Credit Derivatives E-Book

Geoff Chaplin

• Herausgeber: John Wiley & Sons
• Kategorie: Fachliteratur
• Serie: Wiley Finance Series
• Sprache: Englisch

Seitenzahl: 739

Veröffentlichungsjahr: 2010

Beschreibung

The credit derivatives industry has come under close scrutiny over the past few years, with the recent financial crisis highlighting the instability of a number of credit structures and throwing the industry into turmoil. What has been made clear by recent events is the necessity for a thorough understanding of credit derivatives by all parties involved in a transaction, especially traders, structurers, quants and investors.

Fully revised and updated to take in to account the new products, markets and risk requirements post financial crisis, Credit Derivatives: Trading, Investing and Risk Management, Second Edition, covers the subject from a real world perspective, tackling issues such as liquidity, poor data, and credit spreads, to the latest innovations in portfolio products, hedging and risk management techniques.

The book concentrates on practical issues and develops an understanding of the products through applications and detailed analysis of the risks and alternative means of trading.

It provides:

• a description of the key products, applications, and an analysis of typical trades including basis trading, hedging, and credit structuring;
• analysis of the industry standard 'default and recovery' and Copula models including many examples, and a description of the models' shortcomings;
• tools and techniques for the management of a portfolio or book of credit risks including appropriate and inappropriate methods of correlation risk management;
• a thorough analysis of counterparty risk;
• an intuitive understanding of credit correlation in reality and in the Copula model.

The book is thoroughly updated to reflect the changes the industry has seen over the past 5 years, notably with an analysis of the lead up and causes of the credit crisis. It contains 50% new material, which includes copula valuation and hedging, portfolio optimisation, portfolio products and correlation risk management, pricing in illiquid environments, chapters on the evolution of credit management systems, the credit meltdown and new chapters on the implementation and testing of credit derivative models and systems.

The book is accompanied by a website which contains tools for credit derivatives valuation and risk management, illustrating the models used in the book and also providing a valuation toolkit.

Leseprobe

Table of Contents

Dedication

Title Page

Copyright Page

Preface to the First Edition

Preface to the Second Edition

Acknowledgements

Disclaimer

INSTRUCTIONS FOR THE ‘NDB PRICER’ AND THE ‘CDO PRICER’

APPLICATION RESTRICTIONS

Table of Spreadsheet Examples and Software

About the Author

Part I - Credit Background and Credit Derivatives

Chapter 1 - Credit Debt and Other Traditional Credit Instruments

1.1 BONDS AND LOANS; LIBOR RATES AND SWAPS; ‘REPO’ AND GENERAL COLLATERAL RATES

1.2 CREDIT DEBT VERSUS ‘RISK-FREE’ DEBT

1.3 ISSUE DOCUMENTS, SENIORITY AND THE RECOVERY PROCESS

1.4 VALUATION, YIELD AND SPREAD

1.5 BUYING RISK

1.6 MARKING TO MARKET, MARKING TO MODEL AND RESERVES

1.7 THE ‘CREDIT CRUNCH’ AND CORRELATION

1.8 PARTIES INVOLVED IN THE CREDIT MARKETS AND KEY TERMINOLOGY

Chapter 2 - Default and Recovery Data; Transition Matrices; Historical Pricing

2.1 RECOVERY: ULTIMATE AND MARKET-VALUE-BASED RECOVERY

2.2 DEFAULT RATES: RATING AND OTHER FACTORS

2.3 TRANSITION MATRICES

2.4 ‘MEASURES’ AND TRANSITION MATRIX-BASED PRICING

2.5 SPREAD JUMPS AND SPREAD VOLATILITY DERIVED FROM TRANSITION MATRICES

2.6 ADJUSTING TRANSITION MATRICES

Chapter 3 - Asset Swaps and Asset Swap Spread; z-Spread

3.1 ‘PAR-PAR’ ASSET SWAP CONTRACTS

3.2 ASSET SWAP SPREAD

3.3 MATURITY AND z-SPREAD

3.4 CALLABLE ASSET SWAPS; ‘PERFECT’ ASSET SWAPS

3.5 A BOND SPREAD MODEL

Chapter 4 - Liquidity, the Credit Pyramid and Market Data

4.1 BOND LIQUIDITY

4.2 THE CREDIT PYRAMID

4.3 ENGINEERED AND SURVEY DATA

4.4 SPREAD AND RATING

Chapter 5 - Traditional Counterparty Risk Management

5.1 VETTING

5.2 COLLATERALISATION AND NETTING

5.3 ADDITIONAL COUNTERPARTY REQUIREMENTS FOR CREDIT DERIVATIVE COUNTERPARTIES

5.4 INTERNAL CAPITAL CHARGE

Chapter 6 - Credit Portfolios and Portfolio Risk

6.1 VaR AND COUNTERPARTYVaR

6.2 DISTRIBUTION OF FORWARD VALUES OF A CREDIT BOND

6.3 CORRELATION AND THE MULTI-FACTOR NORMAL (GAUSSIAN) DISTRIBUTION

6.4 CORRELATION AND THE CORRELATION MATRIX

Chapter 7 - Introduction to Credit Derivatives

7.1 PRODUCTS AND USERS

7.2 MARKET PARTICIPANTS AND MARKET GROWTH

Part II - Credit Default Swaps and other Single Name Products

Chapter 8 - Credit Default Swaps; Product Description and Simple Applications

8.1 CDS PRODUCT DEFINITION

8.2 DOCUMENTATION

8.3 CREDIT TRIGGERS FOR CREDIT DERIVATIVES

8.4 CDS APPLICATIONS AND ELEMENTARY STRATEGIES

8.5 COUNTERPARTY RISK: PFE FOR CDS

8.6 CDS TRADING DESK

8.7 CDS CONTRACT AND CONVENTION CHANGES 2009

Chapter 9 - Valuation and Risk: Basic Concepts and the Default and Recovery Model

9.1 THE FUNDAMENTAL CREDIT ARBITRAGE - REPO COST

9.2 DEFAULT AND RECOVERY MODEL; CLAIM AMOUNT

9.3 DETERMINISTIC DEFAULT RATE MODEL

9.4 STOCHASTIC DEFAULT RATE MODEL; HAZARD AND PSEUDO-HAZARD RATES

9.5 CALIBRATION TO MARKET DATA

9.6 CDS DATA/SOURCES

9.7 MODEL ERRORS AND TESTS

9.8 CDS RISK FACTORS; RESERVES AND MODEL RISK

Chapter 10 - CDS Deal Examples

10.1 A CDS HEDGED AGAINST ANOTHER CDS

10.2 INTRODUCTION TO BOND HEDGING

10.3 HEDGE AND CREDIT EVENT EXAMPLES

Chapter 11 - CDS/Bond Basis Trading

11.1 BOND VERSUS CDS: LIQUIDITY

11.2 BOND REPO COST

11.3 BOND SPREAD MEASUREMENT - z-SPREAD NOT ASSET SWAP SPREAD

11.4 BOND PRICE IMPACT

11.5 EMBEDDED OPTIONS IN BONDS AND LOANS

11.6 DELIVERY OPTION IN CDSs

11.7 PAYOFF OF PAR

11.8 TRIGGER EVENT DIFFERENCES

11.9 EMBEDDED REPO OPTION

11.10 PUTTING IT ALL TOGETHER

Chapter 12 - Forward CDS; Back-to-Back CDS, Mark to Market and CDS Unwind

12.1 FORWARD CDS

12.2 MARK-TO-MARKET AND BACK-TO-BACK CDS

12.3 UNWIND CALCULATION; OFF-MARKET TRADE VALUATION AND HEDGING

12.4 ‘DOUBLE-TRIGGER CDS’

Chapter 13 - Credit-Linked Notes

13.1 CLN SET-UP; COUNTERPARTY OR COLLATERAL RISK

13.2 EMBEDDED SWAPS AND OPTIONS

13.3 COSTS

13.4 APPLICATIONS

13.5 CLN PRICING

13.6 CAPITAL GUARANTEED NOTE

Chapter 14 - Digital or ‘Fixed Recovery’ CDS

14.1 PRODUCT DESCRIPTION

14.2 PRICING, HEDGING, VALUATION AND RISK CALCULATIONS

14.3 TRIGGER EVENT DIFFERENCES

Chapter 15 - Spread Options, Callable/Puttable Bonds, Callable Asset Swaps, ...

15.1 PRODUCT DEFINITIONS

15.2 MODEL ALTERNATIVES AND A STOCHASTIC DEFAULT RATE MODEL FOR SPREAD OPTION PRICING

15.3 SENSITIVITIES AND HEDGING

Chapter 16 - Total Return Swaps

16.1 PRODUCT DEFINITION AND EXAMPLES

16.2 APPLICATIONS

16.3 HEDGING AND VALUATION

Chapter 17 - Single Name Book Management

17.1 RISK AGGREGATION

17.2 CREDITVaR FOR CDSs

Chapter 18 - CDS and Simulation

18.1 THE POISSON MODEL AND DEFAULT TIMES

18.2 VALUATION BY MONTE CARLO SIMULATION

18.3 SENSITIVITY

Part III - Portfolio Products

Chapter 19 - Portfolio Product Types

19.1 NTH-TO-DEFAULT BASKETS

19.2 ‘SYNTHETIC’ CDOs

19.3 CASHFLOW CDOS

19.4 CREDIT SECURITISATIONS

19.5 RATING

19.6 ALTERNATIVE LEVERED CREDIT PORTFOLIO PRODUCTS

Chapter 20 - The Normal Copula and Correlation

20.1 DEFAULT TIME CORRELATION

20.2 NORMAL COPULA

20.3 CORRELATION

Chapter 21 - Correlation in Practice

21.1 TRANCHE CORRELATION

21.2 BASE CORRELATION

21.3 CORRELATED RECOVERIES

21.4 CORRELATION REGIME CHANGE AND OTHER MODELLING APPROACHES

Chapter 22 - Valuation and Hedging

22.1 VALUATION EXAMPLES

22.2 SENSITIVITY CALCULATION AND HEDGING

22.3 PRICING MORE COMPLEX STRUCTURES

22.4 MODEL ERRORS AND TESTS; ALTERNATIVE MODELS

Chapter 23 - Alternative Copulas

23.1 STUDENT’S t-DISTRIBUTION

23.2 COPULAS IN GENERAL

23.3 ARCHIMEDEAN COPULAS: CLAYTON, GUMBEL

23.5 MODEL RISK

Chapter 24 - Correlation Portfolio Management

24.1 STATIC AND DYNAMIC HEDGES

24.2 CORRELATION BOOK MANAGEMENT

24.3 CREDITVaR AND COUNTERPARTYVaR

Part IV - Default Swaps Including Counterparty Risk

Chapter 25 - ‘Single Name’ CDS

25.1 NON-CORRELATED COUNTERPARTY

25.2 100% CORRELATION

25.3 CORRELATED COUNTERPARTY: PRICING AND HEDGING

25.4 CHOICE OF COPULA

25.5 COLLATERALISED DEALS AND CDS BOOK MANAGEMENT

Chapter 26 - Counterparty CDSs

26.1 PRICING

26.2 COUNTERPARTY CDS (CCDS) BOOK MANAGEMENT

Part V - Systems Implementation and Testing

Chapter 27 - Mathematical Model and Systems Validation

27.1 TESTING PROCEDURES

27.2 IMPLEMENTATION AND DOCUMENTATION

Chapter 28 - System Implementation

28.1 ANATOMY OF A CDO

28.2 MANAGEMENT

28.3 VALUATION

28.4 IT CONSIDERATIONS

Part VI - The Credit Crisis

Chapter 29 - Cause and Effect: Credit Derivatives and the Crisis of 2007

29.1 THE CREDIT MARKETS PRE-CRISIS

29.2 THE EVENTS OF MID-2007

29.3 ISSUES TO BE ADDRESSED

29.4 MARKET CLEARING MECHANISMS

29.4.1 Central Credit Counterparty

29.4.2 Centralised Clearing and Systemic Risk

29.4.3 A Dedicated CCP for CDSs Alone

29.4.4 Conclusions

Appendix - Markit Credit and Loan Indices

References

Index

For other titles in the Wiley Finance series please see www.wiley.com/finance

This edition first published 2010

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ISBN 978-0-470-68644-7

Typeset in 10/12pt Times by Aptara Inc., New Delhi, India

To my parents, my partner and my children

Preface to the First Edition

This book arose out of several courses and training sessions given to finance professionals. Those courses, and this book, are aimed at traders in the credit and other derivatives, quants wishing to develop some product knowledge in this area, risk managers including the corporate treasurer, investors and others requiring both product knowledge and a grasp of valuation and risk. The aim is to develop an understanding of the various credit derivatives products. In order to achieve this it is necessary for the reader to have a certain amount of financial background - and a certain level of maturity when it comes to understanding structured products and the management of a portfolio of risks. I have tried to cover key credit background in Part I of the book. Readers with experience in this area can skip through most of the topics covered here - though generally the role of repo, and the difference between asset swap spreads and z-spreads, do not appear to be well understood, and the reader is advised to study these sections at least. Other areas that may be unfamiliar are the calibration of transition matrices to market data (spreads and volatilities), and the generation of correlated spread moves using the Normal ‘Copula’.

The most common credit derivative product is the credit default swap (CDS), and this is covered along with other ‘single name’ credit derivatives in Part II. A relatively simple ‘deterministic’ pricing model is described, together with a stochastic model in section 9.4. A model is of little use without data, and various data-related issues are discussed, together with detailed analysis of model shortcomings, potential improvements, and the setting of reserves. Chapters 10 to 12 analyse a variety of trades: credit curve trades, cross-currency hedges, spread and event risk, valuation and unwind of default swap positions, and a detailed breakdown of the basis between CDS premiums and bond spreads. Non-vanilla default swaps, options, and total return swaps are also covered in Part II. Chapter 17 draws together many of the aspects of valuation, risk calculation and hedging with a discussion of book management applied to the single name credit risk book. Finally we end this section with a look at pricing single name default swaps via default time simulation. This introduces some of the techniques used in analysing portfolio products in the relatively simple context of default swaps.

Part III looks at nth-to-default products and CDOs. The latter form a very extensive topic in their own right: I have given a descriptive analysis of many of the different types of CDO structure, and product applications though, when it comes to using the pricing model, we concentrate on synthetic static CDOs (including standard CDO products such as iTraxx). Nevertheless the model described is appropriate to both the standardised products and the more complex waterfall structures seen in some cashflow CDOs. Examples are given to develop an intuitive understanding of the correlated ‘default time’ modelling approach, and of default time correlation. This part quotes valuation results extensively to illustrate important points. In addition, software provided allows the user to reproduce quoted results and to experiment with different portfolios, structures and situations to learn more about both the product and the consequences of the modelling approach. Product risks and model deficiencies are discussed in detail as is the question of reserves. A section is devoted to the correlation matrix. The preferred interpretation of the model is as an ‘interpolation rule’, driven by market prices - giving rise to ‘implied correlation’ - and then applied to non-traded products. Implied (‘compound’) correlation and ‘base’ correlations are covered here. Alternative Copulas are discussed and applied. This part concludes with a discussion of the management of a portfolio of ‘correlation’ products and high- and low-risk hedging techniques.

We revisit the topic of ‘single’ name credit default swaps in Part IV where we take account of counterparty risk. In addition, the topic of protection on counterparties is briefly covered.

In writing the book I have been conscious that CDO products, and methods of analysis, have been developing rapidly. Although the book is illustrated using some of these products, their precise form has changed in the past and will continue to change in the future. I have not been able to cover all aspects of credit derivatives - neither equity default swaps nor constant maturity default swaps have been discussed, and some model approaches have not been covered (for example ‘claim of market value’ models, and ‘blended correlation’). I have chosen to concentrate on core products, general principles and features that are likely to continue to apply in the future. Nevertheless the book reflects the situation at the time of writing (late 2004): products, methodologies and interpretations will evolve.

Throughout the book I have tried to illustrate ideas and principles using real world examples. In addition there are a variety of thought experiments and exercises to encourage the reader to test theories against real life, and to develop a deeper understanding of the various concepts through their application. A prerequisite to effective application and management of credit derivative positions is an understanding of the products, and an ability to discuss those products in a meaningful way. This book is intended to help in this process and is therefore not aimed solely at the trader, or the risk manager, or the investor, but is intended to be read by all. Consequently I have tried to separate the mathematics from product discussion and application. Of course a good understanding of one cannot be achieved without some understanding of the other. To this end I have used as simple a version of the theory as possible, yet one which is capable of capturing the key aspects of the problem. Generally simplifications amount to ignoring the details of premium payment (frequency and day-count conventions) and often using a constant hazard rate curve. I have also given many calculated results to illustrate points arising out of the theory, and provided software to allow the reader to examine the implications of the model.

Preface to the Second Edition

In writing the second edition I have tried to keep to the objectives and style of the first edition while updating the content and, additionally, broadening the standpoint from which the book is written. Specifically, the examples have been updated and enhanced, and Parts I and II have been amended where relevant to bring them up to date. Additional text has been included to cover syndicated loans, LCDS, up-front premiums, and other changes to the CDS market, and further background has been included in Part I on terminology, investment perspectives and the ‘credit crunch’. Part III has been restructured and largely rewritten, including material on LCDOs, cashflow CDOs and structurings as well as significantly enhanced descriptions of the standard and bespoke CDO products and valuation and risk methodology. New sections (Part V and VI) have been added covering aspects of model testing and implementation, and the credit crisis respectively.

Acknowledgements

In preparing this second edition I have also asked for input and different perspectives from several individuals, and their contributions have been included. In particular, my thanks go to my colleagues Robert Reoch and Robert Baker (Reoch Credit Partners LLP), Darren Smith (WestLB) and Onur Cetin (Calypso) for considerable contributions included as separate chapters or sections in the book and attributed to them. I would also like to thank Markit Group Limited for their substantial assistance, and, again, Bloomberg, Moody’s and ISDA. Contributed material and helpful comments in the first edition continue to be reflected in this edition and again I thank JP Morgan, Mizuho International, Morgan Stanley, Richard Flavell, Phil Hunt, Lee McGinty (JP Morgan) and Chris Finger (RiskMetrics Group). I would also like to thank Dr Jonathan Staples (fund manager, Charles Taylor Investment Management Company Limited) and Rafik Mrabet (Structured Credit Trading, Mizuho International) for their significant help in collecting data and research material and in reviewing parts of the book.

Remaining errors are my responsibility and the views expressed are entirely my own.

Disclaimer

1. Use of company names in examples and illustrations in no way indicates the actual involvement of that company in the example deal or any similar deal. Use of such names is purely for a sense of realism.

2. Software is provided to illustrate certain points in the text only. It is in no way intended for commercial use and, indeed, commercial use of the software, examples, or underlying code in any form is strictly forbidden.

Most examples are in Excel (2002) spreadsheets, some requiring the dll.

A few applications are in MathCad1 sheets (and run under MathCad 11). Sometimes the code and the application are in separate sheets. MathCad users will need to copy these sheets to a single directory and re-establish the reference to the code sheet in the application sheet. For non-MathCad users the sheets are also copied in rtf format so that the code can be read and converted to their preferred 4GL.

INSTRUCTIONS FOR THE ‘NDB PRICER’ AND THE ‘CDO PRICER’

• These spreadsheets require the ‘Xlcall32.lib’ and the ‘bookdemo.xll’ to be installed in the directory in which the spreadsheets are run. The user will also need to create the directory ‘“C://CDO intermediate output/’.

• All other Excel applications should be closed down and calculation should be set to manual (with no ‘recalculate on save’). Do not subsequently open up any other spreadsheets unless they are also set to manual calculation - doing so can cause Excel to recalculate the CDO sheet and this will lead to a crash.

• Begin by opening the xll and enabling macros.

• Open the spreadsheet (either CDS or NDB) enabling macros, and go to the ‘CDS calibration’ sheet. Enter data in the yellow areas only - Name, Currency, Seniority, Notional Size, rating, CDS premium, and recovery rate. Hit shift-F9 to recalculate the sheet. Implied default rates and survival probabilities are shown in AN-BF.

NDB Pricer

CDO Pricer

APPLICATION RESTRICTIONS

The software is intended only to illustrate points made in the text. To restrict its possible commercial application the following limitations are imposed.

1. Only 0 or 4% flat interest rate curves

2. Only flat hazard rate curves

3. Only continuous CDS and portfolio product premiums

4. No standard maturity or coupon dates have been catered for

5. The basket pricer will only handle two or three name baskets, and the CDO pricer will only handle 100 reference names

6. The following are disabled

a. Hedge calculations (except for the default event hedge of 22.2)

b. VaR and tranche option calculations

c. Semi-closed form pricing - crude Monte Carlo simulation only is used.

Table of Spreadsheet Examples and Software

About the Author

Geoff Chaplin studied mathematics at Cambridge (MA 1972) and Oxford (MSc 1973, DPhil 1975) and trained as an actuary (FFA 1978) while working in a life insurance company. He moved to the City in 1980 and has worked for major banks (including HSBC, Nomura International, and ABN AMRO) as well as consulting to hedge funds, corporate treasurers, and institutional investment funds. He has been involved in the credit derivatives market since 1996 and has both traded portfolio products and developed risk management systems for these products. In addition to consulting and training work for the major financial institutions, Geoff has maintained strong academic interests and was a visiting (emeritus) professor at the University of Waterloo (Canada) from 1987 until 1999. He has also published many articles (in Risk, the Journal of the Institute and Faculty of Actuaries, and others), speaks regularly at conferences on credit derivatives, and recently co-authored Life Settlements and Longevity Structures in the Wiley Finance series. Geoff continues to be actively involved in the credit derivatives market as a consultant to a wide range of institutions, as an expert witness in dispute resolution, and is a Partner in Reoch Credit.

[email protected]

Part I

Credit Background and Credit Derivatives

1

Credit Debt and Other Traditional Credit Instruments

The reader is assumed to be familiar with government bonds, with the LIBOR market and to have had some familiarity with traditional credit instruments (bonds and loans). The following sections briefly review these areas and develop some techniques for the analysis of credit risk - particularly in relation to credit portfolios.

1.1 BONDS AND LOANS; LIBOR RATES AND SWAPS; ‘REPO’ AND GENERAL COLLATERAL RATES

1.1.1 Bonds and Loans

A bond (Bloomberg definition) is a certificate of debt issued by a government or corporation with the promise to pay back the principal amount as well as interest by a specified future date. A loan is a broader concept than a bond - it is a sum of money lent at interest. In practice bonds are usually traded instruments (at least in principle) whereas loans are often private agreements between two or more parties (usually a corporate entity and a bank). There has been a growing market recently in syndicated loans as opposed to bilateral loans. Traditionally loans have been bilateral agreements between the borrower (typically a corporate entity) and a lender (typically a bank but often a private individual) the terms of which can be very varied including various options and restrictions on the borrower’s use of the money or financial performance. A syndicated loan is offered by a group of lenders (a ‘syndicate’) who work together to provide funds for a single borrower and share the risk - unlike bilateral loans. Typically there is a lead bank or underwriter of the loan, known as the ‘arranger’, ‘agent’, or ‘lead lender’. The term ‘leveraged loan’ simply means a high-yield loan (high-risk borrower) and apart from the higher spread on such loans there are no other new features. There has also been growing market in secondary trading of bilateral loans - where the cashflows under the loan agreement are assigned (sold) to a third party. For this to be possible the loan agreement has to allow this transfer to take place; such loans are assignable loans. Many loans - although this is becoming less common than historically - are non-assignable (the ownership of the cashflows cannot be transferred). Some loans are non-assignable except in default. (We see later that these variations have implications for the credit derivatives market - they potentially restrict the deliverability of some debt into default swap contracts.) The syndicated loan market has sought standardisation of loan terms and restrictions - this is covered in more detail later, but syndicated loans are usually immediately callable by the borrower, contain a wide range of restrictions on the borrowers financial performance (covenants) and are often but not always issued through CLO structures to investors.

When we speak of a credit bond (or loan) we are explicitly recognising the risk that the payments promised by the borrower may not be received by the lender - an event we refer to as default (we discuss this further below).

1.1.2 BBA LIBOR and Swaps

According to the British Bankers’ Association, ‘LIBOR stands for the London Interbank Offered Rate and is the rate of interest at which banks borrow funds from other banks, in marketable size, in the London interbank market’. BBA LIBOR rates are quoted for a number of currencies and terms up to one year, and are derived from rates contributed by at least eight banks active in the London market. (See http://www.bba.org.uk for more information.) LIBOR rates clearly refer to risky transactions - the lending of capital by one bank to another - albeit of low risk because of the short-term nature of the deal and also the high quality of the banks contributing to the survey.

An interest rate swap contract (Bloomberg) is ‘a contract in which two parties agree to exchange periodic interest payments, especially when one payment is at a fixed rate and the other varies according to the performance of a reference rate, such as the prime rate’. Typically interest rate swaps are for periods of more than a year, and usually the reference rate is LIBOR. The swap itself is a risky deal although on day one the value of the fixed flows equals the value of the floating payments, so the risk is initially zero. Risk emerges as interest rate levels change, affecting the value of the floating and fixed payments differently. (Expected risk at a forward date will not be zero if the interest rate curve is not flat though will typically be small in relation to the value of one of the legs of the swap.) Swaps are low risk, although swap rates themselves are risky rates largely because the reference floating rate itself (LIBOR) is a risky rate.

1.1.3 Collateralised Lending and Repo

Banks often use listed securities as collateral (assets pledged as security) against cash they borrow to meet other needs. Such lending (of securities) and borrowing (of cash) is referred to as collateralised borrowing and the rate of interest applicable to generic collateral is the general collateral rate (GC). Typically such collateralised lending agreements are for short terms (they may be on a rolling overnight basis) and the GC rate itself is usually a few (2-7) basis points below LIBOR rates.

The reader should note several things at this point.

1. Any structured product created by a bank can in principle be securitised and used as collateral to obtain the cash required to finance the transaction. General collateral rates are therefore key in determining the cost of any structured deal (including credit derivatives).

2. GC rates are not readily available - they are known by the repo trading desk but not made publicly available. Some US repo rates are published on Reuters. The European Banking Federation sponsors the publication of EUREPO, a set of GC repo rates relating to European government bonds. However, LIBOR rates are very easy to obtain and are also close to GC.

3. Investment banks typically mark their positions to market by discounting off the LIBOR and swap curve (because LIBOR is the financing cost, and for arbitrage reasons - see, for example, Chapter 9).

LIBOR (and swap) rates are therefore key to the development of the pricing of credit derivatives.

We shall look at the deal underlying collateralised lending (a ‘repo’) in detail since an understanding of this issue is required later. A ‘repo’ or repurchase agreement is a contract giving the seller of an asset the right, and the obligation, to buy it back at a predetermined price on a predetermined date. The borrower of cash (‘lender’ of the asset) sells the asset to a counterparty under the repo contract and receives cash (equal to the market value of the asset2). Prior to termination of the deal any cashflows generated by the asset are passed on to the original owner of the asset, and the borrower of cash pays interest at a rate specific to that asset - the repo rate. On the termination of the deal the borrower of cash repays the loan and receives the asset back. In the event of default of the asset, the end date of the repo would be accelerated - the asset passes back to the original owner and the debt is repaid. (See Figures 1.1-Figures 1.3.)

Figure 1.1 Repo deal - initial, capital and asset flows.

We can see that, although legal ownership of the asset passes from the original owner, ‘economic ownership’ remains with the original owner (i.e. the original owner of the asset receives all the cashflows from the asset in any eventuality as if he owned that asset).

Typically repo deals are short term - from overnight to a few months.

Figure 1.2 Repo deal - ongoing cashflows.

GC rates are tiered according to the class of asset. There are different GC rates for Government bonds depending on the country of issue; GC rates for corporate bonds are determined by the rating. However, a particular asset - for example, a certain bond - may go ‘special’ on repo. An institution may need to borrow a particular asset (for example, it may have sold the asset short) and may be prepared to ‘pay’ in order to receive that asset. Under the repo deal the institution pledges (lends) cash against the asset it borrows and, if the asset were not special, would receive interest at the GC rate. But if the asset goes special, the repo rate for that asset falls below the GC rate - and may fall to zero or even become negative. (‘Special’ repo rate have an impact on the basis between bonds and default swaps (see Part II).) The borrower of the bond lends cash and therefore receives a sub-LIBOR return on that cash.

Figure 1.3 Repo deal - final capital and asset flows.

1.1.4 Repo as a Credit Derivative

A repo is not traditionally regarded as a credit derivative even if the collateral is a credit risky asset, although it is almost identical to a ‘Total Return Swap’ (see Part II) which is usually classed as a credit derivative. Furthermore, there is a significant and complex embedded credit risk if there is a correlation between the borrower of cash and the reference entity of the collateral (see Parts III and IV). A default of the borrower may cause a sudden drop in the value of the collateral in this case, so the lender of cash is taking on a non-trivial credit derivative risk. A similar risk exists in default swap contracts where the counterparty and the reference entity are correlated: indeed we can view a repo as an outright purchase of the collateral with a forward sale back to the original owner, plus a purchase of default protection on the reference entity from the borrower of cash. The question of the embedded counterparty credit risk is analysed in detail in Parts III and IV.

1.2 CREDIT DEBT VERSUS ‘RISK-FREE’ DEBT

We often talk of a risk-free bond - one where there is no risk of default - and usually identify this with government debt of certain countries issued in their own currency.3 The yield curve associated with government debt is then often called the risk-free curve. This identification is, in the first instance, only approximate - any government can default on its own debt. The risk may be remote but it still exists in principle. The second problem with the identification of risk-free rates with the government curve is that trading in government debt is only a part of debt trading, and special factors - such as a heavy issuance program, a buy-back program, or regulatory requirements on banks (for example) to hold government debt - can distort the price of government debt and separate the government yield curve from ‘underlying’ risk-free rates.

We shall see later (Part II) that, whatever ‘risk-free’ rates may be, they are irrelevant to the pricing of credit derivatives.

1.3 ISSUE DOCUMENTS, SENIORITY AND THE RECOVERY PROCESS

1.3.1 Issue Documents and Default

A bond is subject to a legally binding document (the ‘issue document’) which is usually substantial (100 pages or so) and describes the parties involved in the issue of the bond, the borrower, legal jurisdiction, etc., together with payment information such as the cashflow dates and amounts. Loans are subject to corresponding loan documentation.

One item defined in the documentation is the ‘grace period’ or the number of ‘days of grace’ for the bond cashflows. If a scheduled payment is not made on the due date this does not constitute default - the borrower is allowed a period of time in which to make the payment. This is intended to cover administrative errors and omissions, and other events which might make payment impossible in the very short term.

The documentation also specifically covers what constitutes default, and what recourse the lender has in the event of a default.

Typically, default is defined as the failure to pay a significant promised cashflow. Generally it is not just failure to pay a cashflow on that bond which causes default; failure of the borrowing entity to pay any significant loan cashflow usually triggers default on all bonds and loans issued by that entity (i.e. a cross default clause usually applies).

1.3.2 Claim Amount

In the event of default the lender usually has the right to claim a sum of money from the borrower, and this sum of money is usually par plus accrued coupon up to the date of default. The amount the lender can claim from the borrower is referred to as the claim amount or claim value, and in the above example the claim amount is ‘par plus accrued’ (usually we just say ‘par’ - the accrued is implicit).

The claim amount is a key element in the valuation of credit derivatives (including bonds) so we shall introduce some notation and capture the above in a formula. Define C(t) to be the claim amount at time t, then the above paragraph tells us that

(1.1)

where A(t) is the accrued on the bond at time t. (We shall work in par amounts of 1 rather than 100 or 1,000.)

Variations in the claim amount occur. For example, deep discount debt may have a claim amount which rises from the issue price to par at maturity, according to some formula or printed schedule. This is not necessarily the case - for example, convertible bonds usually have a low coupon but have a claim amount of par.

Some issue documents (usually only a few loan documents) say that the borrower can claim the promised cashflows in the event of default. Thus the future cashflows are not replaced by a single immediately payable sum. However, at the default date we can value these cashflows to get a financially equivalent amount. We refer to such a claim amount as ‘treasury’, meaning that it is financially equivalent to the value of a series of bond cashflows.

An alternative claim model is sometimes useful for risk calculations (particularly for sovereign debt - see below) so we show the formula for this case and make a few further comments. It is notationally easier to work in continuous time4 but in this case we shall show the formula both in continuous and discrete time. Of course, the formula for continuous time can be made to reproduce the formula for discrete time by making the continuous cashflows have a certain function form (a sum of ‘Dirac delta functions’5).

First, the continuous time version, let c(t) be the rate of cashflow promised under the bond at time t for t < T the bond maturity, and let d(t) be the discount factor for a payment at time t.6 Then the claim amount is given by

(1.2a)

Second, in the discrete time case, let the cashflow at time ti be ci, where i is a counter over the payment dates. For a bond, these cashflows are just the coupon payments at an ordinary coupon date, and the coupon payment plus maturity amount at the maturity date. Then we can express the claim amount as

(1.2b)

where n is the number of cashflows. In the typical case - claim amount of par (plus accrued) - on the default event, all debt becomes an immediately due cash amount. Thus a bond with a one year outstanding life and a bond with a 30-year outstanding life will both have exactly the same market value after default (assuming zero accrued for simplicity) - both have the same claim value (par) which is due immediately.

1.3.3 The Recovery Process and Recovery Amount

The issue document also covers the question of how a claim on one bond relates to claims on other debt. This is a complicated issue involving the law generally - usually requiring that back taxes are settled before anything else, and that employees get back pay before banks get repayment of debt etc. Settlement of claims on debt will be described in the issue document which will usually (in the case of corporate and bank borrowers) refer to the seniority of debt, and the order in which different seniorities are to be recompensed. Some debt may be secured on a specific asset - for example a property. In the event of default then settlement on this specific asset is related to the amount which the specific asset realises on sale. Typically debt is secured on residual assets of the firm generally. Terminology varies but debt commonly found in the market and in bank portfolios generally falls into one of three further levels of seniority: loans, senior secured, and subordinated (or junior) debt. It is generally the case that ‘loans’ differ from bonds in that they are usually more senior in addition to other points discussed in section 1.1.1.

Once a corporate entity defaults the administrators of the company seek to realise maximum value from the assets of the company. When the value of these assets is realised then cash is used in the prescribed order until it is used up. If cash is available after paying (in full) the most senior creditors (such as the tax man, accountant’s fees, back pay etc., and debt secured on specific assets) then cash is applied to claims on loans being the most senior debt. If this can be met in full, then the remaining cash is applied to senior unsecured bonds and, if these can be repaid in full, it moves on to junior debt. If, at any point, cash is insufficient to cover the claims of that seniority in full, then all claims receive the same proportion (the recovery rate) of the claim amount. The cash is then used up. Thus in a corporate default where the cash is insufficient to cover all the claims, one seniority level will receive a partial recovery; more senior levels of debt receive 100% of the claim amount, and more junior levels receive zero. Any deviation from the legal framework and the legally binding issue documents can be challenged in the courts by the creditors.

We state here that the condition (1.4b) only applies when we are looking at the actual amounts finally recovered (‘ultimate recovery’ - see Chapter 2). Recovery as applicable to credit derivative products is a different concept from the above (being the one-month post-default bond price) and conditions (1.4b) no longer apply (although (1.4a) remains: see Chapter 2 for further details).

The amount 1 − Rs is often referred to as the ‘loss given default’ (LGD).

We shall address the question of how we estimate R prior to the default event in the following section. At present formula (1.3) and following conditions apply to a specific defaulted entity - the recovery numbers for that entity are not the same as the recovery numbers for another entity.

1.3.4 Sovereign versus Corporate Debt

Issue documents for sovereign debt (e.g. Argentinean debt in USD) are very similar to those of corporates. Typically the claim amount is also par. Usually there is only one level of seniority for sovereigns. The major difference between corporate and risky government debt is in the recovery process itself. Firstly there is no ‘wind up’ process via the courts as in the case of corporates. A ‘defaulting’ sovereign typically restructures its debt and investors lose value compared with their promised cashflows. The government may offer terms which are very different from the recovery levels one might expect from the issue document - typically long dated debt recovers a smaller proportional of notional than short-dated debt. However the lenders have no court they can go to in order to seek a strict implementation of the process described in the issue document. In practice this means that recovery for sovereign debt is not at the same rate for all bonds7 - instead it is typically high for short-dated debt and low for long-dated debt.

1.4 VALUATION, YIELD AND SPREAD

Bonds are bought and sold based on a price. Often the price quoted is a clean price, and the consideration paid also takes into account the accrued interest. Given the market price of the debt, and the cashflow schedule, we can calculate the yield (internal rate of return), and we can also calculate the ‘spread’. There are various ways of measuring spread (see Chapter 5 for further discussion) - for the moment we shall define spread as the difference between the bond’s yield and the interpolated yield off the LIBOR/swap curve (interpolated to the maturity date of the bond).

High-grade debt may trade close to swap rates - even sub-LIBOR for very high grade borrowers which are perceived to be less risky than the banks which define the LIBOR rate. Typically investment grade debt (debt rated BBB or better by the rating agencies) trades up to 300 bp over the swap curve depending on the name and varying with time and the economic cycle, sentiment, etc. Sub-investment grade debt typically trades wider - to 10 000 bp or 100 000 bp above the swap curve.

For investment grade names the market will usually talk in terms of spread rather than price. The reason for this is that the price of a 5-year Unilever bond (for example) will change moment by moment as interest rate futures tick up or down. However, the spread on the bond typically changes much more slowly and may even be static for days or even weeks.

For sub-investment grade debt the market usually talks in price terms. Where spreads are high (and bond prices may be 50% below those of low-risk debt) the main determinant of price is the perceived default risk, not interest rate levels.

1.5 BUYING RISK

A buyer of a credit bond is taking on the default risk of the underlying entity - the investor is not only buying an asset but also buying risk. Imagine an insurance policy which insures the par value of the bond in the event of default of the underlying name. The buyer of the insurance policy is the buyer of protection, and the writer of the policy is the seller of protection. We can also talk in terms of risk - the seller of protection is taking on risk, similar to the buyer of the bond itself, while the buyer of protection is also the seller of risk.

In the credit derivative market both sets of terminology are used - buyer or seller of protection or of risk. The word ‘buyer’ on its own conveys nothing - the buyer of protection is the seller of risk and vice versa. It is essential to be clear whether one is talking about risk or protection. Often ‘selling’ means selling protection when talking about single-name default swaps, but selling a tranche of a CDO usually means buying protection. Investors (asset managers, hedge funds, pension funds, etc) often talk in terms of buying and selling risk to reflect what is in effect happening if they invest in a corporate bond. Banks may use both terminologies depending on the area within the bank - traders often talk in terms of protection whereas structurers will talk in terms of risk.

In this book we shall generally talk in term of buying or selling protection when we talk about credit derivatives.

1.6 MARKING TO MARKET, MARKING TO MODEL AND RESERVES

When it is required to value a deal - whether for the purpose of calculation profit to date, for accounting, for regulatory or other reasons - the best approach in principle is to obtain a bid for the asset held. This is easy for liquid bonds for example, and is called ‘marking to market’.

For many other assets - such as credit derivatives, structured products and many option contracts - this is generally not practical. For example, consider a portfolio of equity call options of various maturities and strikes. Typically some maturities and strikes on each name trade sufficiently frequently that market prices for these maturities and strikes are easily available. We can obtain these prices and interpolate for other maturities and strikes on the same underlying asset. Usually this interpolation uses a pricing model (such as the Black-Scholes model) and an intermediate variable (volatility) is obtained. Interpolation on this variable is performed (perhaps involving a further model such as a volatility smile model) and the interpolated variable put back into the model in order to get an estimated market price for the asset. This is referred to as ‘marking to model’.

Reserves

Reserves are set against the value for products marked to model to give rise to a ‘conservative’ valuation of the portfolio in the institutions accounts. Even when products are marked to market they may be marked to mid: in this case a ‘bid-offer’ reserve will usually be held against the value to reflect the realisation value achievable on an asset.

Example 1 Suppose we mark to mid (and mid prices are easily available). Suppose we are long a unit of Asset1, mark it to mid-price P1, and S1 is the estimate of (half) the offer-bid difference. Then the value in the books would appear as P1- S1.

Exercise 1 If we sell the Asset1 to a market maker, how many trades does the market maker do? The answer is (at least) two: the trade with us and a hedging trade or trades with another party. A key feature of the market maker’s role is to minimise risk and lock in profits arising from bid-offer spreads rather than take views on the market direction. A market maker generally does not run unhedged positions - this role is left to other traders (‘proprietary’ or ‘prop traders’) who generally do not deal directly with investors.

Exercise 2 Suppose we are long Asset1 above and short Asset2, which is a very good hedge for all the risks in Asset1. Should we mark to P1 - S1 - P2 + S2?

Generally if we have a hedged ‘trade’ (made up of several ‘deals’ - Asset1 and Asset2 in the above) the costs and the risks to the market maker of taking on the position are less, so the bid-offer spread on the trade will be tighter than on a single unhedged asset. The trade in Exercise 2 would generally be marked to a better overall price than the less-well-hedged asset. Similar arguments apply to other reserves mentioned below and later in the book.

If deals are marked to model there are uncertainties involved in arriving at the mid price. In the equity option example above, uncertainties arise from

i. The interpolation routine to interpolate for the maturity and the strike of the actual option held

ii. The uncertainty in the validity and accuracy of the model being used (other traders may use a different model). [This is not really an issue for the equity option example but might apply if we were pricing exotic options.]

Marking to model generally gives rise to additional reserves, which will depend on the product and model being used, and there may also be a general ‘model reserve’. We shall discuss these in the context of credit derivatives products in Parts II and III.

Traders’ P&L

Reserves may also apply to the calculation of traders’ P&L - but not necessarily the same figures. The institution may take a cautious view of its books for a variety of reasons. On the other hand, reserves may be set against traders’ P&L in order to avoid a situation where a trader can take a large mark-to-market profit (and bonus) on Day 1 then, subsequently, the deal turns out to be worth far less than anticipated.

Reserves have little impact on a portfolio with high turnover: a deal done on Day 1 with high reserves, and take off on Day 2, will release the reserves and the actual profit can be calculated on the basis of the buy and sell prices. Typically, a high turnover portfolio usually faces low reserves, and the case where reserves tend to be high is a buy-and-hold book. Faced with a certain level of reserves the trader may take the view that, if he puts a deal on, the level of reserves against that position may eat too far into the P&L accumulated so far. The deal will not be done.

Another problem in setting reserves is that some sort of approximation is made. This may imply that the reserves are low on one version of the trade and high on another, with the result that the trader deals only on one type of trade (where the reserves are too low).

Institutions take a variety of views regarding traders’ P&L. At one extreme the traders’ and the institution’s reserves are the same; at the other extreme the trader may face an estimated bid-offer reserve only, and the institution takes the risk on other valuation estimates (and on the trader).

1.7 THE ‘CREDIT CRUNCH’ AND CORRELATION

A key element of the risk management process within a bank or an asset manager is understanding the risk on a portfolio of assets. In particular, how does asset class A (e.g. corporate bonds) perform when asset class B (e.g. equities) is falling? The answer to the question is often captured in the concept of asset correlation, but there is no single correlation that correctly captures the relationship and risk. During ‘normal’ times the correlation between asset classes may be low (say 20%) but during abnormal times (generally times of crisis of some form) the correlation rises sharply. Correlation itself is a variable and can indeed vary over a wide range. This aspect of correlation is generally not well captured in financial models and the problem is often tackled by looking at the tails of a value distribution (e.g. in VaR analysis). We shall see later that correlation is a key driver to the value of tranches of risk in CDO structures, and a large rise in correlation can have a devastating effect on the value of what in ‘normal’ times would be regarded as high-quality low-risk investments.

The credit crunch of 2008 was not a unique event. A credit crunch occurs when the banking system itself is compromised for some reason, interbank rates rise sharply relative to government rates and the banking system largely fails to function - lending to corporate borrowers is sharply reduced in a sudden change of risk appetite by the banks. This withdrawal of liquidity is rapidly felt throughout the financial system - for example, wealthy individuals often having to use their own financial resources to support companies in which they are closely involved rather than investing in peripheral activities - with the consequence that available cash for investment is sharply reduced. In such circumstances all asset prices fall as demand collapses - in other words, previously uncorrelated assets suddenly become highly correlated. Within the credit markets the crunch results in sharply higher spreads, and sharply higher actual default rates as bank lending to corporates is withdrawn, and has a dramatic impact on the pricing of structured credit as we shall see later.

CDO structures played a role in the credit crunch, the valuation aspects of which will become clear later in the book. It is important for the reader to realise the wide application of the term ‘CDO’ and the range of assets covered by the term. For example, a CDO referencing corporate credits may allow investors easy access to the corporate credit market. The impact of the new source of capital can drive the reward for risk (corporate credit spread) down but will have limited impact on the corporate’s appetite for new borrowing. On the other hand, a CDO of mortgages can allow a bank to reduce its book of loans to individuals thus giving it scope to make additional lending, and the borrowers (individuals) are much more likely to take on additional loans, escalating the amount of debt in existence. This is a very real risk and has been one of the contributors to the size of the credit crunch starting in 2008.

1.8 PARTIES INVOLVED IN THE CREDIT MARKETS AND KEY TERMINOLOGY

Borrowers: Corporate and sovereign entities (through loans and bonds) and individuals (through mortgages, consumer loans, lease contracts, credit card debt).

Lenders: Generally the same institutions as above - banks, corporate entities, sovereigns, investment institutions and high-net-worth individuals.

Traders and hedging: Individuals within various institutions. Traders may be ‘front book’ traders whose role is to make a market (set bid and offer prices at which he or she is prepared to deal) and run a very limited risk generally offsetting deals done with a customer with other transactions (‘hedges’) done with other customers or the ‘market’. ‘Prop[rietary]’ traders are taking a view of the market [direction] using their institution’s funds. Such trades may be hedged but the meaning of the term ‘hedge’ may be very loose [for example, the trader may take the view that equities currently form a very good hedge for a bond position because of unusual circumstances at the time] but generally prop trades have a much higher risk profile than hedged trades.

Investors: Asset managers, pension and insurance funds, high-net-worth individuals. The investor is typically seeking risk reduction through diversification across different assets, asset classes, maturity, etc.

Economic equality: For example, a single-name CDS contract and an insurance contract on corporate debt may be ‘economically equal’ in the sense that the cashflows under the two contracts are equal. This does not mean that the two contracts are identical - in this case documentation may be written differently, as there may be requirements on the insured to make a claim under the contract, and the writers of the contract may be differently regulated.

Natural measure versus risk-neutral measure: The natural measure is often used by investors: What is this ‘really’ worth? What income do I anticipate I will receive? The risk-neutral measure is typically used by front-book traders: What is the cost to me of hedging this transaction so that I can lock in a known profit with minimal risk? The two measures are widely used in the ‘valuation’ of different assets but may provide widely different answers. [We shall return to this point several times in this book, and it is important to understand that the mathematical models we develop are often equally applicable under either measure - the difference being the data put into the model (or its ‘calibration’) rather than the sums being done.]

Funded instruments versus CDSs: In the past the term CDS meant a single-name default swap, but is now more commonly used to mean a contract written like a single-name default swap (or interest rate swap) with periodic payments of premium and a contingent capital payment. A ‘funded instrument’ is more like a bond - the investor pays a large sum up front and receives larger periodic payments plus all, or a proportion of, the investment at maturity or default. The terms are often used to distinguish how tranches of risk of a CDO are documented and sold. In particular it is now common usage for the term CDS to mean any credit derivative (from a single-name risk to a complex non-standard structure referencing non-traded risks) which is documented as an unfunded trade.