Semi-Markov Migration Models for Credit Risk Stochastic Models for Insurance Set coordinated by Jacques Janssen Volume Semi-Markov Migration Models for Credit Risk Guglielmo D’Amico Giuseppe Di Biase Jacques Janssen Raimondo Manca First published 2017 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc Apart from any fair dealing for the purposes of research or private study, or criticism or review, as permitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced, stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers, or in the case of reprographic reproduction in accordance with the terms and licenses issued by the CLA Enquiries concerning reproduction outside these terms should be sent to the publishers at the undermentioned address: ISTE Ltd 27-37 St George’s Road London SW19 4EU UK John Wiley & Sons, Inc 111 River Street Hoboken, NJ 07030 USA www.iste.co.uk www.wiley.com © ISTE Ltd 2017 The rights of Guglielmo D’Amico, Giuseppe Di Biase, Jacques Janssen and Raimondo Manca to be identified as the authors of this work have been asserted by them in accordance with the Copyright, Designs and Patents Act 1988 Library of Congress Control Number: 2017931483 British Library Cataloguing-in-Publication Data A CIP record for this book is available from the British Library ISBN 978-1-84821-905-2 Contents Introduction ix Chapter Semi-Markov Processes Migration Credit Risk Models 1.1 Rating and migration problems 1.1.1 Ratings 1.1.2 Migration problem 1.1.3 Impact of rating on spreads for zero bonds 1.1.4 Homogeneous Markov chain model 1.1.5 Migration models 1.2 Homogeneous semi-Markov processes 1.2.1 Basic definitions 1.2.2 The Z SMP and the evolution equation system 1.2.3 Special cases of SMP 1.2.4 Sojourn times and their distributions 1.3 Homogeneous semi-Markov reliability model 1.4 Homogeneous semi-Markov migration model 1.4.1 Equivalence with the reliability problem 1.4.2 Transient results 1.4.3 Asymptotic results 1.4.4 Example 1.5 Discrete time non-homogeneous case 1.5.1 NHSMPs and evolution equations 1.5.2 The Z NHSMP 1.5.3 Sojourn times and their distributions 1.5.4 Non-homogeneous semi-Markov reliability model 1 10 10 14 16 19 21 23 23 24 26 28 33 33 34 36 37 vi Semi-Markov Migration Models for Credit Risk 1.5.5 The non-homogeneous semi-Markov migration model 1.5.6 A non-homogeneous example 38 39 Chapter Recurrence Time HSMP and NHSMP: Credit Risk Applications 51 2.1 Introduction 2.2 Recurrence times 2.3 Transition probabilities of homogeneous SMP and non-homogeneous SMP with recurrence times 2.3.1 Transition probabilities with initial backward 2.3.2 Transition probabilities with initial forward 2.3.3 Transition probabilities with final backward and forward 2.3.4 Transition probabilities with initial and final backward 2.3.5 Transition probabilities with initial and final forward 2.3.6 Transition probabilities with initial and final backward and forward 2.4 Reliability indicators of HSMP and NHSMP with recurrence times 2.4.1 Reliability indicators with initial backward 2.4.2 Reliability indicators with initial forward 2.4.3 Reliability indicators with initial and final backward 2.4.4 Reliability indicators with initial and final backward and forward 51 52 53 53 55 57 58 60 61 63 63 66 70 73 Chapter Recurrence Time Credit Risk Applications 79 3.1 S&P’s basic rating classes 3.1.1 Homogeneous case 3.1.2 Non-homogeneous case 3.2 S&P’s basic rating classes and NR state 3.2.1 Homogeneous case 3.2.2 Non-homogeneous case 3.3 S&P’s downward rating classes 3.3.1 An application 3.4 S&P’s basic rating classes & NR1 and NR2 states 3.5 Cost of capital implications 80 81 86 90 91 106 120 122 127 134 Chapter Mono-Unireducible Markov and Semi-Markov Processes 137 4.1 Introduction 4.2 Graphs and matrices 4.3 Single-unireducible non-homogeneous Markov chains 137 138 145 Contents vii 4.4 Single-unireducible semi-Markov chains 4.5 Mono-unireducible non-homogeneous backward semi-Markov chains 4.6 Real data credit risk application 158 160 Chapter Non-Homogeneous Semi-Markov Reward Processes and Credit Spread Computation 165 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 5.5.1 Data 5.5.2 Results 165 166 168 170 173 173 176 Chapter NHSMP Model for the Evaluation of Credit Default Swaps 183 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 184 188 196 198 Chapter Bivariate Semi-Markov Processes and Related Reward Processes for Counterparty Credit Risk and Credit Spreads 205 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 7.4.1 Pricing a risky CDS and CVA evaluation 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 152 206 208 220 224 227 230 233 247 249 259 Chapter Semi-Markov Credit Risk Simulation Models 267 8.1 Introduction 8.2 Monte Carlo semi-Markov credit risk model for the Basel II Capital at Risk problem 267 267 viii Semi-Markov Migration Models for Credit Risk 8.2.1 The homogeneous MCSM evolution with D as absorbing state 8.3 Results of the MCSMP credit model in a homogeneous environment 273 Bibliography 279 Index 297 269 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 x Semi-Markov Migration Models for 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 Introduction xi 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 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 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208, 224, 228 credit rating, 3, 4, 10, 167 risk, 267, 269 spread, 205, 249–252, 261 D, F, G, H default probability, 2, 5, 83, 84, 121 swap, 183, 200, 201 finance, 80, 134, 135, 145, 161 graph, 138, 150 homogeneous, 1, 7, 8, 10, 12, 16, 21, 23, 28, 33–37 M, N migration models, 267–278 mono-unireducibility, 158, 159 Monte Carlo methods, 267, 268 non-homogeneity, 162, 183 non-homogeneous, 1, 10, 28, 33, 36–41 P, R point wise availability function, 22, 37 rating, 1–5, 8–10, 13, 15, 16, 24, 25, 29, 34, 35, 38, 39, 41, 46, 47, 49 reliability, 63-76, 85, 88–90, 96, 105, 113, 119, 121, 124–130, 132–134, 162, 183, 186, 198 function, 22, 24, 37 reward, 167–169, 171, 172 process, 205, 207, 208, 235, 236, 239, 246, 249, 254, 256, 257 risk, 79, 126, 128, 134–137, 148, 154, 160 S semi-Markov models, 88, 124, 125, 131 processes, 51–55, 59, 184, 190, 198, 267–269 sojourn times, 19, 20, 36 spread, 3, 5–7, 25 stochastic processes, 160 recovery rate, 183, 184, 188, 194, 202 Semi-Markov Migration Models 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(0.77% – 0.34%)/(1 – 0.34%) = 0.43% 8 Semi- Markov Migration Models for Credit Risk 1.1.5 Migration models 1) Credit risk and reliability problems Homogeneous semi- Markov processes (HSMPs) were defined