# A dimension reduction approach for loss valuation in credit risk modelling

Jian He 111Corresponding author<EMAIL_ADDRESS>Korteweg-de Vries Institute,
University of Amsterdam, Amsterdam, The Netherlands  0000-0002-5372-0541 Asma
Khedher  Korteweg-de Vries Institute, University of Amsterdam, Amsterdam, The
Netherlands  0000-0003-1528-3923 Peter Spreij  Korteweg-de Vries Institute,
University of Amsterdam, Amsterdam, The Netherlands  IMAPP, Radboud
University, Nijmegen, The Netherlands  0000-0002-6416-6320

###### Abstract

This paper addresses the “curse of dimensionality” in the loss valuation of
credit risk models. A dimension reduction methodology based on the Bayesian
filter and smoother is proposed. This methodology is designed to achieve a
fast and accurate loss valuation algorithm in credit risk modelling, but it
can also be extended to valuation models of other risk types. The proposed
methodology is generic, robust and can easily be implemented. Moreover, the
accuracy of the proposed methodology in the estimation of expected loss and
value-at-risk is illustrated by numerical experiments. The results suggest
that, compared to the currently most used PCA approach, the proposed
methodology provides more accurate estimation of expected loss and value-at-
risk of a loss distribution.
keywords: Bayesian filter, credit risk, loss valuation
2020 Mathematics Subject Classification: 62P05, 91G40

###### Contents

  1. 1 Introduction
    1. 1.1 Problem description and background
    2. 1.2 Contribution
    3. 1.3 Organization of the paper
  2. 2 Set up and background
    1. 2.1 Overview of loss calculations in credit risk
    2. 2.2 Valuation model for the credit losses
    3. 2.3 Valuation grid for losses valuation
    4. 2.4 valuation grid pre-processing and loss simulation
  3. 3 A dimension reduction approach: Bayesian filter projection
    1. 3.1 State space model
    2. 3.2 Bayesian filter and smoother
      1. 3.2.1 Bayesian filter recursion
      2. 3.2.2 Bayesian smoother recursion
    3. 3.3 Bayesian filter/smoother projection
    4. 3.4 Application examples
      1. 3.4.1 Credit Risk Economic Capital
      2. 3.4.2 Incremental Risk Charge
      3. 3.4.3 International Financial Reporting Standard 9
      4. 3.4.4 Market risk
      5. 3.4.5 Interest rate risk in the banking book
  4. 4 Numerical studies
    1. 4.1 Transition model
    2. 4.2 LGD model
    3. 4.3 Experiments
      1. 4.3.1 Transition probabilities
      2. 4.3.2 Expected PD simulation
      3. 4.3.3 Experiments for the projection approach for expected PD
      4. 4.3.4 Expected LGD: Black-Scholes formula v.s. Monte Carlo
      5. 4.3.5 Experiments for the projection approach for risk measures of the loss distribution
  5. 5 Conclusions
  6. A Analytic formula for expected loss-given-default

##  1 Introduction

###  1.1 Problem description and background

In this paper we apply the grids methodology to construct an approximation of
the valuation function which is used to evaluate potential credit losses of
loans. To tackle the “curse of dimensionality” problem that the grids
methodology heavily suffers from, we propose a dimension reduction technique
based on the Bayesian filter and smoother.

Computing risk metrics in credit risk for large portfolios is computationally
intensive. This is especially true when the analytical loss distribution is
not available. In general, Monte Carlo simulation is required to estimate the
loss distribution. In such a case, one needs to simulate the multi-year
default probabilities (PD) and realize loss-given default (LGD) of each single
loan or each pool of loans, and then evaluate the potential losses. The
simulation of the multi-year PD and realized LGD for large and complex
portfolios requires extremely large computational resources. Therefore, in
practice a valuation grid is usually used to approximate the portfolio
valuation functions to avoid multi-year simulations. Valuation grids are a
commonly applied computational tool in science and engineering (for example,
Thompson et al. (1998)) and have also been applied in finance. For instance,
Broadie and Glasserman (2004) uses grids to speed up the valuation of high
dimensional American options. Recently, the grids method are also used
together with the neural networks or Gaussian process regression to obtain
better precision and faster calculations in valuations, see Horvath et al.
(2021), Pagnottoni (2019), Evans and Nair (2018). The idea of using grids to
measure portfolio risk is used in the stratified sampling methodology by
Jamshidian and Zhu (1996) to approximate directly the probability density
function of a portfolio value.
The grids methodology that applies to approximate the valuation function needs
only a small number of exact valuations, which are used to construct a grid.
The constructed grid is a fast approximation of the valuation function that
can then be used during a simulation. The key to applying this method is to be
able to express the valuation function as a function of a small number of risk
factors, usually not larger than three, since the grids methodology heavily
suffers from the “curse of dimensionality”, Bellman (1957). The “curse of
dimensionality” describes the extraordinarily rapid growth of computation time
as the number of the dimension increases. When the dimensionality of the risk
factors is large, a large number of valuations is required to construct the
interpolation pricing function and hence the grid methodology becomes
impractical. Nevertheless, in credit risk modelling, a higher factor model is
preferred to better describe the transition probabilities. Therefore, a
dimension reduction approach is needed to map the higher dimensional factors
to lower dimensional factors that can be used in the valuation grids.
Currently, the most used dimension reduction approach is Principal Component
Analysis (PCA), Jolliffe (1990). It transforms the original risk factors into
a new risk factor space composed of orthogonal principal directions. For
example, five dimensional risk factors result in five uncorrelated principal
components. Movements of the five dimensional factors can thus be described by
the principal components. The principal components are ordered from the
largest to the least in terms of explaining the total variance of the five
dimensional risk factors. In practice, one needs to decide on the number of
principal components to use. Assume one decides to use the first two principal
components. Then the original five dimensional risk factors are ”reduced” to
two dimensional risk factors, which are a linear combination of the first two
principal components. However, using the PCA approach in credit risk modelling
has the following drawbacks.

  * •

_Low interpretability_ : principal components are linear combinations of the
risk factors, but they are not easy to interpret. For example, usually the
risk factors need to be standardized before PCA decomposition. Consequently,
it is difficult to tell which are the most important risk factors after
computing principal components. Therefore one loses the economic
interpretation when analyzing the impact of the principal components.

  * •

_Sensitivity to outliers_ : PCA is a method based on the correlation or
covariance matrix, which can be very sensitive to outliers. Consequently,
outliers can completely mess up a classical PCA analysis and yield a PCA model
with an arbitrarily bad fit to the genuine part of the data. Indeed, PCA
cannot account for outliers in the risk factors that are outside the range of
the selected components and such outliers usually need to be removed before
performing PCA. However, in credit risk modeling, these outliers are actually
desired since they might be located in the tail of the loss distribution and
hence contribute to the Value-at-Risk (VaR).

  * •

_Assumption on the correlations_ : PCA assumes correlations between the risk
factors. If the risk factors are not correlated, PCA will not provide any
additional insight.

As an alternative to the PCA approach, in this paper we propose a dimension
reduction methodology.

###  1.2 Contribution

Our first and main contribution is a dimension reduction methodology based on
the Bayesian filter and smoother. The Bayesian filter and smoother estimate
the distribution (or value) of a latent process, given the observed data. We
interpret the Bayesian filter and smoother as a projection from the observed
data to unobserved latent factors. More precisely, this feature is applied to
project the _higher_ dimensional risk factors onto _lower_ dimensional
factors.

Compared to the commonly used Principal Components Analysis (PCA) approach,
the proposed dimension reduction methodology has the follow advantages.

  * •

_Economic interpretability_ : The proposed approach projects the higher
dimensional factors to lower dimensional factors in such a way that the lower
dimensional factors capture the most importance features in the model
maintaing an economic interpretation. For instance, if one applies the
proposed approach to a transition model in which a certain rating migration
constitutes the majority of observations (i.e. the majority of clients), this
rating migration is an importance risk driver and the lower factors will tend
to capture the feature of this rating migration.

  * •

_Optimality_ : The proposed approach is optimal in the sense that the Bayesian
filter is optimal on the criterion of Bayes risk of minimum mean square error
(MMSE), see Chen (2003). Intuitively, the proposed approach is optimal in that
it seeks the posterior distribution of the lower dimensional factors which
integrates and uses all of available information.

  * •

_It is generic and robust_ : This approach is generic in that it applies to a
wide range of risk or pricing models, and insensitive to outliers.

Our second contribution is an application of our proposed dimension reduction
approach to credit loss estimation. We review the important concepts and
models used in the credit risk modelling, like transitions, loss given default
(LGD), exposure at default (EAD) and loss valuation models. We also derive a
closed form formula of Black-Scholes type for the expected LGD of collateral
LGD models. We conduct numerical experiments to illustrate the performances of
the proposed methodology on the credit loss estimation and we assess the
accuracy of the estimation on PD curves, expected losses, and the VaR at
different levels. We compare the performance of the valuation grid based on
the proposed methodology and the PCA approach. The results indicate that the
proposed approach convincingly outperforms the PCA approach.

###  1.3 Organization of the paper

Section 2 provides the background knowledge for loss calculations in credit
risk modelling. We present the valuation model and the valuation grid approach
for the loss calculation. This section also contains brief reviews of
transition, LGD and EAD models and specifies the models used in this paper.
The proposed dimension reduction methodology is presented in Section 3, in
which the Bayesian filter and smoother are also presented. In Section 4 we
provide numerical experiments to show the performance of the proposed
approach. Finally, Section 5 is devoted to the conclusions. In the Appendix we
provide a Black-Scholes formula for the expected LGD.

Here follows some notations and other conventions used in this paper. All
random processes are defined on a fixed probability space
$(\Omega,\mathbb{F},\mathbb{P})$. Time is assumed to be discrete. We will use
a filtration $\\{\mathcal{F}_{k},k=0,\ldots,n\\}$, where $\mathcal{F}_{k}$
summarizes the information up to time $k$ and $n$ is the final time. We denote
by $\cdot^{\top}$ the transpose of a vector or a matrix.

##  2 Set up and background

In this section we outline the set up, we pose the problem, give a brief
survey of models used for loss calculations in credit risk modeling, Bayesian
filters (including Kalman, and particle filter) and address the “curse of
dimensionality” problem for loss calculations using Bayesian filters.

###  2.1 Overview of loss calculations in credit risk

One of the main tasks in credit risk is to forecast and quantify future losses
of a portfolio at a certain time horizon. The time horizon is a future time
period over which the credit risk is assessed. For instance, in the
regulations prescribed by the Basel Committee on Banking Supervision Basle
Committee (1999) there is a one-year time horizon across all asset classes.
The forecast risk can be quantified, for example, by expected losses or value
at risk, depending on the purpose. In the International Financial Reporting
Standards (IFRS) extensive reports of the expected losses for different
obligors and sectors are required and for Economic Capital (EC) the extreme
quantile of the loss distribution (usually the $99.95\%$ quantile) needs to be
calculated.

Three risk parameters are essential in the process of calculating the
potential losses: the probability of default (PD), the loss given default
(LGD) and the exposure at default (EAD). The PD is the probability that a
borrower will fail to pay back a debt. The LGD is the fraction of a loan that
is lost when a borrower defaults and the EAD is the predicted amount of the
loss, the borrower may be exposed to, when a debtor defaults on a loan. They
are generally estimated using the available historical information and are
assigned to operations and customers according to their particular
characteristics. In this context, the credit rating tools (ratings and
scorings) assess the risk for each transaction or customer according to their
credit quality by assigning them a score. This score is then used in assigning
risk metrics, together with additional information such as transaction
seasoning, loan to value ratio, customer segment, etc. The increase in the
number of default events in the current economic situation contributes to
reinforce the soundness of the risk parameters by adjusting their estimates
and by refining methodologies. The incorporation of data from the years of
economic slowdown is particularly important for refining the analysis of the
cyclical behavior of credit risk. The effect on the PD estimates and on the
credit conversion factor is immediate.222The credit conversion factor
calculates the amount of a free credit line and other off-balance-sheet
transactions (with the exception of derivatives) to an EAD amount and is an
integral part in the European banking regulation since the Basel II accords,
see Basle Committee (2006). An analysis of the impact on the LGD, however,
requires waiting for the outcome of the recovery processes associated with the
default events. Below we present a brief review to the models of the three
risk parameters, i.e. PD, LGD and EAD.

#### Transition models

A model for the PD, or for transition matrices that govern rating migrations,
depends on the rating system. A credit rating system uses a limited number of
rating grades to rank borrowers according to their default probabilities.
Ratings are assigned by rating agencies such as Fitch, Moody’s and Standard &
Poor’s, but also by financial institutions themselves. Rating assignments can
be based on a qualitative process or on default probabilities estimated with a
scoring model. To translate default probability estimates into ratings, one
defines a set of rating grades depending on the default probabilities. For
example, borrowers are assigned grade AAA if their probability of default is
lower than $0.02\%$, to grade AA if their probability of default is between
$0.02\%$ and $0.05\%$ and so on. Once the ratings are defined, the concern is
to determine the probability with which the credit risk rating of a borrower
decreases or increases by a given degree and a transition (PD) model is
required. These probabilities that a credit risk rating of a borrower
decreases or increases, from one period to the next one, are usually collected
in a transition matrix. Namely, the transition matrix measures the
probabilities of the migrations between different credit ratings over specific
time intervals. In the transition matrix, the element located at the $i$-th
row and $j$-th column represents the probability of a borrower migrates from
the $i$-th rating to the $j$-th rating. These are called transition
probabilities and are often related to macroeconomic variables such as
interest rates, inflation, gross domestic product (GDP), unemployment, etc.
Alternatively, transition probabilities can also be modelled by using certain
abstract latent processes.
The transition (PD) models in general can be divided into two main classes:
structural and reduced form models.

Structural models link the up- or down-grade (or default) probabilities of a
firm to the value of its assets and liabilities. The origin of structural
credit risk models is the Merton model, Merton (1974), and then further
developed by, for instance, Hull and White (2001), Avellaneda and Zhu (2001),
Duffie and Singleton (2004), Albanese et al. (2003), and Jeanblanc and
Rutkowski (1999). In these structural models, the latent credit process is
called a credit index $(X_{t})_{t\geq 0}$, reflecting unobservable credit
quality over time which is driven by firm-specific variables such as the asset
values. A default occurs if the credit index crosses a default barrier. The
dynamics of the credit index are specified by the model and the default
barriers can be calibrated to historical migration or market data. Structural
models can be generalized to describe not only defaults, but also rating
migrations. This requires a mapping from the latent credit index to the rating
states at each time, which corresponds to specifying an interval for the
credit index that corresponds to each rating class. The intervals are defined
by rating barriers, and rating changes occur when the credit index passes a
barrier value.

Reduced form models, see for example, Jarrow and Turnbull (1995), Duffie and
Singleton (2004), Frey and McNeil (2002), Wendin and McNeil (2006), Koopman et
al. (2008), Jeanblanc and Li (2020), and Jeanblanc and Lecam (2008), give a
description for the latent credit quality of an entity by assuming that it is
driven by exogenous factors. They are based on a linear factor model for the
latent variable, or equivalently, for the hazard rate of a firm. The key
assumption of reduced form models is that the latent process or hazard rate
process is driven by systematic external factors. For instance, in a reduced
form model the latent credit process $(X_{t})_{t\geq 0}$ can be decomposed
into two independent parts: a systematic risk factor and a idiosyncratic risk
factor. The systematic risk factor includes factors that are relevant for the
business environment of the firms, such as the state of the economy, dummy
variables for the industry sectors and regions, or other variables that are
related to credit quality. The idiosyncratic risk factor is the firm-specific
term. As in the structural model, in reduced form model a default occurs if
the latent variable $(X_{t})_{t\geq 0}$ falls below a certain threshold at a
certain time. Same concept can be generalized to credit ratings by defining a
mapping from the credit variable $(X_{t})_{t\geq 0}$ to the rating states, by
using threshold parameters for all ratings. These threshold parameters give
the limits for the latent variable to move from current rating to other
ratings, see for instance Nickell et al. (2000), Frey and McNeil (2002), Wei
(2003), Duffie and Singleton (2004). Given the systematic factors, the default
(transition) probabilities depends on the cumulative distribution function of
the idiosyncratic factors. This distribution function is usually referred to
as the response function. The most common choices for the response function
are the probit function (standard normal distribution function) and the logit
function. Compared to the structural model, the reduced form models are not
explicitly based on on the company’s balance sheet. It only requires
information generally available in financial markets and hence historical
estimation methods can be easily used. But the consequence, and drawback, is
that the models do not explain the economic reasons for default.

In this paper we use a reduced form approach to model the transition
probability. We will describe this in more detail in Section 4.1.

#### LGD models

Loss Given Default (LGD) is one of the key determinants of the credit risks of
loans and other credit exposures, which measures the severity of the loss when
a debtor defaults. It is the share of an asset that is lost when a borrower
defaults. An client may not end up with a loss if it is in default since it
could be cured. Hence it is always worth noting that the LGD estimate is
sensitive to the definition of default, which can substantially change the
implied Recovery rate. Moreover, the LGD is client-specific because such
losses are generally understood to be influenced by key transaction
characteristics such as the presence or the value of a collateral. For
example, in mortgages the houses themselves are usually the collateral. That
means that if the borrower defaults, the bank could sell the house to
compensate for the loss from missing payments from the borrower. Obviously,
the LGD could be stochastic over time. Despite its stochastic nature,
statistical modeling of the LGD has been challenging in the academic
literature and in banking practice. Therefore many credit risk models used in
practice assume that LGD is a deterministic proportion of the exposures
subject to impairment and ignore the fact that LGD can fluctuate according to
the economic cycle. For example, Altman (2011) and Altman et al. (2005) show
that default rates and recovery rates are strongly negatively correlated and
measure a correlation of 0.75 between yearly average default rates and loss
rates in the United States. They provide strong correlation evidence between
macroeconomic growth variables (such as gross domestic product) and recovery
rates and test the impact of correlated defaults and LGD. There are various
models proposed to describe the LGD. Frye (2000a) and Frye (2000b) propose a
structural model with a systematic risk factor representing the state of the
economy that drives both defaults and LGD. The dependence of the default
indicator and LGD on the common risk factor gives rise to a strong correlation
between the two, which is in line with empirical evidence. Hillebrand (2006)
introduces dependent LGD modeling into a multifactor latent variable
framework, providing a good fit to corporate bond data. Marginal distributions
for indicator functions and LGD can be specified. Frontczak and Rostek (2015)
show that for the retail sector it is important to include collateral and
suggested improvements for modelling LGD for this sector. We adopt this latter
approach to model the LGD in this paper. The details are presented in Section
4.2.

#### EAD models

Exposure at Default (EAD) is the predicted amount of loss that may be exposed
to when a debtor defaults on a loan. In many cases, such as residential
mortgages and personal loans, the EAD are deterministic and can be simply
taken from the current on-balance amount. For credit cards though, the
revolving nature of the credit line poses challenges with regards to
predicting the exposure at default time. As credit card customers may borrow
more money (off balance) in the months prior to default, simply taking the
current balance for non-defaulted customers would not produce a conservative
enough estimate for the amount drawn by the time of default. Examples to model
the EAD with an off balance amount can be found in Gumbo (2013) and Tong et
al. (2016). In this paper, we treat the EAD as deterministic and equal to the
on-balance amount. Finally, given the risk parameters we next explain how the
portfolio losses are determined.

#### Portfolio losses

Given the PD, LGD and the EAD models, the realized losses of the obligors can
be determined by simulating the risk factors and the migrations. If the
simulated rating of an obligor becomes default at a certain time $t$, the loss
of this obligor can be simply computed as
$\mathrm{LGD}_{t}\times\mathrm{EAD}_{t}$, the notation should be obvious.
After discounting and aggregating all the losses at different times, one can
obtain the aggregated loss of the obligor. Then the portfolio loss for each
realization is computed by summing the aggregated losses from all obligors.

By simulating a lot of scenarios, one obtains the distribution of the
portfolio losses and hence determines the desired risk metrics, such as
expected loss or value-at-risk. Figure 2.1 is presented as a simple
illustration to determine the loss distribution of the portfolio. A number of
scenarios are simulated until maturity. In this case the maturity is 30
periods and the horizon is one period. In each scenario, the migrations of all
obligors are simulated over time and the losses at each time point are
computed for the obligors that move to default at that time point. Aggregating
all the discounted losses from all time points for all obligors, one obtains
the portfolio losses for this specific scenario. In the end the loss
distribution can be obtained by repeating the calculations for all simulated
scenarios.

Figure 2.1: An illustration to determine the loss distribution

This procedure is very time consuming when dealing with a huge portfolio,
since it requires simulations of the migrations for all the obligors along
each scenario. Hence, in practice, a valuation model together with a valuation
grid method is needed to simplify the loss calculations. In the next
subsections we introduce the valuation model we use in this paper and we
introduce our valuation grid approach to compute the risk metrics.

###  2.2 Valuation model for the credit losses

In general, a portfolio’s credit loss is defined as the difference between the
portfolio’s current value and its future value at the end of some time horizon
see Basle Committee (1999). Therefore a valuation model can be used to compute
the value of the loans and hence to capture the risks. For example, such a
model is used to capture the risk of a one-year loan and a ten-year loan. If,
over the one-year horizon, the one-year loan is downgraded but does not
default, then it survives and there is no loss. If the ten-year loan is
downgraded but does not default, then the _economic_ value of this loan should
be lower than if this loan was not downgraded. The loan values are based on a
net present value (NPV). Specifically the NPV is computed as the expected
discounted value of all future net cash flows:

$\displaystyle\mathrm{NPV}=\mathbb{E}\big{(}\sum_{k=1}^{n}\delta_{k}\mathrm{CF}_{k}\big{)},$
(2.1)

where $n$ is the maturity of the loan, the $\delta_{k}\in\mathbb{R}^{+}$ are
the risk-free discounting factors and the $\mathrm{CF}_{k}$ are the future
cash flows at time $k$. It is very important to note, while it is correct to
use risk-free rates to discount risk-free cash flows, we have to discount
risky cash flows due to potential defaults. This issue can be resolved in two
ways: find risk-adjusted discount rates or transform the probabilities to risk
adjusted (risk neutral) probabilities. The former way requires finding an
appropriate discount rate for each (type of) obligor or loan, which would
introduce a lot of parameters in the model. Therefore the latter approach to
transform the physical default probabilities into risk-neutral default
probabilities is usually used. More details about the risk neutral default
probability can be found in Delianedis and Geske (2003). To reduce the
complexity of the model in practice, the risk-free discounting factors are
usually derived from deterministic curves, for instance those obtained from
the Libor rates or Overnight Indexed Swaps (OIS). In this paper, we do not
distinguish between the physical measure and the risk neutral measure for
simplicity, and hence the expectation in equation (2.1) is interpreted as a
risk neutral expectation.
Consider a loan with coupon payments $s_{k}$ at time $k=1,2,\ldots,n$ and a
principal repayment $\mathrm{PC}$ at maturity $n$. Since the default depends
on the rating of the loan, we denote the default time of a loan with rating
$r$ by $\tau_{r}$, a non-negative real random variable. If the loan defaults
in period $k$ (between time ${k-1}$ and $k$), denoted by the indicator
$\mathbbm{1}_{\tau_{r}\in(k-1,k]}$, the cash flow equals the coupon until time
$k-1$ plus the recovery $1-\mathrm{LGD}_{k}$. If the loan does not default,
the cash flow equals all the coupons plus the principal. Hence, one obtains
the value of the loan with rating $r$ at initial time $k=0$, denoted by
$V_{0}^{(r)}$ as

$\displaystyle V_{0}^{(r)}$
$\displaystyle=\mathbb{E}\left[\sum_{k=1}^{n}\mathbbm{1}_{\tau_{r}\in(k-1,k]}\big{(}\sum_{j=1}^{k-1}\delta_{j}s_{j}+\delta_{k}(1-\mathrm{LGD}_{k})\mathrm{EAD}_{k}\big{)}+\mathbbm{1}_{\tau_{r}>n}\big{(}\sum_{k=1}^{n}\delta_{k}s_{k}+\delta_{n}\mathrm{PC}\big{)}\right]$
$\displaystyle=\underbrace{\mathbb{E}[\sum_{k=1}^{n}\delta_{k}\mathbbm{1}_{\tau_{r}\in(k-1,k]}}_{\mathrm{lifetime}\,\mathrm{PD}}(1-\mathrm{LGD}_{k})\mathrm{EAD}_{k}]+\underbrace{\mathbb{E}\left[\sum_{k=1}^{n}\delta_{k}\mathbbm{1}_{\tau_{r}>k}s_{k}\right]}_{\mathrm{coupon\,payments}}+\underbrace{\mathbb{E}\left[\delta_{n}\mathrm{PC}\mathbbm{1}_{\tau_{r}>n}\right]}_{\mathrm{principle}}.$

Hence, standing at horizon $h$ and suppose the sigma-algebra $\mathcal{F}_{h}$
contains the information up to time $h$,333In practice, the specific
definition of $\mathcal{F}_{h}$ depends on the models used for PD, LGD, and
EAD. For instance, if factor models are used, the sigma-algebra
$\mathcal{F}_{h}$ is usually defined as the sigma-algebra that is generated by
the factors upto time $h$. the resulting value of a performing loan with
rating $r$, denoted by $V_{h}^{(r)}$, is

$\displaystyle V_{h}^{(r)}$
$\displaystyle=\mathrm{RV}_{h}+\sum_{k=h+1}^{n}\delta_{k}\mathbb{E}\left[\mathbbm{1}_{\tau_{r}\in(k-1,k]}(1-\mathrm{LGD}_{k})\mathrm{EAD}_{k}\mid\mathcal{F}_{h}\right]$
(2.2)
$\displaystyle+\sum_{k=h+1}^{n}\delta_{k}s_{k}\mathbb{E}\left[\mathbbm{1}_{\tau_{r}>k}\mid\mathcal{F}_{h}\right]+\delta_{n}\mathrm{PC}\,\mathbb{E}\left[\mathbbm{1}_{\tau_{r}>n}\mid\mathcal{F}_{h}\right],$

where $\mathrm{RV}_{h}$ is the summation of the cash flow until horizon, i.e.

$\mathrm{RV}_{h}=\sum_{i=1}^{h}\delta_{i}s_{i}.$

Note that $V_{h}^{(r)}$ is $\mathcal{F}_{h}$ measurable.

If there is no default after horizon $h$, the value of the loan is just the
summation of all the discounted cash flows,

$V_{h,\mathrm{ND}}^{(r)}=\mathrm{RV}_{h}+\sum_{k=h+1}^{n}\delta_{k}s_{k}+\delta_{n}\mathrm{PC}.$

Therefore, one obtains the valuation model for the loss of a performing loan
with rating $r$ at horizon $h$, conditioned on information at $h$, as follows.

$\displaystyle L_{h}^{(r)}$
$\displaystyle=V_{h,\mathrm{ND}}^{(r)}-V_{h}^{(r)}$ (2.3)
$\displaystyle=\sum_{k=h+1}^{n}\delta_{k}s_{k}\mathbb{E}\left[\mathbbm{1}_{\tau_{r}\leq
k}\mid\mathcal{F}_{h}\right]+\delta_{n}\mathrm{PC}\,\mathbb{E}\left[\mathbbm{1}_{\tau_{r}\leq
n}\mid\mathcal{F}_{h}\right]$
$\displaystyle\qquad-\sum_{k=h+1}^{n}\delta_{k}\mathrm{EAD}_{k}\,\mathbb{E}\left[\mathbbm{1}_{\tau_{r}\in(k-1,k]}(1-\mathrm{LGD}_{k})\mid\mathcal{F}_{h}\right].$

Moreover, note that in the last equality we consider the $\mathrm{EAD}_{t}$ as
the on-balance amount at time $t$, as in most cases, and hence it is
deterministic. One observes that the loss at time $k$ is from the future cash
flows after the defaults, but compensated by the recovery
$(1-\mathrm{LGD}_{k})\mathrm{EAD}_{k}$. It is worth noticing that the
definition of loss in this valuation model is in the _economic_ sense. That
means once a loan defaults, the losses not only come from the outstanding
exposures at default, but also from missing interest payments from the future
coupons. Note that the loss in Equation (2.3) is path-dependent. The
probability of default depends on the value of the risk factors in the
transition model and the $\mathrm{LGD}_{k}$ may also depend on risk factors in
the LGD model.

###  2.3 Valuation grid for losses valuation

There is usually no closed-form expression for the conditional distribution of
the cumulative PD or the conditional joint distribution of PD and LGD.
Accordingly it is not possible to derive analytical formulas for the
conditional expectations $\mathbb{E}[\mathbbm{1}_{\tau_{r}\leq
k}\mid\mathcal{F}_{h}]$ and
$\mathbb{E}[\mathbbm{1}_{\tau_{r}\in(k-1,k]}\mathrm{LGD}_{k}\mid\mathcal{F}_{h}]$
in Equation (2.3). As alternative, a valuation grid is usually used to
approximate the aforementioned conditional expectations. The valuation grid
methodology, see for instance, Chishti (1999) and Gibson and Pritsker (2000),
requires only a small number of valuations, which are used to construct a fast
approximation of the valuation function (in our case the conditional
expectations in (2.3)) when doing the simulation. The basic idea behind the
valuation grid approach is simple. Consider a conditional expectation in which
the value depends on a small number of risk factors (say two or three).
Firstly, the conditional expectation values are evaluated, by Monte Carlo for
instance, on a small, pre-selected set of points which forms a grid in the
risk factor space. Then, during the simulation to construct the loss
distribution, the conditional expectation values under different scenarios are
quickly calculated by using the previously computed values on the grid points
through interpolation or more sophisticated tools such as Gaussian process
regression, see Rasmussen and Williams (2006) or neural networks, see Bishop
(1994) and Angelini et al. (2008).

We apply the valuation grid to every conditional expectation in (2.3), that is
to $\mathbb{E}[\mathbbm{1}_{\tau_{r}\leq k}\mid\mathcal{F}_{h}]$ and to
$\mathbb{E}[\mathbbm{1}_{\tau_{r}\in(k-1,k]}\mathrm{LGD}_{k}\mid\mathcal{F}_{h}]$
for $k=h+1,\dots,n$. These conditional expectations are also called _bullets_
since they can be seen as the loss of a unit bullet loan444A unit bullet loan
is a loan with unit notional where a payment of the entire principal of the
loan, possibly the principal and interest, is due at the end of the loan
term.. In the rest of our paper, we adopt this terminology and refer the
conditional expectations as bullets. The key to applying the valuation grid
method for valuation is to be able to express the bullet as a function of a
small number of risk factors. However, these risk factor inputs are often high
dimensional, since high dimensional factors are usually needed to describe the
transition probabilities. The high dimensionality will result in big
challenges in the implementation of the valuation grid, since the number of
grid points to maintain a ‘minimum density’ grows too fast. This can be
understood intuitively with a simple example. Suppose five risk factors are
used to determine the value of a loan. If one wants to create a valuation grid
with ten points in each of the five dimensions, then $10^{5}$ exact valuations
would be required to create the grid. To tackle this problem, we propose a
dimension reduction approach to project a higher factor model onto a lower
factor model for the valuation. The advantage of the proposed projection
approach is that the higher factor model is kept to model the transition
probabilities, whereas a corresponding lower factor model is used for the
valuation. Further details of the proposed projection approach will be
presented in Section 3.

###  2.4 valuation grid pre-processing and loss simulation

Suppose factor models are assigned to model the transitions and LGDs. Denote
the risk factors at time $k$ by $x_{k}$. The filtration
$\\{\mathcal{F}_{s},s=0,1,...,n\\}$ is the information set in which
$\mathcal{F}_{s}$ is the sigma-algebra generated by $\\{x_{0},...,x_{s}\\}$.
For every time $k$, every initial rating and possibly every initial loan-to-
value (depending on the LGD model), the values of the bullets
$\mathbb{E}\left[\mathbbm{1}_{\tau_{r}\leq k}\mid\mathcal{F}_{h}\right]$ and
$\mathbb{E}\left[\mathbbm{1}_{\tau_{r}\in(k-1,k]}\mathrm{LGD}_{k}\mid\mathcal{F}_{h}\right]$
in Equation (2.3) are pre-computed on certain pre-selected points of risk
factors. Then given time $k$, initial rating $r$ and possibly initial loan-to-
value, the set of bullet values and the risk factor points form a valuation
grid. The pre-computation of the bullets are usually done by Monte Carlo
simulation.

Suppose the default indicator $\mathbbm{1}_{\tau_{r}\in(k-1,k]}$ and
$\mathrm{LGD}_{k}$ are conditionally independent given $\mathcal{F}_{k}$. Then
one derives

$\displaystyle\mathbb{E}\left[\mathbbm{1}_{\tau_{r}\in(k-1,k]}\mathrm{LGD}_{k}\mid\mathcal{F}_{0}\right]$
$\displaystyle=\mathbb{E}\left[\mathbb{E}\left[\mathbbm{1}_{\tau_{r}\in(k-1,k]}\mathrm{LGD}_{k}\mid\mathcal{F}_{k}\right]\mid\mathcal{F}_{0}\right]$
$\displaystyle=\mathbb{E}\left[\mathbb{E}\left[\mathbbm{1}_{\tau_{r}\in(k-1,k]}\mid\mathcal{F}_{k}\right]\mathbb{E}\left[\mathrm{LGD}_{k}|\mathcal{F}_{k}\right]\mid\mathcal{F}_{0}\right]$
$\displaystyle=\mathbb{E}\left[\mathbb{P}(\tau_{r}\in(k-1,k]\mid\mathcal{F}_{k})\mathbb{E}\left[\mathrm{LGD}_{k}|\mathcal{F}_{k}\right]\mid\mathcal{F}_{0}\right]$
$\displaystyle=\mathbb{E}\left[\left(\mathrm{PD}_{r}(k\mid\mathcal{F}_{k})-\mathrm{PD}(k-1\mid\mathcal{F}_{k})\right)\mathbb{E}\left[\mathrm{LGD}_{k}|\mathcal{F}_{k}\right]\mid\mathcal{F}_{0}\right],$

with $\mathrm{PD}_{r}(k\mid\mathcal{F}_{k})$ the cumulative PD of loan with
initial rating $r$, at time $k$ conditioned on the information
$\mathcal{F}_{k}$. This result suggests that, in the case that
$\mathbbm{1}_{\tau_{r}\in(k-1,k]}$ and $\mathrm{LGD}_{k}$ are conditionally
independent given $\mathcal{F}_{k}$, one only needs to simulate transition
(default) probabilities and expected LGD when using Monte Carlo simulations to
estimate
$\mathbb{E}\left[\mathbbm{1}_{\tau_{r}\in(k-1,k]}\mathrm{LGD}_{k}\mid\mathcal{F}_{0}\right]$.
The additional simulations of the indicator $\mathbbm{1}_{\tau_{r}\in(k-1,k]}$
or the realized $\mathrm{LGD}_{t}$ are not necessary. Therefore, the valuation
function (2.3) at horizon $h$ can be reformulated as

$\displaystyle L_{h}^{(r)}$
$\displaystyle=\sum_{k=h+1}^{n}\delta_{k}s_{k}\mathbb{E}\left[\mathrm{PD}(k\mid\mathcal{F}_{k})\mid\mathcal{F}_{h}\right]+\delta_{n}\mathrm{PC}\mathbb{E}\left[\mathrm{PD}(n\mid\mathcal{F}_{n})\mid\mathcal{F}_{h}\right]$
(2.4)
$\displaystyle\qquad-\sum_{k=h+1}^{n}\delta_{k}\mathrm{EAD}_{k}\mathbb{E}\left[\left(\mathrm{PD}(k\mid\mathcal{F}_{k})-\mathrm{PD}(k-1\mid\mathcal{F}_{k})\right)(1-\mathrm{LGD}_{k})\mid\mathcal{F}_{h}\right].$

When constructing the loss distribution, firstly the paths of the risk factors
need to be simulated until the horizon. Then based on the simulated risk
factors, the rating transitions and the LGDs can be simulated (computed)
according to the transition and LGD models. If the loan defaults or expires
before the horizon, the losses are already obtained by the simulation. If the
loan is still performing (not defaulted or expired) at the horizon, the
valuation grids are then used to valuate the bullets and consequently the loss
after the horizon according to the valuation model. Aggregating the losses of
all the loans, one obtains the distribution of the aggregated loss. In Figure
2.2 a simple illustration is presented that shows how the valuation grid is
used to approximate the loss distribution. Scenarios and migrations are
simulated only until the time horizon. Then given each simulation, the
realised loss is computed for each loan. Aggregating the losses from all the
loans, the loss distribution is obtained by collecting the portfolio losses
from all the scenarios.

Figure 2.2: An illustration to determine the loss distribution using valuation
grids

As stated, the losses happened before or on the horizon can be directly
computed using the simulated transitions and LGDs and only the losses that
happened after the horizon will be evaluated by the valuation grid. Therefore,
in this paper we ignore the pre-horizon losses and only focus on the losses
after the horizon.

##  3 A dimension reduction approach: Bayesian filter projection

This section on our projection method is the core of the paper and the
approach is motivated as follows. The desired advantage of the proposed logit
transition model, see (4.2) and (4.4), is that it allows a high dimensional
multi-factor model which describes the transition data accurately. However,
the challenge of implementing a high dimensional transition model in credit
risk modeling is the computational complexity in the expected loss valuation
using valuation grid. Ideally, the high dimensional transition model is used
to simulate the risk factors and determine the transitions, while lower
dimensional factors are used for the valuation grid. Note that the simulated
higher dimensional factors that are used to determine the transitions and the
lower dimensional factors used in the valuation grid need to be consistent.
Therefore a (dimension reduction) projection approach is required to map the
higher dimensional factors to lower dimensional factors. The most used
dimension reduction method approach to tackle this problem is Principal
Component Analysis (PCA). But this approach is not optimal to capture the
transition probabilities in a lower dimensional space. Hence the performance
of this approach is not assured, see the experiments in Sections 4.3.1, 4.3.3
and 4.3.5. In this section, we propose a dimension reduction method, which
link the higher dimensional factors with the lower dimensional factor through
the Bayesian filter (smoother). Since the Bayesian filter (smoother) achieves
minimum mean square error (MMSE) Chen (2003), the proposed dimension reduction
method is optimal for the criterion of the Bayes risk, i.e. MMSE, to project
the simulated higher dimensional factors to lower dimensional factors. The
Bayesian filter is a general probabilistic approach for estimating the
(distribution) of the latent states in a state space model. The state space
model and the Bayesian filter will be described in Sections 3.1 and 3.2
respectively. The proposed dimension reduction algorithm is described in
Section 3.3.

###  3.1 State space model

State-space models deal with dynamic time series problems that involve
unobserved variables or parameters that describe the evolution in the state of
the underlying system. The general state space model, defined on some
probability space $(\Omega,\mathcal{F},\mathbb{P})$, is as follows

$\displaystyle x_{k}$ $\displaystyle=f_{k}(x_{k-1},u_{k}),\,x_{0}$ (3.1)
$\displaystyle y_{k}$ $\displaystyle=h_{k}(x_{k},v_{k})\,,\quad
k\in\mathbb{N}^{+}\,,$

where $f_{k}:\mathbb{R}^{d}\times\mathbb{R}^{p}\rightarrow\mathbb{R}^{d}$,
$h_{k}:\mathbb{R}^{d}\times\mathbb{R}^{q}\rightarrow\mathbb{R}^{m}$ are given
Borel measurable functions, $\\{u_{k}\\}_{k\in\mathbb{N}^{+}}$ are
$p$-dimensional and $\\{v_{k}\\}_{k\in\mathbb{N}^{+}}$ are $q$-dimensional
white noise processes both independent of the initial condition $x_{0}$, and
mutually independent as well. Parameters in the functions $f_{k}$ and $h_{k}$,
together with the covariance of $u_{k}$ and $v_{k}$ can be seen as the
parameters of the state space model, to which we collectively refer to as
$\psi$.

We are interested in estimating the (latent) state process
$\\{x_{k}\\}_{k\in\mathbb{N}^{+}}$, but only have access to the process
$\\{y_{k}\\}_{k\in\mathbb{N}^{+}}$ which represents the observations. Because
of the existence of the white noise in the data, estimating the values, or
even the distributions, of the latent states
$\\{x_{k}\\}_{k\in\mathbb{N}^{+}}$ by the observations
$\\{y_{k}\\}_{k\in\mathbb{N}^{+}}$ is not trivial. There are different
methodologies in the literature to estimate the latent process, see e.g. Press
(2003); Chui and Chen (2017); Arulampalam et al. (2002), and the Bayesian
filter (smoother) is one of the mostly used approach in practice. Especially,
the Bayesian filter (smoother) can be considered as a projection from the
observation space to the state space. The Bayesian filter and smoother will be
explained in Section 3.2.

The state space model is widely used in many different areas such as economics
and finance, statistics, computer science, and electrical engineering and
neuroscience. Particularly in finance, it can be used to identify the latent
variables including business cycles, expectations of certain economic
variables, permanent income streams, reservation wages, etc. The state space
model is also often applied to factor models to describe, for instance,
interest rate and credit risk driving factors.

###  3.2 Bayesian filter and smoother

The Bayesian filter and Bayesian smoother, see e.g. Särkkä (2013); Press
(2003); Robert (2007) for an overview, are used to estimate the distribution
of the latent states $\\{x_{k}\\}_{k\in\mathbb{N}^{+}}$ in (3.1) given the
parameters, which we denote $\psi$. Often, but not always, we emphasize the
dependence of densities on the parameters in the notation.

We let the initial density function of $x_{0}$ be $p_{0}$. The methodology in
Bayesian filtering consists of two parts: prediction and update. At every time
point $k$, the prediction part computes (estimates) the prior distribution
(density) of $x_{k}$ given the past observations up to time $k-1$),

$p(x_{k}\mid y_{1:k-1},\psi),$

and the update part computes (estimates) the posterior distribution (density)
of $x_{k}$ given the past up to time $k$,

$p(x_{k}\mid y_{1:k},\psi).$

The purpose of the Bayesian smoother555This definition actually applies to the
fixed-interval type of smoothing. For fixed-leg smoothing, one can consult
Anderson and Moore (2012). is to compute the marginal posterior distribution
of the state $x_{k}$ at the time step $k$ after receiving the measurements up
to a time step $T$ with $T>k$,

$p(x_{k}\mid y_{1:T},\psi).$

The difference between filters and smoothers is that that the filter uses the
information from the previous and current period to estimate the posterior
distribution of current state $x_{k}$, while the smoothers uses also the
observations after the current period and hence produces optimal estimations.

It follows that the state space model (3.1) satisfies the properties of a
stochastic system, i.e. at every (present) time $k\geq 1$ the future states
and future observations $(x_{j},y_{j})$, $j\geq k$, are conditionally
independent from the past states and observations $(x_{j},y_{j-1})$, $j\leq
k$, given the present state $x_{k}$, see van Schuppen (1989). It then follows
that $\\{x_{k}\\}_{k\in\mathbb{N}}$ is a Markov process, and for every $k\geq
1$ one has that $y_{k}$ and $y_{1:k-1}$ are conditionally independent given
$x_{k-1}$, in terms of densities,

$p(x_{k}\mid x_{k-1},y_{1:k-1})=p(x_{k}\mid x_{k-1}).$ (3.2)

Similarly, due to Markov property, $x_{k}$ and $y_{k+1:T}$ are independent
given $x_{k+1}$, which gives

$p(x_{k}\mid x_{k+1},y_{1:T})=p(x_{k}\mid x_{k+1},y_{1:k}).$ (3.3)

Moreover, one also has, for every $k\geq 1$, that $y_{k}$ and $y_{1:k-1}$ are
conditionally independent given $x_{k-1}$, in terms of densities,

$\displaystyle p(y_{k}\mid y_{1:k-1},x_{k})=p(y_{k}\mid
x_{k})\,,\quad\mbox{for}\quad k\in\mathbb{N}^{+}\,.$ (3.4)

These three equations, (3.2), (3.3) and (3.4) are used in the next sections in
the derivation of Bayesian filter and smoother recursions.

####  3.2.1 Bayesian filter recursion

Using Bayes’ rule and Equation (3.2), we deduce that the density function of
the prior distribution is given by

$\displaystyle p(x_{k}\mid y_{1:k-1},\psi)$ $\displaystyle=\int p(x_{k}\mid
x_{k-1},y_{1:k-1},\psi)p(x_{k-1}\mid y_{1:k-1},\psi)\,\mathrm{d}x_{k-1}$
$\displaystyle=\int p(x_{k}\mid x_{k-1},\psi)p(x_{k-1}\mid
y_{1:k-1},\psi)\,\mathrm{d}x_{k-1}\,.$ (3.5)

Note that when $k=1$, the posterior distribution $p(x_{k-1}\mid
y_{1:k-1},\psi)$ is defined as the initial density $p_{0}$.

The purpose of the Bayesian algorithm is to sequentially compute the posterior
distribution $p_{k}(\mathrm{d}x_{k}\mid\psi)=p(\mathrm{d}x_{k}\mid
y_{1:k},\psi)$. Again using Bayes’ rule, (3.4) and (3.5), we obtain the
posterior

$\displaystyle p(x_{k}\mid y_{1:k},\psi)$ $\displaystyle=\frac{p(y_{k}\mid
x_{k},y_{1:k-1},\psi)p(x_{k}\mid y_{1:k-1},\psi)}{p(y_{k}\mid
y_{1:k-1},\psi)}$ $\displaystyle=\frac{p(y_{k}\mid x_{k},\psi)p(x_{k}\mid
y_{1:k-1},\psi)}{\int p(y_{k}\mid x_{k},\psi)p(x_{k}\mid
y_{1:k-1},\psi)\,\mathrm{d}x_{k}}.$ (3.6)

If we assume the marginal likelihood function $p(y_{k}\mid x_{k},\psi)$ and
the transition probability $p(x_{k}\mid x_{k-1},\psi)$ are known, then given
the posterior density $p(x_{k-1}\mid y_{1:k-1},\psi)$ at time $k-1$, we can
use Equations (3.5) and (3.6) to compute the posterior density $p(x_{k}\mid
y_{1:k},\psi)$ at time $k$. In this way the posterior distributions can be
computed recursively given the initial distribution $p_{0}$.

####  3.2.2 Bayesian smoother recursion

According to Bayes’s rule and Equations (3.2) and (3.3), the density
$p(x_{k}\mid y_{1:T})$ can be expressed as

$\displaystyle p(x_{k}\mid y_{1:T},\psi))$ $\displaystyle=\int p(x_{k}\mid
x_{k+1},y_{1:T},\psi)p(x_{k+1}\mid y_{1:T},\psi)\,\mathrm{d}x_{k+1}$
$\displaystyle=\int p(x_{k}\mid x_{k+1},y_{1:k},\psi)p(x_{k+1}\mid
y_{1:T},\psi)\,\mathrm{d}x_{k+1}$ $\displaystyle=\int\frac{p(x_{k},x_{k+1}\mid
y_{1:k},\psi)}{p(x_{k+1}\mid y_{1:k},\psi)}p(x_{k+1}\mid
y_{1:T},\psi)\,\mathrm{d}x_{k+1}$ $\displaystyle=\int\frac{p(x_{k+1}\mid
x_{k},y_{1:k},\psi)p(x_{k}\mid y_{1:k},\psi)}{p(x_{k+1}\mid
y_{1:k},\psi)}p(x_{k+1}\mid y_{1:T},\psi)\,\mathrm{d}x_{k+1}$
$\displaystyle=p(x_{k}\mid y_{1:k},\psi)\int\frac{p(x_{k+1}\mid
x_{k},\psi)}{p(x_{k+1}\mid y_{1:k},\psi)}p(x_{k+1}\mid
y_{1:T},\psi)\,\mathrm{d}x_{k+1}.$ (3.7)

According to Equation (3.7), given that the prior and posterior distribution
are known from the Bayesian filter recursions, the Bayesian smoother
estimation $p(x_{k}\mid y_{1:T},\psi)$ can be computed using a backward
recursion.
While the mathematical derivations of the Bayesian filter and smoother are
straightforward, in practice it is always a big challenge to compute the
integrals in Equations (3.5), (3.6) or (3.7). Although there are some special
cases (Gaussian linear cases) where theoretical formulas are available for the
integrals, such as Kalman filter (see Welch and Bishop (1995) for instance),
in most cases one has to find approximations or numerical algorithms to
compute these integrals. For the extended Kalman filter, see, for example,
Einicke and White (1999) and Wan and Nelson (2002) and for the unscented
Kalman filter, see Wan and van der Merwe (2002), the prior and the posterior
distributions are approximated by Gaussian distributions. Durbin and Koopman
(2012, Section 10.6–10.7) proposes a mode estimation approach to compute the
maximum-a-posterior (MAP) estimate of the posterior distribution. A more
general approach to estimate the prior and posterior distribution is the
particle filter, see e.g. Arulampalam et al. (2002); Cappé et al. (2007);
Doucet and Johansen (2011), in which prior and posterior distributions are
estimated by a Monte Carlo method. For the implementation of the particle
filter one can also refer to Chopin et al. (2013); Crisan and Míguez (2018);
He et al. (2021).

###  3.3 Bayesian filter/smoother projection

As described in Section 3.2, the Bayesian filter and smoother can be seen as a
projection from the observations $y_{1:k}$ to the (distribution of) state
variables, given the model parameters $\psi$. From this perspective, the
Bayesian filter (smoother) can be used as an approach to project higher
dimensional factors to lower dimensional factors. In this section we will
illustrate the Bayesian filter (smoother) projection approach in the credit
risk environment specified in Sections 2.2. However, note that this approach
can also easily be applied or extended to other models for different risk
types, such as market risk or interest rate risk, see also the examples in
Section 3.4.
Suppose transition models, see Section 2.1, with higher and lower dimension
are calibrated based on the same data, and denote the calibrated parameters of
higher and lower dimensional model by $\psi_{H}$ and $\psi_{L}$, respectively.
The higher dimensional factor model is used to simulate the transition
probabilities until the horizon according to the transition model, for
instance, a simulation based on Equations (4.4) and (4.2) in Section 4. Denote
the simulated higher dimensional factors by $x_{k,H}$ and the corresponding
transition probabilities by $T(x_{k,H})$, for $k=1,\ldots,h+1$. One would like
to use a lower factor model for a loss calculation. It is then required to
project the higher dimensional factors $x_{k,H}$ to the lower dimensional
factors, denoted by $x_{k,L}$. Note that the simulated transition
probabilities $T(x_{k,H})$, based on the higher dimensional factors $x_{k,H}$,
can be used as the observations of the lower factor model. Therefore, using
the Bayesian filter and smoother iterations, one can obtain a projection from
$x_{k,H}$ to $x_{k,L}$, as described in Algorithm 3.1.

######  Algorithm 3.1 (projection).

Given are the calibrated parameters of the higher and lower factor model
$\psi_{H}$ and $\psi_{L}$. Suppose $x_{1:h+1,H}$ and $T(x_{1:h+1,H})$ are the
simulated higher dimensional factors and the corresponding transition
probabilities from time $k=1$ to horizon $h+1$.

  1. 1.

Specify the initial density $p(x_{0}\mid\psi_{L})$ for the lower dimensional
factors.

  2. 2.

Using the transition probabilities $T(x_{1:h+1,H})$ as the data,

    1. a.

for $k=1,\dots,h+1$, compute the prior and posterior distribution $x_{k,L}$
according to the Bayesian filter iterations, see Section 3.2.1, i.e.

$\displaystyle p(x_{k}\mid T(x_{1:k-1,H}),\psi_{L})$ $\displaystyle=\int
p(x_{k}\mid x_{k-1},\psi_{L})p(x_{k-1}\mid
T(x_{1:k-1,H}),\psi_{L})\,\mathrm{d}x_{k-1}\,$ $\displaystyle p(x_{k}\mid
T(x_{1:k,H}),\psi_{L})$ $\displaystyle=\frac{p(T(x_{k,H})\mid
x_{k},\psi_{L})p(x_{k}\mid T(x_{1:k-1,H}),\psi_{L})}{\int p(T(x_{k,H})\mid
x_{k},\psi_{L})p(x_{k}\mid T(x_{1:k-1,H}),\psi_{L})\,\mathrm{d}x_{k}};$

    2. b.

given prior and posterior distributions computed in step 22.a, for
$k=h,\ldots,1$, compute the Bayesian smoother using the backward recursion,
see Section 3.2.2,

$\displaystyle p(x_{k}\mid T(x_{1:h+1,H}),\psi_{L})$
$\displaystyle=p(x_{k}\mid T(x_{1:k,H}),\psi_{L})\times$
$\displaystyle\int\frac{p(x_{k+1}\mid x_{k},\psi_{L})}{p(x_{k+1}\mid
T(x_{1:k,H}),\psi_{L})}p(x_{k+1}\mid
T(x_{1:h+1,H}),\psi_{L})\,\mathrm{d}x_{k+1};$

    3. c.

determine the lower dimensional factor $x_{k,L}$ projected from $x_{k,H}$ as
equal to

$\mathbb{E}\left[p(x_{k}\mid T(x_{1:h+1,H}),\psi_{L})\right],\,k=1,\ldots,h.$

We are in particular interested in the value of the lower dimensional factor
at horizon $h$, i.e. $x_{h,L}$, since it is finally used as the input of the
valuation grid to evaluate the losses after this horizon. Note that in order
to obtain the optimal estimate of $x_{h,L}$, the simulation is generated until
$h+1$ so that the Bayesian smoother, step 22.b in Algorithm 3.1, can be used
for $x_{h,L}$. Figure 3.1 presents the diagram of the projection approach.

Figure 3.1: High level description of projection algorithm

######  Remark 3.1.

An important input for the iteration of the projection is the initial
distribution of $x_{k,L}$, i.e. $p(x_{0}\mid\psi_{L})$. In credit risk, there
are two methodologies to specify the initial distribution (or starting point)
of the simulation, the point-in-time (PiT) or through-the-cycle (TtC). In PiT,
all the simulations start at the same point and this point is defined as the
‘state of today’, i.e. the value of the risk factor at the last period used in
the calibration. In TtC, instead of defining a single starting point, an
initial distribution is assigned to sample the starting point of each path.
This initial distribution is usually defined as the equilibrium distribution
of the risk factors process.666In practice, constraints are always imposed to
make sure that the risk factor process is stationary and hence the equilibrium
distribution exists. Therefore, in both PiT and TtC cases, the initial
distribution of $x_{k,L}$ can be easily defined.

Integrating the projection algorithm into the calibration, simulation and
valuation algorithms, we have our engine for the risk calculation, illustrated
by Figure 3.2, and described in the following Algorithm 3.2 , formulated in
high level terms.

######  Algorithm 3.2 (calibration, training, simulation, projection and
valuation).

  * •

Calibration

    1. 1.

The transition matrices are modeled by a higher factor model, which is
calibrated using historical data. One obtains $\psi_{H}$, the calibrated
parameters of the higher dimensional model.

    2. 2.

A lower factor model is also calibrated using the historical data and one
obtains $\psi_{L}$, the calibrated parameters of the lower dimensional model.

  * •

Training of the interpolation grid
Define the higher dimensional grid, perform then for each rating the following
steps.

    1. 1.

Starting at the points on the defined grid, use the higher factor model to
simulate the scenarios of the risk factors and determine the expected losses.

    2. 2.

Project the points (factors) of the higher dimensional grid to obtain lower
dimensional points (factors) using the projection approach as in Algorithm
3.1.

    3. 3.

Use the lower dimensional points and the simulated expected losses to build
the valuation grid, using interpolation or other techniques like Gaussian
process regression, Neural networks, etc.

  * •

Simulation

    1. 1.

Using the high factor model, generate the factors and hence the transition
matrices until the horizon.

    2. 2.

Based on the generated transition matrices, simulate the rating migrations of
the obligors until the horizon.

  * •

Projection and valuation

    1. 1.

Project according to the Projection Algorithm 3.1 the higher dimensional
factors simulated in Step 2 of the Simulation part above to lower dimensional
factors.

    2. 2.

Use the valuation grid to evaluate the expected losses, given the ratings at
the horizon and the lower dimensional factors.

    3. 3.

The desired risk metrics, such as expected loss and value-at-risk, can finally
be obtained by aggregating the expected losses.

Figure 3.2: High level description of the algorithm

###  3.4 Application examples

The proposed Bayesian projection approach is a generic dimension reduction
method which can be used for different types of credit risk valuations. It is
originally designed for the credit risk valuations, but it can also be applied
to other grid based valuation models where the dimension reduction approach is
needed. We list some possible directions for further applications of the
proposed Bayesian projection approach in the following subsections.

####  3.4.1 Credit Risk Economic Capital

Economic Capital, mainly for financial service forms, is a metric to determine
the amount of capital a firm must hold to protect itself against economic
risks (such as market risk, credit risk, and operational risk), given its
mixture of assets and liabilities and its strategy to risk appetite. The risks
are measured by the unexpected losses through client defaults and the asset
value changes through the movements in the market parameters. Typically the
losses are estimated based on a time horizon and a confidence level. The time
horizon is usually one year and the confidence level is usually $99.9\%$ or
$99.95\%$, implying that the Economic Capital should cover the losses that
would arise in case an adverse event would occur with a probability of $0.1\%$
or $0.05\%$ over one year horizon. In particular, the Credit Risk Economic
Capital (CREC) is to measure the internal capital required to cover the
unexpected credit losses under ECB ICAAP Guidelines, see European Central Bank
(2018).

To measure the CREC, one particularly looks into two statistics: the expected
loss (EL) and the $99.9\%$ or $99.95\%$ quantile of the portfolio loss. The
CREC covers both default risk and migration risk. The default risk captures
the losses from the default event within the time horizon, while the migration
risk concerns the potential economic losses associated with non-default credit
rating migrations. Although a non-default transition may not result in a
direct loss under accrual accounting at the time horizon, it may result in an
indirect loss in terms of the expected future cashflows, interest, principal
and recoveries. According to this nature, the calculation of the CREC risk
measures can be divided into two parts: the realised default losses within the
time horizon and the potential losses after the time horizon. A Monte Carlo
approach can be directly used to estimate the default losses within the time
horizon, while a valuation grid is needed to compute the potential losses
after the time horizon. Higher dimensional factors are desired for the
simulation of the defaults and rating migrations up to the horizon, due to the
higher precision. Meanwhile, a dimensional factors are used as the inputs of
the grids to compute the potential losses after horizon. Therefore, the
proposed Bayesian dimension reduction approach can be implemented to project
higher dimensional factors to lower dimensional factors for the valuation of
the potential losses after the horizon.

####  3.4.2 Incremental Risk Charge

The Incremental Risk Charge (IRC), see European Banking Authority (2012), is a
complement to the traditional stressed Value-at-Risk measured in Market Risk
Economic Capital. It is an estimate of default and migration risk of
unsecuritized credit products in the trading book. The concept and calculation
of IRC is in general similar to CREC. The IRC also measures the default and
migration risks at a $99.9\%$ level of confidence over a one-year horizon.
However, the CREC model mainly applies to the banking book, including
exposures to government, institutions, corporates, retail exposures secured by
real estate, and revolving retail exposures. It also covers the (one-year)
counterparty credit risk of derivative instruments, securities financing
transactions and hold-to-maturity assets in the trading book. By comparison,
the IRC mainly concerns the credit risks of the bond and equity exposures in
the trading book. Different from the booking book, in which the positions are
held to maturity, the positions in the trading book are continuously bought
and sold before their maturities. Therefore, unlike CREC, the IRC model
differentiates the underlying traded instruments by liquidity horizon, with a
minimum of three months777Note that the capital horizon is still one year.. A
constant level of risk assumption 888The constant level of risk means the risk
of the portfolio remains to be its initial risk level over the one-year
horizon. In practice, This can be achieved by holding the constant portfolio
over the liquidity horizon, rebalancing any default, downgraded, or upgraded
positions at the beginning of each liquidity horizon, and rolling over any
matured positions at the beginning of each horizon. is also imposed in IRC and
ensures that all positions in the IRC portfolio are evaluated over the full
one-year time horizon. Since the calculation of IRC also requires
migration/default simulations for every liquidity until a one year capital
horizon, and the migration risks need to be evaluated according to certain
valuation function, the proposed Bayesian dimension reduction approach can be
applied to make the calculations more efficient.

####  3.4.3 International Financial Reporting Standard 9

The International Financial Reporting Standard 9 (IFRS 9), see for example
International Accounting Standards Board (2016), Board (2019), Gornjak (2020),
is published by the International Accounting Standards Board (IASB). It
addresses the accounting for financial instruments and it replaced the
International Accounting Standard (IAS) 39 with regard to the methodology used
to compute impairment provisions on financial instructions. IFRS 9 introduces
a new impairment model based on the Expected Credit Loss (ECL), which is
different from the Incurred Credit Loss (ICL) model under IAS 39. Under IFRS
9, institutions, as reporting entities, will have to recognize not only
incurred credit losses but also losses that are expected in the future. In
IFRS 9, two different ECLs are required, depending on the stages of
impairment, see Bank for International Settlements (2017). In stage 1, the
ECLs result from the default events that are possible within the next 12
months (12-month ECL). While in stage 2 or 3, lifetime ECLs are recognised. To
obtain a 12-month ECL, a Monte Carlo simulation can be used based on the given
PD and LGD models. While, for the lifetime ECL, a grid approach is
required999Sometimes the 12-month ECL is used to construct a proxy for the
lifetime ECL in the simplified approach, see European Banking Authority
(2021).. Since the valuation of life time ECL usually involves a large number
of factors from the PD and LGD models, the proposed Bayesian projection
approach can be applied to reduce the dimensionality of the valuation grid for
ECL and hence improve the efficiency of the computations.

####  3.4.4 Market risk

Market risk can be defined as the risk of losses in on and off-balance sheet
positions arising from adverse movements in market prices. From a regulatory
perspective, market risk stems from all the positions included in the trading
book as well as from commodity and foreign exchange risk positions in the
whole balance sheet. In market risk calculations, the VaR needs to be
determined according to the future Profit-and-Loss (P&L) distribution of the
portfolios. The value of the portfolio depends on the future values of the
asset prices, equity market volatilities, interest rates, etc. Therefore, to
obtain the P&L distribution, one has to simulate the future scenarios of (the
risk factors of) the relevant asset prices, equity market volatilities or
interest rates curves, and re-evaluate the portfolio values given the
simulated scenarios. In order to efficiently re-evaluate the portfolio values,
a valuation grid is usually used, see for instance Gibson and Pritsker (2000),
Zamani et al. (2022). The valuation grid links the risk factors and the
derivative values. However, many derivatives involve multiple underlyings, for
instance the multi-asset options (Leentvaar and Oosterlee (2007), Ekedahl et
al. (2007)). Therefore, the proposed Bayesian projection approach can be used
in pricing such derivatives to avoid the “curse of dimensionality”.

####  3.4.5 Interest rate risk in the banking book

Interest rate risk in the banking book (IRRBB) refers to the current or
prospective risk of the bank’s capital and earnings arising from adverse
movements in interest rates that affect the bank’s banking book positions.
When interest rates change, the present value and timing of future cash flows
change. This in turn changes the underlying value of a bank’s assets,
liabilities and off-balance sheet items and hence its economic value. Changes
in interest rates also affect a bank’s earnings by altering interest rate-
sensitive income and expenses, affecting its net interest income (NII). The
calculation of the VaR for IRRBB is similar to the calculation of VaR for
market risk. The P&L is generated by simulating the (risk factors of) interest
rates curves and re-evaluating the balance sheet given the simulated
scenarios. The re-evaluation of the balance sheet can be done by using the
valuation grid method. In the past years since the credit and liquidity crisis
of 2007, a multi-curve framework of interest rates was developed to replace
the single-curve framework. The single-curve framework bootstrapped a single
risk-free curve which represented both the cost of funding future cash ﬂows
and forward rates. This unique curve is used for both discounting and
forecasting, while in the multi-curve framework, the discount curves and the
forecasting curves are distinguished. As expected, the different forecasting
tenor curve forecasts the future cashﬂows of different tenors and the
discounting curve values these cashﬂows to today. More details regarding the
single- and multi-curve framework can be found in, for instance, Henrard
(2014), Pallavicini and Tarenghi (2010). The multi-curve framework introduces
more curves in pricing the derivatives. For example, three curves are needed
to price the Constant Maturity Swaps or the Basis Swaps, i.e. one discounting
curve and two forecasting curves. If each curve requires a two-factor model,
in total six factors are needed to construct a valuation grid for these
derivatives. To decrease the dimensionality of the valuation grid, one can
apply the proposed Bayesian projection approach.

##  4 Numerical studies

In the sequel we illustrate the performance of the projection approach by
examples. We aim to forecast the bullets in Equation (2.4), and hence forecast
the loss distribution of a loan. The chosen loan in the experiment is a unit
bullet loan, of which the notional is 1. Consequently the principal repayment
$\mathrm{PC}=0$ and $\mathrm{EAD}_{k}=1$ for $k=1,\ldots,n$. The coupon
payments $s_{k}$ are set to be equal to $s=\frac{1}{n},k=1,\ldots,n$. For
convenience and without loss of generality, discounting is not considered in
this experiments. Hence Equation (2.4) simplifies to

$\displaystyle L_{1}^{(r)}$
$\displaystyle=\sum_{k=2}^{n}s\mathbb{E}\left[\mathrm{PD}(k\mid\mathcal{F}_{k})\mid\mathcal{F}_{h}\right]+\mathbb{E}\left[\mathrm{PD}(n\mid\mathcal{F}_{n})\mid\mathcal{F}_{h}\right]$
(4.1)
$\displaystyle\qquad-\sum_{k=h+1}^{n}\mathbb{E}\left[\left(\mathrm{PD}(k\mid\mathcal{F}_{k})-\mathrm{PD}(k-1\mid\mathcal{F}_{k})\right)(1-\mathrm{LGD}_{k})\mid\mathcal{F}_{h}\right].$

In all experiments we assume the maturity of the unit loan after a horizon of
30 periods, i.e. $n=30$.
The value of the bullets depends on the transition and LGD models, therefore,
we next introduce the transition and LGD models used in the experiments.

###  4.1 Transition model

The transition model we use belongs to the reduced form model. Let
$k=1,\ldots,n$. Suppose we have $R>2$ credit states. We denote by $T_{ij,k}$
the probability at time $k$ of the transition from a credit state
$i=1,\ldots,R$ to a credit state $j=1,\ldots,R$. We model the transition
probabilities (or observed transition rates) using a logit function as
follows,

$T_{ij,k}=\frac{g_{ij}\exp(\theta_{ij,k})}{\sum_{j=1}^{R}g_{ij}\exp(\theta_{ij,k})}\,,\qquad
i,j=1,\ldots R\,,\quad k=1,\ldots,n\,.$ (4.2)

The $\theta_{ij,k}+\log{g_{ij}}$ are usually referred to as the log-odds for
the transition from a credit state i to a credit state j, in which the
$g_{ij}$ are the parameters which indicate the level of the transition
probabilities and $\theta_{ij,k}$ are the signals that describe the dynamics
of the transition probabilities. Suppose, for clients in the $i$-th rating,
that in $N_{i,k}$ independent trials the number of observed migrations of
these clients to the $j$-th credit state is $m_{ij,k}$ for $j=1,\ldots,R$. It
is easy to see that $m_{i,k}=[m_{i1,k},\ldots,m_{iR,k}]$ has a multinomial
distribution, with log-density given by

$\log p(m_{i,k}|T_{k})=\sum_{j=1}^{R}m_{ij,k}\log T_{ij,k}+\log C_{i,k}\,,$
(4.3)

where

$C_{i,k}=\frac{N_{i,k}!}{\prod_{j=1}^{R}m_{ij,k}!}.$

Plugging Equation (4.2) into Equation (4.3), one can rewrite the log-
likelihood function (4.3) at time $k$ as

$\log p(m_{i,k}|\theta_{ij,k})=\sum_{j=1}^{R}m_{ij,k}(\theta_{ij,k}+\log
g_{ij})-N_{i,k}\log\big{(}\sum_{j=1}^{R}g_{ij}\exp(\theta_{ij,k})\big{)}+\log(C_{k})\,.$

We assume that the migrations from a state $i$ to a state $j$ are driven by
the latent common factors $x$ (hence the migrations are also correlated). The
latent process $x$ is assumed to evolve linearly over time with a Gaussian
error $\eta$, and therefore our transition model is given by:

$\displaystyle m_{i,k}$ $\displaystyle\sim
p(m_{i,k}|\theta_{ij,k})\,,\quad\theta_{ij,k}=Kx_{k},\quad k=1,\ldots,n\,,$
(4.4) $\displaystyle x_{k}$
$\displaystyle=Ax_{k-1}+\eta_{k}\,,\quad\eta_{k}\sim\mathcal{N}(0,Q),$

where $x_{k}$ is a $d\times 1$ random vector, $K$ is a $(R-1)\times d$-matrix,
and $A$ and $Q$ are $d\times d$-matrices. The $g=\\{g_{ij},i,j=1,\ldots,R\\}$,
$K$, $A$ and $Q$ are referred to as the parameters of the transition model.
The initial distribution of $x_{k}$ is assumed to be Gaussian with mean
$a_{0}$ and (co)variance $P_{0}$, i.e. $x_{0}\sim\mathcal{N}(a_{0},P_{0})$.

This transition model belongs to the class of _state space models_ of Section
3.1. The latent states $x_{t}$ and the model parameters are usually estimated
by the Bayesian filter introduced in Section 3.2. Specially, we estimates the
latent states $x_{t}$ by using the method of _mode estimation_ (Durbin and
Koopman (2012, Section 10.6)) based on the Kalman filter.
We suppose to have four different ratings with three performing ratings (P1,
P2 and P3) and one default rating (D). We further assume that the default
state is absorbing, which means that the probabilities of the transition from
D to P1, P2 or P3 are zero. The set-up of the transition model in this example
is the same as in Equations (4.4) and (4.2). In this experiment we have three
models: one benchmark model and two testing models. The benchmark model is
chosen to be a four factor model, i.e. the dimensionality of $x_{k}$ is four.
This model is treated as the ‘true’ model and used to generate the loss
distribution which serves as a benchmark for comparison. The two testing
models are chosen to be a two factor model and an one factor model. These two
models are used to assess the performance of the proposed dimension reduction
approach. All models are calibrated using the same migration data with a
horizon of 120 periods. These migration data are simulated by using the
benchmark model with parameters as follows. The parameter values are obtained
from further classified real data, which we are not allowed to disclose.

$\displaystyle A$ $\displaystyle=\mathrm{diag}([0.6,0.95,0.9,0.5]),$
$\displaystyle Q$ $\displaystyle=\mathrm{diag}([0.6,0.1,0.1,0.7]),$
$\displaystyle K$
$\displaystyle=\begin{pmatrix}k_{11,1}&k_{11,2}&k_{11,3}&k_{11,4}\\\
k_{12,1}&k_{12,2}&k_{12,3}&k_{12,4}\\\ k_{13,1}&k_{13,2}&k_{13,3}&k_{13,4}\\\
k_{14,1}&k_{14,2}&k_{14,3}&k_{14,4}\\\ k_{21,1}&k_{21,2}&k_{21,3}&k_{21,4}\\\
k_{22,1}&k_{22,2}&k_{22,3}&k_{22,4}\\\ k_{23,1}&k_{23,2}&k_{23,3}&k_{23,4}\\\
k_{24,1}&k_{24,2}&k_{24,3}&k_{24,4}\\\ k_{31,1}&k_{31,2}&k_{31,3}&k_{31,4}\\\
k_{32,1}&k_{32,2}&k_{32,3}&k_{32,4}\\\ k_{33,1}&k_{33,2}&k_{33,3}&k_{33,4}\\\
k_{34,1}&k_{34,2}&k_{34,3}&k_{34,4}\\\ k_{41,1}&k_{41,2}&k_{41,3}&k_{41,4}\\\
k_{42,1}&k_{42,2}&k_{42,3}&k_{42,4}\\\ k_{43,1}&k_{43,2}&k_{43,3}&k_{43,4}\\\
k_{44,1}&k_{44,2}&k_{44,3}&k_{44,4}\\\ \end{pmatrix}=\begin{pmatrix}0&0&0&0\\\
0.2&0.04&0.06&0.1\\\ 0.12&-0.36&0.12&-0.04\\\ 0.38&-0.28&-0.26&-0.08\\\
-0.17&0.34&0.18&-0.01\\\ 0&0&0&0\\\ 0.02&-0.27&0.08&-0.01\\\
-0.07&-0.14&0.16&-0.05\\\ -0.22&-0.11&0.01&0.12\\\ -0.03&-0.2&0.01&0.22\\\
0&0&0&0\\\ -0.08&0.09&-0.05&-0.03\\\ 0&0&0&0\\\ 0&0&0&0\\\ 0&0&0&0\\\
0&0&0&0\\\ \end{pmatrix},$ $\displaystyle G$
$\displaystyle=\begin{pmatrix}g_{11}&g_{12}&g_{13}&g_{14}\\\
g_{21}&g_{22}&g_{23}&g_{24}\\\ g_{31}&g_{32}&g_{33}&g_{34}\\\
g_{41}&g_{42}&g_{43}&g_{44}\\\
\end{pmatrix}=\begin{pmatrix}0.95&0.03&0.0198&0.0002\\\ 0.05&0.9&0.04&0.01\\\
0.05&0.12&0.78&0.05\\\ 0&0&0&1\\\ \end{pmatrix}.$

###  4.2 LGD model

We adopt ideas from Frontczak and Rostek (2015) and use the collateral to
model the LGD as

$\mathrm{LGD}_{k}=(1-\frac{c_{k}}{\mathrm{EAD}_{k}})^{+},$ (4.5)

where $c_{k}$ is the value of the collateral at time $k$. At time $k$, the
log-return of the collateral value is denoted $\mathrm{LC}_{k}$,
$\mathrm{LC}_{k}=\frac{c_{k}}{c_{k-1}}$, $k\geq 1$. The log-return of the
collateral is assumed to be modeled by an AR(1) process, see also Equation
(A.2),

$\mathrm{LC}_{k}=0.73\,\mathrm{LC}_{k-1}+z_{k},\,z_{k}\sim N(0,0.04^{2}).$

The value of the collateral, recall (4.5), determines the LGD as follows.

$\displaystyle\mathrm{LGD}_{k}$
$\displaystyle=(1-\frac{c_{k}}{\mathrm{EAD}_{k}})^{+}$
$\displaystyle=(1-\frac{c_{k}}{c_{0}}*\frac{c_{0}}{\mathrm{EAD}_{0}}*\frac{\mathrm{EAD}_{0}}{\mathrm{EAD}_{k}})^{+}$
$\displaystyle=(1-\frac{c_{k}}{c_{0}}*\frac{1}{\mathrm{LTV}_{0}}*\frac{1}{n-k+1})^{+}.$

Note that, in order to determine the $\mathrm{LGD}_{k}$, the loan-to-value at
the horizon $\mathrm{LTV}_{0}$ should be defined. Moreover, for convenience,
we further assume that the risk factors for the transition matrices and the
log-returns of the collateral factors are independent, hence the default
probabilities and the LGD are independent. Note that this assumption is not
valid in real life practice. Actually, in practice the correlation between the
PD and LGD is a very important parameter in risk calculations. Under the
independence assumption the loss in Equation (4.1) is simplified to

$\displaystyle L_{1}^{(r)}$
$\displaystyle=\sum_{k=2}^{n}s\mathbb{E}\left[\mathrm{PD}(k\mid\mathcal{F}_{k})\mid\mathcal{F}_{h}\right]+\mathbb{E}\left[\mathrm{PD}(n\mid\mathcal{F}_{n})\mid\mathcal{F}_{h}\right]$
(4.6)
$\displaystyle\qquad-\sum_{k=2}^{n}\mathbb{E}\left[\left(\mathrm{PD}(k\mid\mathcal{F}_{k})-\mathrm{PD}(k-1\mid\mathcal{F}_{k})\right)\mid\mathcal{F}_{h}\right](1-\mathbb{E}\left[\mathrm{LGD}_{k}\mid\mathcal{F}_{h}\right]).$

Given the assumed independence, we only need to approximate the multi-year
expected probability of default $\mathbb{E}[\mathrm{PD}_{k}^{(r)}]$ by Monte
Carlo simulation, while the expected loss given default
$\mathbb{E}[\mathrm{LGD}_{k}]$ is analytically computed by using the Black-
Scholes formula (A.6) of Corollary A.2.
In order to assess the performance of the proposed Baysian projection
approach, a _waterfall_ terminology is used. In the waterfall, the analyses
are done step by step as follows.

  1. 1.

The Bayesian and PCA projection approach are compared when used to describe
the transition probabilities. The two approaches project the four dimensional
factors onto two dimensional factors respectively, and then the transition
probabilities based on the two dimensional factors are compared with the
original transition probabilities from the four dimensional factors. In
particular, for the Bayesian projection approach, Algorithm 3.1 is applied.

  2. 2.

The convergence analysis of the Monte Carlo simulation for expected PD is
conducted to find out how many Monte Carlo scenarios are needed to obtain a
reliable estimate of the multi-year expected PD.

  3. 3.

Then the Bayesian and PCA projection approaches are used in the valuation grid
to estimate the multi-year expected PD and accuracies are compared. In this
case, Algorithm 3.2 is applied.

  4. 4.

The closed-form Black-Scholes formula for the expected LGD is checked by
comparing with the Monte Carlo estimation. The expected LGDs are used to
construct the loss distribution according to Equation (4.1).

  5. 5.

In the end, the proposed Bayesian approach, see Algorithm 3.2, is used to
construct the loss distribution of a unit loan and the risk metrics (expected
losses and value-at-risk) are estimated. The accuracy of the estimation is
compared with the estimation from the PCA approach.

Table 1 summaries the waterfalls in the experiments Section 4.3.

Section | Description | Results
---|---|---
Section 4.3.1 | Comparison of Bayesian and PCA projection approach | Figure 4.1
| on transition probabilities |
Section 4.3.2 | Convergence of Monte Carlo simulation for expected PD | Figure 4.2
Section 4.3.3 | Comparison of Bayesian and PCA projection | Figures 4.3, 4.4
| approach on multi-year PD estimation |
Section 4.3.4 | Black-Scholes formula for expected LGD | Figure 4.5
Section 4.3.5 | Comparison of Bayesian and PCA projection approach | Figures 4.6-4.9
| on loss distribution of the target unite bullet loan |
Table 1: Waterfalls

###  4.3 Experiments

In this section we perform a number of numerical studies to assess and
illustrate our projection approach.

####  4.3.1 Transition probabilities

Suppose the four dimensional model and the two dimensional model are
calibrated on the data. In the experiment of this section, we assess the
potential error introduced by the dimension reduction approaches, when
projecting a higher dimensional transition model, in this case a four factor
model, to a lower dimensional transition model, in this case, a two factor
model. In particular, we apply the proposed projection approach, see Algorithm
3.1 and the PCA approach on the transition probability models and compare the
performance of the two approaches. To achieve this, in total 100 different
scenarios (risk factors and hence the transition probability matrices) are
simulated using the four factor model. Then the proposed projection approach
and the PCA approach are used to project the four dimensional factors onto two
dimensional factors. For the proposed approach, the two factor model above is
used and for the PCA approach the first two principal components are used to
match with the number of factors. Based on the two dimensional factors, the
transition probabilities can be recomputed according to the two factor model.
These transition probabilities from the two factor model are compared with the
original transition probabilities based on the four factor model and the
relative difference between them are calculated. Figure 4.1 shows the relative
difference (error) between the transition probabilities of the benchmark (four
factor model) and the projections from the proposed approach and PCA approach.
One can observe that the differences for the Bayesian projection approach are
significantly smaller than those for the PCA approach.

Figure 4.1: Relative difference of the transition probabilities: the red line
is for the proposed Bayesian projection approach and the blue line is for the
PCA. On the $x$-axis are the 100 simulated scenarios.

####  4.3.2 Expected PD simulation

In order to create the training samples for the expected losses, the expected
cumulative PD up to the maturity need to be approximated using Monte Carlo,
see Equation (4.6). In the experiment of this section we investigate how many
Monte Carlo scenarios are required to obtain a adequate approximation for the
expected cumulative PD. For each rating $P1,P2$ and $P3$, we randomly create
1000 starting points for the simulation. These starting points can be treated
as the points on the grids. Then starting at each point, the risk factor paths
are simulated up to the maturity, in total $30$ periods. The number of risk
factor paths for each starting points varies from 500 to 20000. The expected
cumulative PDs for each starting point are approximated by averaging over the
risk factor paths. For the expected cumulative PD approximation 20000 paths
are used as the benchmark, while the approximation using 500, 1000, 5000 and
10000 paths are used as the test sets. The relative difference of the expected
cumulative PD between the benchmark and the test sets are computed and the
results are presented in Figure 4.2. The results indicate that the expected
cumulative PD can be adequately approximated by using 1000 paths.

Figure 4.2: Maximum relative difference of the expected cumulative PD between
20000 MC paths and 500, 1000, 5000, 10000 paths: the $y$-axis is the maximum
relative difference of the expected cumulative PD for the 1000 different
starting points, on the $x$-axis are the time periods up to 30.

####  4.3.3 Experiments for the projection approach for expected PD

The purpose of the experiments is to assess the performance of the proposed
Bayesian filter dimension reduction approach on valuating the losses, see
Algorithm 3.2. Due to the independent assumption between the PD and LGD, the
loss valuation is simplified such that the interpolation grid is only required
for the expected PD valuation, see Equation (4.1). While the expected LGD can
be easily computed by using the Black- Scholes formula in Appendix A.
Therefore, in this section, we apply Algorithm 3.2 to compute the expected PD
and the performance of the proposed Bayesian projection approach is assessed
by comparing with the PCA approach. The assessment for the loss valuation will
be presented later in Section 4.3.5.

Based on the four and two dimensional transition models, one can train the
interpolation grid by simulating the expected PD for each point on the grid.
As we described in Algorithm 3.2, the simulation of the expected PD uses the
higher dimensional transition model while the grid points are in a lower
dimensional space, obtained by a dimension reduction approach, i.e. the
proposed Bayesian projection approach or the PCA approach. Then the valuation
grid can be trained using the simulated expected PD and the projected factors.
We compare the accuracy of the valuation grid based on the proposed Bayesian
filter projection approach and the PCA approach. To build the valuation grid,
the expected PD are simulated from the four factor model specified in Section
4.1.

The starting points of the simulations (i.e. the higher dimensional factors)
consist of two parts: one contains the pre-determined points to cover the
possible extreme simulated values of the factors, the other contains random
simulated points to fill the density. In this experiments, the pre-determined
points are chosen linearly spaced from minus four sigma to plus four sigma,
i.e. for the $i$-th component of the factor, the linear spaced points are
chosen from interval $[-4\sigma_{i},4\sigma_{i}]$, with $\sigma_{i}$ the
standard deviation of the $i$-th component of the factor. For each dimension,
we chose $15$ points. Moreover, $1000$ more points are randomly simulated from
the joint distribution of the four dimensional factors. Therefore, in total
there are $15^{4}+1000=51625$ points in the valuation grid. Starting from the
factors, each term structure of the expected PD is estimated by Monte Carlo
simulation with $1000$ paths. The term structure is considered from $1$ to
$30$ periods. Meanwhile, the four dimensional factors are projected to lower
dimensional factors using the Bayesian filter projection approach and the PCA
approach. The chosen lower dimensional models are the one-factor and two-
factor model. Once the valuation grids are trained, $1000$ new four
dimensional factors are simulated as the test set to assess the performance of
different valuation grids based on different approaches. Based on these $1000$
new four dimensional factors, the term structures of the expected PD are
estimated by using 20000 Monte Carlo paths. These Monte Carlo estimations are
used as the benchmark for the comparison.

Figures 4.3 and 4.4 show the average (over the 20000 MC paths) relative
difference on the term structure between the valuation grids and the Monte
Carlo benchmarks. Specially, Figure 4.3 presents the comparison between the
Bayesian filter projection using one factor model and the PCA projection using
the first principal component, while Figure 4.4 presents the Bayesian filter
projection of the two factor model and the PCA projection using the first two
principal components. One observes that for one factor projection, in general
PCA approach performs slightly better than the Bayesian filter projection
approach. In particular, the Bayesian filter projection performs better for
the first rating, but worse for rating 2 and rating 3. This is because the one
factor model mainly focuses on the transitions of the first rating and hence
the errors for the other two ratings are large. For the two factor projection,
on the other hand, the Bayesian filter projection performs much better than
the PCA approach. Comparing Figures 4.3 and 4.4, one observes very big
improvements on the relative error for the Bayesian filter projection, while
the improvement for the PCA projection is limited.

Figure 4.3: Average relative error on expected PD: Bayesian filter projection
vs. PCA projection, one factor. Figure 4.4: Average relative error on expected
PD: Bayesian filter projection vs. PCA projection, two factors.

####  4.3.4 Expected LGD: Black-Scholes formula v.s. Monte Carlo

Before proceeding to the experiment on the loss valuation, in the experiment
of this section, we illustrate the Black Scholes formula, the analytical
solution of Equation (A.6) in Corollary A.2, for the expected LGD (ELGD), by
comparing it to the Monte Carlo approximation. Note that the ELGD depends on
the initial LTV and the initial log-return of the collateral. In this case, we
choose the initial LTV to be 0.8, 1, 1.5 and 2, the initial log-return of the
collateral to be -0.1, 0, and 0.1. Figure 4.5 shows the term structure of the
ELGD computed by the Black-Scholes formula and the Monte Carlo simulation with
1 million scenarios. One can easily see that the outcomes of the two
approaches align very well with each other.

Figure 4.5: Term structure of ELGD: Black-Scholes (analytical) vs. Monte
Carlo.

####  4.3.5 Experiments for the projection approach for risk measures of the
loss distribution

These experiments assess the accuracy of the proposed projection approach when
estimating the quantiles of the loss distribution. In order to obtain a deeper
insight about the performance, we investigate the loss distribution per
initial rating, instead of the total loss distribution, which aggregates over
different ratings. In this way we can compare the performance of the
projection approaches for different initial ratings. To be more specific, the
loss distribution per initial rating is constructed as follows.

  1. 1.

Simulate one million scenarios (risk factors and collateral process) using the
four factor model,

  2. 2.

project the simulated risk factors into lower dimensional risk factors,

  3. 3.

for each initial rating, given the lower dimensional factors, estimate the
expected PD per scenario using the trained grid as in Section 4.3.3,

  4. 4.

given the simulated collateral value, compute the expected LGD using the Black
Scholes formula (A.6).

  5. 5.

Note that the PD and LGD are assumed to be independent, therefore the loss per
scenario can be evaluated, according to equation (2.2), by using the estimated
expected PD and the expected LGD.

To illustrate the performance, the loss distribution based on the four factor
model is used as a benchmark. In this benchmark, instead of using the
projection approach and the interpolation grids, the expected PD per scenario
in step 3 mentioned above is estimated directly by Monte Carlo simulation
based on the four dimensional model itself. The chosen risk measures are
expected loss, 95%-, 99%- and 99.9%-quantiles. We compute the absolute
relative difference of these risk measures between the estimates from
projection approaches and from the benchmark. Figures 4.6–4.9 present the
comparison of the relative differences between different projection approaches
for different risk measures: expected loss (EL), 95%-, 99%- and 99.9%-
quantiles, respectively. In each figure, the dotted lines and the solid lines
show the relative difference for the Bayesian projection approach and for PCA
approaches, respectively, while the blue and red lines present the relative
difference of one factor and two factor models (PCA), respectively.
In general, the results are in line with the results in Section 4.3.3. The
Bayesian projection approach with the two factor model performs significantly
better than the other candidates. Specially it has significantly smaller error
than the PCA approach with the first two principal components. By contrast,
the Bayesian projection approach with the one factor model does not outperform
the PCA approach with the first principal component. The reason is that the
Bayesian projection approach with the one factor model focuses on the best
rating, i.e. rating 1, which has the majority number of clients, while the PCA
approach does not have this concentration. Therefore, one observes a better
performance of one-factor Bayesian projection in the best rating but worse
performance in rating 2 and 3. Another important observation is that, the
Bayesian projection approach has a much better performance in the best rating
than the PCA approach when estimating the more extreme quantile of the loss
distribution. in particular one observes in Figure 4.9 that even the one
factor Bayesian projection has comparable performance as the two-factor PCA
projection in the best rating when estimating the 99.9%-quantile. This is a
desired property since in practice most of the exposures are located in the
low-default-probability portfolios.

Figure 4.6: Absolute relative difference on expected loss: Bayesian filter
projection vs. PCA projection. Figure 4.7: Absolute relative difference on
95%-quantile: Bayesian filter projection vs. PCA projection. Figure 4.8:
Absolute relative difference on 99%-quantile: Bayesian filter projection vs.
PCA projection. Figure 4.9: Absolute relative difference on 99.9%-quantile:
Bayesian filter projection vs. PCA projection

##  5 Conclusions

In this paper we developed a dimension reduction methodology based on the
Bayesian filter and smoother. This methodology is designed to achieve a fast
and accurate loss valuation algorithm in the credit risk modelling, but it can
also be extended to the valuation models of other risk types. The proposed
methodology is generic, robust and can easily be implemented. Moreover, we
showed the accuracy of the proposed methodology in the estimation of expected
loss and value-at-risk by numerical experiments. The results suggest that,
compared to the currently most used PCA approach, the proposed methodology
provides more accurate estimation on the expected loss and value-at-risk of a
loss distribution.

##  Appendix A Analytic formula for expected loss-given-default

The LGD at time $k$ conditional on the initial information, the sigma-algebra
$\mathcal{F}_{0}$, is defined by

$\mathrm{ELGD}_{k}=\mathbb{E}\left[\mathrm{LGD}_{k}|\mathcal{F}_{0}\right].$

In the present paper, we assume the LGD depends on the collateral value, see
(4.5). Hence we have

$\mathrm{ELGD}_{k}=\mathbb{E}\left[(1-\frac{c_{k}}{\mathrm{EAD}_{k}})^{+}|\mathcal{F}_{0}\right],$

with $c_{k}$ the value of collateral and $\mathrm{EAD}_{k}$ the exposure-at-
default at time $k$. Since we assume the $\mathrm{EAD}_{k}$ is deterministic,
the $\mathrm{ELGD}_{k}$ is determined by the stochastic nature of the
collateral value $c_{k}$. The loan-to-value ratio is
$\mathrm{LTV}_{0}=\frac{\mathrm{EAD}_{0}}{c_{0}}$, hence we have

$\displaystyle\mathrm{ELGD}_{k}$
$\displaystyle=\mathbb{E}\left[(1-\frac{c_{k}}{c_{0}}\frac{c_{0}}{\mathrm{EAD}_{0}}\frac{\mathrm{EAD}_{0}}{\mathrm{EAD}_{k}})^{+}|\mathcal{F}_{0}\right]$
$\displaystyle=\mathbb{E}\left[(1-\frac{c_{k}}{c_{0}}\frac{1}{\mathrm{LTV}_{0}}\frac{\mathrm{EAD}_{0}}{\mathrm{EAD}_{k}})^{+}|\mathcal{F}_{0}\right]$
$\displaystyle=\mathbb{E}\left[(1-K_{k}\frac{c_{k}}{c_{0}})^{+}|\mathcal{F}_{0}\right],$
(A.1)

with
$K_{k}=\frac{1}{\mathrm{LTV}_{0}}\frac{\mathrm{EAD}_{0}}{\mathrm{EAD}_{k}}$.
We suppose that the log-returns $\mathrm{LC}_{k}=\log\frac{c_{k}}{c_{k-1}}$ of
the collateral follow an AR(1) process,

$\mathrm{LC}_{k}=c+x\mathrm{LC}_{k-1}+\sigma z_{k},\,z_{k}\sim N(0,1),\,k\geq
1,$ (A.2)

with $|x|<1$. Define the numbers $a_{k}$ by the iteration

$a_{k}=1+xa_{k-1},\,a_{0}=0,$ (A.3)

which results in the analytic expression $a_{k}=\frac{1-x^{k}}{1-x}$. We now
have the following result.

######  Proposition A.1.

For given $\mathrm{LC}_{0}$, it holds that

$\log\frac{c_{t}}{c_{0}}\sim N(\mu_{t},\Omega_{t}),$ (A.4)

with

$\displaystyle\mu_{t}$
$\displaystyle=c\sum_{i=1}^{t}a_{i}+a_{t}x\,\mathrm{LC}_{0},$
$\displaystyle\Omega_{t}$ $\displaystyle=\sigma^{2}\sum_{i=1}^{t}a_{i}^{2}.$

###### Proof.

It follows from the recursion (A.2) that all $\mathrm{LC}_{k}$ are normal
(conditional on $\mathrm{LC}_{0}$). Hence for (A.4) to hold we only have to
find mean and variance. We first prove a general result. Namely, for
$s=0,\ldots,k$, the equation

$\log\frac{c_{k}}{c_{0}}+\mathrm{LC}_{0}=c\sum_{i=1}^{s}a_{i}+a_{s+1}\mathrm{LC}_{k-s}+\sum_{i=0}^{k-s-1}\mathrm{LC}_{i}+\sigma\sum_{i=1}^{s}a_{i}z_{k-i+1}$
(A.5)

holds. We now show (A.5) by induction w.r.t. the variable $s=0,\ldots,k$. When
$s=0$,

$\log\frac{c_{k}}{c_{0}}=\sum_{i=1}^{k}\mathrm{LC}_{i},$

which is Equation (A.5) for $s=0$ by definition of the $\mathrm{LC}_{i}$.
Suppose then that Equation (A.5) holds for $s=t$. Using (A.2) with $k$
replaced by $k-t$ and (A.3) with $k$ replaced by $t+2$, one gets starting from
the induction assumption

$\displaystyle\log\frac{c_{k}}{c_{0}}+\mathrm{LC}_{0}$
$\displaystyle=c\sum_{i=1}^{t}a_{i}+a_{t+1}(c+x\mathrm{LC}_{k-t-1}+\sigma
z_{k-t})+\sum_{i=0}^{k-t-1}\mathrm{LC}_{i}+\sigma\sum_{i=1}^{t}a_{i}z_{k-i+1}$
$\displaystyle=c\sum_{i=1}^{t+1}a_{i}+a_{t+1}x\mathrm{LC}_{k-t-1}+\mathrm{LC}_{k-t-1}+\sum_{i=0}^{k-t-2}\mathrm{LC}_{i}+\sigma\sum_{i=1}^{t+1}a_{i}z_{t-i+1}$
$\displaystyle=c\sum_{i=1}^{t+1}a_{i}+a_{t+2}\mathrm{LC}_{k-t-1}+\sum_{i=0}^{k-t-2}\mathrm{LC}_{i}+\sigma\sum_{i=1}^{t+1}a_{i}z_{t-i+1},$

which is indeed Equation (A.5) for $s=t+1$, as was to be shown.

Taking $k=t$ and $s=k$ in Equation (A.5), we have, using (A.3) once more,

$\displaystyle\log\frac{c_{t}}{c_{0}}$
$\displaystyle=-\mathrm{LC}_{0}+c\sum_{i=1}^{t}a_{i}+a_{t+1}\mathrm{LC}_{0}+\sigma\sum_{i=1}^{t}a_{i}z_{t-i+1}$
$\displaystyle=c\sum_{i=1}^{t}a_{i}+xa_{t}\mathrm{LC}_{0}+\sigma\sum_{i=1}^{t}a_{i}z_{t-i+1}.$

Therefore we obtain the distribution of $\log\frac{c_{t}}{c_{0}}$, conditional
on $\mathrm{LC}_{0}$,

$\log\frac{c_{t}}{c_{0}}\mid\mathrm{LC}_{0}\sim N(\mu_{t},\Omega_{t})$

with $\mu_{t}=c\sum_{i=1}^{t}a_{i}+xa_{t}\mathrm{LC}_{0}$ and
$\Omega_{t}=\sigma^{2}\sum_{i=1}^{t}a_{i}^{2}$, which finishes the proof. ∎

As a corollary we obtain a Black-Scholes like formula for the expected loss
given default.

######  Corollary A.2.

For the expected loss given default one has

$\mathrm{ELGD}_{t}=\Phi(-d_{2})-K_{t}e^{\mu_{t}+\frac{\Omega_{t}}{2}}\Phi(-d_{1}),$
(A.6)

with $\Phi$ the standard normal cumulative probability function, and

$\displaystyle d_{1}$
$\displaystyle=\frac{\log(K_{t})+\mu_{t}+\Omega_{t}}{\sqrt{\Omega_{t}}},$
$\displaystyle d_{2}$
$\displaystyle=\frac{\log(K_{t})+\mu_{t}}{\sqrt{\Omega_{t}}}.$

###### Proof.

The proof is completely analogous to the proof of the classical Black-Scholes
formula by realizing that Equation (A.1) gives
$\mathrm{ELGD}_{t}=\mathbb{E}\left[(1-K_{t}\log
Z)^{+}|\mathcal{F}_{0}\right]$, where $Z$ has the conditional normal
distribution with mean $\mu_{t}$ and variance $\Omega_{t}$ as in Proposition
A.1. ∎

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