $0.050$ %
F2 |  $0.260$ % |  $3.180$ % |  $84.420$ % |  $4.660$ % |  $1.370$ % |  $0.100$ % |  $0.090$ %
F3 |  $0.190$ % |  $0.400$ % |  $8.250$ % |  $75.270$ % |  $6.540$ % |  $0.190$ % |  $0.230$ %
B |  $0.000$ % |  $0.030$ % |  $0.460$ % |  $4.040$ % |  $81.890$ % |  $2.780$ % |  $0.950$ %
C |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $31.880$ % |  $42.400$ % |  $10.440$ %
D |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $0.000$ % |  $100.000$ %
Table 19: Default probabilities under measure $\mathbb{Q}$ for $1,3,6,12$ months using Fitch’s data. From To | D $\left(t=\frac{1}{12}\right)$ | D $\left(t=\frac{3}{12}\right)$ | D $\left(t=\frac{6}{12}\right)$ | D $\left(t=\frac{12}{12}\right)$
---|---|---|---|---
F1+ |  $0.042$ % |  $0.127$ % |  $0.253$ % |  $0.505$ %
F1 |  $0.062$ % |  $0.186$ % |  $0.371$ % |  $0.741$ %
F2 |  $0.093$ % |  $0.280$ % |  $0.559$ % |  $1.115$ %
F3 |  $0.314$ % |  $0.939$ % |  $1.870$ % |  $3.704$ %
B |  $0.754$ % |  $2.245$ % |  $4.440$ % |  $8.682$ %
C |  $1.378$ % |  $4.077$ % |  $7.987$ % |  $15.336$ %
D |  $100.000$ % |  $100.000$ % |  $100.000$ % |  $100.000$ %

## Declarations

### Funding

This project has received funding from the European Union’s Horizon 2020
research and innovation programme under the Marie Sklodowska-Curie grant
agreement No 813261 and is part of the ABC-EU-XVA project.

### Conflicts of interests

The authors have no relevant financial or non-financial interests to disclose.

### Data availability

All data generated or analysed during this study are included in this
published article. In particular the code to produce the numerical experiments
is available at
https://github.com/kevinkamm/RatingTriggers.

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