Impact of Default Correlation on Credit Risk Management

UNIVERSITY OF AMSTERDAM

MASTER STOCHASTICS AND FINANCIAL MATHEMATICS

MASTER THESIS

Impact of Default Correlation on Credit

Risk Management

Author: Supervisor:

Douwe Siderius BSc dr. Asma Khedher

Examination Date: Daily Supervisor:

July 15, 2016 dr. Sjoerd C. de Vries

Second Examiner: dr. ir. Eric Winands

Kortteweg- de Vries Institute Credit Model Validation

for Mathematics Rabobank

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Abstract

Financial institutions use credit rating models to determine the creditworthiness of obligors. Most

credit rating models determine the creditworthiness of obligors individually, neglecting possible

correlations. This thesis investigates the impact of neglecting these correlations. We consider a

portfolio consisting of obligors with a parent-subsidiary relationship, since these companies are

assumed to be significantly correlated. After we have determined the creditworthiness of a client,

we should check if the estimated probabilities of default coincide with the observed probabilities of

default. This is done using a backtest. Again, the commonly used backtest neglects any correlation

amongst obligors.

To model the creditworthiness of the clients we propose a model of credit contagion, which allows

parent-subsidiary relationships. We show that this model allows companies to have significant

default correlations. We measure to what extent these correlations and business relations

influence the loss distribution of the portfolio. We used the Value at Risk (VaR) and Expected

Shortfall (ES) as risk measures in order to calculate the increased risk figures when adding

microstructural dependencies (business relations). Furthermore we implement a backtest, for the

calibration of the probability of default induced by this model, that allows default correlations and

compare the results with the ordinary binomial backtest. The incorporation of correlation makes

the rejection policy more optimistic.

Title:

Impact of Default Correlation on Credit Risk Management

Key words:

Credit Risk Management, Contagion, Default Correlation, Copula, Backtest.

Author:

Douwe Siderius BSc

Student number:

10275894

Email:

douwe_siderius@hotmail.com

Supervisor:

dr. Asma Khedher

Daily supervisor:

dr. Sjoerd C. de Vries

Second examiner:

dr. ir. Erik Winands

Examination date:

July 15, 2016

Korteweg-de Vries Institute for Mathematics

University of Amsterdam

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Preface

This thesis has been written to fulfil the graduation requirements of the master Stochastics and Financial Mathematics (SFM). This research project lasted from January 15th till July 15th. Due to my interest in financial mathematics and its practical problems, I have decided to combine my research project with an internship at the Rabobank. This internship provided me with insights on the financial world and gave me the opportunity to using my previous obtained knowledge to solve a practical issue. During my time at the credit model validation department of the Rabobank, I have learned a lot about the credit risk and validation techniques.

First of all, I would like to thank the Rabobank as an organization. There were a lot of changes in the organization during my internship. The Rabobank was executing a reorganization and a lot of departments were under a great amount of pressure. I’m grateful that I was allowed to do my internship during these turbulent times for the bank. Furthermore I would like to thank both Sjoerd de Vries and Asma Khedher for their guidance and support during my project. I would also like to thank the other members of the model validation department for my enjoyable time and useful advices. Last, I would like to thank Misha van Beek for his critical comments and feedback.

I hope you enjoy your reading. Douwe Siderius

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Contents

1 Introduction ... 5

2 Credit Rating Models ... 8

2.1 Explanatory note on the PD-models ... 9

2.2 Group Logic ... 10

2.3 Validation of Credit Models ... 10

3 Methodology PD Model ... 12

3.1 Contagion model ... 12

3.1.1 Rating Dynamics ... 13

3.1.2 Modeling macrostructural factors ... 16

3.1.3 Modeling microstructural factors ... 18

3.1.4 Estimate microstructure dependency ... 21

3.1.5 Approximation of asset process ... 24

3.1.6 Estimate default correlation ... 30

3.1.7 Calibration method for macroeconomic factors ... 31

4 Construction of the transition matrix ... 34

4.1.1 Check Markovian property of the migration matrix... 38

5 Results ... 41

5.1 Calibration of the model parameters ... 41

5.1.1 Calibration of the transition matrix ... 41

5.1.2 Calibration of the macro-economic variables ... 42

5.1.3 Construction of the business matrix ... 42

5.2 Simulation results ... 44 5.3 Significance Test ... 47 5.4 Backtest Methodology ... 49 5.4.1 Backtest results... 53 6 Conclusion ... 55 7 Background information ... 57 7.1 Copulas ... 57 7.1.1 Constructing a copula ... 57 7.1.2 Simulating a copula ... 62 7.1.3 Gaussian copula ... 63 7.1.4 T-Copula ... 69 7.2 Expected Shortfall ... 70

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References ... 73

Appendix A. _{Backtest Methodology………..78}

Appendix B. _{Single Factor Model……….79}

Popular Summary. ……….80

Abbreviations

Backtest Default Rate

Basel Committee on Banking Supervision Credit Model Validation

CRR - Capital Requirement Regulations DR - Default Rate

EAD - Exposure At Default

EBA - European Banking Authority EC EM - - Economic Capital Emerging Markets ES - Expected Shortfall GL IC - - Group Logic Industrialized Countries IRB - Internal Rating Based LGD - Loss Given Default LRT - Likelihood Ratio Test

ODF - Observed Default Frequency

PD - Probability of Default (over a one year period) PIT - Point-In-Time

RC - Regulatory capital RRR - Rabobank Risk Rating RWA - Risk Weighted Assets S&P - Standard and Poor’s TCA - Time Counting Approach TTC - Through-The-Cycle VaR - Value at Risk

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1

Introduction

One major concern for financial institutions is the prediction of the creditworthiness of obligors. The creditworthiness of obligors should be a good reflection of the ability and willingness of this obligor to pay his debts. When a obligor is not able to meet his payment requirements, he goes into default. Financial institutions use credit rating models in order to determine this creditworthiness for clients. There are two possible approaches for the modeling of these credit models. The first approach is the standardized approach, where the financial institution use models that are constructed by the Basel Committee on Banking Supervision (BCBS). This standardized approach could overestimate the risk factors of a financial institution, which eventually leads to a drop in profit. Hence most large financial institutions like to use the second approach. The second approach is the Internal Ratings-Based (IRB) approach, where financial institutions use their own internal credit models. Institutions that want to use a IRB-approach need to have permission from the competent authorities, which is granted under strict conditions. The main restrictions are constant notifications from the financial institutions to the regulators on the performance of the model and the financial institutions should have a sufficient modeling environment.

Recent European regulations, like the CRR [14],aim to promote the adoption of stronger risk management practice within the bank, especially for financial institutions that use the IRB approach. One aspect concerns the use of internal models used by the financial institution to calculate their Regulatory Capital (RC). The RC is the minimum amount of capital that the bank must hold to protect itself against credit losses. In order to calculate the RC under the IRB approach, banks need their own models to calculate risk parameters such as the Probability of Default (PD), Exposure at Default (EAD) and Loss Given Default (LGD). This thesis focuses on the PD models.

Over the last years, amongst other things caused by the financial crisis, risk model validation has become an important aspect of the risk management of banks. The Credit Model Validation (CMV) department basically challenges the internal credit models of a bank and needs a strong methodology to convince its supervisors that the internal credit rating models are performing well. Usually, a bank uses a backtest to assess the performance of a model. A backtest tests whether the estimated risk factors differ significantly from the observed risk factors, i.e. it tests if the calibration of the model parameters is done properly. The credit model should be rejected if there is a significant difference between the two. This thesis compares the approach with a backtest that allows default correlations with one assuming independence.

However, before we can validate a PD-model, we need to construct one. A major concern in the PD- credit models should be the high degree of correlation amongst the obligors. While the process of the estimations of the single PD may be fairly under control, the lack of sufficient statistics puts a burden on the estimation of the joint default probabilities of firms in the portfolio. Due to default correlations, these joint default probabilities are most likely higher than joint default probabilities estimated under the assumption of independent defaults, which could lead to significant default clustering. Note that even random processes, such as the Poisson process, allow default clustering, the probability on such clustering however is usually smaller than the probability of clustering for correlated processes. Consequently, the accuracy of the estimation of the variance of the loss distributions in the portfolio will be less and hence the required RC is hard to calculate.

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Default correlations can enter the model both by macroeconomic factors and by microeconomic factors. The macroeconomic risk factors could for example relate to the common factors of the portfolio, such as the state of the economy or the geographical location. The microeconomic risk factors could for example relate to the business relations amongst clients. There are several techniques to construct PD-models and calculate default correlations in the literature, we discuss a few of them below.

The models suggested by the BCBS are the so-called single factor models, introduced by Vasicek [50]. Besides their simplicity, it has been shown by Gordy [20] that these models are portfolio invariant, which is important to the regulators. However, these models cannot account for correlations due to

parent-subsidiary relations.

This thesis focuses on a portfolio consisting of companies that have a parent-subsidiary relation. This is due to the fact that we expect that these companies have a significant correlation amongst each other based on their business relation. The single factor models have the disadvantage that it is pretty hard to incorporate these business relations in the model. Another approach, that allows some parent-subsidiary relations, models related companies as a merged firm. This idea was introduced by Leland and Skarabot [30].The default process is modeled through the asset returns of that merged firm, and a default occurs when the asset return process reaches a certain threshold. However, there are no closed form solutions for these models and approximation schemes are in order. Hainaut and Deelstra [24] uses a convex upper/lower bound approximation for the asset process. Once this process is constructed the default probabilities are estimated using the standard approach of threshold values. Afterwards, the default correlations between companies can be calculated using a copula approach. Copulas are a mathematical tool which link joint distributions to marginal distributions and vice versa. A good introduction to this approach is given by Mai and Scherer [33]. Their book covers the basics of this approach in a rigorous way.

In this thesis we focus on contagion models. To our knowledge, the first credit contagion models were introduced by Davis and Lo [11] who proposed contagious defaults to account for default clustering. From that moment on there has been a rapid growth in interest in these models. Giesecke and Weber [19] extended the framework by introducing contagion through interacting particle systems. Focardi and Fabozzi [15] extended the framework of Giesecke and Weber using percolation and random graphs. Another extension was made by Egloff et al. [12] by introducing micro-dependencies through the business structure of firms based on expert opinion. They captured these business structures in the so called business matrix. The authors used a Markov Chain model which takes a rating transition matrix and a business matrix as main input. A clear overview of these concepts are given by Davis [10].

This thesis proceeds in a similar way as in Egloff et al. [12]. Additionally, we also introduce a method for the calculation of the transition matrix following Berd [5] and Lando [28].We extend the methodology in Egloff et al. [12] by calculating the default correlations using a copula based approach, by calculating the

convergence of the approximation scheme needed for modeling, and we implementing our own transition matrix instead of using one created by rating agencies. Furthermore, we focus on a different set of

companies, namely companies that have a parent-subsidiary relationship. This selection gives us a better understanding of the business matrix and can be extended to the whole portfolio of the Rabobank if the results are significant. The aim of this thesis is to define a consistent PD-model that allows default correlations, introduced mainly by microstructural dependencies between companies. We calculate the corresponding risk figures in terms of Expected Shortfall (ES) and Value at Risk (VaR) to check the impact of the correlations and business structures. Furthermore, we also want to check if it is sufficient to backtest

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defaults as independent events, when we want to evaluate and manage the bank’s model risk. We use two different backtests and compare the results. At first we use a backtest that allows default correlations, as constructed by Schechtman [43]. Next, we use the standard binomial backtest, that makes the assumption that defaults occur independently. We compare the results of these two backtests in terms of model rejection. To meet these goals we formulate the research question as follows

Can we measure significant default correlations in our portfolio and do they increase the risk profile of the portfolio? Furthermore, should we implement these correlations in the backtest of the model?

We emphasize that this is the public version of the thesis, meaning that we remove all the confidential information. The exact interpretation of models and model calibration can for this reason seem a bit vague. This thesis is structured as follows. Chapter 2 provides us with some extra information on credit rating models of the Rabobank. Chapter 3 and 4 present the methodology of our credit rating model. Chapter 3 discusses a latent variable model for the modeling of the default process for the total portfolio, whereas Chapter 4 discusses the construction of the most import building block of the model, the transition matrix. Chapter 5 shows the results on the available data. Chapter 6 present the conclusion of this thesis. For background information on some techniques used in this paper we refer to Chapter 7 and the Appendix. The available data and calibrated values of the parameters are omitted.

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2

Credit Rating Models

This chapter is based on internal documents of the Rabobank and the Capital Requirement Regulations [14]. First, we roughly describe the construction procedure of rating models, then, in Paragraph 2.1, we describe the rating models of the Rabobank that we use throughout this thesis. Paragraph 2.2 describes Group Logic (GL), which is used later on to define the business relations in the portfolio. Last, Paragraph 2.3 describes the validation procedure of PD credit rating models.

Since the introduction of the IRB approach in the Basel II Accord of 2004 banks are able to use their own internal rating models for risk factors such as PD, EAD, LGD. All these factors are required in order to calculate the Risk Weighted Assets (RWA), the regulation states:

“The calculation of risk-weighted exposure amounts for credit risk and dilution risk shall be based on the relevant parameters associated with the exposure in question. These shall include PD, LGD, maturity and exposure value of the exposure.”

We focus our attention on the PD-models. The PD is calculated to predict the probability that a company will default on its loan within a one year time-horizon. The most common way to calculate the PD of a company is based on a variation of a regression model. The model produces a model score for each individual client and this model score is translated to a PD estimate. Based on this PD estimate, companies are divided in several risk buckets. Companies that are mapped to the same risk bucket are assumed to have a similar risk profile. According to the regulation each class of companies should be rated in a different rating model. These classes can be determined, e.g. by the size or industry of the company. The rating scale should be adapted when a financial institution is concentrated in a certain sector.

The regulators demand that the models of IRB-banks have good predictive power and that all input variables form a reasonable and effective basis for all resulting predictions. Furthermore, to ensure that the models indeed have good predictive power, banks should have a meaningful distribution of exposures of grades on its borrowing-scales. According to the CRR [14] Article 170(1.b) banks must use a rating scale with at least seven rating grades of non-defaulting companies and one rating grade of defaulting companies. This requirement is made in order for the credit models to have a good discriminative power. Less rating grades would result in non-statistical results. The Rabobank uses a so called Masterscale, which we omit in this public version. Institutions should also document the relationship between obligors grades in terms of default risk and criteria to distinguish them. This requirement assures the regulator that models loss characteristics can be validated in a rigorous way.

Once this setup is valid, a financial institution is able to make their credit risk models. We want to emphasize that the PD calculations of banks are usually constructed on a one-year horizon. This convention is used throughout this thesis. Each model should estimate the default probabilities of all obligors of the sector of interest. The Rabobank uses different methodologies to model the internal ratings of clients belonging to different sectors. Each sector has its own specific risk characteristics, which leads to completely different models. Also the availability of data plays an important role in the decision how to model certain events. The Rabobank uses, besides their statistical tools, also expert-based approaches. The statistical models are commonly built using regression. Each methodology results in a final score for a obligor, which is then mapped to the Masterscale. Rabobank uses an average of the lower and upper bucket boundary probabilities of default associated with each rating as pooled probability of default in order to express default risk. This is in line with the CRR [14] Article 179(2).

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In special cases manual overrides can be used. An important step in the calculation of the client specific probability of default is the use of Group Logic. We discuss this procedure later on, since it will be very important in the construction of business relations between companies.

2.1 Explanatory note on the PD-models

This paragraph describes the rating models that we use throughout this thesis. We have chosen these models since they are most frequently used for companies with a parent-subsidiary relation. The Rabobank uses different approaches to develop rating models, most rely on regression techniques. A linear regression can be used under the assumption that the underlying variables have a linear relation with each other. Typically this relation is defined by

= + + ⋯ + +

where is a vector of dependent variables, is a constant value, = ( , … , ) a vector of independent variables and a random noise factor. The simplest approach to solve this equation is by an ordinary least square estimation. In that case the solution to this equation is given by = ( ) . Under some conditions this estimator is the best linear unbiased estimator.

We can also use a generalized regression with for instance a logistic function. With a generalized regression we are able to model binary variables. In the case of generalized linear regression with a logistic function we need to find the vector that maximizes

( ) = 1 +

The ( ) is the probability that the dependent variable has outcome zero, is again a vector of explanatory variables and a vector with coefficients of these explanatory variables. Using a maximum likelihood estimation this vector of coefficients can be estimated. This approach is called logistic regression. The Good-Bad analysis relies on several steps. The creditworthiness information of companies is judged according to the criteria that the company defaults in the next year (bad=0) or still operates (good=1). Note that this one year observation period is chosen because we would like to construct one year PD’s. After this step we perform a logistic regression on all explanatory variables and the good-bad indicators, where the good-bad indicator is the dependent variable. After performing a maximum likelihood maximization a final scorecard can be constructed which can be used for model predictions.

The shadow-bond approach basically maps external rating to internal ratings. This approach uses the same set of explanatory variables as the Good-Bad analysis. However, instead of determining if each observation is good or bad, we use external historical ratings for our analysis. These external ratings can, for instance, be provided by companies like S&P or Moody’s. These rating agencies however, use a different rating scheme than the Rabobank. Therefore the Rabobank has designed a mapping function to overcome this obstacle. The natural logarithm of these PD’s are then regressed to the explanatory variables.

We can’t give the exact derivations and properties of these models. Roughly speaking, for each company the appropriate rating model is chosen on base of the region and turnover of the company. From now on we will refer to the models as , … , . Furthermore, we have to omit the distribution on the parent companies

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for each individual model. However, we will use these distribution on the background in the following chapters.

2.2 Group Logic

In CRR it is stated that, when rating a company that has a parent company registered, the influence of the parent on the subsidiary’s credit quality has to be taken into account. Group Logic (GL) is used in order to comply with this demand. It is a tool that can adjust the rating of subsidiary companies based on the

performance of the parent company. Group Logic influences the PD score for certain companies by adding or subtracting a few notches of the RRR. This influence can both be negative (higher PD) as positive (lower PD), although it turns out that the result of Group Logic mainly has a positive effect. The GL tool can be of interest for this thesis if we want to estimate the strength of the parent-subsidiary relation.

We have tested if it was used in line with the internal documentation of the Rabobank. We performed this test in order to check if it was a proper tool to indicate the strength of a parent-subsidiary relation, if it wouldn’t be in line with the documentation, it would give biased results. In that case it would be

inappropriate to use this tool in this thesis. We tested GL on a subset of clients that used GL in the last five years. GL was used properly and we are thus able to use this in the remaining of this thesis.

2.3 Validation of Credit Models

In CRR [14], Article 185discusses the requirements for the validation procedure of IRB-banks. The CRR regulations require IRB-banks to have a regular cycle of model validation that includes the monitoring of model performance and stability. The internal validation process should be robust, consistent and meaningful. In order to accomplish this banks should have an independent validation team, which means that this team cannot be included in the modeling process. The validation of PD-credit models consists of many steps. Each model requires its own validation procedure. But there are some checks that are applicable to most models.

If, for instance, the observed defaults don’t coincide with the expectations of the model, the validation team should find the reasons behind these deviations. If the realized values continue to be higher than the

expected values, the model should be recalibrated to reflect the observed default experience. The analysis of this problem should be updated annually and reported to the regulators. The validation report should also include the details of a change in the procedure or data, when this occurs. This happens for instance when a model is redeveloped or new data arrived at the financial institution. The model life-time should also be documented, including the involved parties, model approval and model review processes. The on-going validation should be reviewed periodically by a sufficient authority to decide which actions should be taken for the weaknesses in the model. The validation process of the Rabobank is reported to the model

governance committee, who decides whether or not actions should be taken concerning the model weaknesses.

Banks should validate the accuracy/calibration of the model. This can be tested with a backtest, which tests the differences between the predicted probabilities of default and the realized probabilities of default. Backtesting is one of the most important aspects of risk modeling since it provides insights in the adequacy

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of the risk measurement models. The underlying idea of backtesting is checking whether our Null-hypothesis (we have a good model) is consistent with the data. We would like to check if the observed default

probabilities are in line with the observed data. We allow, in contrast to most research, correlations between defaults. Most of the time the modeling of default correlations is based on the conditional independence given some endo/exogenous risk factors.

There are two main aspects during the backtesting. These are the discriminatory power and the calibration of the model. The discriminatory power is the ability of the model to distinguish between borrowers that will go into default and borrowers that won’t go into default. The calibration of the model concerns the accuracy of the model, i.e. the degree to which the model fits the actual data

Banks should validate the discriminative power of the PD models. There are several methods to find the discriminative power of models such as the Cumulative Accuracy Profile (CAP) and Receiver Operator Characteristic (ROC). Roughly speaking both procedures calculate the distance between a perfect

discriminative model and the realized model. We won’t go into detail on these methodologies, since they are out of scope for this thesis. A future research project could investigate the discriminative power of the methodologies described in this thesis.

The backtesting procedure for the calibration of the model is discussed in Paragraph 5.4. We compare two tests, one that allows default correlation with the commonly used binomial backtest. The binomial backtest assumes defaults are independent events.

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3

Methodology PD Model

In this chapter we state the model assumptions and properties, and build a latent variable model for the default process. Under these assumptions we approximate the underlying latent variable model for

computational convenience. We no longer assume that defaults are independent events, and we no longer assume that sector specific risks are contained within each sector, which is the usual practice. We model the creditworthiness of costumers on portfolio level, instead of estimating the default probabilities individually.

Before we continue with the construction of the model we give two examples that outline the danger of the assumption of independent defaults. This risk will be translated in the increasing joint default probability.

Example 3.1

As mentioned before, the assumption of independent defaults could lead to a significant underestimation of the joint default probability of companies in the portfolio. We elaborate this statement with a simple example. Suppose we have a portfolio consisting of two clients, and , which may be correlated. Suppose that company has a default probability = 2% and has a default probability of = 5%.In a similar way we write the probability that both companies go into default by _, , and the conditional default probabilities _| =!",#

!# , | =

!",#

!" . From these definitions we can calculate the correlation between these

companies by

$ , = , −

& (1 − ) (1 − ) .

After rewriting this results in

, = + $ , & (1 − ) (1 − ).

If we assume that default events are independent of each other we would have $ _, = 0 and hence

, = 1%. A slight increase in $ , already results in a significant increase in the joint default probability as

Table 1 below clearly shows.

Correlation coefficient Joint default probability

0 0.1%

0.1 0.4%

0.2 0.7%

0.3 1.0%

Table 1: Impact default correlation on two companies example

Note that the correlation between these companies can’t be larger than 0.62, since in that case the joint probability of default would be larger than the marginal default probability which is not allowed by

definition. □

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We model the behavior of our clients through a credit contagion model as mentioned in the introduction. Credit contagion models account for the fact that one default increases the default probability of another correlated firm which could lead to default clustering, a phenomenon observed in many empirical studies. Since correlation works in two directions we expect a peak in the default distribution for few defaults as well as a (smaller) peak for many defaults. Contagion models pay a lot of attention to microstructural

interdependence structures, in our case parent subsidiary relations, as well as macrostructural dependence structures well-known from the widely used Merton single factor models1. The combination of these factors result in more realistic models. Note that default clustering is not a unique characteristic of the contagion model. Even in a completely random process, such as a Poisson process, default clustering could exist. The chance of this occurrence however is way smaller in these sort of models. Our model setup is based on the paper of Egloff, Leippold and Vanini [12],which we extend with the calculation of default correlation and the addition of our own estimated transition matrix.

Before we construct the model, we give a short overview of the structure of the model, so that the general idea of the model is clear. Egloff, Leippold and Vanini [12] interpret correlations between the firms in a portfolio, which are due to the dependence on some common underlying factors, as macro-structural dependence. These factors could for instance be the state of the economy, the country of risk or the

industry in which a company is active. These macro-structural effects are modeled by a typical latent variable model. Actually this underlying model is similar to the factor models introduced by Merton. Further

dependence structures, e.g. as a result of legal enforceable business links or structural supply arrangements, are represented by the structure which provides a possibility to contagion in the model. These micro-structure factors are modeled by a weighted graph, where the nodes represent the obligors and the edges represent the direction of the business relation. Furthermore there are weights attached to each edge, representing the strength of the business relation. Incorporating these micro-structural information in the latent variable model leads to a change in the idiosyncratic terms. Based on this renewed latent variable model it is possible to quantify the additional risk of default contagion to the portfolio losses. The latent variable model represents the asset return process of the portfolio. This is a common measure to use in credit risk modeling since it has a lot of explanatory power of the default process. Once we have constructed the latent variable model, we are able to simulate the default process for the portfolio of our interest, and check if our portfolio consists significant default correlations amongst obligors. This result can be backtested using techniques described in Paragraph 5.4.

This chapter is ordered as follows. First, we give a short explanatory note on the rating dynamics. Second, we introduce the macro-structural latent variable model and connect this model with the rating dynamics. Next we introduce the micro-structural graphical representation and implement this structure in the macro-structural model. Last, we investigate our constructed latent variable model in more detail and we give an approximation of the asset return process since the theoretical model can be numerically unstable.

3.1.1 Rating Dynamics

The main input of the model that we are constructing is a transition matrix of credit ratings. The usual conventions on such matrices are based on Markovian properties. Definition 3.1 and Definition 3.2 clarify these properties. This paragraph also gives the main assumptions that affecting our model.

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Definition 3.1

A discrete process * = (*₊)_+∈ℕ on a countable state space . is a Markov process if it satisfies the Markov property

ℙ(012 = |01 = 1, 01 = 1 , … = ℙ(012 = |01 = 1

For all 3 ∈ ℕ and , ₁, ₁ , … ∈ ..

A nice property of a (discrete) Markov process is that it is somewhat of a memoryless process. Only the direct past, one discrete time instance before, can influence a Markov process. This condition alone is however not strong enough for the modeling purposes that we are interested in. We would also like the transition matrix to satisfy a time-homogeneous condition.

Definition 3.2

A discrete process 0 = (0_{1 1∈ℕ} on a state space . is a stationary Markov process if ℙ(012 = |01 = = ℙ(01= |01 = = ⋯

for all 3 ∈ ℕ and , ∈ ..

So the time of a transition doesn’t matter in a stationary Markov process, only the time interval is of interest. For further details on Markov processes we refer to Spieksma [47], she gives a good mathematical background in discrete and continuous-time Markov processes.

The model focuses on the process of rating migrations of client of the Rabobank. Assume we have a portfolio consisting of 4 obligors. We suppose that the creditworthiness of obligor 5 at time 3 is fully described by its rating process 0₆(3 , taking values in the set of rating categories used in the internal rating system. The state space of the rating processes for the Rabobank is given by the ordered set . = 70,1, … 8. Included in the state space is the special default state, which is assumed to be absorbing. Once a rating process has entered the default state, it remains there. In this setting 0 corresponds to the Rabobank Risk Rating 9 , 1

corresponds to the Rabobank Risk Rating 9 etc. The last state corresponds to the absorbing default state. For the moment we assume that there is only one default state in contrary to the default definitions given in previous chapters. It is good to notice that better ratings correspond to a lower state number.

We assume that the joint rating dynamics of obligors is given by the discrete time stationary Markov process 0(3 1∈ℕ= :0 (3 , … , 0;(3 <_1∈ℕ. The transition probabilities are summarized in a transition matrix =. The transition probability from state = ( , … _; at time 3 to state = ( , … , _; at time 3 + 1 is given by

=>= ℙ(0(3 + 1 = | 0(3 = 1

Note that each ₆ in and ₆ in are elements of the state space .. We make the usual convention that capital letters represent random variables whereas small letter represent observations of the random variables. Furthermore we note that, since the assumption that the joint rating dynamics are a stationary Markov chain, these transition probabilities are independent of the time 3. Only the length of the interval between two observations is relevant.

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For modeling purposes it is useful if the transition matrix have a product structure. For this reason we assume that, conditional on some random factor ?₁, the ratings are independent of each other.

Assumption 1

Conditional on a latent factor ?₁,with a non-degenerate distribution , the rating dynamics 0 (3 , … , 0_;(3 at the end of period 3 are independent.

This conditional independence structure has the convenience that the transition matrix has a product structure as can be seen in Equation 2. We give the proof of this particular structure in the next lemma. The conditional independence structure is given by

=>= @ AB =6C>C|?1

; 6D

E 2 (

2

where =6_C>C|?₁ = ℙ(0₆(3 + 1 = ₆ | 0₆(3 = ₆, ?₁ is the individual transition matrix of obligor 5. Note that this conditional independence structure reduces the dimensions of the transition matrix =.

Lemma The transition matrix has a product structure conditional on ?₁ Proof

Conditional on the latent factor ?₁, the transitions are assumed to be independent of each other. Thus =>|?1 = ℙ(0(3 + 1 = | 0(3 = , ?1 = ℙ(0 (3 + 1 = , … , 0;(3 + 1 = ;|0 (3 + 1 = , … , 0;(3 + 1 = ;, ?1 = ℙ(0 (3 + 1 = |0 (3 + 1 = , ?1 ⋯ ℙ(0;(3 + 1 = ;|0;(3 + 1 = ;, ?1 = B =6_C>C|?1 ; 6D

If we take the expectation of both sides, we are left with the transition matrix =_> since

@Fℙ(0 = |G = | ? H = I(0 = , G =

for any probability distribution function ℙ 3. Hence Equation 2 follows.

□

For computational convenience and to account for sectoral effects, we divide the 4 debtors in K sectors, where we obviously assume that K < 4. These sectors could be, for instance, the industry in which the companies are active. We represent this division in sectors by the surjective mapping

M: 71, … 48 → 71, … K8 3

2 _{This expectation is the matrix expectation, i.e. taken component-wise. We use this expectation to lose the conditional}

structure on =_>.

16

This simplifies Equation 2 even more, since each company is only influenced by the sector in which it is active, to =>= @ AB = _C>C P(6 _|? 1 ; 6D E 4

In the next two paragraphs we build a causal model behind the rating dynamics. This model incorporates the macrostructure as well as the microstructure mentioned in the introduction. But we must make sure that we incorporate this model with care. We assume that a financial institution is able to correctly asses the rating migration of individual obligors. Therefore it seems reasonable to assume that adding macro- and micro dependency structures shouldn’t alter the individual rating probabilities.

Assumption 2

The addition of macro- and micro-structures do not change the rating dynamics of the portfolio.

3.1.2 Modeling macrostructural factors

We consider a portfolio consisting of 4 clients, indexed by Q = 1, … , 4. The dependence on macro-structural variables is expressed by assuming a conditional independence structure.

Assumption 3

The asset returns of different debtors are independent conditional on some random vector ?.

The random vector ? is assumed to be given by systematic sector specific standard normally distributed risk-factors

?1 = :?1,6<_{6∈ ,…,R}∼ T(U, Λ 5

for every time instance 3 ≥ 0. The covariance matrix Λ = :X_6Y<_{6,Y∈ ,…R} is such that each X_6Y denotes the covariance between the macro-structural risk factors ?₆ and ?_Y. The mean of ?₁ is the K-dimensional zero vector. From now on bold numbers and letters represent vectors. We won’t go into detail on the dimensions of these vectors unless it remains unclear.

For notational convenience we define the macro-structural variables ?₁ on a portfolio level, instead of on sector level. Thus we define

G1 = :?1,P(6 <_{6D ,…,;} ∼ T(U, Γ 6

The process G₁ is defined for every client 5 in the portfolio. Note that many elements of G₁ are equal by the definition of the mapping M in Equation 3. The covariance matrix Γ is extended in a similar way, namely Γ = :XP(6 ,P(Y <_6,Y∈;.

17

We assume that each obligor is only influenced directly by the risk factor of the sector in which he operates. This allows us to define the synthetic asset return process, for each obligor 5 ∈ 71, … , 48 and on every time instance 3 ∈ ℕ, by a univariate Gaussian latent variable4

6,1= [1 − \P(6] ?1,P(6 + \P(6 1,6

7

such that _6,1∼ T(0,1 . The function M(5 is the same as defined in Equation 3. The factors _1,6 represent the idiosyncratic risk factor of obligor 5 at time 3. Idiosyncratic means client specific in this context. The factors _1,6 are assumed to be standard normal distributed random variables for each client 5, and are assumed to be mutually independent of the macro-structural risk factors as well as the other idiosyncratic risk factors. The sensitivity of debtor 5’s asset return to its idiosyncratic part is specified by the sector specific risk weight factor \_P(6 . These factors are assumed to be constant over time. Since we are mainly interested in relative short time periods, read five years, this assumption seems to be valid. Up to know there is no distinction between this model and the typical multi-factor model used by Merton. The distinction will be made in the next paragraph.

We set \ = :\_P(6 <_{6D ,…,;}, as the sector specific risk vectors defined for the total portfolio and ₁ = : 1,6<_{6∈7 ,…,;8} as the portfolio process of idiosyncratic risk factors.

We also introduce the diagonal matrix operator ^. This means that if is a vector then ^( is the matrix with on the diagonal the elements of and on the off-diagonal zeros. From basic linear algebra5 _{we know}

that for any real valued function _ we have ^(_( = _(^( . Furthermore if is a square matrix, then ^( is the matrix with the same diagonal elements as and zeros on the off-diagonal.

With these generalized definitions we deduce that the asset return process of the total portfolio is given by

1 = ^ `&1 − \]a G1+ ^(\ 1

8

Using this framework we can express the conditional rating migration matrices in terms of the latent variable model. In order to do so we associate with each rating class some threshold values

−∞ = c ,deP1 ≤ ⋯ ≤ c , ≤ c , = ∞ 9

We assume that client 5 migrates from one rating class ₆ at time 3 to the next rating class ₆ at time 3 + 1 whenever its asset return process ₆₁ lies between the threshold values c _C>C and c _C>C2 . In other words we have

4 _{In fact we are constructing a Gaussian Copula model, see Paragraph 7.1 for more information}

18 =_CgC

P(6 _{= ℙ(0}

6(3 + 1 = 6|06(3 = 6 = ℙ ` 6,1 ∈ hc C>C, c C>Cij<a

10

First of all, we note that Equation 10 is well defined, since by definition of the thresholds the probability on the right hand side is non-negative. Second, we note, again by definition of the thresholds, that the rating process 0(3 defined in Equation 10 is still an element of . with probability one. Last, we note that the Markov property is still valid.

Lemma The latent variable models respects the stationary Markov properties of the joint rating dynamics. Proof

We recall that conditional on the vector ? ( Assumption 3) the asset return processes of different obligors are independent. Let = ( , … , _; , = ( , … , _; ∈ .;. Then the probability of getting from state to state is given by

ℙ(06(3 + 1 = 6|06(3 = 6, 06(3 − 1 = 6

= ℙ ` 6,1∈ hc C>C, c C>Cij<| 6,1 ∈ hc Ckj C, c Ckj C<a

= ℙ ` 6,1∈ hc C>C, c C>Cij<a

= ℙ(06(3 + 1 = 6|06(3 = 6

The second equality follows by the assumption of independence between _1,6, _{1 ,6}, ?_1,6 and ?_{1 ,6}. The last equality followed by Equation 10. This argument can be repeated such that Equation 10 satisfies Definition 4.1 and Definition 4.2. We state that the latent variable models respect the

stationary Markov property of the rating dynamics.

□

The previous lemma is also justified by Wendin and McNeil [51], since the macro-structural factors ?_{P(6 ,1} are independent for different sectors. Furthermore, Proposition 5.3.2. in the book by Lütkebohmert [32] states that the conditional default probabilities are of the appropriate form for a Markovian model. Thus this latent variable model respects the Markovian setting of the transition matrix and is appropriate for modeling. Note that Assumption 1 translates into Assumption 3 when switching from the rating migration process :0(3 <_1∈ℕ to the asset return process ₆₁, as can be seen from the previous lemma.

So far there are three (high-dimensional) model free parameters, namely the vector of sector specific risk weights \, the threshold parameters c ₆ and the covariance matrix Γ. The parameters will be calibrated in Paragraph 3.1.7 to the data of the Rabobank.

3.1.3 Modeling microstructural factors

This paragraph describes the incorporation of the micro-structural dependencies, mentioned in the introduction of this chapter, in the latent variable model. Recall that the returns defined in Equation 8 only take macroeconomic risk factors into account. This particular setting doesn’t allow parent-subsidiary

relations to influence the asset return process of the individual companies. In order to extend the previously derived model we need to introduce some new variables.

19

Ξ = :m6,Y<_{6,YD ,…,;} 11

which indicates the strength of the business interdependence between two obligors 5 and n. From now on we call this matrix the business matrix in line with Egloff, Leippold and Vanini [12]. In this thesis we only consider positive business relations since we are interested in parent-subsidiary relations. This assumption seems reasonable since companies with a parent-subsidiary relation should be positively correlated in theory. Furthermore we assume that m_6,6 = 0 for all 5 ∈ 71, … , 48. This assumption is made so that an obligor cannot self-affect itself directly, otherwise the model could possibly explode. Furthermore we also assume that each row’s sum in the business matrix is less or equal to one, which basically states that each business relation is a percentage of the total available business relations for that specific company. The business matrix is based on an expert opinion hence this restriction is easily met.

Next to the business matrix we define

η = (p6 6D ,…,; 12

as the weights to the residual risk of each debtor. The residual risk is the risk that is left over after the natural risk is controlled by the risk controls. So in this case, the p measures to what extent the business relation affects the returns of a certain company. The business matrix and the weights p cannot be calibrated to the historical default data, but are instead inferred on forehand on the information available at the Rabobank. We assume from now on that

p6 = 1 − q mY,6 Y

where the sum is taken over every company n that has a business relation with company 5.

The latent variable model that we are building relies on a strong assumption that the previously defined business matrix remains constant over time. This is unfortunately not the case for many real life portfolios. These relations can easily change for instance in times of recessions or due to mergers of firms. For this particular model we however assume stability over time.

Assumption 4

All microstructural risk is captured in the business matrix. Furthermore we assume that this business matrix is constant over time.

It is interesting, for a future research project, to investigate the behavior of the model when this assumption is relaxed. Theoretically it should make the model more realistic.

The definitions of the business matrix and the residual risk factor allow us to incorporate the micro-structure in the purely macro-structural model, using a weighted graph. The definition of a graph is given below. Definition 3.3

20

A graph r(s, t is a mathematical structure used to model pairwise relations between objects. It consists of a number of vertices (s and a set of edges (t that connect a certain subset of the nodes with each other. A weighted graph r(s, t, p assigns a certain weight (p to each of the edges. In addition to this, a graph can be directed or undirected, which indicates if each edge is pointed in a certain direction.

For an elaborate background in Graph Theory we refer to Bundy and Murthy [6]. We mainly use the graph representation for graphical convenience. Some results of the Graph Theory are however quite useful if we want to construct random business matrices. After this short sidebar we are able to construct the micro-structure of our portfolio.

Definition 3.4

The microstructure for a portfolio of 4 obligors is a directed weighted graph r(4, t, p, Ξ . The nodes corresponds to the obligors and the edge represents the direction of a business relation defined in the business matrix Ξ, this information is stored in the set of edges (t corresponding to the non-zero elements of the business matrix. The weights corresponding to the edge from company n to company 5 is given by m_Y6 . The representation of the case 4 = 2, where only company n has a relation with company 5 is shown in

Figure 1 below.

Figure 1: Microstructure between companies

Let us immediately give the first extra feature we observe when we incorporate this micro-structure. In the purely macro-economic setup firm 5 cannot be directly influenced by the macroeconomic effect for sector n nor can firm n be directly influenced by the macroeconomic effects for sector 5, as long as these companies are in different sectors. The only dependence structure is modelled through the correlation matrix Γ (Equation 6). But the previously defined business matrix allows a more sophisticated dependence structure. If we implement this business matrix in the asset return process we are able to model firm 5 such that it can be dependent of the sector specific risk factor for sector n, which makes the model more realistic. Note furthermore that this microstructure allows asymmetric business relationships. This could be useful if, e.g., we assume that the parent company has a greater influence on the subsidiary rating than vice versa. Note that this model can easily be extended for portfolios consisting of firms that don’t have a parent-subsidiary relation. This can be accomplished by extending the business matrix with a row of zeros. However we restrict ourselves to the situation where each company in the portfolio has a parent-subsidiary relation. We must however be careful when we incorporate the micro-structure in the macrostructural model. It seems reasonable to assume that a financial institution is able to assess correctly the rating probabilities of the individual debtors. Therefore must the addition of macro- and micro-structures not change the rating dynamics on portfolio level. Thus we assume the following.

In order to satisfy this assumption, we demand that the dynamics of the synthetic asset return remain the same after we have implemented the micro-structure to this dynamics. Thus the asset return process still

pY p6

21

needs to be standard normally distributed. Even more, the idiosyncratic part needs to be standard normally distributed.

The micro-structural interdependencies can be consistently included in the purely macro-structural model of Equation 8 in the following way

1 = ^u^ `&1 − \]a G1+ ^v^(\ (r, G1, 1 13

We replaced the idiosyncratic risk factor by the function (r, G₁, ₁ which is depending on the microstructure. As a consequence, we also need some normalizing diagonal matrices ^_u, and ^_v. These variables must satisfy the following two technical conditions in order to comply with Assumption 2:

(A):^_vx(y, u₊, v₊ is normalized to standard normal

(B):The marginal distributions of z₊ remain standard normal

Condition (A) follows since the addition of microstructures can’t alter the rating dynamics caused by the idiosyncratic risk part under Assumption 2. This means that the distribution of the idiosyncratic term must remain the same, hence {_| (r, G₁, ₁ must be standard normally distributed.

Condition (B) follows by the same reasoning. The macro- and micro dependence structures cannot alter the rating dynamics thus the latent variable ₁ must remain standard normal. These conditions are also

sufficient since there aren’t any other restrictions on the latent variable model. An appropriate choice for the function (r, G₁, ₁ will be given in Paragraph 3.1.4.

3.1.4 Estimate microstructure dependency

Note that the incorporation of micro-structure in the previous paragraph is a generalization of Equation 8, since the choice ^_u= }, ^_v= } and (r, G₁, ₁ = ₁ leads to Equation 8. This paragraph explains the structure of the microstructure idiosyncratic variable .

In this thesis we explore a structure where the dependency of (r, G₁, ₁ on ₁ is linear. We assume that microstructural idiosyncratic variable is given by

(r, G1, 1 = Ξ 1+ ^(p 1 Substituting this relation in Equation 13 leads to asset return process

1 = ^u^ `&1 − \]a G1 + ^v^(\ (Ξ 1+ ^(p 1 14

Even this simple linear dependence structure of the microstructure on ₁ presented in Equation 14 provides advantages in comparison to the purely macroeconomic model which is widely used. This model can be extended to more complex situations but we restrict ourselves to the current situation. One of the

22

advantages of this model is that firm 5 and firm n can belong to different business sectors. This is usually the case when we observe a parent-subsidiary relation. The parent is commonly a _] company whereas the subsidiary company is more likely to be a _~ or _• cap company.

Another interesting feature is the existence of feedback in this model. This means that a negative jump for a firm can increase the chance of a second negative jump in the near future. The next example provides more insight on this phenomenal.

Example 3.2

Suppose we have a portfolio consisting of three companies, }, € and •. Suppose that all these companies

operate in different sectors, hence their macroeconomic factors are completely independent. Thus these companies cannot affect each other in the purely macro-economic model. Following Equation 14 however the can affect each other. Suppose that company 1 has a business relation with company 2, company 2 has a business relation with company 3 and company 3 has a business relation with company 1. These business relations are shown below in a graphical way.

If company 1 exhibits a devaluation of its assets because of some sort of idiosyncratic shock, this results in a lower asset return. Since the asset return of company 2 is connected to the asset return of company 1 via Equation 14, this could also result in a devaluation of the asset return process of company 2. In a similar way also company 3 can exhibit a devaluation of its asset return process. This is where the feedback kicks in. Since company 3 has a business relation with company 1, the first devaluation of the assets of company 1 can lead to a second devaluation of the asset return process of company 1. This phenomenon is the so called feedback.

□

For sake of notation we rephrase Equation 14 into

1= ‚uG1+ ‚v 1 15

where of course we have

‚u= (ƒ − ^v^(\ Ξ ^u^ `&1 − \]a and ‚v= (ƒ − ^v^(\ Ξ ^v^(\ ^(p

1

2

3

23

The ƒ represents the 4 × 4 identity matrix. Note that ‚_u, ‚_v are well defined since the operator norm of ^v^(\ Ξ is strictly less than one which is proved in the next lemma.

Definition 3.5

The operator norm of a Q × Q …†3‡5 is given by

ˆ| |ˆ_‰!= MŠ ‹ˆ| Œ|ˆ: _•‡ †ŽŽ Œ ∈ ℝ \53ℎ ˆ|Œ|ˆ = 1‘,

where ||. || represents the usual Euclidian norm on vector spaces.

Lemma ‚_u and ‚_v are well defined.

Proof

By definition of the sector specific risk weights, the business matrix and the diagonal matrix ^_v we know that the elements of the matrix ^_v^(\ Ξ are non-negative and strictly smaller than one. This implies that the operator norm is strictly smaller than 1. Thus (ƒ − ^_v^(\ Ξ is non-singular and

we conclude that ‚_u and ‚_v are well-defined.

□

From Equation 15, and the fact that ₁ is a zero-mean Gaussian distributed, we can write the covariance matrix of ₁ as

Σ = ‚uΓ‚u+ ‚v‚v

We still need to check under which circumstances this system fulfils the two imposed Conditions (A) and (B). In order to fulfil Condition (A) we must choose ^_v such that the idiosyncratic term in Equation 14 has unit

variance. As stated in the condition this reduces to the root of inverse of the covariance of the function . Before we can answer this question we must calculate the variance of (r, G₁, ₁ , which is given by

“•Œ: (r, G1, 1 < = @F(Ξ 1+ ^(p 1 ]H

= @FΞA]1Ξ”+ ^](p 1]+ Ξ 1^(p 1+ ^(p 1Ξ 1H = ΞΣΞ + ^(p] _{+ Ξ•–^(p + ^(p ‚vΞ}

The last equality follows from the definition of Σ, Equation 15 and the fact that G₁ and ₁ are independent normally distributed random variables.

Next we need to choose ^_u such that ₁ in Equation 15 has unit variance. These results lead to the two (dependent!) conditions on ^_v and ^_u

—( : ^v€^ `“•Œ: (r, G1, 1 <a = ƒ

( : ^(Σ = ƒ 16

The dependency between these conditions originates from the fact that the ^_v term also appears in the ‚_u expression. Unfortunately, closed form solutions of Equation 16 doesn’t exist (yet?). Thus we need to approximate the asset return process numerically. There arise however a few problems if the portfolio is

24

large. It could turn out that the matrix inversion, (ƒ − ^_v^(\ Ξ , is numerical unstable, since the business matrix is a sparse matrix. Luckily there are some techniques to avoid this problem.

3.1.5 Approximation of asset process

The computational problems originate from the calculation of (1 − ^_v^(\ Ξ when this matrix reaches singularity. The approximation relies on the fact that instead of recursively integrating the microstructure as in Equation 14, we use as starting point the estimation of ₁ based on purely macroeconomic effects. Then we incorporate the micro-structure dependencies in a recursive manner. We calculate the same quantities as before, but this recursive approach leads to solvable equations. This paragraph uses the same

assumptions as before. Hence we, again, assume that the -function is linear dependent on the asset return process.

First, we define the purely macro-structural asset return process as the initialization of the approximation scheme.

1

( _{= ^ `&1 − \]a G}

1+ ^(\ 1 17

From this purely macroeconomic effect we can define the microstructural effect by

( _{(r, G}

1, 1 = Ξ 1( + ^(p 1

= Ξ `^ `&1 − \]_{a G1+ ^(\ 1a + ^(p 1} = Ξ ^ `&1 − \]_{a G1+ :Ξ^(\ + ^(p < 1}

18

For notational convenience we define

˜_u( = Ξ ^ `&1 − \]_a and

˜v( = Ξ^(\ + ^(p

Next, we would like to calculate the covariance of this process, since we must satisfy Condition (A). Since the multivariate normal variables G₁ ∼ T(0, Γ and ₁ ∼ T(0, ƒ are independent, we can easily calculate this covariance. “•Œ ` ( <sub>(r, G1, 1 a = @ ™`š</sub> ›( G1+ ˜v( 1a ] œ = ˜_u( Γ `˜<sub>u</sub>( a + ˜v( `˜v( a 19

25

Note that, contrary to the previous situation, this covariance in this case is independent of the factors ^_v( and ^_u( . This allows us to exactly calculate ^_v( in Equation 16 such that it satisfies Condition (A). Next we define the second approximation step of the asset return process ₁ which follows intuitively from the previous derivations. 1 ( _{= ^} u ( _{^ `&1 − \]a G} 1+ ^v( ^(\ ( (r, G1, 1 20

where ^_u( is chosen such that ₁( satisfies Condition (B). The precise choice of ^_u is however not trivial yet. We proceed just in a similar way as we did for Equation 15, in this setting however, we get an closed analytic solution. For notational convenience we define the matrices

‚_u( = ^_u( ^ `&1 − \]_{a + ^v}( _{^(\ š} › ( and ‚v( = ^v( ^(\ ˜v( such that 1 ( _{= ‚} u ( _G 1+ ‚v( 1 21

Again we are interested in the covariance structure of this asset return process, since we must satisfy Assumption 2. The calculations are basically the same as we did for the non-recursive process and the result is shown below. Σ( _{= “•Œ `} 1 ( _, 1 ( _{a = ‚} u ( _Γ‚ u ( _{+ ‚} v ( _‚ v ( In order to fulfil Condition (B), we must have

s†‡ ` ₁( a = ^:Σ( _{< = ƒ} where ƒ is again the 4 × 4 identity matrix.

Lemma The normalizing matrix ^_u is the solution to a quadratic matrix equation. Proof

The calculations of the next part become a bit tedious, therefore we define for notational convenience the variable

( _{= ^} v

( _{^(\ ˜} u (

Note that • is a symmetric matrix, thus we must have ^( • ≛ ^(• for every matrix of the proportional dimensions. Furthermore we recall that for any real valued function _ and vector Œ we

26

have _:^(Œ < ≗ ^(_(Œ . Hence ^_u( must be chosen such that it satisfies the following, on first sight ugly, quadratic matrix condition

ƒ = ^: ( _< = ^ `‚<sub>u</sub>( •‚<sub>u</sub>( + ‚v( ‚v( a = ^ `‚_u( •‚_u( a + ^ `‚v( ‚v( a = ^ ¡`^_u( ^ `&1 − \]<sub>a +</sub> ( <sub>a • `^</sub> u ( _{^ `&1 − \]a +</sub> ( <sub>a ¢ + ^ `‚} v ( _‚ v ( _a ≛ ^ £`^<sub>u</sub>( ^ `&1 − \]_{a +} ( _a]_{•¤ + ^ `‚</sub> v ( <sub>‚</sub> v ( <sub>a</sub> ≗ ^ £`^u( a ] ^(1 − \] _{• + :} ( _<]_{• + 2^} u ( _{^ `&1 − \]a ( •¤ + ^ `‚} v ( _‚ v ( _a ≗ `^u( a ] ^(1 − \] <sub>^(• + 2^</sub> u ( <sub>^ `&1 − \]a ^: ( •< + ^ `‚} v ( _‚ v ( _{+ ( •} ( _a = ¥(^_u( , ^v( 22

Note that Equation 22 can be explicitly solved for ^_u( . Since ^_u(}is a diagonal matrix, each diagonal

entry can simply be calculated with the ABC-formula.

□

This first approximation step however doesn’t allow feedback. In order for this phenomenon to happen we must have at least two steps in this approximation scheme. Now we have calculated all necessary parameter matrices and we can thus estimate default probabilities.

We could continue in this recursive structure to obtain the Q1¦ order approximation of the asset return process. The following proposition provides the properties of this approximation.

Proposition 3.1

The Q1¦ order approximation of this scheme is given by

1

( _{= ^} u

( _{^ `&1 − \]a G}

1+ ^v( ^(\ ( (r, G1, 1 23

where ( (r, G₁, ₁ = ₁ and ^_u( = ^_v( = ƒ. The idiosyncratic risk effects are modeled by

( _{(r, G} 1, 1 = ˜u( G1+ ˜v( 1 24 with ˜_u( = q § B Ξ ^v( Y ^(\ 6 YD ¨ Ξ^_u(6 ^ `&1 − \] _a 6D and

27 ˜v( = q § B Ξ ^v( Y ^(\ 6 YD ¨ ©(η + B Ξ ^v( 6 ^(\ 6D 6D

The normalizing matrices ^_u( and ^_u( should satisfy the following two conditions

ª( : `^v

( _a]_{= ^:“•Œ(} ( _{(r, G, <} ( : ¥ `^u( , ^v( a = 1

For a proof of this proposition we refer to Egloff, Leippold and Vanini [12], where they also proof that this approximation scheme converges to asset return process of Equation 13. To our knowledge, the rate at which this algorithm converges has never been discussed. We answer this question in Chapter 6, where we show that approximately five steps will be enough for sufficient results for the business matrices in which we are interested. The following example should provide some useful insights of the previous derived

approximation scheme, applied to a portfolio consisting of two clients. Example 3.3

Suppose that we have a portfolio consisting of two companies, a parent company (P) and a subsidiary company (S). For simplicity we assume that the parent company has influence on the asset return process of the subsidiary company but not vice versa. Hence the business matrix Ξ is equal to `0 m«¬

0 0 a , where m«¬ is a

positive number smaller or equal to one. This implies that p = ` 1 − m«¬

1 a. Let us furthermore assume that

the parent company belongs to the _] companies and that the subsidiary company belongs to the _] companies. The risk variables of these sectors are given by ?_1,« respective ?_1,¬. Equation 8 reduces in this situation to 1,¬ ( _{= [1 − \} ¬]?1,¬+ \¬ 1,¬ 1,« ( _{= [1 − \} «]?1,«+ \« 1,«

Thus the total asset return process is given by

1 ( <sub>= ^ `&1 − \]a ?</sub> 1+ ^(\ 1 = -®[1 − \¬ ] <sub>0</sub> 0 [1 − \«]<sub>¯</sub> ° £ ?1,¬ ?1,«¤ + £\ ¬ 0 0 \«¤ ` 1,¬ 1,«a

28

Following the previous paragraph we calculate ( (r, G₁, ₁ and its covariance in order to get the first order approximation of the asset return process. By definition we have

( _{(r, G} 1, 1 = Ξ 1( + ^(p 1 = `0 m«¬ 0 0 a -± ® -®[1 − \¬ ] <sub>0</sub> 0 [1 − \«]<sub>¯</sub> ° £ ?1,¬ ?1,«¤ + £\ ¬ 0 0 \«¤ ` 1,¬ 1,«a ¯ ² ° + `1 − m«¬ 0 0 1a ` 1,¬1,«a = ³0 m«¬[1 − \«] 0 0 ´ £ ?1,¬ ?1,«¤ + `1 − m «¬ m«¬\« 0 1 a ` 1,¬1,«a

and its covariance is given by (recall that X_«¬ is the correlation coefficients between the sectors and the idiosyncratic factors are independent of each other )

“•Œ ` ( <sub>(r, G</sub> 1, 1 a = ³0 m«¬[1 − \«] 0 0 ´ £ 1 X «¬ X«¬ 1 ¤ ³ 0 0 m«¬[1 − \«] 0´ + ` 1 − m«¬ m«¬\« 0 1 a £1 − mm«¬\«¬« 01¤ = £1 − 2m«¬+ 2m«¬] m«¬\« m«¬\« 1 ¤ Define ˜_u( = Ξ ^:√1 − \]< = £0 m«¬&1 − \«]

0 0 ¤ for notational convenience. Since the asset return

process must satisfy (A), we see that

^v( = ¶ 1 [1 − 2m«¬+ 2m«¬] 0 0 1 ·

For notational convenience we write ¸ =

[ ]¹º»2]¹º»¼

. All that is left to calculate is the matrix ^_u( =

£½_{0 ½}0