Table of Contents

Cover

Title

Introduction

1 Semi-Markov Processes Migration Credit Risk Models

1.1. Rating and migration problems
1.2. Homogeneous semi-Markov processes
1.3. Homogeneous semi-Markov reliability model
1.4. Homogeneous semi-Markov migration model
1.5. Discrete time non-homogeneous case

2 Recurrence Time HSMP and NHSMP: Credit Risk Applications

2.1. Introduction
2.2. Recurrence times
2.3. Transition probabilities of homogeneous SMP and non-homogeneous SMP with recurrence times
2.4. Reliability indicators of HSMP and NHSMP with recurrence times

3 Recurrence Time Credit Risk Applications

3.1. S&P’s basic rating classes
3.2. S&P’s basic rating classes and NR state
3.3. S&P’s downward rating classes
3.4. S&P’s basic rating classes & NR1 and NR2 states
3.5. Cost of capital implications

4 Mono-Unireducible Markov and Semi-Markov Processes

4.1. Introduction
4.2. Graphs and matrices
4.3. Single-unireducible non-homogeneous Markov chains
4.4. Single-unireducible semi-Markov chains
4.5. Mono-unireducible non-homogeneous backward semi-Markov chains
4.6. Real data credit risk application

5 Non-Homogeneous Semi-Markov Reward Processes and Credit Spread Computation

5.1. Introduction
5.2. The reward introduction
5.3. The DTNHSMRWP spread rating model
5.4. The algorithm description
5.5. A numerical example

6 NHSMP Model for the Evaluation of Credit Default Swaps

6.1. The price and the value of the swap: the fixed recovery rate case
6.2. The price and the value of the swap: the random recovery rate case
6.3. The determination of the n-period random recovery rate
6.4. A numerical example

7 Bivariate Semi-Markov Processes and Related Reward Processes for Counterparty Credit Risk and Credit Spreads

7.1. Introduction
7.2. Multivariate semi-Markov chains
7.3. The two-component reliability model
7.4. Counterparty credit risk in a CDS contract
7.5. A numerical example
7.6. Bivariate semi-Markov reward chains
7.7. The estimation methodology
7.8. Credit spreads evaluation
7.9. Numerical experience

8 Semi-Markov Credit Risk Simulation Models

8.1. Introduction
8.2. Monte Carlo semi-Markov credit risk model for the Basel II Capital at Risk problem
8.3. Results of the MCSMP credit model in a homogeneous environment

Bibliography

Index

End User License Agreement

List of Tables

1 Semi-Markov Processes Migration Credit Risk Models

Table 1.1. Example of transition matrix of credit ratings (source: [JAN 07])
Table 1.2. Example of withdrawn rating (source: [JAN 07])
Table 1.3. Yearly transition matrix
Table 1.4. Probabilities ϕ_ij (5)
Table 1.5. Probabilities ϕ_ij (10)
Table 1.6. Probabilities to not have a default
Table 1.7. Probabilities to remain in the starting state
Table 1.8. Probabilities ϕ_ij (5)
Table 1.9. Distribution function from 1 to 5
Table 1.10. Distribution function from 6 to 10
Table 1.11. Embedded non-homogeneous MC
Table 1.12. Probability to remain in the starting state without transitions
Table 1.13. Probability φ_ij (0,t)
Table 1.14. Probability φ_ij, (15,t)
Table 1.15. Probability ϕ_ij (0,t)
Table 1.16. Probability ϕ_ij(15,t)
Table 1.17. Reliability R_i(s,t)

3 Recurrence Time Credit Risk Applications

Table 3.1. Transition probabilities with initial backward
Table 3.2. Reliability function with initial backward. For a color version of this table, see www.iste.co.uk/damico/credit.zip
Table 3.3. Reliability as a function of backward time
Table 3.4. Non-homogeneous transition probabilities with initial backward
Table 3.5. Backward reliability from s to t
Table 3.6. Transition probabilities at times 6, 12, 18
Table 3.7. Stay on probabilities until t given the starting rank value
Table 3.8. Next transition probabilities in the default and in the no rating states
Table 3.9. Reliability and maintainability functions given the starting rating
Table 3.10. Backward transition probabilities
Table 3.11. Backward probabilities do not move within t. For a color version of this table, see www.iste.co.uk/damico/credit.zip
Table 3.12. Backward reliability and maintainability functions. For a color version of this table, see www.iste.co.uk/damico/credit.zip
Table 3.13. Transition probabilities at some starting and arrival times s, t
Table 3.14. Probabilities of not leaving the rating occupied at s up to time t
Table 3.15. Next transition probabilities in the default and in the no rating states from s to t
Table 3.16 Reliability and maintainability functions up to t given the rating occupied at s
Table 3.17. Backward transition probabilities at some backward, starting and arrival times s, t
Table 3.18. Backward probabilities do not move from s to t
Table 3.19. Backward reliability and backward maintainability from s to t
Table 3.20. Reliability function ^bR_i (0; t)
Table 3.21. Reliability function ^bR_i(1; t)
Table 3.22. Embedded Markov chain at s = 0
Table 3.23. Semi-Markov transition probabilities
Table 3.24. Reliability with initial and final backward from rating AAA and BB
Table 3.25. Maintainability functions with initial backward from rating NR2 and D
Table 3.26. Rating-dependent cost of capital

4 Mono-Unireducible Markov and Semi-Markov Processes

Table 4.1. Embedded Markov chain for s = 1
Table 4.2. Embedded Markov chain for year s = 11
Table 4.3 Embedded Markov chain for s = 21

5 Non-Homogeneous Semi-Markov Reward Processes and Credit Spread Computation

Table 5.1. Embedded matrix P(5)
Table 5.2. Embedded matrix P(15)
Table 5.3. Mean basis points as a function of the rating
Table 5.4. Complete mean basis points
Table 5.5. Transition matrix Φ(4,5)
Table 5.6. Transition matrix Φ(4,15)
Table 5.7. Transition matrix Φ(10,11)
Table 5.8. Transition matrix Φ(10,21)

6 NHSMP Model for the Evaluation of Credit Default Swaps

Table 6.1. Embedded Markov chain at time 0
Table 6.2. Swap values, starting time 0, deterministic RR = 0.2
Table 6.3. Embedded Markov chain at time 3
Table 6.4. Swap values, starting time 3, deterministic RR = 0.2
Table 6.5. Swap values, starting time 0, deterministic RR = 0.4
Table 6.6. Swap values, starting time 3, deterministic RR = 0.4
Table 6.7. Swap values, starting time 0, stochastic 1-RRR
Table 6.8. Swap values, starting time 3, stochastic 1-RRR

7 Bivariate Semi-Markov Processes and Related Reward Processes for Counterparty Credit Risk and Credit Spreads

Table 7.1. Marginal transition matrices
Table 7.2. Transition matrix of the embedded Markov chain
Table 7.3. Transition matrix of the embedded Markov chain

8 Semi-Markov Credit Risk Simulation Models

Table 8.1. Number of loans for each up rating
Table 8.2. Number of repetitions for each up rating
Table 8.3. Number of computed trajectories for each up ratings
Table 8.4. Mean value of the capital losses because of default and for each year
Table 8.5. Probabilities of mean value of the capital losses because of default and for each year
Table 8.6. Capital at Risk values

List of Illustrations

1 Semi-Markov Processes Migration Credit Risk Models

Figure 1.1. A sample path of an SMP (source: [JAN 07])

2 Recurrence Time HSMP and NHSMP: Credit Risk Applications

Figure 2.1. A semi-Markov process trajectory with initial and final backward and forward. For a color version of this figure, see www.iste.co.uk/damico/credit.zip
Figure 2.2. A semi-Markov process trajectory with initial and final backward

3 Recurrence Time Credit Risk Applications

Figure 3.1. Transition probabilities of the embedded Markov chain at times s = 1 (top-left panel), s = 10 (top-right panel) and s = 20 (bottom-left panel). For a color version of this figure, see http://www.iste.co.uk/damico/credit.zip
Figure 3.2. Transition probabilities of the semi-Markov chain at times l = 0. s = 6, t = 7 (top-left panel), l = 0. s = 6, t = 10 (top-right panel) and l = 0. s = 6, t = 13 (bottom-left panel). For a color version of this figure, see http://www.iste.co.uk/damico/credit.zip
Figure 3.3. Reliability functions for rating BBB (left panel) and B (right panel) for s = 6 and l = 0,2,4,6. For a color version of this figure, see http://www.iste.co.uk/damico/credit.zip
Figure 3.4. Reliability functions for rating A (left panel) and B (right panel) for s = 6,8,10,12 and l = 0 For a color version of this figure, see http://www.iste.co.uk/damico/credit.zip
Figure 3.5. Reliability functions for rating CCC and CCC. For a color version of this figure, see http://www.iste.co.uk/damico/credit.zip
Figure 3.6. Failure rates for rating BBB (left panel) and B (right panel) for s = 6 and l = 0,2,4,6 For a color version of this figure, see http://www.iste.co.uk/damico/credit.zip
Figure 3.7. Reliability functions with initial backward ^bR_i (v, s;s + h): on the x-axis there are the h-values; on the y-axis, there are the probabilities. Top-left panel shows the case v = 8, s = 8, top-right panel shows the case v = 7, s = 8, bottom-left panel shows the case v = 6, s = 8, bottom-right panel shows the case v = 4, s = 5

4 Mono-Unireducible Markov and Semi-Markov Processes

Figure 4.1. Seven classes of single-unireducible Markov matrix states
Figure 4.2. The partial order among the classes
Figure 4.3. Reachability matrix of Markov matrix given in Figure 4.1
Figure 4.4. Conditional waiting time distribution function G_AAAAaa (s, s + k) (dotted lines) and G_AAAA(s, s + k) (continuous lines) for s = 1 a), s = 3 (b), s = 6 c) and s = 9 d). For a color version of this figure, see www.iste.co.uk/damico/credit.zip
Figure 4.5. Reliability function ^bR_A (l,s;s + k) for s = 4 (dotted lines) and s = 8 (continuous lines) for l = 1 a), l = 2 b), l = 3 c) and l = 4 d). For a color version of this figure, see www.iste.co.uk/damico/credit.zip

5 Non-Homogeneous Semi-Markov Reward Processes and Credit Spread Computation

Figure 5.1. Logarithm of the expected sum of basis points paid starting at time 0. For a color version of this figure, see www.iste.co.uk/damico/credit.zip
Figure 5.2. Logarithm of the expected sum of basis points paid starting at time 10 and 15. For a color version of this figure, see www.iste.co.uk/damico/credit.zip
Figure 5.3. Logarithm of mean credit spread starting at time 0. For a color version of this figure, see www.iste.co.uk/damico/credit.zip
Figure 5.4. Logarithm of mean credit spread starting at time 10 and 15. For a color version of this figure, see www.iste.co.uk/damico/credit.zip
Figure 5.5. Expected discount factor v(0,t). For a color version of this figure, see www.iste.co.uk/damico/credit.zip
Figure 5.6. Expected discount factor v(0,t) for s = 10 and s = 15. For a color version of this figure, see www.iste.co.uk/damico/credit.zip

7 Bivariate Semi-Markov Processes and Related Reward Processes for Counterparty Credit Risk and Credit Spreads

Figure 7.1. Unconditional waiting times cumulative distribution functions
Figure 7.2. Reliabilities for speculative (SPE) state as a function of time
Figure 7.3. Expected basis points to be paid at any time starting from ratings (INV) − INV (black line) and (INV) − SPE (gray line)
Figure 7.4. Standard deviation of basis points to be paid at any time starting from ratings (INV) − INV (black line) and (INV) − SPE (gray line)
Figure 7.5. Logarithm in base 10 of the accumulated basis points to be paid at any time starting from ratings (INV) − INV (black line) and (INV) − SPE (gray line)
Figure 7.6. Forward discount factors starting from ratings (INV) − INV (black line) and (INV) − SPE (gray line)
Figure 7.7. Spot discount factors starting from ratings (INV) − INV (black line) and (INV) − SPE (gray line)

8 Semi-Markov Credit Risk Simulation Models

Figure 8.1. Two trajectories without losses
Figure 8.2. Trajectories with default at time 3 and default at time 6
Figure 8.3. Two examples with loss at the last period
Figure 8.4. Number of defaults for each unitary loss at time 5, 10 and 15. For a color version of this figure, see www.iste.co.uk/damico/credit.zip
Figure 8.5. Total value for each possible loss and the related probability. For a color version of this figure, see www.iste.co.uk/damico/credit.zip

Introduction

This book is a summary of several papers that the authors wrote on credit risk starting from 2003 to 2016.

Credit risk problem is one of the most important contemporary problems that has been developed in the financial literature. The basic idea of our approach is to consider the credit risk of a company like a reliability evaluation of the company that issues a bond to reimburse its debt.

Considering that semi-Markov processes (SMPs) were applied in the engineering field for the study of reliability of complex mechanical systems, we decided to apply this process and develop it for the study of credit risk evaluation.

Our first paper [D’AM 05] was presented at the 27th Congress AMASES held in Cagliari, 2003. The second paper [D’AM 06] was presented at IWAP 2004 Athens. The third paper [D’AM 11] was presented at QMF 2004 Sidney. Our remaining research articles are as follows: [D’AM 07, D’AM 08a, D’AM 08b, SIL08, D’AM 09, D’AM 10, D’AM 11a, D’AM 11b, D’AM 12, D’AM 14a, D’AM 14b, D’AM 15, D’AM 16a] and [D’AM 16b].

Other credit risk studies in a semi-Markov setting were from [VAS 06, VAS 13] and [VAS 13]. We should also outline that up to now, at author’s knowledge, no papers were written for outline problems or criticisms to the applications of SMPs to the migration credit risk.

The study of credit risk began with so-called structural form models (SFM). Merton [MER 74] proposed the first paper regarding this approach. This paper was an application of the seminal papers by Black and Scholes [BLA 73]. According to Merton’s paper, default can only happen at the maturity date of the debt. Many criticisms were made on this approach. Indeed, it was supposed that there are no transaction costs, no taxes and that the assets are perfectly divisible. Furthermore, the short sales of assets are allowed. Finally, it is supposed that the time evolution of the firm’s value follows a diffusion process (see [BEN 05]).

In Merton’s paper [MER 74], the stochastic differential equation was the same that could be used for the pricing of a European option. This problem was solved by Black and Cox [BLA 76] by extending Merton’s model, which allowed the default to occur at any time and not only at the maturity of the bond. In this book, techniques useful for the pricing of American type options are discussed.

Many other papers generalized the Merton and Black and Cox results. We recall the following papers: [DUA 94, LON 95, LEL 94, LEL 06, JON 84, OGD 87, LYD 00, EOM 03] and [GES 77].

The second approach to the study of credit risk involves reduced form models (RFMs). In this case, pricing and hedging are evaluated by public data, which are fully observable by everybody. In SFM, the data used for the evaluation of risk are known only within the company. More precisely, [JAR 04] explains that in the case of RFM, the information set is observed by the market, and in the case of SFM, the information set is known only inside the company.

The first RFM was given in [JAR 92]. In the late 1990s, these models developed. The seminal paper [JAR 97] introduced Markov models for following the evolution of rating. Starting from this paper, although many models make use of Markov chains, the problem of the poorly fitting Markov processes in the credit risk environment has been outlined.

Ratings change with time and a way of following their evolution their by means of Markov processes (see, for example, [JAR 97, ISR 01, HU 02]. In this environment, Markov models are called migration models. The problem of poorly fitting Markov processes in the credit risk environment has been outlined in some papers, including [ALT 98, CAR 94] and [LAN 02].

These problems include the following:

– the duration inside a state: actually, the probability of changing rating depends on the time that a firm remains in the same rating. Under the Markov assumption, this probability depends only on the rank at the previous transition;
– the dependence of the rating evaluation from the epoch of the assessment: this means that, in general, the rating evaluation depends on when it is done and, in particular, on the business cycle;
– the dependence of the new rating from all history of the firm’s rank evolution, not only from the last evaluation: actually, the effect exists only in the downward cases but not in the case of upward ratings in the sense that if a firm gets a lower rating (for almost all rating classes), then there is a higher probability that the next rating will be lower than the preceding one.

All these problems were solved by means of models that applied the SMPs, generalizing the Markov migration models.

This book is self-contained and is divided into nine chapters.

The first part of the Chapter 1 briefly describes the rating evolution and introduces to the meaning of migration and the importance of the evaluation of the probability of default for a company that issues bonds. In the second part, Markov chains are described as a mathematical tool useful for rating migration modeling. The subsequent step shows how rating migration models can be constructed by means of Markov processes.

Once the Markov limits in the management of migration models are defined, the chapter introduces the homogeneous semi-Markov environment. The last tool that is presented is the non-homogeneous semi-Markov model. Real-life examples are also presented.

In Chapter 2, it is shown how it is possible to take into account simultaneously recurrence times, i.e. backward and forward processes at the beginning and at the end of the time in which the credit risk model is observed. With such a generalization, it is possible to consider what happens inside the time before and after each transition to provide a full understanding of durations inside states of the studied system. The model is presented in a discrete time environment.

Chapter 3 presents the application of recurrence times in credit risk problems. Indeed, the first criticisms of Markov migration models were on the independence of the transition probabilities with respect to the duration of waiting time inside states (see [CAR 94, DUF 03]). SMP overcomes this problem but the introduction of initial and final backward and forward times allows for a complete study of the duration inside states. Furthermore, the duration of waiting time in credit risk problems is a fundamental issue in the construction of credit risk models.

In this chapter, real data examples are presented that show how the results of our semi-Markov models are sensitive to recurrence times.

Some papers have outlined the problem of unsuitable fitting of Markov processes in a credit risk environment. Chapter 4 presents a model that overcomes all the inadequacies of the Markov models. As previously mentioned, the full introduction of recurrence times solves the duration problem. The time dependence of the rating evaluation can be solved by means of the introduction of non-homogeneity. The downward problem is solved by means of the introduction of six states. The randomness of waiting time in the transitions of states is considered, thus making it possible to take into account the duration completely inside a state. Furthermore, in this chapter, both transient and asymptotic analyses are presented. The asymptotic analysis is performed by using a mono-unireducible topological structure. At the end of the chapter, a real data application is performed using the historical database of Standard & Poor’s as the source.

Chapter 5 presents a model to describe the evolution of the yield spread by considering the rating evaluation as the determinant of credit spreads. The underlying rating migration process is assumed to be a non-homogeneous discrete time semi-Markov non-discounted reward process. The rewards are given by the values of the spreads.

The calculation of the total sum of mean basis points paid within any given time interval is also performed.

From this information, we show how it is possible to extract the time evolution of expected interest rates and discount factors.

In Chapter 6, a discrete time non-homogeneous semi-Markov model for the rating evolution of the credit quality of a firm C is considered (see [D’AM 04]). The credit default swap spread for a contract between two parties, A and B, that sell and buy a protection about the failure of the firm C is determined. The work, both in the case of deterministic and stochastic recovery rate, is calculated. The link between credit risk and reliability theory is also highlighted.

Chapter 7 details two connected problems, as follows:

– the construction of an appropriate multivariate model for the study of counterparty credit risk in the credit rating migration problem is presented. For this financial problem, different multivariate Markov chain models were proposed. However, the Markovian assumption may be inappropriate for the study of the dynamics of credit ratings, which typically shows non-Markovian-like behavior. In this first part of the chapter, we develop a semi-Markov approach to study the counterparty credit risk by defining a new multivariate semi-Markov chain model. Methods are proposed for computing the transition probabilities, reliability functions and the price of a risky credit default swap;
– the construction of a bivariate semi-Markov reward chain model is presented. Equations for the higher order moments of the reward process are presented for the first time and applied to the problem for modeling the credit spread evolution of an obligor by considering the dynamic of its own credit rating and that of a dependent obligor called the counterpart. How to compute the expected value of the accumulated credit spread (expressed in basis points) that the obligor should expect to pay in addition to the risk free interest rate is detailed. Higher order moments of the accumulated credit spread process convey important financial information in terms of variance, skewness and kurtosis of the total basis points the obligor should pay in a given time horizon. This chapter contributes to the literature by extending on previous results of semi-Markov reward chains. The models and the validity of the results are illustrated through two numerical examples.

In Chapter 8, as in the previous chapters, the credit risk problem is placed in a reliability environment. One of the main applications of SMPs is, as it is well known, in the field of reliability. For this reason, it is quite natural to construct semi-Markov credit risk migration models.

This chapter details the first results that were obtained by the research group by the application of Monte Carlo simulation methods. How to reconstruct the semi-Markov trajectories using Monte Carlo methods and how to obtain the distribution of the random variable of the losses that the bank should support in the given horizon time are also explained in this chapter. Once this random variable is reconstructed, it will be possible to have all the moments of it and all the variability indices including the VaR. As it is well known, the VaR construction represents the main risk indicator in the Basel I–III committee agreements.