Technische Universität München Credit as an Asset Class - risklab

Technische Universität München 

Zentrum Mathematik 

Credit as an Asset Class 

Master Thesis 

of 

Barbara Mayer 

Supervisors: Prof. Dr. Rudi Zagst 

Technische Universität München 

Anna Kalemanova 

Date Due: 21/03/2007 

risklab germany GmbH, München

I confirm that I have done this thesis on my own and that I have only used the cited 

sources. 

Munich, 21/03/2007 

i

Danksagungen 

Zuallererst möchte ich mich bei der Firma risklab germany GmbH für die Bereitstellung 

dieses interessanten und aktuellen Themas bedanken. Insbesondere gilt mein Dank Frau 

Anna Kalemanova, die jederzeit für Fragen und Diskussionen offen war und mir stets mit 

wertvollen Anregungen zur Seite stand. Weiterhin möchte ich mich auch bei allen anderen 

Kollegen bei risklab für ihre vielseitige Unterstützung bedanken. 

In besonderem Maße möchte ich mich auch bei Prof. Rudi Zagst für die hervorragende 

fachliche Betreuung während der Diplomarbeit bedanken, aber auch für die sehr lehrreiche 

Heranführung an finanzmathematische Fragestellungen bereits während des Studiums. 

Nicht zuletzt gilt mein Dank auch meiner Familie und meinem Freund Norbert, die 

mich während meines Studiums, vor allem in sehr anstrengenden Phasen, in jeglicher 

Hinsicht voll und ganz unterstützt haben. 

ii

Contents 

1 Introduction 1 

1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 

1.2 Objectives and Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 

2 Credit Risk Transfer 4 

2.1 Credit Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 

2.1.1 Definition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 

2.1.2 Credit Risk and Regulation . . . . . . . . . . . . . . . . . . . . . . . 5 

2.2 Traditional credit related Instruments . . . . . . . . . . . . . . . . . . . . . 6 

2.2.1 Loans . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 

2.2.2 Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 

2.3 Credit Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 

2.3.1 Definition and Functionality . . . . . . . . . . . . . . . . . . . . . . . 10 

2.3.2 Classification according to Risk Category . . . . . . . . . . . . . . . 12 

2.3.3 Reasons for the Utilisation of Credit Derivatives . . . . . . . . . . . 12 

2.3.4 Chances and Risks related to Credit Derivatives Markets . . . . . . 15 

2.3.5 Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 

2.3.6 Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 

2.4 Securitisations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 

2.4.1 Definition and Functionality . . . . . . . . . . . . . . . . . . . . . . . 24 

2.4.2 Classification according to the Underlying Asset . . . . . . . . . . . 26 

2.4.3 Reasons for the Utilisation of Securitisations . . . . . . . . . . . . . 27 

2.4.4 Risks related to Securitisation Markets . . . . . . . . . . . . . . . . . 28 

2.4.5 Transaction Costs . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 

2.4.6 Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 

3 Credit Risk Transfer Instruments 35 

3.1 Derivatives- or Securitisation-based Instruments . . . . . . . . . . . . . . . . 35 

3.2 Credit Default Swap/Credit Default Option . . . . . . . . . . . . . . . . . . 36 

3.3 Total Return Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 

3.4 Credit Spread Option . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 

3.5 Basket Credit Default Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 

3.6 Portfolio Credit Default Swap . . . . . . . . . . . . . . . . . . . . . . . . . . 42 

3.7 Credit Linked Note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 

3.8 Collateralised Debt Obligation . . . . . . . . . . . . . . . . . . . . . . . . . 46 

3.8.1 Traditional CDO (True Sales Securitisation) . . . . . . . . . . . . . . 46 

iii

CONTENTS 

3.8.2 Synthetic CDO (Synthetic Securitisation) . . . . . . . . . . . . . . . 48 

3.8.3 Classification of CDO Transactions . . . . . . . . . . . . . . . . . . . 49 

3.9 CDS Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 

3.9.1 iTraxx Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 

3.9.2 DJ CDX Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 

3.9.3 Motivations for Investors . . . . . . . . . . . . . . . . . . . . . . . . 57 

3.9.4 Market Participants . . . . . . . . . . . . . . . . . . . . . . . . . . . 58 

4 Data Analysis 59 

4.1 A Note on Return Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . 59 

4.2 US Aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 

4.3 Euro-Aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 

4.4 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 

5 Simulation and Pricing Framework 67 

5.1 A Note on Interest-Rate Modelling . . . . . . . . . . . . . . . . . . . . . . . 67 

5.2 Model of Zagst for Economic Scenario Generation . . . . . . . . . . . . . . 69 

5.3 Fitting of the Interest-Rate Curve . . . . . . . . . . . . . . . . . . . . . . . 75 

5.4 Simulation of Correlated Default Times . . . . . . . . . . . . . . . . . . . . 75 

5.4.1 Migration Matrices Approach . . . . . . . . . . . . . . . . . . . . . . 76 

5.4.2 One-Factor Copula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 

5.5 Pricing of Financial Instruments . . . . . . . . . . . . . . . . . . . . . . . . 90 

5.5.1 Zero-Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . 91 

5.5.2 Coupon Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 

5.5.3 Excursus: Default Intensity Model . . . . . . . . . . . . . . . . . . . 93 

5.5.4 Floating Rate Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 

5.5.5 Funded CDS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 

5.5.6 Funded CDS Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 

6 Portfolio Selection 106 

6.1 Traditional Portfolio Selection . . . . . . . . . . . . . . . . . . . . . . . . . . 106 

6.1.1 Mean-Variance Approach . . . . . . . . . . . . . . . . . . . . . . . . 106 

6.1.2 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 

6.2 Conditional Value at Risk Optimisation . . . . . . . . . . . . . . . . . . . . 109 

6.2.1 A Note on Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . . 109 

6.2.2 Conditional Value at Risk . . . . . . . . . . . . . . . . . . . . . . . . 110 

6.3 Performance Measure Omega . . . . . . . . . . . . . . . . . . . . . . . . . . 111 

6.4 Portfolio Selection with Measure Score . . . . . . . . . . . . . . . . . . . . . 112 

7 Simulation and Simulation Results 114 

7.1 US Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 

7.1.1 Calibration of the Economic Scenario Generator . . . . . . . . . . . 114 

7.1.2 Fit to Market Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 

7.1.3 Simulation of Correlated Default Times . . . . . . . . . . . . . . . . 119 

7.1.4 Return Characteristics of Financial Instruments . . . . . . . . . . . 120 


iv

CONTENTS 


7.1.7 Score . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144 

7.1.8 Comparison of Optimal Portfolios . . . . . . . . . . . . . . . . . . . 148 

7.2 US Economy with Higher Default Correlation . . . . . . . . . . . . . . . . . 151 

7.2.1 Return Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . 151 



7.2.4 Score . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 

7.2.5 Comparison of Optimal Portfolios . . . . . . . . . . . . . . . . . . . 173 

8 Summary and Outlook 175 

A Mathematical Preliminaries and Definitions 178 

A.1 Probability Spaces and Stochastic Processes . . . . . . . . . . . . . . . . . . 178 

A.2 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 180 

A.3 Equivalent Measure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 

B Ratings 184 

C Calculation of the Dynamics of the CDS Index 185 

D Transition Matrix for European Union 187 

E List of Abbreviations 188 

Bibliography 189 

v

List of Figures 

2.1 Basic structure of a credit derivative contract . . . . . . . . . . . . . . . . . 11 

2.2 Risks covered by credit derivatives . . . . . . . . . . . . . . . . . . . . . . . 12 

2.3 Global credit derivatives market (in USD billions, excluding asset swaps) . 19 

2.4 Market shares of credit derivatives products . . . . . . . . . . . . . . . . . . 21 

2.5 Market shares of reference entities . . . . . . . . . . . . . . . . . . . . . . . 22 

2.6 Reference entities by credit rating . . . . . . . . . . . . . . . . . . . . . . . . 22 

2.7 Basic structure of a securitisation . . . . . . . . . . . . . . . . . . . . . . . . 25 

2.8 Classification according to the underlying asset . . . . . . . . . . . . . . . . 26 

2.9 Global securitisation issuance (in USD billions) . . . . . . . . . . . . . . . . 31 

2.10 Global securitisation issuance by region (in USD billions) . . . . . . . . . . 32 

2.11 European securitisation issuance by region 2005 . . . . . . . . . . . . . . . . 33 

2.12 US securitisation issuance by collateral 2005 . . . . . . . . . . . . . . . . . . 33 

2.13 European securitisation issuance by collateral 2005 . . . . . . . . . . . . . . 34 

3.1 Derivatives- or securitisation-based credit risk transfer instruments . . . . . 35 

3.2 Credit default swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 

3.3 Total return swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 

3.4 Credit linked note . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 

3.5 Collateralised loan obligation . . . . . . . . . . . . . . . . . . . . . . . . . . 46 

3.6 Collateralised synthetic loan obligation . . . . . . . . . . . . . . . . . . . . . 48 

3.7 iTraxx Europe index family . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 

3.8 Spread history iTraxx Europe, 5 years term . . . . . . . . . . . . . . . . . . 55 

3.9 Spread history DJ CDX.NA.IG, 5 years term . . . . . . . . . . . . . . . . . 57 

4.1 QQ-plots of US aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 

4.2 Histogram of US aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 

4.3 QQ-plots of Euro-aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 

4.4 Histogram of Euro-aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . 65 

5.1 Model of Zagst . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 

5.2 Nelson-Siegel interest-rate curves . . . . . . . . . . . . . . . . . . . . . . . . 76 

5.3 Credit curve (Q (R) (t))0≤t≤400 . . . . . . . . . . . . . . . . . . . . . . . . . . 80 

5.4 Densities of NIG distribution . . . . . . . . . . . . . . . . . . . . . . . . . . 84 

5.5 Correlated default times with Gaussian and NIG one-factor copula model . 88 

5.6 One-factor Gaussian copula vs. one-factor NIG copula . . . . . . . . . . . . 89 

5.7 Compounding of recovery payment . . . . . . . . . . . . . . . . . . . . . . . 91 

vi

LIST OF FIGURES 

5.8 Coupon payment structure of a coupon bond . . . . . . . . . . . . . . . . . 92 

5.9 Possible cash flow structure of a FRN . . . . . . . . . . . . . . . . . . . . . 96 

5.10 Coupon payment structure of a CDS . . . . . . . . . . . . . . . . . . . . . . 98 

5.11 Compounding of recovery payment . . . . . . . . . . . . . . . . . . . . . . . 98 

5.12 Compounding of recovery payments of a CDS index . . . . . . . . . . . . . 104 

6.1 Efficient frontier . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 

7.1 US Treasury Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 

7.2 US credit spreads, rating classes AA, A2, BBB1 . . . . . . . . . . . . . . . 118 

7.3 Model fit of US zero rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 

7.4 Key statistics US economy, Gaussian defaults . . . . . . . . . . . . . . . . . 124 

7.5 Key statistics US economy, NIG defaults . . . . . . . . . . . . . . . . . . . . 125 

7.6 Densities of financial instruments, 1 year time horizon, Gaussian defaults 

(left-hand side) and NIG defaults (right-hand side) . . . . . . . . . . . . . . 126 

7.7 Left tail densities of financial instruments, 1 year time horizon, Gaussian 

defaults (left-hand side) and NIG defaults (right hand side) . . . . . . . . . 127 

7.8 Densities of financial instruments without left tails, 1 year time horizon, 

Gaussian defaults (left-hand side) and NIG defaults (right-hand side) . . . . 127 

7.9 Densities of financial instruments, 3 years time horizon, Gaussian defaults 


7.10 Left tail densities of financial instruments, 3 years time horizon, Gaussian 


7.11 Densities of financial instruments without left tails, 3 years time horizon, 


7.12 Correlation of financial instruments in US economy, Gaussian defaults and 

NIG defaults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130 

7.13 Minimum-variance portfolio with Gaussian defaults (left-hand side) and 

NIG defaults (right-hand side) . . . . . . . . . . . . . . . . . . . . . . . . . 134 

7.14 Results of mean-variance optimisation, 1 year investment horizon with Gaussian 

defaults (left-hand side) and NIG defaults (right-hand side) . . . . . . . . . 135 

7.15 Results of mean-variance optimisation, 3 years investment horizon with 




7.17 Minimum-CVaR portfolio with Gaussian defaults (left-hand side) and NIG 

defaults (right-hand side) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 

7.18 Results of CVaR optimisation, 1 year investment horizon with Gaussian 


7.19 Results of CVaR optimisation, 3 years investment horizon with Gaussian 




7.21 Results of score optimisation, 1 year investment horizon with Gaussian 


vii


7.22 Results of score optimisation, 3 years investment horizon with Gaussian 




7.24 Results for risk-averse and risk-affine investor, 1 year investment horizon, 

Gaussian defaults (top) and NIG defaults (bottom) . . . . . . . . . . . . . . 148 

7.25 Results for risk-averse and risk-affine investor, 3 years investment horizon, 




7.27 Key statistics US economy, Gaussian defaults . . . . . . . . . . . . . . . . . 153 

7.28 Key statistics US economy, NIG defaults . . . . . . . . . . . . . . . . . . . . 154 

7.29 Densities of financial instruments, 1 year time horizon, Gaussian defaults 


7.30 Left tail densities of financial instruments, 1 year time horizon, Gaussian 


7.31 Densities of financial instruments without left tails, 1 year time horizon, 


7.32 Densities of financial instruments, 3 years time horizon, Gaussian defaults 


7.33 Left tail densities of financial instruments, 3 years time horizon, Gaussian 


7.34 Densities of financial instruments without left tails, 3 years time horizon, 


7.35 Correlation of financial instruments in US economy, Gaussian defaults and 

NIG defaults . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159 

7.36 Minimum-variance portfolio with Gaussian defaults (left hand side) and 

NIG defaults (right-hand side) . . . . . . . . . . . . . . . . . . . . . . . . . 161 

7.37 Results of mean-variance optimisation, 1 year investment horizon with Gaussian 






7.40 Minimum-CVaR portfolio with Gaussian defaults (left-hand side) and NIG 

defaults (right-hand side) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 

7.41 Results of CVaR optimisation, 1 year investment horizon with Gaussian 






7.44 Results of score optimisation, 1 year investment horizon with Gaussian 


viii






7.47 Results for risk-averse and risk-affine investor, 1 year investment horizon, 






ix

List of Tables 

2.1 Risks in credit derivatives markets . . . . . . . . . . . . . . . . . . . . . . . 16 

2.2 Market shares of main protection buyers and sellers . . . . . . . . . . . . . 20 

2.3 Risks in securitisation markets . . . . . . . . . . . . . . . . . . . . . . . . . 29 

3.1 iTraxx Europe by industrial sectors . . . . . . . . . . . . . . . . . . . . . . . 52 

3.2 Issuance of iTraxx Europe series . . . . . . . . . . . . . . . . . . . . . . . . 55 

3.3 Fixed spreads and recovery rates of iTraxx Europe series 6 . . . . . . . . . . 55 

3.4 DJ CDX Indices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 

4.1 US Lehman aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 

4.2 Jarque-Bera test for US Lehman aggregates . . . . . . . . . . . . . . . . . . 63 

4.3 Jarque-Bera test for US Lehman aggregates (log returns) . . . . . . . . . . 63 

4.4 Euro-Lehman aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 

4.5 Jarque-Bera test for Euro-Lehman aggregates . . . . . . . . . . . . . . . . . 65 

4.6 Jarque-Bera test for Euro-Lehman aggregates (log returns) . . . . . . . . . 65 

5.1 US average one-year probability of default, 1981 to 2005 . . . . . . . . . . . 77 

5.2 US average one-year transition rates, 1981 to 2005 . . . . . . . . . . . . . . 77 

5.3 Log-expansion of M . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 

7.1 Parameter estimation for processes r and ω . . . . . . . . . . . . . . . . . . 115 

7.2 Parameter estimation for processes s and u for rating classes AA, A2, BBB1116 

7.3 Composition of US CDS index . . . . . . . . . . . . . . . . . . . . . . . . . 116 

7.4 Parameter estimation for process i . . . . . . . . . . . . . . . . . . . . . . . 117 

7.5 Parameters for US equity process . . . . . . . . . . . . . . . . . . . . . . . . 117 

7.6 Nelson-Siegel parameters for US zero rates . . . . . . . . . . . . . . . . . . . 119 

7.7 Parameters of NIG distribution for US economy . . . . . . . . . . . . . . . . 119 

7.8 Characterisation of representative investors . . . . . . . . . . . . . . . . . . 144 

B.1 Standard & Poor’s credit ratings . . . . . . . . . . . . . . . . . . . . . . . . 184 

D.1 European Union average one-year transition rates, 1981 to 2005 . . . . . . . 187 

x

Chapter 1 

Introduction 

1.1 Motivation 

The credit market has experienced an enormous growth over the last years. As part of the 

credit market, the credit derivatives market is the worldwide fastest growing derivatives 

market. 1 It expanded from USD 180 billion of outstanding contracts on a total notional 

amounts basis in 1997 to USD 5,021 billion in 2004 and to estimated USD 8,206 billion in 

2006. 2 After a standardisation of credit derivative contracts and the introduction of CDS 3 

indices, a revolution in terms of liquidity has taken place. The improved liquidity is only 

one reason why credit instruments are very attractive to investors. 

In addition, credit instruments, such as corporate bonds, credit derivatives and securi- 

tisations, often have an appealing risk-return profile, allowing to enhance portfolio return. 

Furthermore, due to the correlation structure of their returns to returns of traditional asset 

classes, such as stocks and government bonds, they offer high potential for diversification. 

Finally, they allow to manage credit risk exposure by hedging credit risk. 

Even knowing the potential benefits and risks of different credit instruments, many 

investors still have to become familiar with these instruments and have to learn how to 

combine them optimally with traditional asset classes, i.e. they have to know the optimal 

proportion of credit instruments, especially for their individual level of risk-aversion. 

1.2 Objectives and Structure 

With this thesis, we want to contribute to a deep understanding of credit related prod- 

ucts, particularly credit derivatives and securitisations. For this purpose, the reader shall 

understand the drivers and the need for credit risk transfer. He also shall understand the 

functionality of various products and the risks inherent in them. Furthermore, we want 

to give an idea of the size and the impressive development of the market. 

Besides, we also aim at analysing credit instruments in a portfolio context with tradi- 

tional asset classes. Therefore, we introduce an asset allocation framework with a scenario- 

1 See [BR04], p. 16. 

2 See [Bri04], p. 5 and [Fit05], p. 1. 

3 A credit default swap (CDS) is a contract in which the protection buyer pays a fixed periodic fee on 

the notional amount to the protection seller over a predetermined period. In exchange, the protection 

buyer receives a contingent payment from the protection seller, triggered by a credit event of a reference 

entity. 

1

CHAPTER 1. INTRODUCTION 

based simulation of risk factors on the one hand and a simulation of default times on the 

other hand. Then, risk factors and default times are used to price various credit instru- 

ments. After having determined the return distribution of the government bonds, the 

equity index and the credit instruments, we are able to calculate optimal asset allocations 

with several optimisation criteria. We do not only apply traditional portfolio optimisation 

according to Markowitz 4 , but also optimisations, based on other criteria that are able to 

better capture the distinct distributional properties of credit instruments. That way, we 

can show that investors benefit from adding credit instruments to their portfolio, i.e. with 

the same level of risk they can generate a higher expected return. 

To reach the objectives listed above, we chose the following structure of the thesis. 

Chapter 2 gives a general introduction into credit risk transfer. After defining credit 

risk and explaining some regulatory aspects, we briefly describe traditional credit related 

instruments, loans and bonds. Then, we introduce the two main possibilities of credit 

risk transfer, credit derivatives and securitisations. We describe their functionality, the 

reasons for using credit risk transfer instruments, and both, chances and risks related to 

these markets. Finally, we give an overview on market evolution and size. 

In Chapter 3, single credit risk transfer instruments are explained in their basic form, 

but also some product variations. In particular, we introduce CDS, total return swaps, 

credit spread options, basket CDS, portfolio CDS, credit linked notes, collateralised debt 

obligations and CDS indices. 

In order to learn more about distributional properties of the returns of credit instru- 

ments, we analyse Lehman aggregate indices, which are fixed income indices, in Chapter 

4. 

In Chapter 5 we introduce a model for scenario generation. It has a cascade form, 

incorporating macro-economic factors to simulate interest-rates and equity indices. To 

realistically price credit instruments, we need to take into account potential defaults. 

The one-factor copula approach for modelling correlated default times between reference 

entities has become very popular. We use this approach not only for a Gaussian one- 

factor copula, but also for a Normal-Inverse-Gaussian one-factor copula, which is able to 

produce more realistic properties for default times than the Gaussian version, such as a 

higher probability of joint defaults of different companies. The last section is dedicated to 

the pricing formulas for the credit instruments in the investment universe. In particular, 

these instruments are zero-coupon bonds, coupon bonds, funded CDS and funded CDS 

indices. 

Chapter 6 deals with portfolio selection. As a first step, the mean-variance approach 

according to Markowitz and its problems due to strict assumptions on the structure of 

asset returns or on utility functions is described. Then, we introduce the optimisation with 

the conditional value at risk as risk measure. Finally, we present the portfolio selection 

with the measure score taking into account an investor’s risk-aversion. In this context, we 

assume two representative investors, one being risk-averse and the other being risk-affine. 

We apply our asset allocation framework to investors in the United States in Chapter 

7. At first, we present the simulation parameters, the return statistics for the instruments 

under consideration and the optimisation results with Gaussian and NIG default times. 

4 See [Mar52]. 

2

CHAPTER 1. INTRODUCTION 

For all selection criteria, we examine optimal asset allocations with different investment 

universes and different investment horizons. As market implied default correlations be- 

tween reference entities as per 30 th of September, 2006 (simulation start date) is rather 

low, we also perform an optimisation for a higher default correlation. The results can be 

found in the second part of the chapter. Particularly, we work out the main characteristics 

and the main differences to the results with the market-implied default correlation. 

fields. 

Chapter 8 summarises the results of this thesis and gives an outlook on further research 

3

Chapter 2 

Credit Risk Transfer 

To obtain a deeper understanding of credit related products, it is necessary to understand 

credit risk as such, and the drivers and the need for credit risk transfer. This chapter 

provides an introduction into credit risk transfer. 

Section 2.1 covers credit risk. After having defined credit risk and its elements, we 

explain how credit risk is embedded into regulation and its importance for regulation. 

The traditional credit related instruments, loans and bonds, are described in Section 2.2. 

They often serve as underlying for credit risk transfer. The two following sections are 

dedicated to the main possibilities of credit risk transfer. Section 2.3 deals with credit 

derivatives. It describes their functionality, the risk categories, which can be covered 

with them, the reasons for their utilisation, and the chances and risks related to these 

markets. A brief paragraph regarding transaction costs can be found as well as an extensive 

description of the size and evolution of the market. Section 2.4 covers securitisation. First, 

a securitisation is described in its basic form. After having given a classification according 

to possible underlying assets and reasons for the utilisation of securitisations, the risks 

related to these markets are explained. Then, a brief paragraph regarding transaction 

costs follows. The section closes with a description of the size and the evolution of the 

securitisation market. 

2.1 Credit Risk 

2.1.1 Definition 

Credit risk can be defined as the ”risk of changes in value associated with unexpected 

changes in the credit quality” 1 of a counterparty in a financial contract. These unexpected 

changes range from a reduction in the market value of the financial contract to the default 

of the counterparty, which is the inability of the obligor to meet payment obligations. 

Considering credit risk in more detail, one can differentiate between the following 

individual risk elements: 

• Default probability – the probability that the counterparty will default on its con- 

tractual obligations to pay back the debt. 

1 See [DS03], p. 3. 

4

CHAPTER 2. CREDIT RISK TRANSFER 

• Recovery rates – the fraction of the nominal amount which may be recovered in case 

of counterparty default. 

• Credit migration – the expected changes in credit quality of the obligor or counter- 

party. 

Furthermore, there are portfolio risk elements: 

• Default and credit quality correlation – the degree of correlation between default or 

credit quality of one obligor and another. 

• Risk contribution and credit concentration – the contribution of one instrument in 

the portfolio to the overall portfolio risk. 2 

2.1.2 Credit Risk and Regulation 

Bearing credit risk is an essential part of banking and a significant source of income 

for financial institutions. Consequently, it is important to monitor and manage the risk 

positions in the portfolio continuously. In order to have a diversified credit portfolio, it 

is crucial to avoid concentration of credit risk vis-à-vis a client or an industry. This is 

vital for each single financial institution in order to reduce its risk of financial distress and 

thus ensuring the stability of the financial system, which is finally the main reason for 

regulation. 

1988 Basel Accord 

According to the 1988 Basel Accord, banks have to hold capital in reserve – the so- 

called regulatory capital – in order to be able to offset losses resulting from credit or 

market risk. It uses only four different risk weights to distinguish between asset categories. 

Thus, the risk structure of the assets in a bank’s balance sheet determines the minimum 

regulatory capital. Vice versa, a given regulatory capital limits risk-bearing credit and 

trading transactions of a bank. Therefore, banks have an incentive to look for capital 

relief, for example by using credit derivatives. 

At its inception, this first approach to a risk-based supervisory process represented an 

innovative and intelligent way of regulation. In the meantime, however, a rapid technolog- 

ical and financial evolution has taken place, undermining the effectiveness of regulation. 

This development demanded a new, modern framework taking into account the increasing 

number of complex financial products, other forms of risk such as operational risk, and 

risk weights better aligned with actual risk inherent in a bank’s assets. 

Basel II Accord 

In 2004, a new Capital Accord for international operating banks, the Basel II Accord, was 

passed. It constitutes a framework based on three pillars: Minimum capital requirements, 

supervisory review process and enhanced disclosure. The Accord has to be implemented 

by the signatory countries by the end of 2006. It can be considered as a milestone for the 

international harmonisation of regulation. It has to overcome weaknesses of Basel I such as 

2 See [BK04], p. 376 and [Sch03], pp. 2-3. 

5


insufficient coverage of new financial instruments and insufficient differentiation according 

to credit quality of the obligor when calculating the regulatory capital. Basel II aims to 

consider new developments and innovations in financial markets. Furthermore, it aims 

to improve risk adjustment of capital requirements and thus alignment of the regulatory 

capital with the economic capital – the capital requirement reflecting the actual economic 

risk a bank faces. This can be accomplished with more sophisticated measurement ap- 

proaches taking into account the credit quality in form of credit ratings of obligors – either 

provided by credit rating agencies or based on bank-internal ratings – to determine the 

risk weights needed for the calculation of regulatory capital. 3 Due to the implementation 

of the new Accord, a bank has a higher incentive to monitor and manage the credit risk 

exposure more precisely, for example, by using a higher proportion of credit derivatives or 

by employing more risk-adjusted pricing when allocating loans and mortgages. 

After having explained credit risk with its components and the importance of credit 

risk for regulation, we now turn to traditional credit related instruments. 

2.2 Traditional credit related Instruments 

In this section we describe loans and bonds that often serve as underlying asset in credit 

derivative contracts. 

2.2.1 Loans 

Loans are contracts between two (or more parties) – the borrower or obligor, for example a 

corporate entity, and the lender or creditor, typically a bank. In the basic form, the parties 

agree upon the lending of money (the principal or notional amount) by the creditor to the 

borrower with the obligation for the latter to pay back the notional amount at maturity. 

In the meantime, the borrower has to make regular interest payments at predefined dates 

in return for the use of someone else’s capital. 

Variations on the Basic Structure 

The payment from the lender to the borrower can be made in one sum, in partial amounts 

or it can be provided in form of a credit line. The borrower can receive the notional 

amount at par (at 100%) or with an agio (> 100%) or a disagio (< 100%). 

Regarding the repayment of the notional amount, we can distinguish between the 

repayment 

• in one sum at maturity (bullet repayment), 

• in regular instalments plus the interest-rate payments for the remaining sum of the 

previous period (amortisable loan), or 

• in constant regular instalments containing repayment and interest rate payment 

(annuity loan). 4 

3 For further reading regarding the different measurement approaches in terms of credit risk, such as the 

Standardized approach and the Internal Ratings Based approaches, we recommend the original framework 

(see [BIS05a]). 

4 See [Sch03], p. 10 

6


Interest-rates can be stipulated fixed or floating. In the case of floating interest-rates, 

there is usually a market interest-rate which is used as benchmark (for example LIBOR 5 ). 

The obligor has to pay the benchmark rate plus a spread. Furthermore, loans can be 

granted unsecured or secured. Real estate or securities often serve as collateral. 

Apart from the presented specifications, there are much more variations in designing 

a loan contract since it is a private agreement and thus very flexible. 

Loans can be classified into defaultable or risky loans and non-defaultable or (default) 

risk-free 6 loans, dependent on the obligor’s credit quality. ”Defaultable” is a term used 

for any kind of loan bearing credit risk, while a non-defaultable loan has no credit risk 

exposure, since the obligor has the highest credit quality. 

Risks related to Loans 

Lenders and obligors are exposed to several risks. Besides credit risk, which is discussed in 

Section 2.1.1, there are more risks inherent in loans. In the following, we give an overview 

on the most important ones. 

• Interest-rate risk– risk of a potential loss that could arise from a change of the market 

interest-rates. 

• Currency risk – risk of a potential loss that could arise from an adverse price change 

of foreign currency. 

• Inflation risk – risk of reduced real purchasing power of future payments driven by 

inflation. 

A loan is the oldest means of payment obligation and it is still an important source 

of funding. Since loan contracts are often non-standardised, it is difficult or impossible to 

trade single loans. In contrast, bonds enable investors to trade debt. They are described 

in the following section. 

2.2.2 Bonds 

A bond is a securitised form of a loan. At the issuing of a bond, the bond holder or lender 

buys the bond with a certain principal amount 7 from the issuer or borrower either at 

par, with an agio or a disagio. 8 By issuing bonds, the borrower commits itself to make 

stipulated payments comprising the repayment of the principal amount at maturity date 

and interest or coupon payments at predefined coupon dates (for example annually or 

semi-annually) to the bond holder. 

Classification of Bonds 

Analogous to loans, bonds can be classified into defaultable or risky bonds and non- 

defaultable or (default) risk-free bonds. ”Defaultable bond” is a term for any kind of 

5 LIBOR is the London InterBank Offered Rate. It represents the interest-rate between banks. 

6 We use the term ”risk-free” if we consider a financial instrument to be default risk-free. 

7 Analogue to loans, the principal amount is the amount which the issuer pays back at maturity and 

it is the basis for calculating the interest payments. We will also denote it by nominal amount, notional 

amount or simply notional. 

8 The meaning of ”at par”, ”agio” and ”disagio” is explained in Section 2.2.1. 

7


bond with the chance of a default. If the issuer defaults on its payment obligations, the 

investor will suffer a financial loss as he only receives a recovery payment. Unlike the 

defaultable bond, the non-defaultable bond pays the coupons and the notional amount 

surely and in time. Hence, ”non-defaultable” means that the bonds involve no credit risk 

exposure since they have the highest credit quality. Nonetheless, the market risk exposure 

remains. Treasury bonds are often considered to be non-defaultable. 9 

There are several types of bonds. In the following, we will explain the most important 

ones which are relevant for the scope of the thesis. 10 

• Fixed-coupon bonds are bonds with a constant coupon payment throughout their 

lifetime. They are the most widespread form of bonds. 

• Zero-coupon bonds do not pay any coupon throughout the lifetime. They are issued 

at a substantial discount from par. 

• Floating rate notes (FRN) have a time-varying coupon which is linked to a bench- 

mark, such as LIBOR, plus a constant spread. 

From an investor’s point of view, buying a FRN primarily means having an exposure to 

credit risk, as the interest-rate risk is nearly eliminated by the time-varying interest-rate. 

Unlike FRN investors, investors buying a fixed-coupon bond or a zero-coupon bond have 

an exposure to both credit and interest-rate risk 11 . 12 

Bond Market Risks 

Bond holders are exposed to similar risks as already described in Section 2.2.1. Besides 

credit risk, which is discussed in Section 2.1.1, there are the following important risks 

inherent in bond markets. 13 

• Market risk/interest-rate risk – risk of a potential loss that could arise from an 

adverse change in the price of a security due to changes of the market interest-rates. 

In bond markets prices vary inversely with market interest-rates: They fall as market 

interest-rate goes up, and vice-versa. 

• Liquidity risk – risk of adversely affecting the market price when trading in large 

size. 

• Currency risk – risk of a potential loss that could arise from an adverse price change 

of foreign currency. 

• Inflation risk – risk of reduced real purchasing power of future payments driven by 

inflation. 

9 

See [BOW03] p. 4. 

10 

In addition, there are for example convertible bonds, callable bonds, high yield bonds, subordinated 

bonds, perpetual bonds, etc.. For a detailed description we refer to [SB00], pp. 133-137, and to [GP95], 

pp. 362-382. 

11 

The bond market risks are explained below. 

12 

See [Bom05], p. 45. 

13 

One can think of many more risks related to certain types of bonds. Describing all of them would go 

beyond the scope of this thesis. 

8

Credit Spread 


The credit spread is measured as excess return of a defaultable bond over the return of 

an equivalent non-defaultable bond (with respect to a certain maturity, currency, etc.). 

In theory, the credit spread represents the credit quality of the issuer that is assessed by 

the market. Hence, it compensates the investor for taking over credit risk. Consequently, 

a higher probability to default requires a larger credit spread of the bond. In practice, 

however, the credit spread may also include non-credit factors such as liquidity risk. 14 

Rating 

Ratings express the opinion of commercial rating agencies such as Standard & Poor’s, 

Moody’s or Fitch on the obligor’s capacity to meet its financial obligations. They are a 

naive measure of an issuer’s credit-worthiness and its probability of default as they reflect 

the average historical frequency of defaults of issuers with the same rating. They are 

named ”naive” as rating agencies do not suppose credit ratings to be a measure of default 

probability over a predefined time horizon. They also reflect qualitative criteria such as 

management quality, transparency, product quality, etc., which are at most indirectly 

related to default probability. Credit ratings are relative measures of credit quality, as 

they are more stable through business cycles than absolute default probabilities. They do 

not reflect the latest information available in the market since they are normally renewed 

once a year. 

Credit ratings express the opinion of the rating agency on the general credit-worthiness 

of an obligor with respect to a particular debt security or a financial obligation. 

Note that a credit rating for a debt title can be better or worse than the issuer rating, 

for example in the case of senior or subordinated debt. Thus, issuer ratings have limited 

explanation power with respect to a certain debt title. 

Credit ratings can roughly be differentiated between investment-grade ratings, which 

mean a high credit quality of the rating object, and non-investment-grade (also subinvestment- 

grade or speculative grade) ratings, which mean a low credit quality of the rating object. 15 

Finally, we want to remark that not only issuers and debt titles can be rating objects, 

but also countries or single equity titles. 

Characteristics of Bonds 

The main difference to loans is the tradeability of bonds which entails the following ad- 

vantages: 

• Access to the debt market for a much higher number of investors. 

• Possibility of small investments. 

• No need to hold investment until maturity. 

• More standardised than loans (in general). 

14 See [DS03], p. 156. 

15 For a detailed description of the credit ratings, we refer to Appendix B. 

9


• Informational function of bond prices: They reflect market’s perception of the 

lender’s credit risk. 

Thus, value is added by using bonds instead of loans. Moreover, the issuer can raise 

capital at better conditions than by individually negotiating with loan creditors. Bond 

issuance, however, is not worthwhile or possible for every kind of borrower. Bonds can 

only be issued by certain institutions like governments, companies, banks, special purpose 

vehicles, etc.. Issuing bonds is a time-consuming process which needs much more resources 

than borrowing from a bank. Therefore, it is only worthwhile for large issue sizes. 

Unlike stock holders, bond holders do not own a part of the issuing institution. They 

are rather lenders to the institution. As bonds have a maturity date with a predetermined 

repayment, bond price volatility is decreasing with less remaining time to maturity. Par- 

ticularly investment-grade 16 bonds are therefore seen as relatively safe investments if they 

are held until maturity. 

Weaknesses of the Bond Market 

Bond markets are an important component of transferring and trading credit risk. Still, 

they involve some weaknesses which are described below. 

• The bond market size is relatively small related to the overall credit risk universe 

accruing from financial intermediation. A large fraction comes from banks acting as 

lenders. 

• Government and sovereign debt constitute a large fraction of the overall bond market. 

Thus, investors have limited possibilities to trade with corporate bonds. 

• The existing corporate bond market enables investors mainly to invest in large cor- 

porations, investment-grade rated. Due to the relatively small market, liquidity 

problems may arise. 

• The available bonds may not meet investor’s requirements, regarding for example 

maturity or currency, as they are issued mainly according to issuer’s needs. 

• Investors in the bond market cannot engage in structured credit risk exposure. 17 

The weaknesses of the bond markets, the increasing interest in credit risk from fixed 

income investors and the banks’ efforts to trade credit risk have been vital for the evolution 

of credit derivatives and securitisations, which are described in the following sections. 

2.3 Credit Derivatives 

2.3.1 Definition and Functionality 

At first, we give a general definition of a derivative security: ”A derivative security, or 

contingent claim, is a financial contract whose value at expiration date T (more briefly, 

expiry) is determined exactly by the price (or prices within a prespecified time-interval) 

16 For more information regarding ratings, we refer to the Rating-paragraph in this section. 

17 See [Das06], pp. 803-804. 

10


of the underlying financial assets (or instruments) at time T (within the time interval 

[0, T ]).” 18 

[Das06] defines credit derivatives as a class of financial instrument, whose value ”is 

derived from an underlying market value driven by the credit risk of private or government 

entities other than the counterparties to the credit derivative transaction itself.” 19 

A credit derivative is a bilateral financial contract which allows to separate and isolate 

credit risk related to a credit-sensitive underlying asset, and thus facilitates the trading of 

credit risk. This leads to a greater efficiency in the pricing and distribution of credit risk 

among market participants, who are now able to better manage it by hedging, transferring 

or replicating it. 

Figure 2.1 shows the basic structure of a credit derivative. 

Figure 2.1: Basic structure of a credit derivative contract 

During the term of the credit derivative contract, the protection buyer or risk seller 

makes one or several premium payments to the protection seller or risk buyer. In return, 

the latter has to make a conditional payment to the protection buyer if a credit event 

occurs during the lifetime of the derivative. A credit event is characterised with respect to a 

reference entity and the reference obligations, which can be any financial instrument issued 

by the reference entity and exposed to credit risk, such as a loan, a bond or a portfolio 

of loans or bonds. The possible credit events might include bankruptcy, deterioration of 

credit quality or changes of the credit spread. The contract terminates either at maturity 

or if a credit event has occurred. Apart from the conditional payment, a credit derivative 

can oblige the protection seller to other payments. For simplicity reasons, we will use the 

terms reference entity and reference obligation synonymously, as most authors also do not 

differentiate between the terms. 

18 See [BK04], p. 2. 

19 See [Das06], p. 669. 

11

2.3.2 Classification according to Risk Category 


Credit derivative contracts can be classified according to different risks they cover. Figure 

2.2 gives an overview. 

Figure 2.2: Risks covered by credit derivatives 

There are credit derivatives that are solely linked to default events. The default risk 

can be covered by default products, such as credit default swaps, credit default options, 

basket default swaps, n th -to-default swaps, etc.. 

Moreover, there are derivatives that are sensitive towards changes in credit quality. 

The credit spread risk can be covered by spread products, such as credit spread swaps or 

options. 

Finally, there are credit derivatives that transfer the total risk related to a bond. 

This can be done by means of total return derivatives like total return swaps. All these 

instruments will be explained in detail in Chapter 3. 

We want to remark that – due to the complexity and variety of credit derivative 

products – it is not possible to give a complete classification. 

One can also think of another classification scheme. Credit derivatives can be grouped 

into single-name and multi-name instruments. Single-name credit derivatives do only have 

one reference entity, while multi-name credit derivatives involve protection against credit 

events in a pool of reference entities, such as a portfolio of bonds or loans. 

2.3.3 Reasons for the Utilisation of Credit Derivatives 

In this section, we will point out the reasons for the utilisation of credit derivatives. At 

first, we present the specific reasons for financial institutions to use credit derivatives, since 

these institutions build the foundation for their development and the enormous growth of 

the market. Then, we will explain general reasons for investors to participate in the credit 

derivatives market both as protection buyer and as protection seller. 

Specific motivations for Financial Institutions to operate in the Credit Deriv- 

atives Market 

As explained in Section 2.1.2, credit risk management is a vital task for banks. An active 

credit risk management has to be in line with an overall risk management strategy. It 

12


aims at managing exposure concentrations and diversifying the portfolio geographically. 20 

Nowadays, it is not only regulation authorities that give a motivation for efficient credit 

risk management, but also other stakeholders, such as credit rating agencies or sharehold- 

ers that increasingly focus on a bank’s risk exposure, risk management and capital use. 

Hence, a financial institution has strong incentives to reduce its credit risk which can be 

accomplished by the use of credit derivatives. That way, a bank can manage its capital 

and reach both an economic and a regulatory capital relief. Economic capital is freed up, 

if the credit derivative contract presents an effective hedge against a position held in the 

portfolio. Regulatory capital can be relieved by the utilisation of derivative products if 

credit risk exposure can be reduced according to regulatory rules. 21 This was one of the 

main drivers for the enormously growing credit derivatives market. 

The use of credit derivatives to manage their credit risk exposure involves several 

advantages for financial institutions. 

• They can use their capital more efficiently. Thus, they have more flexibility entering 

new risk-bearing activities. At the same time, their utilisation enables banks to 

maintain client relationships, as credit derivatives constitute an anonymous and 

confidential way to manage credit risk exposure. 

• In general, using credit derivatives entails lower legal or setup costs than sales and 

securitisation of loans. 

• The use of derivative products to reduce the credit risk exposure can be more tax 

efficient than securitisation of loans. 

• Moreover, it helps banks to improve the management of their credit portfolio and the 

portfolio diversification. On the one hand, banks can act as protection buyers, and 

thus managing their credit portfolio and accomplish capital relief. On the other hand, 

banks can act as protection sellers, also for reasons of portfolio management but also 

for diversification and yield enhancement reasons. That way, banks can diversify 

their portfolio by buying credit risk from other regions or industries and they can 

acquire exposure to certain borrowers that they otherwise hardly could acquire. 

Thus, selling protection can be viewed as an alternative to loan origination. 22 

This paragraph presented the specific reasons for financial institutions to enter into credit 

derivative contracts. The following paragraph will present the motivations for investors, 

in general, to participate in the credit derivatives market. 

Motivation for Investors to participate in the Credit Derivatives Market 

There are various reasons for investors to participate in the credit derivatives market. We 

can differentiate between participants who act as protection buyers or as protection sellers. 

In the following, we describe reasons for acting as protection buyers. 

20 See [Bom05], pp. 29-30, [Fit05], pp. 7-8. 

21 The terms economic and regulatory capital are explained in Section 2.1.2. 

22 See [Das06], p. 671, [Bom05], p. 31, 34, 39, [Deu04], p. 29. 

13


• Buying protection allows investors to mimic a short selling of bonds, particularly of 

corporate bonds, if they expect an issuer’s credit quality to deteriorate and hence, 

bond prices to fall. In general, short selling strategies can be implemented in highly 

liquid financial markets, such as the market for US Treasury securities. But this 

might prove difficult for corporate debt market. By entering into a credit derivative 

contract, the investor can express its view about the future credit quality of the 

reference entity and hence profit from corresponding developments. 

• Buying protection in the credit derivatives market against the default of major pur- 

chasers or suppliers allow corporations to manage their credit risk exposure. Thus, 

they can concentrate on their core business and might even be able to strengthen and 

to extend their market position, for example by granting more vendor financing. 23 

As we have seen, there are many reasons for banks and other market participants to 

buy protection in the credit derivatives market. There are also other market participants 

for whom it is appealing to act as protection sellers for numerous reasons, which are 

presented below. 

• Generally, credit derivatives are ”unfunded” instruments, which means that, at in- 

ception, no capital has to be invested. Usually, there exists an agreement that the 

protection buyer makes (periodic) payments to the protection seller. If no credit 

event occurs and if the counterparty remains solvent, the protection seller receives 

a guaranteed, regular income stream during the term of the derivative without ini- 

tially investing any capital. In contrast, creating a similar income stream from a 

bond would involve the purchase of the bond which is a considerable investment. 

Therefore, acting as protection seller allows investors to effectively leverage their 

credit risk exposure. This kind of investing is particularly interesting for investors 

with high funding costs, which could mean that buying the bond is more expen- 

sive than the income generated from the bond and consequently, a bond investment 

would be unattractive. 

• Some investors, such as insurance companies, consider credit risk to be uncorrelated 

with other risk positions in their portfolio. Therefore, they act as protections sellers 

in order to add value to their portfolio while diversifying their risks. 

• Traditional institutional investors and asset managers are generally not able to invest 

in loan markets. If a certain borrower does not issue bonds but only borrows from 

banks, selling protection against this reference entity enables investors to synthesise 

a long position in that borrower’s debt, and thus synthesise an asset that would 

otherwise not be available to them. 24 

Within the credit derivatives market, there are different types of market participants: 

Hedgers or investors with a demand for risk management or for tailored products, specu- 

lators and arbitrageurs. 

Credit derivatives allow to isolate credit risk from various other types of risks embedded 

in debt instruments, and then transfer it from one counterparty to another. They can be 

23 See [Das06], p. 671, [Bom05], pp. 37-38. 

24 See [Das06], p. 671, [Bom05], pp. 35-36 and [BR04], p. 16. 

14


tailored for the specific needs of a market participant. Thus, they help investors to better 

manage their credit risk exposure. They also allow for synthesising assets that would not 

be available for certain investors or that even do not exist. 

Credit derivatives are also used for the purpose of speculation. A market participant 

does not need to have an exposure to a debt instrument to enter into a credit derivative 

contract. Credit derivatives can rather be used to express one’s opinion about future 

credit quality of a reference entity, and that way to financially participate in its changes 

by trading these instruments. For the investor, the underlying itself is irrelevant. He is 

only interested in generating a profit from trading. 

Another kind of market participant is an arbitrageur, who tries to benefit from pricing 

differences in two or more markets. 

2.3.4 Chances and Risks related to Credit Derivatives Markets 

The credit derivatives markets incorporate both chances and risks, which are described in 

this section. 

Chances related to Credit Derivatives Markets 

On a microeconomic level, credit derivatives are instruments for credit risk transfer, partly 

comparable to traditional instruments for credit risk transfer such as loan sale or credit 

insurance. The characteristics of the derivatives and their possible applications, however, 

go far beyond those of the traditional instruments such as loan sales and credit insurance. 

Credit derivatives allow banks to isolate and to trade credit risks as well as to dynamically 

adjust their credit risks, and thus optimising their risk profile. 

On a macroeconomic level, the utilisation of credit derivatives can contribute to im- 

prove the overall risk allocation within financial systems, which can be accomplished in the 

following ways: First, banks are enabled to perform risk management in a more flexible 

and efficient way. Second, the increased trading activities improve the transparency and 

the quality of the pricing. Third, credit risk is distributed more efficiently. Hence, the 

aggregated risk within a national economy is reduced and market crises can be better ab- 

sorbed as risks and possibly implied damages are less concentrated. All in all, the stability 

of the financial system can be increased. 25 

Risks related to Credit Derivatives Markets 

We examine the risks induced by credit derivative contracts. Table 2.1 gives an overview 

on the possible risks inherent in credit derivatives markets. 

The risks related to the credit derivatives markets can be classified into two groups: 

Risks on contract level and risks on market level. At first, we explain the risks on contract 

level. 

• Counterparty risk – risk that one of the counterparties (protection seller or buyer) 

is unable to fulfil its contractual obligations. 

25 See [DBR04], pp. 1-3 and [Deu04], pp. 36-37. 

15


Risks on contract level Risks on market level 

Counterparty risk Liquidity risk 

Basis risk Risk of high market concentration 

Legal risk/documentation risk Risk of systematic mispricing 

Operational risk Agency risk 

Model risk Risk of regulatory arbitrage 

Risk of market intransparency 

Table 2.1: Risks in credit derivatives markets 

• Basis risk – risk of a potential loss that could arise from an imperfect hedge, for 

example, hedging the credit risk exposure of one reference entity with another highly 

correlated, but different reference entity. 

• Legal risk – risk of potential losses that could arise from not clearly specified contract 

conditions or from unforeseen future events. Furthermore, they can arise from rights 

that are not enforceable. The documentation risk can be viewed as part of the legal 

risk. It is the risk of different notions of a credit event according to the contract. 

The uniqueness of the contract terms is vital for the stability of the financial system. 

Therefore, the International Swaps and Derivatives Association (ISDA) developed 

standards for credit derivatives contracts, which are important for facilitating their 

trading. But the standards cannot exclude any legal risk. 26 

• Operational risk – risk of potential losses or unrealised profits caused by the failure 

of the technical infrastructure. It also involves the risk of losses resulting from 

inadequate or failed internal processes, people or external events. 

• Model risk – risk of losses caused by an under- or overestimation of the fair value 

of a credit derivative contract due to a wrong pricing model or driven by incorrect 

model parameters. 27 

After having seen the risks on contract level, we will discuss the risks involved in credit 

derivatives markets on market level. 

• Liquidity risk – risk of adversely affecting the market price when trading in large 

size. 

• High market concentration – This means that the trading of credit derivatives has 

been limited to a small number of market participants. From the high market 

concentration may also arise an illusion of liquidity. A sudden withdrawal of a large 

market participant may lead to considerable price movements in the market and to 

a serious liquidity risk in the short-term. 28 Furthermore, it might lead to higher 

transaction costs and a suboptimal risk allocation within a financial system. Hence, 

the stability of financial markets could be affected. 29 

26 For more information about the standardisation process we refer to Section 2.3.6 

27 See [Deu04], pp. 37-44, [DBR04], pp. 6-7, [Bom05], pp. 10-15, [BIS05a], p. 140. 

28 This is the result from a poll conducted by Deutsche Bundesbank in autumn 2003 (see [Deu04], p. 

40). Also, the FitchRatings Global Credit Derivatives Survey (see [Fit05], p. 2) highlights the risk for the 

market liquidity in case of the withdrawal of an important market-maker. 

29 This is considered to be relatively unrealistic. There is rather a tendency to an increasing number of 

participants as more participants want to profit from the use of credit derivatives. (See [DBR04], p. 9.) 

16


• Systematic mispricing – Models for pricing credit derivatives are in an early de- 

velopment stage. There is no generally accepted pricing model. Partly, the used 

models are insufficiently able to map complex portfolio products, which can lead to 

an underestimation of the real risk, and thus to undesirably high risk and to a wrong 

allocation of resources with a disequilibrating impact on financial stability. 

• Agency risk – In terms of occurrence probability and expected amount of loss, protec- 

tion sellers are able to better evaluate the transferred credit risk than the protection 

buyers. Therefore, there is a danger that the protection sellers only sell bad risks. 

This phenomenon is also known as adverse selection. A further agency problem may 

accrue from the fact that, after the credit risk transfer, the protection buyer has 

little incentive to monitor the borrower in order to influence the occurrence proba- 

bility and expected amount of loss in case of default. At the granting of a credit, a 

bank might be even less careful, knowing that she will hedge the credit risk. This 

phenomenon is known as moral hazard. 30 

• Regulatory arbitrage – Banks effectively hedge against certain positions in their port- 

folio by transferring the inherent credit risk to counterparties. This is particularly 

done with positions requiring more regulatory than economic capital. Thus, regula- 

tory capital relief is accomplished, while credit risk exposure may then be assumed 

by less regulated market participants. This could result in an inefficient risk al- 

location as risk is systematically taken over by market participants that have less 

knowledge and experience in dealing with risks. Furthermore, they might have a 

smaller capital cushion to absorb unexpected losses as they are not regulated. Reg- 

ulatory arbitrage exploits inconsistencies in capital regulation and is one of the main 

reasons for continually improving capital requirements. 31 32 

• Market intransparency – As the credit derivatives market is an over-the-counter 

(OTC) market, there is an information deficiency. Information regarding the trading 

volume of credit derivatives often has been based on estimations and polls among 

market participants, but not on disclosure requirements. This insufficient trans- 

parency in the market and the lack of insight into firm level exposures and positions 

has prevented an adequate assessment of the risk allocation within a national econ- 

omy. Hence, the possibility to early identify potential systemic risks threatening the 

stability of the financial system is limited. There are efforts to improve the trans- 

parency and disclosure requirements with respect to financial institutions. Not all 

market participants, however, can be included. 33 

To conclude, there are several risks in credit derivative markets that can potentially 

threaten the stability of financial markets. They are particularly important until the 

market is mature. At this stage, some risks have already been reduced successfully, for 

30 

This risk is more likely to be of theoretical nature. In practice, there are several mechanisms that 

prevent these risks such as reputation effects and retention. (See [DBR04], p. 13, [Deu04], p. 42.) 

31 

Regulatory arbitrage is to be prevented through the implementation of Basel II as regulatory capital 

is calculated by taking into account the borrower’s and the counterparty’s credit quality. 

32 

Deutsche Bundesbank considers risk for the stability of financial markets accruing from regulatory 

arbitrage not to be pivotal. (See [Deu04], p. 41.) 

33 

See [Deu04], p. 37-45, [DBR04], pp. 1-14, [BOW03], p. 211, [Fit05], pp. 2-3. 

17


example the documentation risk, or can be viewed as being primarily of theoretical interest, 

such as agency problems. Therefore, credit derivatives contribute today to a better risk 

management. By further reducing the described risks in the future, financial markets can 

increasingly benefit from credit derivatives and gain stability in the long term. 

2.3.5 Transaction Costs 

Credit derivatives involve lower transaction costs than replicating the derivative contract 

with cash instruments. Furthermore, transaction costs have been reduced due to the 

narrowing of the bid-ask spreads. 34 

2.3.6 Market 

The credit derivatives market has grown at an enormous rate over recent years. In this 

section, we provide an overview on the evolution of the market and its size. Furthermore, 

we have a closer look at the market participants, also differentiating between protection 

sellers and protection buyers. Finally, we will consider the market size by instrument, by 

reference entity and by credit quality. 

As already mentioned, credit derivatives are mostly OTC-traded. Therefore, it is 

difficult to analyse the market. There are a few surveys of market participants, conducted 

for example by FitchRatings 35 or British Bankers’ Association (BBA) 36 , that publicly 

provide information with respect to volume, market participants or reference entities. 

These surveys entail some problems that have to be taken into account when using and 

interpreting the results: Only a limited number of market participants can be surveyed. 

Not all of those will answer the survey, some will answer it in a bad quality. Moreover, the 

stated notional amounts overestimate the effective net exposure of the credit derivatives 

contracts since potential recovery rates as well as netting and collateralisation are not 

considered. 37 38 

Evolution and Size 

Figure 2.3 shows the development of the global credit derivatives market with respect to 

the total notional amount. 39 Here, we want to remark that the problems related to such 

surveys and pointed out earlier, should be borne in mind. 

FitchRatings presents similar results with respect to market size in 2004. According 

to FitchRatings, the market expanded to USD 5,290 (= 5.290 · 10 12 ) billion of outstanding 

contracts on a total notional amounts basis in 2004. If cash CDOs are included, the market 

even expanded to USD 5,449 billion. 40 

34 

See [Bom05], p. 5 and [Deu04b], p. 47. 

35 

See [Fit05]. 

36 

See [Bri04]. 

37 

The FitchRatings survey ([Fit05]) incorporates 120 financial institutions (73 banks and broker dealers, 

39 insurance and re-insurance companies, eight financial guarantors) from all over the world. So it covers all 

major institutions with exception of one big broker-dealer. The BBA study ([Bri04]) involves 30 institutions 

from many different countries, which are key player in the credit derivatives market. 

38 

Netting and collateralisation are discussed at the end of Section 2.3.6. 

39 The data is taken from [Bri04], p. 5. 

40 See [Fit05], p. 1 and underlying data file. 

18


Figure 2.3: Global credit derivatives market (in USD billions, excluding asset swaps) 

The credit derivatives market shows an impressing growth. Its size is still small com- 

pared to other derivatives markets. Also in other respects, such as liquidity, transparency, 

standardisation and number of market participants, the credit derivatives market lags 

behind other derivatives markets. Credit derivatives, however, are the worldwide fastest 

growing derivative product and one of the fastest growing securities in the global securities 

market. 41 

Market Participants 

Large commercial and investment banks, securities houses, insurance and re-insurance 

companies, financial guarantors, and hedge funds are the main market participants in the 

credit derivatives market. There are still some more participants who only contribute 

a small share to the overall market such as asset managers, high-net-worth individuals, 

special purpose vehicles (SPV) 42 , and other corporates. 43 

The global banking industry still turns out to be a significant buyer of protection, with 

a net position 44 of USD 427 billion. Hence, this amount is transferred outside the banking 

industry to third parties, such as hedge funds and insurance companies. 45 

Analysing the banking industry in more detail, we detect considerable regional dif- 

ferences. All reported regions acted as net protection buyers in 2004. European banks 

bought USD 294 billion of protection, North American banks and broker-dealers bought 

USD 123 billion, and Australian and Asian banks bought USD 10 billion. Within the 

41 See [Bom05], p. 18, [BR04], p. 16, [Fit05], p. 11. 

42 More information regarding SPVs can be found in Section 2.4.1. 

43 See [Bri04], p. 5, [Fit05], pp. 1-3, [Bom05], p. 23, [BIS05a], pp. 22-23. 

44 The net position is the difference between the amount that was gross sold and the amount that was 

gross bought. Thus, if the amount that was gross bought exceeds the amount that was gross sold, then 

the market participant is a net protection buyer, and vice versa. 

45 See [Fit05], p. 1. 

19


European banks, Germany had by far the smallest net share with approximately USD 15 

billion. In 2003, Germany even was a net protection seller. 46 

There are smaller regional banks within the banking sector acting as protection sellers, 

since they consider credit derivatives as an alternative to enhance the yield on their capital, 

and they view them as an alternative or additional means of originating loans. 47 

Insurance, re-insurance and financial guarantors mainly act as protection sellers. The 

overall net position amounts to USD 556 billion, thereof USD 319 billion taken over by 

the insurance sector. The insurance industry considers protection selling as profitable 

for several reasons: Corporate default is viewed as being nearly uncorrelated with the 

risks in their portfolio. Thus, by selling protection, both portfolio diversification and 

yield enhancement of capital can be achieved. Financial strength of insurance companies 

makes them appealing counterparties for institutions looking for protection buying, as 

counterparty risk is very small. 48 

The difference of approximately USD 130 billion is held by market participants that 

are not covered by the survey, such as asset managers or hedge funds. 49 

Hedge funds have emerged as players in the credit derivatives market for both protec- 

tion buying and selling. They are estimated to have a share of approximately 25%-30% of 

the overall credit derivatives trading volume. Hedge funds have become more and more 

important to financial markets, as they contribute to reduce or even to eliminate mis- 

pricing of financial instruments by pursuing arbitrage opportunities. Probably the most 

important characteristic of hedge funds is their ability to inject significant liquidity into 

financial markets not only in periods of normal market conditions but also – particularly 

more important – in periods of stress. As they are largely exempted from regulation and 

able to easily implement investment strategies (often realised by taking short positions to 

a great extent), they can supply the market more easily and effectively with liquidity than 

other financial institutions. 50 

Table 2.2 is taken from [Bri04] and it gives an overview on the market shares of main 

buyers and sellers of protection for 2003 and forecasts the shares for 2006. 

Protection Buyer 2003 2006e 

Banks 51% 43% 

Securities Houses 16% 15% 

Hedge Funds 16% 17% 

Protection Seller 2003 2006e 

Banks 38% 34% 

Insurance Companies 20% 21% 

Securities Houses 16% 14% 

Hedge Funds 15% 15% 

Table 2.2: Market shares of main protection buyers and sellers 

From the table, we can see a forecasted declining share of banks acting both as pro- 

tection sellers and buyers. The other shares are forecasted to remain relatively stable. 

46 

See [Fit05], pp. 4-6, [Fit04], p. 4. 

47 

See [Fit05], p. 6, [Bom05], p. 24. 

48 

See [Fit05], p. 2,6, [Bom05], p. 24. 

49 

See [Fit05], p. 2. 

50 

See [Bri04], p. 5, [Fit05], p. 7, [Gre05], p. 5. 

20

Market Breakdown 


The credit derivatives market can be classified according to different aspects, such as 

instrument type, reference entity, credit quality. In this section, we give an overview on 

the market in these aspects. 

By Instrument From a global perspective, single name CDS dominate the market. 

With gross sold USD 3,626 billion in 2004, they account for two-third of all gross sold 

credit derivatives positions. Compared to 2003, they grew by 88%. Portfolio products 

represent the second largest share of credit derivatives with gross sold USD 1,119 billion. 

This category contains single tranche CDOs, n th -to-default swaps, etc.. The third largest 

gross sold position is a category denoted ”Other” with gross sold USD 469 billion, which 

contains traded CDS indices 51 , such as DJ iTraxx Europe CDS index and the DJ North 

American High-Yield CDS index, as well as index-related products. During 2004 it grew 

by even 425%. In FitchRating’s opinion, it is supposed to continue its rapid growth in the 

future, as the indices represent a standard benchmark by which credit risk can be hedged 

and traded. In terms of gross sold positions follow the categories cash CDOs, total return 

swaps and credit linked notes. In 2003, the product order with respect to size was the 

same. 52 53 

Figure 2.4 shows the market breakdown in 2004, which is described above. 54 

Figure 2.4: Market shares of credit derivatives products 

According to the BBA study, CDS had a market share of 53% in 2003, which is 

forecasted to decline to approximately 46% in 2006. The portfolio/synthetic CDOs had 

a market share of 16% in 2003, which supposed to remain stable. Indices had a share of 

11% in 2003 and are supposed to have a share of 20% in 2006. Credit linked notes, total 

return swaps, basket products and credit spread products had a share of smaller than 8% 

in 2003. The forecasts also see them remaining smaller than 8% in 2006. 55 

51 For more information regarding CDS indices we refer to Section 3.9. 

52 See [Fit05], pp. 1-4, [Fit04], p. 2-3. 

53 The study also gives more insight into breakdown by instruments within the banking, insurance and 

financial guarantors sector. We refer the interested reader to [Fit05]. 

54 See [Fit05], pp. 1-4. 

55 See [Bri04], p. 6. The numbers are adjusted to the needs of the thesis, as our notion of credit derivatives 

does not comprise asset swaps or equity linked products in accordance with most of the literature. 

21


By Reference Entity The breakdown of reference entities in 2004 in a global consid- 

eration is presented in Figure 2.5. 56 

Figure 2.5: Market shares of reference entities 

Corporates have a dominant share of 62% of gross sold positions in 2004. They are 

followed by financials with 14%, sovereigns only have a share of 6%. Compared to 2003, the 

shares remained relatively stable. The portfolio breakdown reveals the needs of financial 

institutions to meet the funding demands of large corporate customers, while reducing 

large corporate exposures. 57 The share of sovereign entities dramatically decreased since 

1996, from an estimated share of 54%. 58 

By Credit Quality The global credit derivatives exposures in 2004 by rating is given 

in Figure 2.6. 59 

Figure 2.6: Reference entities by credit rating 

56 The data is taken from [Fit05]. 

57 See [Fit05], p. 8, [Fit04]. 

58 See [Bri04], p. 6. The document does not give more insight into the breakdown by reference entity. 

59 The data is taken from [Fit05]. 

22


Within the recent years, a shift in credit quality of the reference entities has taken 

place. While in 2002 the share of AAA-rated reference entities was 22%, it decreased to 

14% in 2004. A similar reduction on a relative basis can be observed for AA- and A-rated 

reference entities. In contrast, the share of BBB-rated reference entities increased from 

28% in 2002 to 32% in 2004. The share of below investment-grade rated reference entities 

increased from 8% in 2002 to 24% in 2004. These developments have various reasons: 

Partly, the shift in the high-rated reference entities can be explained by a negative ratings 

migration. But there is also a change in investors’ needs and in the structure of the market 

participants. There is a growing demand for high-yield CDS indices and index products, 

which is driven to a large extent by hedge funds. They are looking for an alternative way 

to trade credit risk, with a focussed interest in high-yield assets. 60 

Standardisation of Documentation 

Credit derivatives are legally binding contracts between two counterparties, which may 

involve legal risk 61 and enormous transaction costs (including setup, negotiation and ad- 

ministration costs), if every contract is negotiated individually. The counterparties have 

to agree upon key items, such as maturity, premium, reference entity, credit event, etc.. 62 

In order to standardise the contracts and to facilitate trading, the International Swaps 

and Derivatives Association (ISDA) 63 has pursued a steady convergence of documentation 

standards for basic credit derivatives contracts. In 1997, ISDA published the first set of 

credit derivatives documents. In 1999, the association issued the ”Master Agreement” 

and the ”Credit Derivatives Definitions”, of which a new version was published in 2003. 

The former is a contract of general nature between the two counterparties on which is 

agreed upon before the first credit derivative contract is signed. It contains legal aspects, 

procedures in case of default of one counterparty, etc.. The latter defines key terms, such 

as possible credit events or possible settlement procedures. ISDA in cooperation with 

its members has continuously improved these publications in order to meet the changing 

needs of market participants. 64 

The ISDA standard contract specifies all obligations and rights of the counterparties 

as well as key definitions, such as reference entity, credit event and its verification, etc. It 

contains some fixed core terms and definitions, some clauses, however, are variable. 

Several events can be included in the definition of a credit event, for example bank- 

ruptcy, obligation acceleration, failure to pay, moratorium, restructuring. 65 

Credit derivatives can be settled either physically or cash. Physical settlement means 

that in case of default, the protection buyer has to deliver the designated obligation to the 

protection seller in return for the par value of the reference obligation. Cash settlement 

means that if a credit event occurs, the protection seller has to pay the difference between 

the par value and the market value or the recovery value of the reference obligation to the 

protection buyer in cash. 

60 

See [Fit05], p. 7, 10. 

61 

The legal risk involved in credit derivatives contracts is described in Section 2.3.4. 

62 

See [Bom05], p. 26, [Cha05], p. 57. 

63 

ISDA is an association composed of leading swap and derivatives market participants. 

64 

See [Bom05], p. 26, 286, [BR04], p. 17. 

65 

For a description of the events, we refer for example to [Das06], pp. 712-717 or to [Bom05], pp. 

289-290. 

23


More than 90% of the market participants use contract specifications in line with the 

ISDA publications. But there are still a few countries where the ISDA framework coexists 

with local documentation frameworks. The nearly marketwide adoption of standardised 

contracts has led to a commoditisation of the credit derivatives market, and thus entailed 

a number of advantages. Price discovery for these products was improved, legal risk was 

reduced and transaction costs were substantially lowered. Therefore, the standardisation 

of documentation has considerably contributed to the rapid growth of the credit derivatives 

market. 66 

Collateralisation and Netting 

With a growing credit derivatives market, the exposures of market participants to each 

other have grown, and therefore also the counterparty credit risk. In order to reduce the 

latter, the credit risk exposures are netted, i.e. if two counterparties have exposures to 

each other, they are offset and only the difference is considered as exposure - the total net 

exposure. Often, a collateral is posted against the net exposure, which is called collateral- 

isation. According to the FitchRatings survey, market participants have their hedge fund 

exposures at least partially collateralised. To sum up, netting and collateralisation have 

helped to reduce counterparty risk. 67 

2.4 Securitisations 

2.4.1 Definition and Functionality 

Securitisation denotes the transfer of an asset pool into tradeable securities. In its basic 

form, a securitisation is a financial transaction where assets exposed to credit risk are 

pooled and sold to a bankruptcy remote and autarkic special purpose vehicle (SPV) 68 , 

which in turn refinances in the money or capital market by issuing tradeable securities – 

the asset backed securities (ABS). 

A substantial characteristic of an ABS is the separation from the bank’s credit-worthiness 

of the securities’ credit-worthiness. As the asset pool serves as principal collateral, the 

credit-worthiness of the SPV does not play an important role either. The ABS investors 

receive all cash flows from the asset pool minus service fees. Thus, the risk and return 

profile almost exclusively depends on the performance of the asset portfolio. 

Assets opposed to a large number of homogeneous borrowers with low default proba- 

bilities and a clearly defined cash flow structure are best suited for a securitisation. 

Figure 2.7 shows the basic structure of a securitisation. It is not intended to be 

exhaustive but rather gives an overview. 69 

The main involved parties are the originator, the SPV, the investor, the servicer, the 

rating agency and the trustee. 

The originator generates the asset pool, for example by granting loans to borrowers. 

The asset pool is transferred to the SPV which is founded only for the reason of taking over 

66 See [Bom05], p. 26, [Fit05], p. 3, [BIS03], p. 1, 4. 

67 See [Bom05], p. 27, [Fit05], p. 7. 

68 An exact definition of a SPV or special purpose entity (SPE) can be found in [BIS05], p. 118. 

69 See [Oes04], p. 11. 

24


Figure 2.7: Basic structure of a securitisation 

the asset pool or its risk, funded by the issuance of ABS. The originator can often serve 

as servicer, who is responsible for the management of the asset pool, partly including the 

regular collection of instalments from the borrowers, and for regular reportings for investors 

and rating agencies in return for a servicing fee. The trustee represents the investors’ 

interests. He supervises the regularity of the SPV and the asset pool management. In 

addition, he also supervises the accomplishment of regular reports and verifies losses. 

Sometimes, he is responsible for the payments to the investors such as shown in Figure 

2.7. These tasks are often performed by an independent chartered accountant. 

In general, ABS are issued in different risk classes, so-called tranches. In order to be 

able to place the securities in the capital market, it is necessary to get the tranches rated 

by a large rating agency. The agency regularly monitors the risk involved in the single 

tranches and adjusts the rating if necessary. 

Ratings can be influenced and improved by so-called credit enhancements. Credit 

enhancements are a distinctive feature of securitisations, unlike conventional corporate 

bonds, which are often unsecured. Through a credit enhancement, a security’s or tranche’s 

credit quality is raised above that of the underlying asset pool. We can distinguish between 

a variety of internal and external enhancements that are used to limit specific risks 70 

accruing from a securitisation and to decrease the probability of default in the tranches. 

The internal enhancements arise from the asset pool or the involved parties, examples 

are subordination, overcollateralisation, excess spread and cash reserves. External credit 

enhancements are provided by a third party. Examples are financial guarantees, letters of 

credit, credit insurances and cash collateral accounts. 71 

70 These are explained in Section 2.4.4. 

71 For a detailed description of these credit enhancements, we refer for example to [Oes04], pp. 22-23, 

25


The asset pool might be classified into ”senior tranche”, ”junior tranche” and ”first- 

loss-piece” (or ”equity tranche”). The involved risk increases in that order. However, 

there might also be more tranches. The first-loss-piece is often held by the originator for 

several reasons: First, it can be interpreted as a signal for investors about the quality of 

the asset pool. Second, it reduces incentives accruing from agency risks due to asymmetric 

information. Finally, demand for this tranche is often not sufficient. 

Repayments are made according to the ”waterfall principle”: If cash flows have to 

be distributed to investors, at first, claims of senior tranche investors are satisfied, then 

junior tranche investors receive payments, and finally equity tranche investors. Conversely, 

the most risky tranche takes over the first losses. If there are still losses to distribute, 

then the junior tranche has to carry them, and at last the senior tranche. This is called 

”reverse order of seniority”. For taking over more risk, investors of the riskier tranches 

are compensated with a higher return. 

2.4.2 Classification according to the Underlying Asset 

After the introduction of securitisations, we present a classification according to the un- 

derlying asset. Figure 2.8 gives an overview. 

Figure 2.8: Classification according to the underlying asset 

Collateralised debt obligations (CDO) comprise collateralised loan obligations (CLO), 

which are the securitised form of a loan portfolio, and collateralised bond obligations 

(CBO), which consist of a bond portfolio. Often, the borrowers of the underlying assets 

are corporations. 

Mortgage backed securities (MBS) are assets that are collateralised by real estate. 

Here, one can differentiate between mortgages granted to retail banking customers, so- 

called residential mortgage backed securities (RMBS), or to corporations, so-called com- 

mercial mortgage backed securities (CMBS). 

Finally, there are asset backed securities (in a narrow sense) which cannot be classified 

into one of the other categories. They contain for example credit card receivables, leasing 

receivables and consumer loans. 

For the scope of the thesis, we will concentrate on CDOs since they have the same 

underlying assets as many credit derivatives, and thus a similar risk exposure. In contrast, 

[Bay06], pp. 13-17 and [Das06], pp. 859-860. 

26


MBS for example have only little credit risk, as they are highly collateralised; in case of 

default, the recovery value is very high. 

2.4.3 Reasons for the Utilisation of Securitisations 

In this section, we illuminate the reasons for investors to use securitised products. We 

differentiate between the specific reasons for financial institutions, as they are the main 

market driver, and general reasons for investors to engage in these products. 

Motivation for Financial Institutions to use Securitisations 

For banks, the utilisation of securitisations is appealing for several reasons. 

• Risk diversification – The risk transfer of a securitisation can be used to restructure 

the loan portfolio of a bank under risk-return considerations and to deliberately re- 

duce risk or to enter into new risks. This is reasonable for example, if a bank has 

considerable concentrations of credit risk in a certain region or vis-à-vis a certain in- 

dustry. Such concentrations can be reduced by securitising a fraction of the portfolio 

or by investing in securitised products concentrating on other regions or industries. 

Hence, securitisation can be used to manage credit risk exposure. 

• Liquidity – By selling parts of their loan portfolio, banks can raise liquidity for 

potentially better conditions than raising it over the capital market as debt. This 

might be particularly interesting for banks that do not have a first-class credit rating. 

• Capital relief – Transferring credit risk to third parties, can imply both an economic 

and a regulatory capital relief. Thus, flexibility is created for other risk-bearing 

activities. 

• Management of balance sheet and cost structure – Through securitisation, banks can 

improve their risk and cost structure for example by generating nearly risk-free com- 

mission earnings while reducing total assets. Hence, they achieve an improvement 

of performance figures. 72 

Due to harmonisation and internationalisation of markets, these motivations will prob- 

ably become more and more important for banks and so lead to an increasing use of 

securitisation. 

However, from a bank’s point of view, there are some drawbacks with respect to 

securitisation: 

• Large parts of a bank’s portfolio might be unsuitable for securitisation in case of non- 

funded types of exposure, for instance revolving credits, lines of credit or derivatives. 

• Securitisations might be subject to restrictions. Notifications or approval by bor- 

rowers can be required which might damage the client relationship. 

• The economic and regulatory capital relief might be limited as banks often hold a 

large proportion of the equity tranche. Then, only a small part of the credit risk 

exposure is transferred. 

72 See [Oes04], p. 9, [Das06], pp. 851-852, [Bom05], pp. 30-31 and [Deu97], p. 58. 

27


• Securitisations may be time-consuming and costly regarding necessary legal steps 

and the structuring of the portfolio, for example the notification and approval of the 

borrower, the rating process for the tranches, etc.. 73 

After explaining the motivations for financial institutions to use securitisations, we will 

discuss the reasons for investors to engage in securitised products. 

Motivation for Investors to buy Securitised Products 

By securitising asset pools, and structuring them in several tranches, credit risk is isolated 

from the original obligation and transferred to investors in different tranches according 

to individual risk-return requirements. There are several reasons for investors to buy 

securitised products. 

• Performance – Securitised products provide credit spreads that are potentially higher 

than returns received from investments with a comparable risk profile and rating. In 

addition, the default history is very good, which is the reason why many institutional 

investors take part in the market. 

• Diversification – Securitised products constitute a way for investors to obtain a more 

diversified portfolio by adding a credit risk exposure to it. 

• Structured exposure – The products can enable investors to achieve a credit risk 

exposure that is not available in the market. Furthermore, the investor has access 

to structured credit risk exposure where he can invest in different tranches. So, he 

can determine the level of credit risk exposure he wants to have. 

• Liquidity – Often, ABS markets are more liquid than the corporate bond market. 

• Possibility to participate in credit derivatives market – Some fixed income investors 

are – for regulatory, investment policy or administrative reasons – not allowed or not 

able to invest in unfunded credit derivatives. By investing in funded 74 structures, 

an investor has exposure to the sort of credit risk he is interested in. 75 

2.4.4 Risks related to Securitisation Markets 

In this section, we examine the specific risks induced by securitisations, not, however, the 

kind of risk transferred by the securitisation. Note that these specific risks are similar to 

the risks related to credit derivative contracts. Table 2.3 gives an overview on the possible 

risks inherent in securitisations which will be described in more detail below. 

The risks related to securitisations can also be classified into two groups: Risks on 

contract level and risks on market level. At first, we will to explain the risks on contract 

level. 

73 See [Das06], pp. 863-864 and [Bom05], pp. 30-31, 139. 

74 ”Funded” means that the protection seller has to make an up-front funding when entering into the 

contract. The protection seller, for example, buys a fraction of the securitised portfolio, and thus takes 

over credit risk. (See [BIS03], p. 5.) 

75 See [Das06], pp. 805, 852, 864-867 [Bay06], p. 3, [Deu97], p. 58. 

28


Risks on contract level Risks on market level 

Counterparty risk Liquidity risk 

Legal risk Market risk/interest-rate risk 

Operational risk Agency risk 

Liquidity risk/prepayment risk Risk of regulatory arbitrage 

Table 2.3: Risks in securitisation markets 

• Counterparty risk – As seen in the Figure 2.7, there are numerous cash flow streams 

involved in a securitisation. All of those are exposed to the inability of a counterparty 

to fulfil its payment obligations, which is denoted ”counterparty risk”. To reduce 

this kind of risk, credit enhancements 76 can be used. 

• Legal risk – There are several facets regarding legal risk associated with securitisa- 

tions. 

– In order to achieve the desired status of a securitisation, a financial institution 

has to consider many aspects such as regulatory, tax and legal aspects. 

– Realising a securitisation, it should be ensured that in case of a default of an 

involved party, the claims are enforceable. 

– For an effective risk management, an optimal availability of information is neces- 

sary. Therefore, legal aspects regarding bank secret, protection of data privacy 

and disclosure requirements have to be taken into account. 

• Operational risk – A securitisation involves numerous parties with mutual contrac- 

tual obligations. This entails risks of potential losses from the failure of the technical 

infrastructure, inadequate or failed internal processes or people. 

• Liquidity risk/prepayment risk – From the SPV’s point of view, liquidity risk emerges 

if there is a mismatch of the time of income streams and the scheduled payoff streams. 

This also includes the prepayment risk which arises if a borrower makes use of his 

right to cancel and pays back the loan before maturity. Then, investors have to 

accept foregone interest payments. 77 

After having addressed the risks on contract level, we will consider the risks involved 

in securitisations on market level. Apart from the general risks in bond markets, which 

have already been described in Section 2.2.2, there are more securitisation-specific risks. 

• Liquidity risk – On a market level, liquidity risk means for the SPV the inability to 

place the securitised bonds completely in the market. This can lead to a liquidity 

squeeze for the SPV. But there is also a liquidity risk for the investor, who might be 

unable to trade the securitised bond in the market at any time or without adversely 

affecting the price. 

• Market risk/interest-rate risk – Market risk describes the risk for an investor to incur 

a potential loss from an adverse change of the security price due to changes of the 

market interest-rate. 

76 For further information on credit enhancements, we refer to Section 2.4.1. 

77 See [Oes04], pp. 21-34, [Deu04], pp. 37-45. 

29


• Agency risk – A securitisation involves several parties with mutual contractual oblig- 

ations which – in combination with information asymmetry – results in potential 

agency risk. As already explained earlier we differentiate between adverse selection 

and moral hazard. 78 

• Regulatory arbitrage – As already explained in Section 2.3.4 regulatory arbitrage is 

used by financial institutions to achieve a regulatory capital relief where the regula- 

tory capital is higher than the economic capital due to inefficient regulatory rules. 

A major tool to realise this type of arbitrage is securitisation. Under the Basel I 

Accord, banks have a strong incentive to securitise loans towards highly rated bor- 

rowers, so-called ”good risks”. Then, banks would primarily keep the low rated 

loans, and would thus increase their own probability to default resulting in a threat 

for the stability of the financial system. Such effects are to be prevented by the 

implementation of the Basel II Accord, demanding a more risk-adjusted regulatory 

capital. 79 

On the one hand, there are concerns from regulatory authorities that the stability of 

the financial system might be endangered by an increasing securitisation volume, as more 

and more credit risk is assumed by non-regulated investors. Moreover, from securitisations 

a sustainable interference of the relationship between the bank and the borrower might 

accrue. 

On the other hand, there are nearly no examples of failures of ABS transactions so far. 

ABS can rather be considered as save investment delivering comparatively high returns. 

2.4.5 Transaction Costs 

As already mentioned, the transaction costs involved in securitised products may be very 

high. Costs to be covered comprise service fees, fees for the trustee, the rating agency, etc.. 

There is a danger of fee maximisation by the involved parties, resulting in a considerable 

reduction of the available cash flows for investors. 80 

2.4.6 Market 

The securitisation market has grown in recent years. In the following, we provide an 

overview on the evolution of the market and its size. Then, we have a closer look at the 

market by region and by collateral. 81 

Evolution and Size 

Figure 2.9 shows the development of the global securitisation issuance. 82 

78 

For an explanation of these terms, we refer to Section 2.3.4. 

79 

See [Oes04], pp. 21-34, [Deu04], pp. 37-45, [Gre98], pp. 163-166, [Deu97], pp. 60-61. 

80 

See [Oes04], p. 27. 

81 

The information for this section is mainly taken from [Int06] and [Eur06]. The International Financial 

Services, London, also publishes datafiles on their website for all charts and tables that are contained in 

its report. 

82 

The data is taken from [Int06] and the underlying datafiles. We rearranged the data, also including the 

Emerging Markets, comprised of Latin America, Asia, Eastern Europe, Middle East & Africa (EEMEA). 

30


Figure 2.9: Global securitisation issuance (in USD billions) 

The issuance volume of 2005 amounts to USD 3,630 billion and is nearly five times as 

high as that of 1996. It can be observed that the issuance volume is volatile. This can 

be partly attributed to the interest-rate level, as particularly MBS issuance is inversely 

correlated with interest-rate level over time; if interest-rates rise, MBS issuance decreases. 

Other factors influencing the issuance activity are investors’ demand, development of the 

underlying market and general market environment. Especially in recent years, the securi- 

tisation market was driven by a broader investor base, including hedge funds, institutional 

investors, asset managers, retail funds and private banks. These investors are looking for 

higher yields relative to other fixed-income markets. 83 

Market Breakdown 

The securitisation market can be classified according to different aspects. In this section, 

we give an overview on the market with respect to region and collateral. 

By Region In Figure 2.10 we see the development and the size of securitisation issuance 

in different regions over time. 84 

The US market clearly dominates the global securitisation issuance with a volume 

of USD 3,023 billion or 83% of the whole issuance volume in 2005, followed by Europe 

with 11%, Japan and Australia with 2% and Emerging Markets with 1%. While all 

other markets almost exceptionally increased since 1996, the US market experienced both 

increases and decreases. 85 

Figure 2.11 provides information regarding European securitisation issuance. 86 

83 

See [Int06], pp. 1-2, 5, [Eur06], p. 1, [BIS05a], pp. 22-23. 

84 

The data is taken from [Int06] and the underlying datafiles. 

85 

See [Int06], pp. 1-2. 

86 

The data is taken from [Int06] and the underlying datafiles. 

31


Figure 2.10: Global securitisation issuance by region (in USD billions) 

UK was the largest European issuer in 2005 with a volume of USD 181 billion or 45%, 

followed by Spain with 13%, the Netherlands with 11%, Italy with 10%, and Germany 

with 7%. Still, Germany increased its share from 2004 to 2005 by more than 300%, up to 

USD 22 billion. That way, German banks open more funding sources and keep pace with 

global trends. 87 

By Collateral Now, we give an overview on securitisation issuance in the US and in 

Europe with respect to their collateral. 88 Figure 2.12 shows the breakdown of securitisation 

by collateral in the US. 89 

In 2005, MBS had by far the largest share of securitisation issuance (66%), followed 

by home equity (14%), private ABS, auto, credit card and student loan. The report does 

not explain the meaning of ”Other” or private ABS. Furthermore, it is unclear how big 

87 See [Eur06], pp. 2-3, [Deu04], p. 36. 

88 No global breakdown by collateral was available. 

89 The data is taken from [Int06] and the underlying datafiles. We rearranged the data to meet our needs. 

32


Figure 2.11: European securitisation issuance by region 2005 

Figure 2.12: US securitisation issuance by collateral 2005 

the proportion of CDOs is. The only information regarding CDOs is that they increased 

their market share of outstanding value of ABS in the United States from 12% in 2000 to 

15% in 2005 and are so the third largest sector overall. 90 

Figure 2.13 represents a breakdown of securitisation issuance in Europe by collateral 

in 2005. 91 

The most dominant share of total securitisation issuance in Europe in 2005 had MBS 

with a proportion of 57% (thereof RMBS 45% and CMBS 12%). In terms of size, the 

CDO market had the second-largest issuance volume with 15%. Compared to 2004, its 

size increased by 86%. But the fastest growing product sector in 2005 was the CMBS 

sector, which more than doubled its volume in 2005 compared to 2004 holding market 

share of 12%. 92 

In this chapter, we learned a lot about credit risk transfer in general. We considered 

credit risk as such, traditional credit instruments and the possibilities of credit risk trans- 

fer. Our explanations included a description of the functionality of credit risk transfer 

instruments, the chances and risks inherent in these markets and the market evolution 

and size. In the following chapter, we explain important credit risk transfer instruments 

90 See [Int06], p. 3. 

91 The data is taken from [Eur06]. 

92 See [Eur06], p. 2. 

33

in detail. 


Figure 2.13: European securitisation issuance by collateral 2005 

34

Chapter 3 

Credit Risk Transfer Instruments 

Numerous instruments can be used for credit risk transfer. We are particularly interested 

in derivatives- or securitisation-based instruments. In the previous chapter, we learned 

a lot about credit derivatives and securitisation in general. This chapter describes many 

of the possible credit risk transfer instruments. Section 3.1 categorises derivatives- or 

securitisation-based instruments, according to the dimensions unfunded/funded or single- 

name/multi-name instruments. These instruments will be described during the course 

of the chapter. The instruments covered, are credit default swaps (CDS)/credit default 

options, total return swaps, credit spread options, basket CDS, portfolio CDS, credit linked 

notes, collateralised debt obligations and CDS indices. The covered instruments, however, 

are not meant to be exhaustive, as there are many product variations and continuous 

innovations. 

3.1 Derivatives- or Securitisation-based Instruments 

Figure 3.1 gives an overview on derivatives- or securitisation-based credit risk transfer 

instruments, which will be described in this chapter. 

Figure 3.1: Derivatives- or securitisation-based credit risk transfer instruments 

One can differentiate between the dimensions unfunded/funded or single-name/multi- 

name instruments. When a protection buyer enters into a funded credit risk transfer 

35

CHAPTER 3. CREDIT RISK TRANSFER INSTRUMENTS 

contract, he receives funds from the counterparty at inception. In contrast, he does not 

receive any funds at inception when he enters into an unfunded contract. As already ex- 

plained, a single-name instrument has one reference entity, while a multi-name instrument 

has a portfolio of reference entities. 

The overview does not contain CDS indices. The reason is that they cannot be classified 

into this scheme, since they are indices on synthetic reference portfolios, either being issued 

unfunded or funded. 

Besides, the overview does not contain credit risk transfer instruments that are not 

derivatives- or securitisation-based, such as guarantees, letters of credit, or insurances, as 

they are out of scope of this thesis. 

3.2 Credit Default Swap/Credit Default Option 

Functionality 

A credit default swap (CDS) in its basic form is a bilateral contract in which the protection 

buyer pays a fixed periodic (typically quarterly paid) fee (CDS swap spread or premium), 

generally expressed in basis points (bps) on the notional amount, over a predetermined 

period (maturity) to the protection seller. In exchange, the protection buyer receives a 

contingent payment from the protection seller, triggered by a credit event of a reference 

entity. Figure 3.2 illustrates the functionality of a CDS. 

Figure 3.2: Credit default swap 

The standard CDS contract is based on ISDA Master Agreements. Moreover, it uses 

ISDA credit derivatives definitions and the Short Form Confirmation for individual CDS 

including the following key terms and conditions. 1 

• Reference entity. 

• Reference obligation. 

1 See [BIS03], p. 35, [Das06], pp. 708-709. 

36


• Maturity. 3, 5, 10 years are the most common maturities. The CDS maturity does 

not need to match that of a particular debt obligation. 

• Credit events. The standard CDS contract allows for the following credit events: 

Bankruptcy, obligation acceleration, failure to pay, restructuring of debt, and in 

case of sovereign reference entities debt moratorium or debt repudiation. 2 

• Settlement method. In case of default, CDS can be settled in cash (more common in 

Europe) or physically (more common in the US). 3 

If a credit event occurs, the contract terminates, i.e. the protection buyer stops paying 

the periodic fee, and the protection seller has to compensate the protection buyer according 

to the stipulated settlement method. 

CDS can be viewed as a kind of debt insurance or guarantee. Nonetheless, there are 

three significant differences between CDS and insurance-based products. The notion credit 

event that triggers payment is broader for credit derivatives than for a guarantee. The 

protection buyer of a CDS does not need to prove that he actually has suffered a loss in 

order to obtain a contingent payment. Finally, there is a standardised documentation for 

CDS, which facilitates and promotes trading. 

As already seen, CDS are the most widespread instrument within the credit derivatives 

market, and they are the most important instrument in this market. 4 This development 

can be attributed to the standardisation of documentation, and to low transaction costs 

due to a narrowing of bid-ask-spreads for CDS. Thus, CDS are a cost efficient alternative to 

insurance-based products. Furthermore, CDS build the basis for more complex structured 

products, such as synthetic CDOs, which will be explained later in this chapter. 

The main reasons for using CDS are the same as for credit derivatives in general. 5 . 

A credit default option is very similar to a CDS contract. The central difference between 

them is that the protection buyer of the option does not pay a periodic fee. He rather 

pays an up-front fee or option premium once. 

Product Variations 

There are numerous product variations on the basic structure of a CDS, which is described 

earlier in this section. No variation is, however, as liquid and widespread as the basic CDS. 

Here, we only want to show a few variations. 

A binary CDS (also called fixed-recovery CDS or digital CDS) differs from vanilla 

CDS contracts in the recovery value in case of default. It is stipulated in the contract in 

advance. Hence, the uncertainty about the recovery rate is eliminated. 

For CDS contracts written on reference entities with a low credit quality, often two 

fees are quoted – an up-front fee and a periodic fee. The former has to be paid from the 

protection buyer to the protection seller at inception of the contract. The latter has to be 

paid periodically, analogous to a vanilla CDS. 6 

2 

For a description of these events we refer to [Bom05], p. 290, [Das06], pp. 713-717. 

3 

The settlement methods are described in the paragraph ”Standardisation of Documentation” in Section 

2.3.6 

4For 

the size of the market we refer to 2.3.6. 

5 

For a discussion, we refer to Section 2.3.3. 

6 

For a more detailed description of product variations we refer for example to [Das06]. 

37


A CDS can also be issued in a funded version. The protection seller (investor) buys a 

FRN, which pays a coupon of the 3 month LIBOR plus the fixed CDS spread quarterly. 

If no credit event occurs, the coupon is paid until maturity. Then, the investor receives 

the notional amount of the FRN. However, if a credit event of the reference entity takes 

place, the protection seller receives the recovery value and the contract terminates. This 

can also be considered as special case of a credit linked note as described in Section 3.7. 

CDS as Market Indicators 

Some studies indicate, that market information is more quickly incorporated in CDS prices 

than in bond prices. The main reason is that the market for CDS is more liquid than the 

bond market, particularly the corporate bond market. But there are more reasons why 

CDS prices and bond credit spreads are not absolutely high-correlated, for example tax 

aspects or limited possibility of short-selling in bond markets. 7 

Other studies come to the conclusion that CDS show a price leadership and anticipate 

credit rating changes and downgradings of the reference entity in particular. 8 In contrast 

to bonds, CDS prices cannot only be viewed as indicators for a certain reference entity, but 

they can also be seen as indicators of market sentiment regarding credit risk in general. 

Altogether, CDS prices contain substantial information for an early detection of potentially 

critical developments within the financial system. 9 

3.3 Total Return Swap 

Functionality 

A total return swap (TRS) (or total rate of return swap) is a bilateral agreement between 

the total return receiver and the total return payer. 10 The former receives all cash flows 

associated with a reference asset without owning the asset. In return, the latter receives 

periodic payments which are based on LIBOR plus a fixed spread with respect to the 

same notional amount. The payments from the total return receiver are not related to the 

credit-worthiness of the reference asset. 

The basic structure of a TRS, as described above, is displayed in Figure 3.3. 

A TRS contract can terminate in two ways, either if the reference entity defaults, or at 

maturity. If no default occurs, the difference in the reference asset’s market value between 

maturity and inception is calculated at maturity. If it is positive, the total return receiver 

obtains the difference. If it is negative, he has to pay the difference to the total return 

payer. 

In case of a default of the reference entity, however, the total return receiver has to 

bear the change in the value of the underlying asset by paying the difference between the 

market value of the reference asset at inception and its recovery value to the total return 

payer. Then, the contract terminates. 11 

7 

See [Bom05], pp. 39-40, [Deu04b], pp. 1, 50-51. 

8 

The study of Zhu ([Zhu04]) confirms the price leadership of the credit derivatives market compared to 

the bond market. [Deu04b] shows a significant widening of CDS spreads before a rating downgrade. 

9 

See [Bom05], pp. 39-40, [Deu04b], pp. 1, 50-51, [Zhu04], p. 15. 

10 

The terms are taken from swap markets. 

11 

See [Bom05], p. 85, [BOW03], pp. 212-213, [BR04], p. 21. 

38


Figure 3.3: Total return swap 

From a receiver’s perspective, a TRS is a means to replicate the total performance of a 

reference asset. Thus, it is used to synthetically create the cash flows of a reference asset, 

which for example cannot be bought by the investor, such as a loan, or does not exist, for 

instance with respect to maturity. Consequently, TRS can be used to synthesise assets in 

order to meet the investor’s individual needs. 

From a payer’s perspective, a TRS transfers the full risk of a reference asset to the 

receiver, including default risk and credit spread risk accruing from a change in credit 

quality of the reference entity. Thus, TRS can be considered as hedging vehicle, while 

maintaining the client relationship. 

Motivations for investors to enter into TRS contracts are the same as for credit deriv- 

atives in general, as discussed in Section 2.3.3. One of the main reasons, however, is the 

ability to leverage credit risk exposure, since no initial investment is necessary at inception. 

The counterparty risk accruing from TRS contracts can be reduced by collateralisation 

or netting agreements. Furthermore, TRS are particularly interesting for investors which 

incur high funding costs, because they can act as total return receivers. That way, they 

can synthetically own the reference asset while avoiding the funding disadvantage, as the 

payments to the total return payer are lower than financing a purchase of the reference 

asset. But highly rated total return payers can benefit from a TRS contract too, as they 

can thereby earn a certain spread over their funding costs. 

A TRS contract is usually based on ISDA Master Agreement, but, in contrast to CDS, 

there is no standard confirmation for these contracts. 12 


In this paragraph, we give an idea of different variations on the basic structure of TRS. It 

is, however, not meant to be exhaustive. 

A standard TRS terminates at default of the reference entity. Though, there is a 

variation in which the contract would continue until maturity. 

12 See [BIS03], p. 35, [Bom05], p. 85. 

39


In another variation, the TRS does not only refer to one reference asset, but to a 

portfolio or bond index as reference asset. 

Furthermore, the total return payer might not receive floating payments based on 

LIBOR but fixed interest payments on the notional amount. 13 

3.4 Credit Spread Option 

Functionality 

A credit spread option is an option on the spread of a defaultable reference asset (e.g. 

a bond) over a reference yield (e.g. LIBOR). We can differentiate between credit spread 

call or credit spread put options. The call (put) option gives the buyer – in return for a 

premium payment to the option seller – the right but not the obligation to buy (sell) the 

reference asset at maturity (European) or until maturity (American) at the price implied 

by the strike spread and the reference yield. 

Thus, a buyer of a call option benefits from an increase in credit quality. Then, he can 

buy the reference asset for a price implied by the strike spread plus the reference yield, 

which is lower than the market value of the reference asset. In contrast, a buyer of a put 

option benefits from a decrease in credit quality, as he can sell for a higher price than the 

market value. If a default occurs, a put option is normally exercised immediately. 

There is no standard confirmation for credit spread products according to ISDA. 14 


Premiums for a credit spread product can be paid up-front or in swap form. 

A credit spread put option may be knocked-out on default of the reference entity, which 

means that the option is worthless immediately. 15 

The strike spread can be represented by a fixed spread over a reference yield or by a 

fixed spread between two credit sensitive assets. 16 

A credit spread option cannot only be settled physically, but also in cash. If the option 

is exercised, the seller has to pay the difference between the price of the reference asset 

implied by the strike spread plus the reference yield, and the market price to the buyer. 17 

3.5 Basket Credit Default Swap 

Functionality 

A basket credit default swap is a credit derivative, comparable to a single-name CDS, 

that is linked to a basket or portfolio of assets of more than one reference entity. It is a 

bilateral contract in which the protection buyer pays a fixed periodic fee (swap spread) over 

a predetermined period to the protection seller. In return, the protection buyer receives 

a contingent payment in case of the k th default of any reference entity in a basket of n, 

13 For a more detailed description of product variations, we refer for example to [Bom05], pp. 89-90. 


15 See [Cha05], p. 158. 


17 See [BR04], p. 25, [Cha05], p. 158. 

40


n ≥ k, during the life of the contract. Such a product is also denoted as k th -to-default 

swap or k th -to-default basket. The most common form of basket CDS are first-to-default 

(FtD) swaps. Basket CDS typically include five to ten reference entities. 

A basket credit default swap can terminate in two ways, either if the k th default in the 

basket occurs during the life, or at maturity. In case of a default of the k th reference asset, 

the protection seller has to pay the difference between the notional amount of the reference 

asset and the recovery value. Such a payment, however, does not affect the validity of 

other potential basket CDS that reference to the same portfolio. 

The parties to the contract cannot only agree on cash settlement in case of default, 

but also on physical settlement, which is described in the paragraph ”Standardisation of 

Documentation” in Section 2.3.6. 

Besides the general reasons for investors to enter into credit derivatives contracts, as 

described in Section 2.3.3, there are also specific reasons for using basket CDS. 

For the protection buyer, a basket CDS is a possibility to buy protection for a basket 

of credit risks in a single transaction. Moreover, basket CDS are normally less expensive 

than purchasing protection for each reference entity through single-name CDS. The reason 

is that a k th -to-default swap only protects the protection buyer from the k th default. 

Consequently, he would have to bear default related losses accruing from defaults other 

than the k th default if he did not buy further protection. 

For the protection seller, a basket CDS is a possibility to get exposed to a basket of 

credit risks in a single transaction. 

First-to-Default Swap As already mentioned, the FtD-swaps are the most important 

form of basket CDS. Therefore, we describe the characteristics and challenges related to 

these instruments in more detail. 

Basket CDS can be viewed as a means of yield enhancement, while limiting the down- 

side risk. The premium for a FtD-swap is generally higher than any single-name CDS 

premium of the individual reference entities in the basket since the basket can have a 

lower credit quality than the ”average” credit quality of the reference entities contained in 

the basket. This fact can be explained by two reasons, default correlation and leverage. 

• Default correlation: If the probabilities of default of the basket reference entities 

are independent, the fair premium for a FtD-swap is supposed to be approximately 

the sum of the premiums over all reference entities. If, however, the probabilities 

of default are highly correlated, the premium of a FtD-swap is approximately the 

premium of the reference entity with the highest probability of default. 18 

• Leverage: As he is exposed to a basket of credit risks, the protection seller of a 

FtD-swap has a leveraged exposure but, his maximum loss is limited to the highest 

notional amount of the portfolio’s reference asset. 19 

18 See [Das06], p. 778, [Bom05], p. 101, [BOW03], p. 218. 

19 See [Das06], p. 778, [Bom05], p. 101. 

41



Variations on basket CDS include, for example, loss limits. Then, the loss for the protec- 

tion seller is limited, for instance to 50% of the notional amount, in case of default. This 

structure is similar to a binary CDS 20 . 

A Note on the Pricing of Multi-Name Credit Derivatives 

The pricing of basket CDS, but also the pricing of multi-name credit derivatives in general, 

is challenging in practice, as there are a number of determinants for the premiums. We 

describe the most important determinants in the following. 

• Number of reference entities in the basket: The more reference entities contained in 

the basket, the higher the probability of default of any reference entity and hence, 

the higher the premium will be. 

• Credit quality and expected recovery rate of each reference entity: The worse the 

credit-worthiness of a reference entity and the lower the expected recovery rates, the 

higher the premium will be. 

• Default correlation 21 between reference entities. As already explained above for 

FtD-swaps, default correlation has a substantial influence on the premium. The 

higher the default correlation between the reference entities, the lower the premium 

for a FtD-swap, but the higher the premium for a k th -to-default swap, k ≥ 2. 

This seems counterintuitive at first. A closer look at the nature of these contracts 

proves otherwise. With low default correlation between the reference entities, the 

protection seller of a FtD-swap is exposed to nearly uncorrelated sources of risks. On 

the contrary, with a high default correlation, the protection seller of a FtD-swap is 

mainly exposed to one source of risk, therefore the premium is lower. When default 

correlation becomes very high, the premium will be similar to the premium for a 

single-name CDS on the reference entity with the highest probability to default. 

Considering a k th -to-default swap with k ≥ 2, a higher default correlation increases 

the probability that defaults occur together. Hence, the protection sellers of the k th - 

loss face a higher risk of having to cover a loss and therefore need to be compensated 

with a higher premium. 22 

3.6 Portfolio Credit Default Swap 

Functionality 

A portfolio credit default swap is similar to a basket CDS. It is also a credit derivative that 

is linked to a portfolio of assets of more than one reference entity. The main difference, 

however, is that the portfolio is subdivided into loss-pieces (the first-loss piece, the second- 

loss piece, etc.), which are prespecified in size. The first-loss piece, for instance, carries a 

20 A binary CDS is described in the paragraph ”Product Variations” in Section 3.2 

21 A very popular way to model correlated default times for a portfolio of credit risks is the one-factor 

copula approach as described in Section 5.4.2. 

22 See [Bom05], pp. 101-105, 255-256. 

42


certain percentage, for example 10%, of default-related losses of the portfolio. Hence, the 

protection seller is exposed to a prespecified size of default-related losses, but not to the 

number of defaults. For obtaining this protection, the protection buyer of a portfolio CDS 

makes periodic payments to the protection seller. 

In case of default, the investor, who is accountable for the respective loss-piece, has to 

pay the difference between the par values and the recovery values to the protection buyer 

as long as all payments do not exceed the original size of his loss piece. Otherwise, the 

protection seller of the next loss piece has to cover default-related losses. After a payment 

from the protection seller to the protection buyer, the loss piece is reduced accordingly, and 

usually also the premium payment which is made by the protection buyer. If a protection 

seller has covered the maximum amount of losses, the contract terminates immediately. 

Otherwise, it terminates at maturity. 

The contract parties cannot only agree on cash settlement, but they can also agree on 

physical settlement. 23 

Besides the general reasons for investors to enter into portfolio CDS, as described in 

Section 2.3.3, there are also specific reasons which are very similar to the reasons for using 

basket CDS. 

For both, the protection buyer and the protection seller, portfolio CDS allow to transfer 

the credit risk related to a portfolio or to get exposed to it in one single transaction. This 

is a more cost efficient way than realising a similar protection structure by several single- 

name CDS. 

From the protection buyer’s perspective, portfolio CDS are a possibility to obtain 

partial or complete protection against default-related losses, according to his needs. 

For a protection seller, portfolio CDS are a possibility to achieve a leveraged credit 

risk exposure with limited downside risk, since he is exposed to a complete portfolio while 

the maximum loss is limited. Furthermore, portfolio CDS allow with one transaction 

to have a credit risk exposure to a diversified portfolio. Protection sellers who want to 

carry less risk, can enter into a higher-order-loss product, for example a second-loss piece. 

Hence, a portfolio CDS gives the investor the possibility to invest according to his own 

risk propensity. 

Here, we remark that the structure of portfolio CDS build the foundation for synthetic 

collateralised debt obligations (CDO), which experienced an enormous growth in recent 

years. 24 


Portfolio CDS are less standardised than, for example, single-name CDS. Therefore, they 

appear in numerous variations. 

Settlement in case of default can be agreed as immediate or deferred settlement. The 

premium payment made by the protection seller may not be reduced after a default-related 

payment. 

It is not necessary that the reference portfolio consists of bonds or loans. It can rather 

contain only single-name CDS. This could be attractive for market participants who sold 

23 This is described in paragraph ”Standardisation of Documentation” in Section 2.3.6 

24 CDOs in general, and synthetic CDOs in particular are described in Section 3.8. 

43


protection with CDS but then want to transfer a part or the complete credit risk accruing 

from the CDS contracts via portfolio CDS. 

3.7 Credit Linked Note 

Functionality 

A credit linked note (CLN) is a coupon paying security with an embedded credit derivative. 

Thus, it is a funded way to mimic cash flows of a credit derivative contract. The CLN 

investor buys a CLN from the CLN issuer, that can be a bank, a SPV 25 or a corporation. 

The issuer in turn enters into a credit derivatives contract (often a CDS contract) with a 

protection buyer referring to the same reference entity, amounting to the notional amount 

of the CLN and having the same maturity as the CLN. He invests the notional amount 

in a high-grade collateral, for example a government bond, with the same or a similar 

maturity as the CLN. The CLN investor receives coupon payments compensating him for 

the premium payment from the credit derivative contract, the interest-rate payment from 

the high-grade collateral, net of administration fees for the issuer. The coupon rate can 

be fixed or floating. In the latter case it is expressed, for example, as LIBOR plus a fixed 

spread. 

The basic structure of a CLN, as described above, is displayed in Figure 3.4. 26 

Figure 3.4: Credit linked note 

A CLN either terminates at maturity or if a credit event of the reference entity takes 

place. If no default occurs, both the credit derivative contract and the CLN terminate at 

25 More information regarding SPVs can be found in Section 2.4.1. 

26 This figure is following a similar figure in [Bom05], p. 124. 

44


maturity and the investor receives the last coupon payment and the notional amount of 

the CLN. 

Though, if a default occurs, the investor has to carry the default related loss. After the 

issuer paid the difference between the notional amount of the derivative contract and the 

recovery value from the high-grade collateral to the protection buyer, the investor receives 

the remaining sum from the high-grade investment, which is approximately the recovery 

value. Then, both the CLN and the credit derivative contract terminate. 

From an investor’s point of view, a CLN investment can be compared to direct in- 

vestment in a debt instrument issued by the reference entity, where the investor also has 

to carry the whole loss and the debt instrument does not exist any longer in case of de- 

fault. Otherwise, he receives the stipulated coupon payments and the notional amount at 

maturity. But CLNs bear more risks than a direct debt investment, for instance there is 

counterparty risk towards an additionally involved party, the SPV. 

CLNs, however, can also be rated by a rating agency. The major agencies mainly use 

an approach considering both the credit risk of the issuer and of the reference entity. Thus, 

the rating also takes into account the counterparty risk involved in a CLN. 27 

For investors, CLNs offer several advantages: 

• They enable investors, who only want to enter into funded investments as they are 

familiar with that kind of investing, to engage in credit derivatives. 

• CLNs enable investors, who are – for regulatory restrictions or due to internal in- 

vestment policies – not allowed to enter into credit derivative contracts, to invest in 

them anyway. 

• A CLN allows investors to enter into credit derivative contracts while avoiding ad- 

ministrative issues of derivative transactions, as there are for example high system 

requirements and the need for an ISDA Master Agreement. Hence, documentation 

obligations and setup costs can be reduced. 

• By employing CLNs, tradable products can be created which are otherwise not 

available to certain investors, such as corporate loans. 28 

By means of CLNs, the range of market participants in the credit derivatives market 

has been expanded. Now, it is also possible, for example for institutional investors and 

mutual funds, to enter into such contracts. 

From issuers’ perspective, a CLN is an additional means of hedging their exposure in 

credit derivatives positions. 


There are many product variations of CLN. The type explained above is the most common 

type, using a CDS as a credit derivative contract. Even on that structure, there are many 

variations, which include a callable structure (the issuer is allowed to call the note prior to 

maturity) or a guaranteed principal structure (the investor receives a guaranteed principal 

27 See [Das06], pp. 808-809. 

28 See [Bom05], pp. 123-125, [Das06], pp. 805-806 

45


payment in case of default). As already mentioned, other credit derivatives can also be used 

to build up a CLN. There are total return swap linked notes, credit spread linked notes, 

first to default notes and CLN using a portfolio CDS as embedded credit derivative. 29 

CLN with a credit derivative that has a portfolio as underlying asset allows the investor 

to be exposed to a pool of (diversified) reference entities in a single transaction. 

3.8 Collateralised Debt Obligation 

Collateralised Debt Obligations (CDOs) are instruments which redistribute the credit risk 

of a portfolio into tranches with different risk and return characteristics. There are two 

main types of CDOs – the traditional CDO, with a portfolio securitisation and the synthetic 

CDO, that uses credit derivatives to mimic the cash flows of a traditional CDO. Both are 

described in detail below. 

3.8.1 Traditional CDO (True Sales Securitisation) 

Functionality 

Traditional CDOs usually have a loan or a bond portfolio as underlying asset. Corre- 

spondingly, they are denoted collateralised loan obligation (CLO) or collateralised bond 

obligation (CBO). Though, the functionality is the same. Therefore, we only describe it 

exemplary for a CLO. The basic structure of a traditional CLO and the main involved 

parties are shown in Figure 3.5. The functionality is very similar to that of a securitisation 

as explained in Section 2.4.1. 

Figure 3.5: Collateralised loan obligation 

29 For a detailed description of the functionality of these products and their variations, we refer to [Das06], 

pp. 811-827. 

46


The initiative for a CLO transaction is taken by the originator, which is usually a 

bank selling a loan portfolio to a SPV 30 . In order to finance the purchase of the loan 

portfolio, the SPV issues securities which constitute a claim for the investors to interest and 

instalment payments (net of administration fees) from the loans. The SPV in turn issues 

securities in different tranches that reflect a different level of risk and seniority. Figure 3.5 

shows three tranches. A CLO, however, can have more or less tranches. At the coupon 

payment dates, the cash flows are distributed according to the waterfall principle 31 , i.e. 

the investors receive payments in an order according to seniority of their tranche beginning 

with the most senior tranche. Consequently, the losses are distributed in the reverse order. 

Accordingly, the investors receive a higher coupon for less senior tranches to compensate 

the higher risk. 

The CDO tranches are rated by a rating agency. This procedure is necessary to find 

investors. Ratings inform about the credit-worthiness and risk of the different tranches. 

This is important due to the lack of the investor’s information regarding the underlying 

asset pool. There are several criteria taken into account by rating agencies. They comprise 

the asset quality, a cash flow analysis, involved market and legal risk and the skills of the 

asset manager. 32 

By rating the tranches, the liquidity of the securities can be increased. This is why 

rating agencies are often involved in the structuring process of the securities to determine 

the required credit enhancements 33 in order to reach the target rating. Typically, the 

equity tranche is not rated. It is often taken over (at least in parts) by the originator as 

a further signal for the credit-worthiness of the asset pool. 

The trustee represents investors’ interests and supervises the asset pool management 

and the correctness of cash inflows and outflows. 

The main driver for banks and for investors to use CDOs are the same as already 

explained in Section 2.4.3. For banks, they comprise risk diversification, liquidity, capital 

relief, management of the balance sheet and cost structure. 

From an investor’s point of view, securitisations entail possible higher returns, diversi- 

fication, a structured credit exposure and possibly more liquidity than the corporate bond 

market. 34 In addition, particularly CLOs allow investors to participate in markets where 

they were traditionally excluded. 


Product variations, also taking into account different drivers for CDO transactions, are 

described in Section 3.8.3 for both true sales and synthetic securitisation. 

30 

More information regarding SPVs can be found in Section 2.4.1. 

31 

The waterfall principle is explained in more detail in Section 2.4.1. 

32 

For more information regarding the rating criteria for CDO tranches, we refer to [Sta02] and to [Das06], 

pp. 858-862. 

33 

Typical forms of credit enhancements include for example subordination, overcollateralisation, cash 

reserves and financial guarantees. For more details we refer to Section 2.4.1. 

34 

For a detailed explanation we refer to Section 2.4.3. 

47


3.8.2 Synthetic CDO (Synthetic Securitisation) 

Functionality 

Having seen the basic structure of traditional CDOs, it is not difficult to understand 

the structure of synthetic CDOs. Figure 3.6 shows the basic structure of a synthetic 

securitisation exemplary for a synthetic CLO. 

Figure 3.6: Collateralised synthetic loan obligation 

Unlike a traditional CLO, the originator does not sell the loan portfolio to the SPV. 

The originator rather enters as protection buyer into a credit derivative contract (usually 

a series of single-name CDS or a portfolio CDS) with the SPV. Hence, only the credit 

risk associated with the portfolio is transferred while keeping the loans in the balance 

sheet. The SPV issues CLNs in different tranches. The SPV invests the proceeds of the 

CLNs in a high-grade collateral, for example AAA-rated assets, with the same or a similar 

maturity as the CLN. The CLN investors receive coupon payments compensating them for 

the premium payment from the credit derivative contract and the interest-rate payment 

from the high-grade collateral, net of administration fees for the issuer. 

If there are no defaults in the underlying portfolio, the CLN investors receive their 

regular coupon payments. At maturity, the credit derivative contract terminates and the 

investors receive the last coupon payment and the notional amount. 

If, however, a default occurs, the CDO investors have to absorb all default related 

losses, starting with the most junior tranche investors. Then, the SPV liquidates a part 

of the high-grade investment to cover the loss in the underlying portfolio and the no- 

tional amount of the corresponding CLN tranche is reduced accordingly while the spread 

(expressed in bps of the notional amount) remains the same. 

At the coupon payment dates, the cash flows are distributed according to the waterfall 

48

principle. 35 


Synthetic CDO tranches can also be rated. Typically, they are rated by a major rating 

agency to increase the liquidity of the tranches. 36 

From an originator’s perspective a synthetic securitisation is the same as to enter into 

a credit derivative contract. Therefore, the main drivers for the loan originator to use 

a synthetic securitisation are the same as the specific reasons for financial institutions 

to enter into a credit derivative contract as explained in Section 2.3.3. They comprise 

economic and regulatory capital management while maintaining the client relationship. 

Furthermore, it may save costs compared to a traditional CDO because of high legal costs 

involved in a true sales securitisation and high costs for borrower approval and borrower 

notification process. 

For an investor, investing in synthetic CDOs is rather similar to investing in traditional 

CDOs. Therefore, the motivations are similar to buy securitised products as already 

described in Section 2.4.3 for securitisation in general and in Section 3.8.1 for traditional 

CDOs in particular. Contrary to a traditional CDO, synthetic CDOs do not involve 

prepayment risk. 


Besides fully funded synthetic securitisations, there are also partially funded structures, 

where the SPV acts as protection buyer of a super senior swap. The investor in that swap 

acts as protection seller and receives premium payments in return. The remaining credit 

risk in the underlying portfolio is securitised in different tranches via CLNs and sold to 

investors as described above. 

Furthermore, there are product variations taking into account different drivers for 

CDO transactions. They are described in Section 3.8.3 for both true sales and synthetic 

securitisation. 

3.8.3 Classification of CDO Transactions 

There are several classification schemes for CDO transactions. One classification was 

already introduced, namely the differentiation between a funded or a synthetic structure 

with respect to the underlying portfolio as explained in the Sections 3.8.1 and 3.8.2. In 

the following, we will briefly describe more classifications. 

Balance Sheet versus Arbitrage Structure 

Balance sheet structures are driven by the originator who wants to transfer credit risk 

for various reasons. The reasons may comprise capital relief, credit risk management or 

funding and reduction of balance sheet size (only in case of a true sales securitisation). 

This kind of structure is mainly used for CLOs. 

Arbitrage structures are generally initiated by investment banks or asset managers who 

want to profit from price differences between the market price of the asset pool and the 

35 For an explanation of the waterfall principle, see for example 3.8.1. 

36 For more information regarding the rating process, we refer to Section 3.8.1. 

49


price for securitised risk in a structured form. Furthermore, the CLN issuer benefits from 

management and administration fees. 

Cash Flow versus Market Value Structure 

Cash flow structures are CDO transactions where payments to the CDO investors are 

secured and made by the cash flows generated by the underlying asset pool. This structure 

is usually used in static transactions with a predetermined asset pool of either loans or 

bonds. 37 

Market value structures are similar to cash flow structures. However, they are used for 

managed (or dynamic) CDO structures, where the portfolio credit risk is actively managed 

by a portfolio manager who has to optimise the return of the portfolio by trading the 

underlying assets. Therefore, the SPV does not issue tranched CLNs based on the static 

underlying asset pool. It rather issues tranched CLNs referring to an actively managed 

asset pool, which is marked to market periodically (daily or weekly). Investors in different 

tranches have to absorb default related losses in reverse order of seniority. Within a market 

value structure, the investor is exposed to both credit risk and the trading strategy of the 

portfolio manager. These structures are generally used for CBOs. 

3.9 CDS Indices 

The formerly competing two major families of CDS indices, the TRAC-X and the iBoxx, 

began trading a few years ago and merged in 2004 to one index family. The CDS indices are 

supported by licensed market makers. For names from North America and the emerging 

markets, the indices are denoted by DJ CDX and for Europe and Asia by iTraxx 38 . The 

broadest and most actively traded investment-grade indices are the CDX.NA.IG for North 

America and the iTraxx Europe for Europe. Both contain 125 equally weighted names. 

Besides the geographical segmentation of the indices, the broad market indices are also 

segmented by credit quality (for example investment-grade, high yield), by sector (for 

example financials, consumers, industrials, etc.) and by maturity (for example 5 years, 10 

years). 

All indices have in common that they include large, liquid reference entities in partic- 

ular market segments. At the moment, these indices are traded OTC in either funded (as 

CLNs) or unfunded form (as credit derivatives). 

Several CDS indices are available in tranched form, where every tranche is related to 

a specific segment of default related losses, comparable to CDO tranches. 

For a better understanding we will explain the functionality of CDS indices in detail. 

This is done exemplarily for the iTraxx indices in the next section. Thereafter, we will 

briefly describe the DJ CDX indices as the functionality is rather similar. 

37 A cash flow structure is described in Sections 3.8.1 and 3.8.2. 

38 Up to series 2, the names of the iTraxx indices had the word ”DJ” at their beginning, for example DJ 

iTraxx Europe. 

50

3.9.1 iTraxx Indices 


The iTraxx indices are owned, managed and published by the International Index Com- 

pany Limited (IIC). IIC considers itself as independent index provider aiming at improve- 

ment of market efficiency and transparent markets. 39 Guaranteeing for the independence, 

a broad shareholder base is necessary. 40 

In the following, we have a closer look at the iTraxx index family, in particular the 

iTraxx Europe. As already mentioned, the iTraxx index family covers the regions Europe 

and Asia. For the Asian segment, there is the iTraxx Asia with its sub-indices iTraxx 

Australia, iTraxx Japan and iTraxx Asia ex-Japan. 

Composition of the iTraxx Europe The structure of the European index family is 

shown in Figure 3.7. 41 

Figure 3.7: iTraxx Europe index family 

The European index family can be categorised according to benchmark indices, sector 

indices and derivatives. Within the benchmark indices, the iTraxx Europe is the master 

index being composed of the the so-called iTraxx Sector Indices as displayed in Table 3.1. 

One more sector index exists, the iTraxx Subordinated Financials. It is composed 

of the same reference entities as the iTraxx Senior Financials but containing lower rated 

CDS. 

There is a further benchmark index and sub-index of the DJ iTraxx Europe – the iTraxx 

Europe HiVol, containing the 30 entities from the iTraxx Europe non-financials index with 

the widest 5-year CDS spreads. Moreover, we see the iTraxx Europe Crossover, containing 

39 See [Itr06a], p. 28. 

40 Shareholders are ABN Amro, Barclays Capital, BNP Paribas, Deutsche Bank, Deutsche Börse, Dresdner 

Kleinwort, Goldman Sachs, HSBC, JP Morgan, Morgan Stanley, UBS Investment Bank. (See website 

www.itraxx.com). 

41 Inspired by [Itr06a], p. 6. 

51


Non-Financials 100 entities Financials 25 entities 

Autos 10 Financials Senior 25 

Consumers 30 

Energy 20 

Industrials 20 

Telecom, Media, Technology (TMT) 20 

Table 3.1: iTraxx Europe by industrial sectors 

the most liquid 45 non-financial entities with a rating not better than BBB- (in S&P 

classification or an equivalent rating from another rating agency), a negative outlook, and 

with more than EUR 100 million publicly traded debt. Besides, there exist more products. 

first-to-default baskets are available for the single sectors, but also in the categories HiVol, 

Crossover and Diversified. The latter contains one reference entity of each sector. 

Finally there are derivatives of the iTraxx indices. The iTraxx Europe is available in 

five standard tranches: 0-3%, 3-6%, 6-9%, 9-12% and 12-22%, where the 0-3% tranche 

is to absorb the first 3% of default related losses in the pool of reference entities (equity 

tranche), the 3-6% tranche absorbs the losses from 3% to 6% etc.. Investors can buy 

iTraxx options on the spread movement of iTraxx indices. And in the future, it is planned 

to introduce iTraxx futures contracts to trade exposure in the iTraxx Europe. 

Rules-based Process All index series are constructed according to a rules-based process, 

which is described in the following paragraph. At first, we introduce general rules. The 

roll date for each index is the 20 th of March and the 20 th of September. New indices 

start on the roll date or the following business day if the roll date is not a business day. 

iTraxx Europe and iTraxx HiVol have maturities of 3, 5, 7 and 10 years, while the iTraxx 

Crossover and the iTraxx Sector Indices are traded with maturities 5 and 10 years. The 

maturity date for the March roll is always the 20 th of June and for the September roll it 

is the 20 th of December. 

The reference entities in every index are equally weighted. Weighting adjustments 

(+/ − 0.01%) are made in alphabetical order if the number of index components cannot 

be divided equally to two decimal places. 42 

Now, we explain the index rules exemplarily for the iTraxx Europe. This index is 

composed of the 125 most liquid CDS of investment-grade rated European reference enti- 

ties with respect to the previous six months, following dealer liquidity polls. Each market 

maker has to submit a list of 200 to 250 reference entities based on the following criteria: 

• The reference entity is incorporated in Europe. 

• The listed names had the highest CDS trading volume over the previous 6 months. 

• The market maker’s internal transactions are excluded from the volume statistics. 

• If several CDS of one reference entity are traded, the total volume of one name is 

calculated as sum of the single trading volumes. 

42 See [Itr06b], p. 3. 

52


We denote the consolidated list of all market makers, which is sorted by liquidity of the 

reference entity, as master list. 

ria: 43 

IIC then determines the members of the new indices according to the following crite- 

• Each entity has to be rated investment-grade by Fitch, Moody’s or S&P. If an entity 

has several ratings, the lowest rating is considered. 

• The remaining entities are assigned to their appropriate iTraxx sector (as listed in 

Table 3.1) and then ranked within the sector according to the liquidity ranking of 

the market makers. The remaining list is denoted by sector list and is part of the 

overall master list. 

• At the beginning, the index composition is set to be the same as the previous series. 

• Entities that defaulted, changed sector, merged or were downgraded are excluded 

from the index. Entities ranking in the top 50% of the number of entities in a sector 

list that are not yet in the index are added to the index. 44 Entities in the index 

with a lower rank than 125% of the predetermined number of entities in a sector, i.e. 

in the case of the energy sector ranked lower than 25 as the predetermined number 

is 20, are eliminated and replaced by the next liquid entity not yet in the index. 

Entities ranked below 150 in the overall master list are removed from the index and 

replaced by the most liquid entity in this sector which is not yet in the index, unless 

the potential entity is not less liquid. 

• The iTraxx Europe comprises 125 reference entities which are selected according to 

the highest ranks in each sector. The sector weights are given in Table 3.1. The 

names in the index are all equally weighted with 1 

125 

or 0.8%. 

Every index has a fixed spread, which represents a kind of average spread of all reference 

entities. Analogously, every index has a fixed recovery rate in the case of default of a 

reference entity, independent from the actual recovery rate. Both, the fixed spread and 

the fixed recovery rate, are determined after having agreed upon the composition of the 

new index series. Then, IIC initiates a telephone poll with all market makers to determine 

the fixed spread of each index and the recovery rates. The market makers quote a spread 

for every index. The spread which is accepted by a majority of the participating market 

makers is rounded to the nearest 5 bps. Recovery rates are determined analogously, but 

rounded to the nearest 5%. 45 

Considering tranched iTraxx Europe indices, the fixed spread for each tranche within 

one series is determined at inception analogous to the process for untranched indices. 

43 

We enumerate the most important criteria. There are some more which can be found in [Itr06b], pp. 

4-5. 

44If 

for example a reference entity of the energy sector is ranked in the top 10 in the sector list, but it is 

not in the index, it is included. At the same time, the lowest ranking entity in this sector list is eliminated 

from the index. 

45 

For more information regarding the coupons and recovery rates for the series 6, we refer to paragraph 

”History” in this section. 

53


Trading iTraxx Indices There are two ways of trading iTraxx indices – unfunded or 

funded. Trading the index in a funded way, the protection seller (investor) buys a FRN, 

which pays a coupon of the 3 month LIBOR plus the fixed index spread quarterly. If no 

credit event occurs, the coupon is paid until maturity. Then, the investor receives the 

notional amount of the FRN. However, if a credit event of a reference entity takes place, 

the protection seller receives the recovery value but the notional amount of the FRN is 

reduced according to the weight of the reference entity (for example from 100% to 99.2%) 

and future coupon payments are based on the new notional amount. Though, the coupon 

rate remains unchanged. 

Trading the index in an unfunded way, the protection seller receives the quarterly 

spread premium determined at inception. In case of no credit event, the cash flows remain 

the same until maturity. If, however, a credit event occurs, the default related loss is 

settled physically. Then, the protection buyer delivers bonds issued by the reference 

entity, amounting to the weight of the defaulted reference entity in the index (for example 

0.8%) multiplied by the notional amount of the contract between the involved parties. In 

return, he receives the nominal value of the bonds. Afterwards, the nominal value of the 

contract between protection seller and protection buyer is reduced accordingly while the 

spread premium (in bps) remains unchanged. 

During the term of a series, the development of the quoted spread depends on supply 

and demand, while the fixed spread for the series remains the same. Therefore, a compen- 

sation for the difference between the quoted and the fixed spread has to take place. We 

want to explain this fact with an example. 46 

Let us assume an investor who wants to act as protection seller in an unfunded CDS 

index with a notional amount of EUR 10,000,000 at t. Let the fixed spread for the series 

be 30 bps and the quoted spread 28 bps at t. Then, the protection seller has to pay the 

present value of 2 bps for the remaining term of the CDS index to the protection buyer. In 

return, he receives 30 bps per annum quarterly on the notional amount. If, on the other 

hand, the quoted spread is higher than 30 bps, say 31 bps, the protection buyer has to pay 

the present value of 1 bp for the remaining term of the CDS index to the protection seller. 

Furthermore, the protection seller receives 30 bps per annum quarterly on the notional 

amount from the protection buyer for the remaining term of the CDS index. 

Regarding tranched iTraxx Europe CDS indices, market makers also quote daily spreads 

for the tranches. In addition, the investor of the equity tranche receives an upfront payment 

which is quoted in the market and an annual spread of 500 bps (i.e. 125 bps quarterly). 

History The first series was launched at the 21 st of June, 2004, and the second at the 

20 th of September, 2004. Since then, every six months (at the 20 th of March and at the 

20 th of September), a new series of iTraxx Europe has been issued. Table 3.2 gives an 

overview over the issuances. 

Exemplarily for the iTraxx Europe series 6, we show the fixed spread for the different 

maturities and the recovery rates. This is done in Table 3.3. 47 

We see an increasing fixed spread with longer maturities. The recovery rate is 40% for 

46 See [Itr06a], p. 14. 

47 See [Itr06a], p. 26. 

54


Series Issuance 

1 21.06.2004 

2 20.09.2004 

3 21.03.2005 

4 20.09.2005 

5 20.03.2006 

6 20.09.2006 

Table 3.2: Issuance of iTraxx Europe series 

Term in years Maturity Fixed Spread Recovery Rate 

3 20.12.2009 20 40% 

5 20.12.2011 30 40% 

7 20.12.2013 40 40% 

10 20.12.2016 50 40% 

Table 3.3: Fixed spreads and recovery rates of iTraxx Europe series 6 

all maturities. A recovery rate of 40%, however, is not given for all iTraxx indices. The 

iTraxx Subordinated Financials of the series 6, for example, has a recovery rate of 20%. 48 

Figure (3.8) gives an impression of the spread history of the iTraxx Europe with a 5 

years term at issuance. We take this term as it is the most liquid index. 49 

Figure 3.8: Spread history iTraxx Europe, 5 years term 

From the figure, we observe a term structure of the quoted spreads. It becomes obvious 

that a new series always starts with a higher spread than a previous series, i.e. the quoted 

48 See [Itr06a] p. 26. 

49 The data is taken from Bloomberg. The ticker symbols are ITRXEB51 Curncy, ITRXEB52 Curncy, 

ITRXEB53 Curncy, ITRXEB54 Curncy, ITRXEB55 Curncy and ITRXEB56 Curncy, where the first number 

denotes the term in years at issuance (here 5), and the last number indicates the number of the 

series. 

55


spread is lower for a series with a lower remaining time until maturity. 

Furthermore, we observe one period that is very striking. In spring 2005, the American 

car producers General Motors and Ford were in a deep crisis. After announcing high 

losses and negative future outlooks, their bonds were downgraded to non-investment-grade 

bonds. This development was also anticipated by CDS indices. 

3.9.2 DJ CDX Indices 

As already mentioned, we briefly describe the DJ CDX indices since their functionality is 

rather similar to the iTraxx indices. The DJ CDX indices cover the regions North America 

and emerging markets. Table 3.4 gives an overview on DJ CDX indices. 

Index No. of entities Characteristics 

DJ CDX.NA.IG 125 Investment-grade reference entities, 

domiciled in North America 

DJ CDX.NA.IG.HVOL 30 Investment-grade reference entities 

with a wide CDS spread, domiciled 

in North America 

DJ CDX.NA.XO 35 Rating BBB or BB (in S&P classification 

or an equivalent rating from 

another rating agency), entities that 

are domiciled in North America or 

have a majority of their outstanding 

bonds and loans denominated in 

USD 

DJ CDX.NA.HY 100 Non-investment-grade 

domiciled in North America 

entities 

DJ CDX.NA.EM 14 Sovereign issuers from three regions 

(Latin America; Eastern Europe, 

the Middle East and Africa; Asia) 

DJ CDX.NA.EM.DI- 

VERSIFIED 

40 Sovereign and corporate issuers 

from three regions (Latin America; 

Eastern Europe, the Middle East 

and Africa; Asia) 

Table 3.4: DJ CDX Indices 

Analogous to the iTraxx Europe, the DJ CDX.NA.IG has several sub-indices: DJ CDX 

IG Consumer, DJ CDX IG Energy, DJ CDX IG Financials, DJ CDX IG HighVol, DJ CDX 

IG Industrials, DJ CDX IG TMT. Like the iTraxx indices, the DJ CDX indices are also 

composed of equally weighted reference entities. 

The CDX.NA.IG is available in five tranches which are slightly different from the 

iTraxx Europe tranches. They comprise 0-3% (equity tranche), 3-7%, 7-10%, 10-15% and 

15-30%. 

We do not want to describe the rules for constructing the DJ CDX indices as they are 

similar to the iTraxx construction rules. But we refer the interested reader to [Dow05]. 

However, we give an impression of the spread history of the DJ CDX.NA.IG indices. 

This is shown in Figure 3.9. We also take the DJ CDX.NA.IG with a 5 years term at 

issuance. 50 

50 The data is taken from JP Morgan Portal and represents the DJ CDX.NA.IG mid spread quoted by 

56


Figure 3.9: Spread history DJ CDX.NA.IG, 5 years term 

From this figure, the term structure is not that obvious. We can see that series 3 and 

4 have the same level at issuance of series 4. Series 5 even starts with a lower spread, 

which might be assigned to a new index composition with higher rated CDS. At issuance, 

series 6 starts with a higher spread than series 5 has at that date. This is what we saw for 

every series in the iTraxx Europe. In Spring 2005, we observe again the crisis of General 

Motors and Ford. 

3.9.3 Motivations for Investors 

Compared to single credit derivative contracts, CDS indices offer several advantages, that 

fuelled the growth of the market. 51 

• They provide narrow bid-ask-spreads. 52 

• They provide a diversified credit risk exposure in one single transaction. 

• Formerly, the CDS market was mainly an interbank market. By the introduction of 

CDS indices, however, investors who were formerly excluded from that market are 

now more easily able to take part in it. 

• They facilitate the implementation of investment, hedging and trading strategies in 

the field of credit risk. 

JPMorgan. 

51 

For more information regarding the size and the evolution of the market, we refer to Section 2.3.6, 

paragraph market breakdown by instrument. 

52 

The bid-ask-spread for the iTraxx Europe with a maturity of 5 years within one year was approximately 

0.6 bps, for 10 years approximately 1.1 bps, for 3 years approximately 1.5 bps and for 7 years approximately 

1.6 bps. 

57


• They help that the credit market have become more liquid, efficient and transparent. 

• The sectoral sub-indices can be used to limit or to extend credit risk exposure towards 

a certain industry. 

Therefore, in the meantime CDS indices provide a standard benchmark in the credit 

market. 

3.9.4 Market Participants 

The main market participants are asset managers, hedge funds, insurance companies, 

corporate treasury and bank proprietary desks. 

The asset managers try to diversify their portfolio by adding credit risk. Moreover, 

they regard the CDS indices as hedging tools. 

Hedge funds often want to implement relative value trading, for example name vs. 

sector, sector vs. sector, sector vs. benchmark. This can be realised by CDS indices. 

Generally, the sector indices can represent a hedging tool for corporates, if they are 

exposed to the credit risk of one branche in particular. Furthermore, CDS indices give 

access to diversified credit risk. Adding CDS indices to a portfolio can be reasonable from 

an asset allocation perspective. 

For banks, CDS indices allow for a better management of credit risk. In addition, the 

indices are also used to bet on credit views. 

So far, we learned a lot about credit risk transfer and the respective instruments in 

theory. In the following chapter, we analyse return data to better understand the distinct 

return properties of credit instruments. 

58

Chapter 4 

Data Analysis 

In this chapter, we aim at giving an idea of distributional properties of credit instruments. 

The analyses are exemplarily done for returns of Lehman aggregates. 

We analyse the Lehman aggregates for the United States in Section 4.2 and for 

the Euro-zone in Section 4.3. The US aggregate index 1 covers the USD-denominated, 

investment-grade, fixed-rate, taxable bond market, while the Euro-aggregate index 2 tracks 

Euro-denominated, investment-grade, fixed-rate securities. Lehman aggregates are con- 

sidered as representatives for credit instruments in general, since they are portfolios of 

credit instruments and since the data is easily accessible. 3 We are particularly interested, 

whether the return distributions follow a normal distribution, because the assumption of 

normally distributed returns is often made to facilitate portfolio selection. 4 If the as- 

sumption does not hold true, the optimal asset allocation, however, is not more than an 

approximation and other selection criteria are more appropriate. 

For our further analyses we use discrete daily return data. Though, before analysing 

the data, we make a few remarks on return definitions in Section 4.1. 

4.1 A Note on Return Definitions 

Return is a very important concept in financial modelling. There are different ways to de- 

fine return. We introduce the concept of simple or discrete returns and of log or continuous 

returns. 5 

1 

See [Leh06], p. 174. 

2 

See [Leh06], p. 200. 

3 

The data is easily available on the Lehman Brothers Lehman Live Portal (www.lehmanlive.com). In 

contrast, for many credit instruments, such as CDS indices, there is no return data available. Provider 

such as Bloomberg only provide quoted CDS index spreads (see Figures 3.8 and 3.9). Spreads alone, 

however, do no not allow to make return calculations (The return components of (funded) CDS and CDS 

indices, as well as their pricing is explained in Sections 5.5.5 and 5.5.6.). For example, we do not have 

information about defaults, which obviously have a significant influence on the return of a CDS index. 

Furthermore, there are new series twice a year, with the latest series being the most liquid one. For this 

reason, considering the return of a certain index series entails a distortion, due to a lower liquidity 6 months 

after issuance. All these facts prevent us from reasonably calculating the returns of CDS indices. 

4 

For more background regarding portfolio selection and various selection criteria, we refer the reader 

to Chapter 6. 

5 

[Dor02] provide an extensive discussion of dealing with the application of discrete and continuous 

returns. 

59

Definition 4.1 (Simple Return) 

CHAPTER 4. DATA ANALYSIS 

Let the price of an asset or an index at time t denote by Pt. Then, the simple or discrete 

return between the dates t − 1 and t is defined by 

Rt = Pt 

− 1. (4.1) 

Pt−1 

Let us assume, that there are n assets to invest in with returns Ri,t, i = 1, ..., n and 

let xi be the portfolio weight of asset i with 

n� 

xi = 1. (4.2) 

i=1 

Then, we denote the portfolio by x := (x1, ..., xn) T , and we calculate the simple return of 

a portfolio of n assets according to 

Rt(x) = 

n� 

i=1 

xiRi,t 

(4.3) 

From this equation, it becomes obvious that the linear relationship allows for a simple 

return calculation in a portfolio context. 

Definition 4.2 (Log Return) 

Let the price of an asset or an index at time t denote by Pt. Then, the log or continuous 

return between the dates t − 1 and t is defined by 

� � 

Pt 

rt = ln = ln (1 + Rt) . (4.4) 

Pt−1 

In equation (4.4) we have already seen that we can express the log return in terms 

of a discrete return. Rewriting the equation, the discrete return in terms of log return is 

calculated by 

Rt = exp (rt) − 1. (4.5) 

The main difference between simple returns and log returns is their domain. Log 

returns rt can attain values in R, while simple returns Rt can only have values between 

[−1, ∞). The approximation errors between log and simple returns are negligible when 

short periods of time are considered. 6 

According to Dorfleitner, simple returns are to use in a portfolio context 7 . That is 

why we use discrete returns for return calculations of single instruments and of portfolios. 

In the following, we will use the term ”return” for a simple return. If we use a log return, 

we will write ”log return”. 

4.2 US Aggregates 

At first, we analyse daily discrete returns of the US Lehman Brothers aggregates listed in 

Table 4.1. 

6 See [Dor02], p. 4. 

7 See [Dor02], p. 20. 

60


Aggregate Name Period Time Series No. of Issuers per 

29.09.2006 

US Aggregate: Aaa 07.11.2002 – 29.09.2006 3560 

US Aggregate: Aa 07.11.2002 – 29.09.2006 759 

US Aggregate: A 07.11.2002 – 29.09.2006 1404 

US Aggregate: Baa 07.11.2002 – 29.09.2006 1215 

US Aggregate: Corporates 07.11.2002 – 29.09.2006 2721 

Table 4.1: US Lehman aggregates 

An easy way to compare the sample distribution to the normal distribution 8 is a 

quantile-quantile plot (QQ-plot) which is shown in Figure 4.1 for all considered US ag- 

gregates. There, the empirical returns are plotted against the theoretical quantiles, in 

this case the quantiles of a normal distribution. 9 If the data points build a line, we can 

conclude that the empirical and the theoretical distributions are the same. However, if 

there are significant deviations from the theoretical distribution, this is an indication for 

a different empirical distribution. 

Figure 4.1: QQ-plots of US aggregates 

This figure indicates a potential non-normality of the return data, as the tails signifi- 

cantly deviate from the theoretical quantiles. 

Figure 4.2 shows the histograms for the US aggregates and the density of the normal 

distribution (solid line) having the same mean and standard deviation as the data. 

Also here, we have the impression that the returns do not exactly follow a normal 

distribution. The empirical distribution seems to be fat tailed. 

8 Formally, the normal distribution is given in Definition 5.16. 

9 For more background regarding QQ-plots, we refer for example to [Har95], pp. 847-849. 

61

Figure 4.2: Histogram of US aggregates 


Although graphical examination of data is a useful tool for becoming aware of de- 

viations from normality, it is too imprecise for statistical inference. In order to obtain 

statistically significant statements on the return distributions, we perform the Jarque- 

Bera test 10 for normality. 

Definition 4.3 (Jarque-Bera Test for Normality) 

Let be X1, ..., Xm random variables with sample mean 

¯X = 1 

m 

m� 

Xi. (4.6) 

i=1 

Let Zi be the standardised random variable defined as 

Zi := 

� 1 

m 

Xi − ¯ X 

� m 

i=1 (Xi − ¯ X) 2 

. (4.7) 

Then, the the sample skewness S(Z) and sample excess kurtosis K excess are defined ac- 

cording to 

10 See, for example, [Gre03], p. 225. 

S(Z) := 1 

m 

K excess (Z) := 1 

m 

m� 

i=1 

Z 3 i , (4.8) 

m� 

Z 4 i − 3. (4.9) 

i=1 

62

and the test statistic T is given by 


� 

S(Z) 2 

T = m · 

6 + Kexcess (Z) 2 � 

. (4.10) 

24 

Under the null hypothesis of normal distribution, T is asymptotically χ 2 2 -distributed. 

Hence, the critical value for a significance level of α = 1% is 9.2103 and α = 5% it is 

5.9915. 

Table 4.2 summarises the results of the Jarque-Bera test. 

Aggregate Name m S(Z) K excess (Z) T p-value 

US Aggregate: Aaa 975 -0.1191 1.7622 128.46 0.00% 

US Aggregate: Aa 975 -0.2802 1.5500 110.36 0.00% 

US Aggregate: A 975 -0.1635 1.4726 92.45 0.00% 

US Aggregate: Baa 975 -0.0356 1.3836 78.09 0.00% 

US Aggregate: Corporates 975 -0.1838 1.2390 67.86 0.00% 

Table 4.2: Jarque-Bera test for US Lehman aggregates 

We can see that T is in every case much higher than the critical values for α = 0.01% 

and α = 0.05%. Then, the p-value 11 equals 0.00% for every aggregate, which means that 

the null hypotheses of normally distributed returns have to be rejected on a significance 

level of α = 0.01% and α = 0.05%, respectively. Furthermore, we observe exclusively 

negatively skewed return distributions. 

We also performed the test with log returns. 12 The result is summarised in Table 4.3. 

Aggregate Name T p-value 

US Aggregate: Aaa 129.47 0.00% 

US Aggregate: Aa 113.66 0.00% 

US Aggregate: A 94.50 0.00% 

US Aggregate: Baa 78.43 0.00% 

US Aggregate: Corporates 70.04 0.00% 

Table 4.3: Jarque-Bera test for US Lehman aggregates (log returns) 

The test statistics T are even higher than for discrete returns. Therefore, we reject the 

null hypotheses of normally distributed returns. 

4.3 Euro-Aggregates 

Now, we analyse the corresponding Euro-Lehman Brothers aggregates listed in Table 4.4 

analogous to the US Lehman aggregates. 

Both, the QQ-plots (Figure 4.3) and the histograms (Figure 4.4) indicate a clear non- 

normality of the return data. Particularly the left tail seems to be heavily tailed. 

Therefore, we perform the Jarque-Bera test for normality, again, as defined in 4.3. So, 

we want to find statistically significant statements regarding the return distribution. The 

test results are shown in Table 4.5. 

11 The p-value is the the lowest significance level allowing to reject the null hypothesis. Thus, a p-value 

lower than the significance level implies a rejection of the null hypothesis. 

12 See Definition 4.2. 

63


Aggregate Name Period Time Series No. of Issuers per 

29.09.2006 

Euro-Aggregate: Aaa 03.08.1998 – 29.09.2006 1054 

Euro-Aggregate: Aa 03.08.1998 – 29.09.2006 690 

Euro-Aggregate: A 03.08.1998 – 29.09.2006 559 

Euro-Aggregate: Baa 03.08.1998 – 29.09.2006 279 

Euro-Aggregate: Corporates 03.08.1998 – 29.09.2006 1093 

Table 4.4: Euro-Lehman aggregates 

Figure 4.3: QQ-plots of Euro-aggregates 

For the Euro-aggregates the test statistics are even higher than for the US aggregates. 

The p-value is 0.00% for every aggregate. Consequently, we clearly have to reject the 

null hypotheses of normally distributed returns. Besides, we observe again exclusively 

negatively skewed return distributions. 

We compare the test results of discrete returns to log returns. These results are 

summarised in Table 4.6. 

Again, we see higher test statistics T for the log return and therefore a clear evidence 

for non-normality of log returns. 

64

Figure 4.4: Histogram of Euro-aggregates 


Aggregate Name m S(Z) K excess (Z) T p-value 

Euro-Aggregate: Aaa 2065 -0.5003 2.4611 607.31 0.00% 

Euro-Aggregate: Aa 2065 -0.4686 3.4010 1070.79 0.00% 

Euro-Aggregate: A 2065 -0.4752 2.0234 430.00 0.00% 

Euro-Aggregate: Baa 2065 -0.3132 11.5625 11228.74 0.00% 

Euro-Aggregate: Corporates 2065 -0.4969 2.8251 771.70 0.00% 

Table 4.5: Jarque-Bera test for Euro-Lehman aggregates 

Aggregate Name T p-value 

Euro-Aggregate: Aaa 629.02 0.00% 

Euro-Aggregate: Aa 1095.21 0.00% 

Euro-Aggregate: A 445.17 0.00% 

Euro-Aggregate: Baa 11255.58 0.00% 

Euro-Aggregate: Corporates 800.73 0.00% 

Table 4.6: Jarque-Bera test for Euro-Lehman aggregates (log returns) 

65

4.4 Summary 


Particularly the Euro-Lehman aggregates are heavily tailed in the left-hand side of the 

distribution, i.e. for negative returns. This might be attributed to defaults within the 

aggregates. The reason why this can be better observed for the Euro-Lehman aggregates 

compared to the US Lehman aggregates is the aggregate size. The European aggregates 

consist of less issuers than the US aggregates. Therefore, a default produces more extreme 

returns within European aggregates. 

We want to consider corporate bonds and CDS that only have one reference entity and 

CDS indices with 125 reference entities, which is still considerably lower than the number 

of reference entities in the Lehman aggregates. Consequently, defaults will play a more 

important role for the investment universe in this thesis resulting in even fatter left tails. 

To conclude, we know that in credit portfolios defaults may occur, which result in 

heavily tailed return distributions. In order to obtain such distributions, we use a sim- 

ulation approach that, on the one hand, simulates the development of interest-rates and 

credit spreads and, on the other hand, default times for issuers with different ratings. This 

approach is introduced in Chapter 5. 

66

Chapter 5 

Simulation and Pricing Framework 

The previous chapters pointed out that returns of credit instruments are not normally dis- 

tributed. This fact can partly be attributed to a positive probability of default for several 

instruments. In order to take into account these distinct characteristics, we introduce a 

simulation and pricing framework in this chapter. 

First, we explain some important aspects of interest-rate modelling in Section 5.1, 

which build a basis for the following sections in this chapter. In Section 5.2, we introduce 

the model of Zagst, which is used to simulate zero-rates, credit spreads for several rating 

classes, a CDS index spread and the return of an equity index. Section 5.3 briefly illus- 

trates an approach going back to Nelson-Siegel to describe a complete interest-rate curve 

with only four parameters. This approach is used to fit the model interest-rate curve to the 

actual interest-rate curve at simulation start date. To take into account the distinct return 

characteristics of credit instruments, we simulate correlated default times. We describe the 

procedure in Section 5.4: For the simulation, we use the popular one-factor copula model, 

which is described in Section 5.4.2. This model is used not only for Gaussian one-factor 

copula, but also for Normal Inverse Gaussian one-factor copula, since the latter is able to 

produce more realistic properties of default times. In order to determine the correlated 

default times, it is necessary to know their distribution function, which is calculated with 

migration matrices. The procedure can be found in Section 5.4.1. The last section intro- 

duces the pricing framework. We price zero-coupon bonds and coupon bonds issued by a 

sovereign or a corporation. Furthermore, we price funded CDS and funded CDS indices, 

applying the constant default intensity model for expected loss calculations. 

5.1 A Note on Interest-Rate Modelling 

In this section we give a brief introduction in interest-rate modelling. We follow primarily 

[Zag02]. 1 

It is obvious that getting one Euro today is better than getting one Euro in the future. 

But what should be paid today or at time t for a guaranteed payment of one Euro at 

time T , T ≥ t? This question is answered by a zero-coupon bond as already described 

in Section 2.2.2. Before defining this product formally, we need to introduce a formal 

definition of interest-rates. 

1 For more theoretical background regarding interest-rate modelling, we refer for example to [Zag02]. 

67

Definition 5.1 (Interest-Rate) 

CHAPTER 5. SIMULATION AND PRICING FRAMEWORK 

An interest-rate (also spot rate or zero rate) R(t, T ) denotes a guaranteed growth of 

one unit of money at time t for the period [t,T]. There are different kinds of interest rates: 

(a) A linear interest-rate Rl(t, T ) is given, if the growth of one unit of money at time 

t to time T is computed as 1 + Rl(t, T )(T − t). 

(b) A discrete interest-rate Rd(t, T ) is given, if the growth of one unit of money at time 

t to time T is computed as [1 + Rd(t, T )] (T −t) . 

(c) A continuous interest-rate Rc(t, T ) is given, if the growth of one unit of money at 

time t to time T is computed as exp(Rc(t, T )(T − t)). 

As a convention, we will use continuous interest-rates throughout the thesis, if not 

stated otherwise and denote them by R(t, T ) = Rc(t, T ). 

Definition 5.2 (Interest-Rate Curve) 

The mapping T → R(t, T ) is called interest-rate curve at time t (T > t). 

Interest-rate curves can take different forms. Most often, we see increasing (normal) 

interest-rate curves, but there are also falling (inverse) interest-rate curves or interest-rate 

curves changing their sign of the slope. 

bond. 

Having defined the interest-rate curve, we can give a formal definition of a zero-coupon 

Definition 5.3 (Zero-Coupon Bond) 

A zero-coupon bond is a financial contract which guarantees the payment of a nominal 

amount L at maturity T . Then, using Definition 5.1, the price of a zero-coupon bond 

P (t, T ) at time t ≤ T is given by 

P (t, T ) is also denoted by discount factor, if L=1. 

Remark 5.4 

P (t, T ) = L · exp(−R(t, T )(T − t)). (5.1) 

In the following we make a few remarks regarding zero-coupon bonds. 

(a) In the following, we assume the notional amount L to equal one, if we do not mention 

anything else. 

(b) Definition 5.3 assumes the payment of the notional amount to be certain and thus 

to be risk-free 2 . Normally, sovereigns like the USA or Germany are considered to be 

non-defaultable or (default) risk-free. 

(c) The price of a zero-coupon bond can also be calculated with discrete or linear interest- 

rates. This, however, leads to the same price. 3 

2 

To be precise, if we use the term risk-free in connection with interest-rates, we think of non-defaultable 

or default risk-free, i.e. not bearing credit risk. 

3 

The interest-rates are slightly different using discrete or linear rates, as well as the way to discount. 

For arbitrage reasons, these effects cancel out each other. 

68

Proposition 5.5 (Zero Rate) 


Given the price of a zero-coupon bond P (t, T ), we can easily calculate the zero rate by solv- 

ing for R(t, T ) 

R(t, T ) = − 

ln (P (t, T )) 

. (5.2) 

T − t 

It is evident that modelling an interest-rate market is not easy, as there are numerous 

different interest-rates, due to numerous different terms. One way to avoid the modelling 

of all interest rates is to model just one interest-rate which is able to describe the complete 

interest-rate curve – the so-called short-rate. 

Definition 5.6 (Short-Rate) 

The short-rate r(t) at time t is the interest-rate for an infinitesimal small time to matu- 

rity. Formally, we define 

ln P (t, t + ∆t) 

r(t) := R(t, t) := − lim 

= − 

∆t→0 ∆t 

∂ 

∂T ln P (t, T )|T =t, (5.3) 

assuming that all derivatives and limits exist. 

On this basis, we can now introduce the model of Zagst. 

5.2 Model of Zagst for Economic Scenario Generation 

In recent years, many models have been developed to describe stock and interest-rate 

markets. The most famous model for describing stock markets is the model of Black and 

Scholes, that uses a geometric Brownian motion to model the movement of stocks. In the 

field of interest-rate markets short-rate models are very popular. 4 Given the short-rate, 

the complete term structure of interest-rates can be derived. This is explained later in 

this section for the Zagst model. 

Often, models for stocks and interest-rates changes are one-factor models, such as the 

Vasicek model 5 . They only take into account their own absolute level, while neglecting 

the influence of, for example, macroeconomic factors like economic growth and inflation. 

The model of Zagst is built up in a cascade form taking into account both micro- and 

macroeconomic factors. 6 Figure 5.1 shows the structure of this model. 

Cascade 1 represents economic factors like economic growth (for example represented 

by the growth of the Gross Domestic Product (GDP)) and inflation rate (for example 

represented by the growth of the Consumer Price Index (CPI)). This cascade provides 

input factors for cascade 2 representing the yield curve with the treasury yield curve and 

credit spreads. The input factors for cascade 3 (equity and property returns) are factors 

of both cascade 1 and cascade 2. In cascade 4, sector returns are modelled with input 

factors of cascade 3. The latter is out of scope of this thesis. The parameter estimation 

has to be done in the order of the cascades beginning with cascade 1. 

4 We defined the short-rate in Section 5.1. 

5 For a description of the Vasicek model we refer for example to [Zag02], p. 126. 

6 The model of Zagst is an extension of the extended model of Schmid and Zagst (see [SZAI06]), providing 

an integrated modelling of stock and bond markets, see [ZMH06] or [Mey05]. 

69


Figure 5.1: Model of Zagst 

The cascade form enables an integrated modelling of stock, interest-rate and credit 

markets, allowing for a high model complexity while ensuring an easy interpretability of 

the model. 

The processes in the cascades are modeled with stochastic differential equations (SDE) 

which are described in the following. 7 Within the thesis, we use the GDP as single eco- 

nomic factor, driving the yield curve. The inflation rate, however, is used as additional 

economic factor to derive equity returns. Then, the resulting SDEs for cascade 1 and 2 

can be found in [SZAI06]. 

Note that all SDEs in the model of Zagst for cascade 1 and 2 have the mean reversion 

property, i.e. the modelled processes tend towards their long-term mean. If, for example 

in terms of the short-rate, its value is lower than the mean reverting level, the short-rate 

is pulled up. If, on the other hand, the short-rate is higher than its mean reverting level, 

it is pulled down. This property can actually be observed in the market. 8 

Cascade 1 At first, we introduce the SDEs for cascade 1. The dynamics of the GDP 

growth rate ω is described by a Vasicek model. It is given by 

dω(t) = (θω − aωω(t)) dt + σωdWω(t), (5.4) 

where aω, σω are positive constants, θω is a non-negative constant and dWω(t) is a standard 

Brownian motion 9 . 

The mean reverting level (MRL) for the GDP growth rate is given by θω 

, where aω 

aω 

determines the reversion rate. 

7 SDEs are a major tool to describe the behavior of financial assets and derivatives. For a definition, see 

Section A.2. For more theoretical background regarding SDEs, we refer for example to [BK04] or [Zag02]. 

8 See [Zag02], p. 133. 

9 Brownian motion is defined in A.7. 

70

with 


The dynamics of the short inflation i is given by 

di(t) = (θi − aii(t)) dt + σidWiω(t), (5.5) 

Wiω = 

� 

1 − ρ 2 iω Wi + ρiωWω. 

Here ai, σi are positive constants, θi is a non-negative constant, ρiω denotes the linear 

correlation coefficient 10 between the processes i and ω, with −1 ≤ ρiω ≤ 1, dWi(t) and 

dWω(t) are a standard Brownian motions. 

The mean reverting level (MRL) for the short inflation is given by θi 

ai . 

Cascade 2 The dynamics of the non-defaultable nominal short-rate r is described by a 

two-factor Hull-White model, which is given by 

dr(t) = (θr(t) + brω(t) − arr(t)) dt + σrdWr(t), (5.11) 

where ar, br, σr are positive constants, θr(t) 11 is a continuous, deterministic function and 

dWr(t) is a standard Brownian motion. 

The mean reverting level for the non-defaultable nominal short-rate is given by 

MRLr(t) = θr(t)+brω(t) 

. ar 

As one can see, the growth rate of the GDP at time t has a positive influence on the 

mean reverting level of the short-rate, i.e. a higher GDP leads to a higher long-term mean 

of the short rate. 

Now, we define the second process within cascade 2 – the short-rate credit spread. Its 

dynamics is described by a three-factor model, where one driving factor is the so-called 

uncertainty index u. It can be interpreted as an aggregation of all available information 

10 Let X be a random variable with density function f. Then, the mean E(X) of a random variable X 

is defined by 

E(X) := 

� +∞ 

−∞ 

The variance V(X) of a random variable X is defined by 

Then, its standard deviation is given by 

x · f(x)dx. (5.6) 

V(X) := E (X − E(X)) 2¡ . (5.7) 

σ(X) := � V(X). (5.8) 

Let X1, . . . , Xn be random variables with E(X 2 ) < ∞, k = 1, . . . , n. For V(Xi) > 0 and V(Xj) > 0 the 

Pearson correlation coefficient of Xi and Xj is defined by 

ρ(Xi, Xj) = ρi,j := 

Cov(Xi, Xj) 

, i, j = 1, . . . , n, (5.9) 

σ(Xi) · σ(Xj) 

with covariance of Xi and Xj, Cov(Xi, Xj) given by 

� 

� 

Cov(Xi, Xj) := E (Xi − E(Xi)) · (Xj − E(Xj)) , i, j = 1, . . . , n. (5.10) 

11 An explicit expression of θr(t) and its derivation is given for example in [Mey05], pp. 40. θr(t) is 

determined in such a way that the interest-rate curve at simulation start date fits the market curve. More 

information about θr(t) and its connection to the interest-rate curve can be found in 5.3. 

71


about the quality of the firm. The higher the value of the uncertainty process the lower 

the firm’s quality. 12 

The dynamics of the short-rate spread is described by 

du(t) = [θu − auu(t)] dt + σudWu(t), (5.12) 

ds(t) = [θs + bsuu(t) − bsωω(t) − ass(t)] dt + σsdWs(t), (5.13) 

where au, σu, bsu, bsω, as, σs are positive constants, θu, θs are non-negative constants and 

dWu(t), dWs(t) are standard Brownian motions. 

The mean reverting level for the short-rate spread is given by 

MRLs(t) = θs+bsuu(t)−bsωω(t) 

. 

as 

The mean reverting level and the drift of the short-rate spread s depend on two more 

factors besides s, namely ω and u. This can be interpreted as follows. If the GDP grows 

at a higher rate, spreads usually tighten as the probability of default of a firm gets smaller, 

and vice versa. This negative correlation is expressed by a negative sign. u represents the 

firm specific risk and has a positive influence on credit spreads, i.e. a higher risk implies 

a higher credit spread. 13 

Cascade 3 Continuous stock returns RE(t) are also modelled with a SDE. The dynamics 

of RE(t) for the period [0, t] is expressed according to 

dRE(t) = [αE + bEωω(t) − (bER + bEi)i(t) + bERr(t)] dt + σEdWE(t), (5.14) 

where αE, bEω, bEi, bER, σE are positive constants and dWE(t) is a standard Brownian 

motion. 14 

The process for the stock returns does not have the mean reverting property. Regarding 

equation (5.14) it becomes obvious that both, the short-rate and the GDP growth rate, 

have a positive influence on the stock returns. 

Processes under the Equivalent Martingale Measure So far, we saw the dynamics 

of the SDEs (given by the equations (5.4) – (5.13)) under the real measure Q. Though, as 

we are interested in zero-coupon bond prices as well as interest-rates and credit spreads 

for different terms, we need to know the parameters for the SDEs under the risk-neutral 

equivalent Martingale measure ˆ Q. 15 

As shown for example in [SZAI06], the ˆ Q-dynamics of ω, r, u and s are given by 

dω(t) = (θω − âωω(t)) dt + σωd ˆ Wω(t), (5.15) 

dr(t) = (θr(t) + brω(t) − ârr(t)) dt + σrd ˆ Wr(t), (5.16) 

12 

See [SZAI06], p. 9. 

13 

According to [SZAI06], p. 13, this is in line with literature. 

14 

See [ZMH06]. 

15 

For an introduction into risk neutral valuation, we refer for example to [BK04]. 

72


du(t) = [θu − âuu(t)] dt + σud ˆ Wu(t), (5.17) 

ds(t) = [θs + bsuu(t) − bsωω(t) − âss(t)] dt + σsd ˆ Ws(t), (5.18) 

where âi = ai + λiσ 2 i , i = ω, r, u, s. λω, λr, λu, λs are obtained when changing measure 

from Q to ˆ Q by applying Girsanov’s Theorem 16 . 17 

Now, the foundation is built to determine zero-coupon bond prices in the Zagst model 

for both non-defaultable and defaultable zero-coupon bonds, which in turn are the basis 

for calculating the zero rates. 

Theorem 5.7 (Price of a non-defaultable zero-coupon bond) 

The time t price of a non-defaultable zero-coupon bond with maturity T is given by 

with 

and 

Proof: 

A(t, T ) = 

P (t, T ) = P (t, T, r(t), ω(t)) = e A(t,T )−B(t,T )r(t)−D(t,T )ω(t) , 

D(t, T ) = br 

� T 

t 

âr 

B(t, T ) = 1 

âr 

� 

1 − e−âω(T −t) 

âω 

� 

−âr(T 

1 − e 

−t)� 

, 

+ e−âω(T −t) − e 

âω − âr 

−âr(T −t) 

� 

1 

2 (σ2 rB(l, T ) 2 + σ 2 ωD(l, T ) 2 � 

) − θr(l)B(l, T ) − θωD(l, T ) dl. 

See [SZAI06]. � 

From zero-coupon bond prices the implicit rate of return can easily be calculated 

according to the following theorem. 

Theorem 5.8 (Zero Rate) 

The zero rate (or spot rate) of a non-defaultable zero-coupon bond at time t with maturity 

T is given by 

R(t, T ) = −1 

[A(t, T ) − B(t, T )r(t) − D(t, T )ω(t)] (5.19) 

T − t 

with A(t, T ), B(t, T ), D(t, T ) as in Theorem 5.7. 

Proof: 

The zero rate R(t, T ) can be calculated plugging P (t, T ) into equation (5.2). � 

Theorem 5.9 (Price of a defaultable zero-coupon bond) 

The price of a defaultable zero-coupon bond at time t < min(τ, T ), with maturitiy T and 

default time τ, is given by 

P d (t, T ) = P (t, T, r(t), ω(t), u(t), s(t)) = e Ad (t,T )−B(t,T )r(t)−D d (t,T )ω(t)−E d (t,T )u(t)−F d (t,T )s(t) , 

16 Girsanov Theorem is given in A.18. 

17 See [SZAI06] pp. 12-13. 

� 

73

with 


Proof: 

A d (t, T ) = 

D d (t, T ) = br 

− bsω 

âs 


âr 

E d (t, T ) = bsu 

� T 

t 

� T 

+ 

B(t, T ) = 1 

âr 

� 

1 − e−âω(T −t) 

âω 

� 

1 − e−âω(T −t) 

âs 

âω 

� 

1 − e−âu(T −t) 

âu 

F d (t, T ) = 1 

âs 

� 

−âr(T 

1 − e 

−t)� 

, 

+ e−âω(T −t) − e 

+ e−âω(T −t) − e 

âω − âs 

âω − âr 

−âs(T −t) 

+ e−âu(T −t) − e 

âu − âs 

� 

−âs(T 

1 − e 

−t)� 

, 

−âr(T −t) 

� 

−âs(T −t) 

� 

1 

2 (σ2 rB(l, T ) 2 + σ 2 ωD d (l, T ) 2 + σ 2 uE d (l, T ) 2 + σ 2 sF d (l, T ) 2 � 

) dl 

� 

−θr(l)B(l, T ) − θωD d (l, T ) − θuE d (l, T ) − θsF d � 

(l, T ) dl. 

t 

See [SZAI06]. � 

Theorem 5.10 (Defaultable Zero Rate and Credit Spread) 

The zero rate (or spot rate) of a defaultable zero-coupon bond at time t < min(τ, T ), with 

maturity T and default time τ, is given by 

R d (t, T ) = − 1 

� 

A 

T − t 

d (t, T ) − B(t, T )r(t) − D d (t, T )ω(t) 

−E d (t, T )u(t) − F d � 

(t, T )s(t) . 

Furthermore, the credit spread is given by 

S(t, T ) = −1 

� 

A 

T − t 

d (t, T ) − A(t, T ) − (D d (t, T ) − D(t, T ))ω(t) 

−E d (t, T )u(t) − F d � 

(t, T )s(t) , 

with A(t, T ), B(t, T ), D(t, T ), A d (t, T ), D d (t, T ), E d (t, T ), F d (t, T ) as in Theorem 5.7 and 

5.9, respectively. 

Proof: 

For calculating the zero rate R d (t, T ) see proof of Theorem 5.8. 

The credit spread can be easily calculated by S(t, T ) = R d (t, T ) − R(t, T ). � 

The model of Zagst is used for economic scenario generation for the following analyses. 

risklab germany GmbH implemented an economic scenario generator (ESG), based on the 

� 

� 

74


model described above. Before we can start the simulations, we need to calibrate the ESG 

to real data. This is described in Section 7.1.1. In the following section, we show how to 

fit the interest-rate curve at simulation start date to market data. 

5.3 Fitting of the Interest-Rate Curve 

We already saw the function θr(t) in the dynamics of the short-rate process (see equation 

(5.11)). θr(t) has to be fitted in such a way that at simulation start date, the current 

interest-rate curve is described as good as possible. We need to know the complete interest- 

rate curve at that date. From Bloomberg, for example, we get zero rates for different terms 

(3 months, 6 months, 1 year, 2, 3, 4, 5, 6, 7, 8, 9, 10, 15, 20 and 30 years). A very popular 

method to obtain the complete interest-rate curve from single data points goes back to 

Nelson and Siegel 18 . They describe the curve by 

T −t 

1 − exp(− β3 

R(t, T ) = β0 + (β1 + β2) · ) 

T − t 

− β2 · exp(− ), (5.20) 

T −t 

β3 

β3 

using only four parameters. The limits can be calculated as 


lim 

(T −t)→0 R(t, T ) = β0 + β1, (5.21) 

lim 

(T −t)→∞ R(t, T ) = β0. (5.22) 

From equation (5.22) we see that β0 describes the interest-rate curve for long terms, 

while short-term interest-rates are determined by the sum of β0 and β1 (see equation 

(5.21)). β2 and β3 are shape factors, on the contrary. All common interest-rate structures 

(normal, flat, inverse, convex, concave) can be produced with a curve described by equation 

(5.20), which is shown in Figure 5.2. 19 The figure shows different outcomes of interest-rate 

curves with parameters β0 = 5%, β1 = −1%, β2 ∈ {−6%, −3%, 0%, 3%, 6%, 9%, 12%} and 

β3 = 100%. 

With the ESG, we simulate the government interest-rate curve, the credit spread curve 

and the curve of the quoted CDS indices which are the basis for pricing various financial 

instruments. We price these instruments taking into account correlated defaults of the 

reference entities. One way how to obtain correlated default times, is described in the 

next section. 

5.4 Simulation of Correlated Default Times 

For the pricing of different financial instruments, particularly corporate-related instru- 

ments, it is necessary to also consider defaults. We use a one-factor copula approach to 

generate correlated default times. On the one hand, we use the Gaussian copula, which 

is quite popular in practice due to an easy implementation and the properties of a stan- 

dard normal distribution, such as the stability under convolution. On the other hand, 

18 For details we refer to [NS87]. 

19 Inspired by [NS87]. 

75


Figure 5.2: Nelson-Siegel interest-rate curves 

we use the normal inverse Gaussian (NIG) copula, which is able to overcome the mod- 

elling deficiencies of the Gaussian copula while remaining the property of stability under 

convolution. To determine the correlated default times, it is necessary to determine a 

distribution function for the default time τ. For this reason, we use a migration matrices 

approach, which is described in the following section. 

5.4.1 Migration Matrices Approach 

In this subsection, we primarily follow [Blu03]. 20 

It is common sense and also confirmed for example by Standard & Poor’s that there is 

a clear correlation between credit quality and default remotedness: the higher the issuer’s 

rating, the lower its probability of default, and vice versa. 21 

We assume a set of ratings 22 R ∈ {AAA, AA, A, BBB, BB, B, CCC/C}, where AAA 

denotes the best credit quality, CCC/C denotes the worst non-defaulted credit quality 

and R D ∈ {AAA, AA, A, BBB, BB, B, CCC/C, D}, where D denotes the state of default. 

Finding generalisations of the following results for finer rating scales is straightforward. 

Each rating class can be mapped to a unique probability of default. Table 5.1 shows 

the US average one year probability of default for each rating class. 23 

The aim of this section is to find a distribution function for the default time τ for any 

given rating R. On our way, we need to calibrate a credit curve for each element of R. 

20 See [Blu03], pp. 12. 

21 See [Sta06], p. 16. 

22 For more information regarding ratings we refer to the corresponding paragraph in Section 2.2.2 and 

Appendix B. 

23 The underlying data is taken from [Sta06] p. 15. 

76


Rating One-Year PD 

AAA 0.00% 

AA 0.01% 

A 0.04% 

BBB 0.30% 

BB 1.20% 

B 6.05% 

CCC/C 31.49% 

Table 5.1: US average one-year probability of default, 1981 to 2005 

Definition 5.11 (Credit Curve) 

A credit curve for a rating R is a mapping 

t ↦→ Q (R) (t) = P[R(t) = D | R(0)] (t ≥ 0; R(0) ∈ {AAA, AA, ..., CCC/C}), 

where D denotes the default state and R(t) the rating at time t. 

Thus, Q (R) (t) is the probability that an issuer with a current rating R(0) defaults 

within the next t years. Therefore, Table 5.1 displays the credit curve with Q (R) (1). 

There are several ways to calibrate a credit curve. One way is to use a Markov chain 24 

approach, based on one-year migration matrices published by rating agencies 25 . 

A migration or transition matrix is a quadratic matrix describing the probabilities of 

changing from one state to another. 

Table 5.2 shows the transition matrix containing US average one-year transition rates 

from 1981 to 2005, where issuers who withdrew their rating were removed and the row 

sums were normalised so that they sum up to one. 26 

AAA AA A BBB BB B CCC/C D 

AAA 92.24% 7.07% 0.55% 0.04% 0.09% 0.00% 0.00% 0.00% 

AA 0.62% 90.71% 7.69% 0.70% 0.07% 0.16% 0.03% 0.01% 

A 0.06% 1.95% 91.22% 5.97% 0.52% 0.19% 0.04% 0.05% 

BBB 0.02% 0.18% 4.01% 89.68% 4.85% 0.82% 0.14% 0.30% 

BB 0.05% 0.07% 0.34% 6.03% 82.68% 8.80% 0.83% 1.20% 

B 0.00% 0.07% 0.23% 0.34% 5.83% 82.63% 4.86% 6.05% 

CCC/C 0.00% 0.00% 0.42% 0.62% 1.56% 10.81% 55.10% 31.49% 

D 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 100.00% 

Table 5.2: US average one-year transition rates, 1981 to 2005 

This matrix has to be read as follows. The probability for an issuer who is AAA-rated 

in t to become AA-rated in t + 1 is 7.07% or to become A-rated in t + 1 is 0.55%. 

The migration matrix M = (mij)i,j=1,...,8 from Table 5.2 is needed for the following 

theorem. 

24 For some background regarding Markov chains we refer for example to [Nor98]. 

25 See for example [Sta06]. They provide tables for transition rates by region for a certain year (p.14), 

or global average one-year transitions rates for a certain period (pp. 15-16). 

26 The average one-year transition matrix for the European Union from 1981 to 2005 is given in Appendix 

D. 

77

Theorem 5.12 (Log-Expansion) 


Let I be the identity matrix and M = (mij)i,j=1,...,n a migration matrix which is strictly 

diagonal dominant, i.e. mii > 1 

2 

Õn = 

for every i. Then, the log-expansion 

n� 

k+1 (M − I)k 

(−1) 

k 

k=1 

converges to a matrix Õ = (oij)i,j=1,...,n satisfying 

1. � n 

j=1 õij = 0 ∀i = 1, ..., n, 

2. exp( Õ) = M.27 

The convergence Õn → Õ is geometrically fast. 

Proof: 

(n ∈ N) 

See [Blu03]. � 

Remark 5.13 

The generator of a time-continuous Markov chain is given by a matrix O, O = (oij)1≤i,j≤n, 

satisfying the following properties: 

1. � n 

j=1 oij = 0 ∀i = 1, ..., n, 

2. −∞ < oii ≤ 0 ∀i = 1, ..., n, 

3. oij ≥ 0 ∀i = 1, ..., n with i �= j. 

Theorem 5.14 is a standard result from Markov chain theory. 

Theorem 5.14 

The following properties are equivalent for a matrix O ∈ R n×n . 

(a) O satisfies properties 1 to 3 in Remark 5.13. 

(b) exp(tO) is a stochastic matrix ∀ t ≥ 0. 

Proof: 

See [Nor98], Theorem 2.1.2. � 

Using Theorem 5.12, Remark 5.13 and Theorem 5.14 we can construct credit curves 

for every time t and every rating class R. At first, we calculate the log-expansion Õ = 

(õij)i,j=1,...,8 of the adjusted one-year migration matrix M = (mij)i,j=1,...,n with Theorem 

5.12. The resulting matrix Õ is displayed in Table 5.3. 

In order to be a generator matrix, 

Õ has to satisfy the properties enumerated in 

Remark 5.13. Property 1 is satisfied because it is guaranteed by Theorem 5.12. Obviously, 

27 The matrix exponential function is defined analogously to the ordinary exponential function: 

Let X be a n × n matrix, the exponential of X, denoted by exp(X), is defined as 

exp(X) = �∞ X 

k=0 

k 

k! . 

78



AAA -8.10% 7.73% 0.28% 0.01% 0.10% -0.01% 0.00% 0.00% 

AA 0.68% -9.86% 8.45% 0.49% 0.04% 0.17% 0.03% 0.00% 

A 0.06% 2.13% -9.43% 6.59% 0.41% 0.16% 0.04% 0.03% 

BBB 0.02% 0.15% 4.43% -11.24% 5.61% 0.65% 0.14% 0.24% 

BB 0.06% 0.06% 0.22% 7.00% -19.61% 10.60% 0.82% 0.84% 

B 0.00% 0.07% 0.23% 0.12% 7.02% -19.96% 7.16% 5.37% 

CCC/C 0.00% -0.01% 0.54% 0.78% 1.68% 15.88% -60.27% 41.41% 

D 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 

Table 5.3: Log-expansion of M 

property 2 is also satisfied. But property 3 is hurt twice by õAAA,B and by õ CCC/C,AA. 

Bluhm suggests to set these values to zero and in return, to decrease the diagonal elements 

of the corresponding rows. The resulting matrix is the generator matrix O = (oij)i,j=1,...,8. 

Bluhm justifies this manipulation since the implied error, calculated as euclidian distance 28 

of M and exp(O) 

is negligible. 

||M − exp(O)||2 

Now, we can calculate the credit curve 29 for every t ≥ 0 by 

Q (R) (t) = (exp(tO)) i(R),8, 

where i(R) denotes the transition matrix row corresponding to the given rating R. Figure 

5.3 shows the credit curve calculated on a quarterly basis for 400 years. 

It becomes obvious that the probability for a default within the next t years is the higher 

the worse the credit quality is. Furthermore, it can be observed that the sub-investment- 

grade ratings slow down their growth, because – conditional on having survived for some 

time – the probability for further survival improves. 

If we assume the credit curves Q (R) (t) to be correct and really to give us the cumulative 

default probabilities for any rating R over any time interval [0, t], there is only one way to 

define the distribution of default time τ for an R-rated issuer, also stated in Lemma 5.15. 

Lemma 5.15 

Given a credit curve (Q (R) (t))t≥0 for a rating R, there exists a unique default times dis- 

tribution for R-rated obligors. 

Proof: 

Let us define a random variable τ (R) with τ (R) ∈ [0, ∞). Then the distribution function 

for τ (R) is given by 

� 

Fτ (R)(t) = P τ (R) � 

≤ t = Q (R) (t). (5.23) 

28 The euclidian distance is the length of a vector x ∈ R n and is defined as follows: 


||x||2 := �� n 

i=1 x2 i . 

79


Figure 5.3: Credit curve (Q (R) (t))0≤t≤400 

Equation (5.23) is used when we generate correlated default times with one-factor 

copula models as described in Section 5.4.2. 

5.4.2 One-Factor Copula 

As described in detail in Chapter 3, there are single-name credit derivatives, such as credit 

default swaps and total return swaps based upon a single underlying credit risk, and multi- 

name credit derivatives that are associated with a portfolio of credit risks. Central to the 

pricing of multi-name credit derivatives is the problem of default correlation. 

In reality, during a recession more firms default than during a booming period. This 

implies that each firm is subject to the same set of macroeconomic environment, and that 

there exists a dependence among the firms. 

The one-factor copula approach for modelling correlated default times has become very 

popular. It is assumed that defaults of different titles in the portfolio are independent, 

conditional on a common market factor. We begin with an introduction of the one-factor 

Gaussian copula. 

One-Factor Gaussian Copula 

In this section we follow primarily [HW04] and [KSW05]. After having defined the normal 

distribution 30 , we define the one-factor Gaussian copula. 

Definition 5.16 (Normal Distribution) 

A random variable X follows a normal distribution with mean µ and variance σ 2 , if its 

density function has the form 

30 The terms ”normal distribution” and ”Gaussian distribution” can be used synonymously. 

� 

80


fN (x) = 

� 

1 

(x − µ)2 

√ · exp − 

2πσ2 2σ2 � 

. (5.24) 

We denote the distribution function by FN (x) and we write X ∼ N (µ, σ 2 ). 

Definition 5.17 (One-Factor Gaussian Copula) 

Consider a portfolio of m credit instruments. The standardised asset return up to time t 

of the i th issuer in the portfolio, Ai(t), is assumed to be of the form 

Ai(t) = aiM(t) + 

� 

1 − a2 i Xi(t), (5.25) 

where M(t) and Xi(t), i = 1, ..., m are independent standard normally distributed random 

variables. 

Let us assume that default occurs when the asset return of obligor i falls below the threshold 

Ci(t), i.e. Ai(t) ≤ Ci(t). Using this copula model, the variable Ai(t) is then mapped to 

default time τi of the i th issuer with a percentile-to-percentile transformation 31 , i.e. 

P[τi ≤ t] = P[Ai(t) ≤ Ci(t)]. (5.26) 

Note that – due to the stability of normal distributions under convolution – the asset 

return Ai(t) is also standard normally distributed. 

The factor M represents the systematic common market factor and Xi represents 

firm-specific factors. Equation (5.25) defines a correlation structure between the random 

variables Ai(t). The correlation between asset returns of the issuers i and j is given by 

aiaj. Conditional on M the asset returns of different issuers are independent. 

Let the probability of the issuer i to default before time t be denoted by 

Qi(t) = P [τi ≤ t] , (5.27) 

where we use for Qi(t) the distribution function for default times derived in Section 5.4.1. 

Let n denote the number of scenarios and m the number of correlated default times 

for one scenario. m is the number of reference entities that we want to consider, i.e. for 

every single-name instrument, such as a bond, we generate one default time, for a multi- 

name instrument, such as a CDS index, we generate for example 125 default times if the 

reference portfolio consists of 125 entities. We can generate the correlated default times 

according to Algorithm 5.1. 

Algorithm 5.1 (Generation of correlated default times with a one-factor Gaussian 

copula model) 

1. Simulate M ∼ N (0, 1). 

2. Simulate Xi ∼ N (0, 1). 

� 

3. Compute Ai = aiM + 1 − a2 i Xi. 

31 In words: A percentile-to-percentile transformation means that the k-percentile point in the probability 

distribution for Ai is transformed to the k-percentile point in the probability distribution of τi. 

81


4. Determine the correlated default times for all i ∈ {1, ..., m} according to 

t = Q −1 

i [FN (Ai)]. 

One technique that simplifies the analysis of relatively homogeneous credit portfolios 

with many reference entities, such as CDS indices, is the so-called large homogeneous 

portfolio approximation method, introduced by Vasicek ([Vas87]). This approximation is 

an asymptotic version of the one-factor Gaussian model of correlated defaults. 

Definition 5.18 (Large Homogeneous Portfolio (LHP)) 

The large homogeneous portfolio is a portfolio consisting of a sufficiently large number of 

issuers having the same characteristics: 

• the same portfolio weight, 

• the same default probability Q(t), 

• the same recovery rate REC, 

• the same correlation to the market factor a. 

From this definition it becomes evident that the correlation between issuer i and j 

reduces to a 2 , as the correlation between the addresses in the LHP approach is assumed 

to be constant and equal, i.e. ai = a. 

One-Factor Normal Inverse Gaussian Copula 

In this section, we first define and describe the Normal Inverse Gaussian (NIG) distribu- 

tion. Thereafter, we describe the one-factor NIG copula model, which was introduced by 

Kalemanova et. al 32 . 

Definition and Properties of the Normal Inverse Gaussian Distribution The 

NIG distribution is a mixture of normal and inverse Gaussian distributions. It is a four 

parameter distribution with very interesting properties: It can produce fat tails and skew- 

ness, it is stable under convolution (under certain conditions), and the density function, 

the distribution function and the inverse distribution function can be computed sufficiently 

fast. 33 

Definition 5.19 (Inverse Gaussian Distribution) 

A non-negative random variable X has an Inverse Gaussian (IG) distribution with para- 

meters α > 0 and β > 0 if its density function has the form 

⎧ 

⎪⎨ 

fIG(x; α, β) = 

⎪⎩ 

32 See [KSW05]. 

33 See [KSW05] and [Tem05]. 

α 

√ x 

2πβ −3/2 � 

exp − 

� 

(α − βx)2 

2βx 

, if x > 0 

0 , if x ≤ 0. 

82

The corresponding distribution function is 

⎧ 

⎪⎨ 

FIG(x; α, β) = 

⎪⎩ 

Then, we write X ∼ IG(α, β). 


α 

√ 2πβ 

� x 

0 

z −3/2 � 

� 

(α − βz)2 

exp − dz , if x > 0 

2βz 

0 , if x ≤ 0. 

The four central statistics – mean 34 , variance 35 , skewness 36 and kurtosis 37 – are given 

by the following lemma. 

Lemma 5.20 

Mean, variance, skewness and kurtosis of an Inverse Gaussian distributed random variable 

X ∼ IG(α, β) are 

Proof: 

E(X) = α 

β 

V(X) = α 

β 2 S(X) = 3 

√ α 

K(X) = 3 + 15 

α . 

See [Tem05]. � 

Definition 5.21 (Normal Inverse Gaussian distribution) 

A random variable X follows a Normal Inverse Gaussian (NIG) distribution with parame- 

ters α, β, µ and δ if 

X | Y = y ∼ N (µ + βy, y) 

Y ∼ IG(δγ, γ 2 ) with γ := � α 2 − β 2 , 

with parameters satisfying the following conditions: 0 ≤ |β| < α and δ > 0. Then, we write 

X ∼ N IG(α, β, µ, δ) and denote the density and probability functions by fN IG(x; α, β, µ, δ) 

and FN IG(x; α, β, µ, δ) correspondingly. 

34 

The mean is defined in equation (5.6) in footnote 10. 

35 

The variance is defined in equation (5.7) in footnote 10. 

36 

We already defined the sample skewness in equation (4.8). Now we define the continuous version. Let 

E(X) and σ 2 (X) denote the mean and the variance of the random variable X. For E(|X| 3 ) < ∞, the 

skewness S(X) of a random variable X is defined as 

�� 

3 

X − E(X) 

S(X) = E 

. (5.28) 

σ(X) 

37 

We already defined the sample excess kurtosis in equation (4.9). Now we define the continuous version 

of the kurtosis. Let E(X) and σ 2 (X) denote the mean and the variance of the random variable X. For 

E(|X| 4 ) < ∞, the kurtosis K(X) of a random variable X is defined as 

�� 

4 

X − E(X) 

K(X) = E 

. (5.29) 

σ(X) 

83


The density of the NIG distribution is then 

fN IG(x; α, β, µ, δ) = 

and the distribution function 

FN IG(x; α, β, µ, δ) = 

� x 

� ∞ 

0 

−∞ 

fN (x; µ + βy, y) · fIG(y; δγ, γ 2 )dy (5.30) 

� ∞ 

0 

fN (z; µ + βy, y) · fIG(y; δγ, γ 2 )dydz (5.31) 

with fN (x; µ, σ 2 ) the density function of the Gaussian distribution as given by Definition 

5.16. 

Figure 5.4 gives an impression of various forms of the density function with NIG 

distribution, depending on different parameter sets. Note that in every figure we only 

change one of the parameters, while the others remain fixed. Additionally, we plot the 

density function of the standard normal distribution (solid red line) in every sub-figure to 

gain a deeper understanding of the forms of the NIG distributions. 

Figure 5.4: Densities of NIG distribution 

From Figure 5.4 we see that the NIG distribution can produce fat tails and skewness. 

The distribution is symmetric if β = 0. The central statistics of the distribution are given 

by the following lemma. 

84

Lemma 5.22 


Mean, variance, skewness and kurtosis of a random variable X ∼ N IG(α, β, µ, δ) are: 

Proof: 

E(X) = µ + δ β 

γ 

β 

S(X) = 3 

α · √ δγ 

V(X) = δ α2 

γ3 � � � � 

2 

β 1 

K(X) = 3 + 3 1 + 4 

α δγ . 

See [Tem05]. � 

The main properties of the NIG distribution are the scaling and the convolution prop- 

erties which are given by the following lemma. 

Lemma 5.23 

(a) Let X be a NIG distributed random variable (X ∼ N IG(α, β, µ, δ)) and let c be a 

scalar, then cX is NIG distributed with parameters 

This is called scaling property. 

� � 

α β 

cX ∼ N IG , , cµ, cδ . (5.32) 

c c 

(b) Let X and Y be independent random variables with X ∼ N IG(α, β, µ1, δ1) and 

Proof: 

Y ∼ N IG(α, β, µ2, δ2). Then, their sum is also NIG distributed with parameters 

X + Y ∼ N IG (α, β, µ1 + µ2, δ1 + δ2) . (5.33) 

See [KSW05]. � 

The convolution property (5.33) does not hold for arbitrary NIG random variables X 

and Y . X and Y must rather have the same α and β. 

One-Factor NIG Copula Model Now, we use the one-factor copula model of corre- 

lated defaults with a NIG distribution. For this reason, we partly follow [KSW05] in this 

section. 

Definition 5.24 (One-factor NIG copula) 

Consider a homogeneous portfolio of m credit instruments. The standardised asset return 

up to time t of the i th issuer in the portfolio, Ai(t), is assumed to be of the form 

with independent random variables 

Ai(t) = aM(t) + � 1 − a 2 Xi(t), (5.34) 

� 

M(t) ∼ N IG 

α, β, − βγ2 γ3 

, 

α2 α2 � 

, (5.35) 

85



�√ √ √ 

1 − a2 1 − a2 1 − a2 Xi(t) ∼ N IG α, β, − 

a a 

a 

where γ = � α 2 − β 2 . 

βγ2 , 

α2 √ 1 − a 2 

a 

γ3 α2 � 

, (5.36) 

Let us assume that default occurs when the asset return of obligor i falls below the threshold 

Ci(t), i.e. Ai(t) ≤ Ci(t). Using this copula model, the variable Ai(t) is then mapped to 

default time τi of the i th issuer with a percentile-to-percentile transformation, i.e. 

P[τi ≤ t] = P[Ai(t) ≤ C(t)]. (5.37) 

The distribution parameters for M(t) and Xi(t) are chosen in such a way that the 

asset return Ai(t) also follows a NIG distribution with the parameters 

� 

α 

Ai ∼ N IG 

a 

β 

, , −1 

a a 

βγ2 1 

, 

α2 a 

γ3 α2 � 

. (5.38) 

This can be easily verified by applying the scaling property (5.32) and the stability under 

convolution (5.33). Furthermore, Ai(t) has zero mean and unit variance. � 

To simplify the notation, we denote the distribution function FN IG x; sα, sβ, −s βγ2 

α2 , s γ3 

α2 � 

by F N IG(s)(x). Then, the distribution function of the factor M is F N IG(1)(x), of the factor 

Xi, it is F √ 

1−a2 N IG( ) a 

(x) and of Ai, it is F 1 

N IG( a )(x). 

Let the probability of default before time t again be denoted by 

Q(t) = P [τi ≤ t] , (5.39) 

where we use for Q(t) the distribution function for default times derived in Section 5.4.1. 

Again, let n denote the number of scenarios and m the number of correlated default 

times for one scenario. 38 

times. 

Analogous to Algorithm 5.1, we can then generate a set of m × n correlated default 

Algorithm 5.2 (Generation of correlated default times with a one-factor NIG 

copula model) 

1. Simulate M, distributed according to equation (5.35). 

2. Simulate Xi, distributed according to equation (5.36). 

3. Compute Ai = aM + √ 1 − a 2 Xi. 

4. Determine the correlated default times for all i ∈ {1, ..., m} according to 

t = Q −1 [FN IG(Ai)]. 

38 The procedure is the same as already explained in Section 5.4.2. 

86


We want to demonstrate the difference between correlated default times generated with 

Gaussian and NIG copula. For this reason, we simulate correlated default times for 5000 

pairs of equally rated firms with the migration matrices approach, described in Section 

5.4.1, with correlation between the reference entities of 0.1, 0.3, 0.5 and 0.7. The results 

are displayed in Figure 5.5. The orange squares represent default times, generated with 

the NIG copula, the grey squares represent default times generated with the Gaussian 

copula. We are particularly interested in default times occurring in the next 10 years after 

generation, since this is a medium-term investment horizon. 

We make three main observations: 

• With a lower rating, pairwise default times increase significantly within the first 10 

years. 

• Pairwise default times increase significantly with a higher correlation between the 

reference entities within the first 10 years. 

• Pairwise defaults occur more frequently within a 10-year time horizon, if the default 

times are simulated with one-factor NIG copula. 

Furthermore, we show the outcome of step 3 in the Algorithms 5.1 and 5.2 for rating 

class AA with correlation of 0.3, from the simulation above. The result is shown in Figure 

5.6. 

Here, it becomes obvious that the one-factor NIG copula produces considerably more 

negative asset returns of several issuers than the one-factor Gauss copula. This implies a 

higher potential of joint defaults, which in turn has implications on the pricing of financial 

instruments, particularly CDO tranches. 39 

39 See [KSW05] for the fit of CDO tranches with one-factor Gauss and one-factor NIG copula. 

87


Figure 5.5: Correlated default times with Gaussian and NIG one-factor copula model 

88


Figure 5.6: One-factor Gaussian copula vs. one-factor NIG copula 

89


5.5 Pricing of Financial Instruments 

Before pricing, we simulate 1000 scenarios with the ESG on a quarterly basis for 5 years. 

Therefore, we have 21 timesteps for each scenario in the end, as the first one is the start 

date 30 th of September, 2006. As output 40 from the simulation we obtain for each scenario 

and for every timestep 

• a zero curve for government bonds, R(t, T ), which is supposed to be non-defaultable, 

• a credit spread curve for the rating classes R ∈ {AA, A, BBB}, SR(t, T ) (i. e. credit 

spreads for different terms), 

• a curve for the quoted spread of the CDS index 41 , SCDS−index(t, T ), 

• return of an equity index. 

Then, we have to generate correlated default times for each simulated scenario. For 

this reason, we apply a migration matrices approach 42 , which generates correlated default 

times with both, Gaussian and NIG copula, as described by the Algorithms 5.1 and 5.2, 

respectively. 

The ESG provides us with interest-rate curves, credit spread curves and the curve 

for the CDS index with certain terms. For pricing issues, however, we will also need 

arbitrary maturities not necessarily coinciding with simulated maturities. 43 Besides, we 

will occasionally need the interest-rate curve at an arbitrary time t, which is not simulated. 

For this reason, we conduct a linear interpolation of the curves, if necessary. 

We price the instruments at every simulated timestep tk, k ∈ {0, ..., m} 44 , with 

t0 < t1 < ... < tm = T sim , 

where T sim denotes the end date of the simulation. As we conduct quarterly simulations, 

the stepsize of the simulated timesteps, denoted by ∆tk, is in our case ∆tk = tk − tk−1 = 

0.25. 

Remark 5.25 

The pricing for all considered instruments is done under the conditions listed below. 

(a) It is always assumed that the cash flows occurring before maturity T are reinvested 

in the considered instrument. 

(b) The cash flows at maturity T or at default time τ are invested in a risk-free cash 

account with the default risk-free 3 month government rate, if T or τ occur before 

T sim , i.e. if min(τ, T ) < T sim . 

(c) The pricing is always done for an initial investment of one unit. 

40 

All output values are determined according to equations given in Section 5.2. 

41 

The dynamics for quoted CDS spreads is calculated using Itô’s Lemma. The calculation can be found 

in Appendix C. 

42 

See Section 5.4.1. 

43 

For example, think of the pricing day half a year after simulation start date. A zero-coupon bond with 

an original term of 5 years, now has a remaining term of 4.5 years. To price the bond, we need to know 

the interest-rate for a term of 4.5 years. 

44 

n = 21 in our case. 

90

5.5.1 Zero-Coupon Bonds 


We consider both – government and corporate zero-coupon bonds. 

The price of a government zero-coupon bond at tk with an initial term of T years is 

given by 45 

P (tk, T, ) = exp(−R(tk, T )(T − tk)). (5.40) 

As we consider the government bonds to be non-defaultable, we do not have to deal with 

default times. 

In the case of a corporate zero-coupon bonds, we need to deal with default times 

τ. Of course, the procedure is exactly the same for default times generated with either 

a one-factor Gaussian or NIG copula. If a default occurs before ˆ T (i.e. τ < ˆ T ), with 

ˆT = min(T, T sim ), the price of the zero-coupon bond at τ reduces to the recovery value 

RV , which is calculated as RV = 1 · REC 46 assuming a fixed recovery rate REC = 40% 

of the notional for every zero-coupon bond. 47 We need to compound the recovery value to 

the next simulation date tk. Then, we compound the value of the defaulted bond at every 

remaining pricing day with the 3 month government interest-rate. If no default occurs, 

the pricing works analogously to a government bond. In Figure 5.7, we illustrate what 

happens at default. 

Figure 5.7: Compounding of recovery payment 

If a default occurs between tk−1 and tk, we assume the recovery payment RV to be 

received at τ. As this happens without any uncertainties, we compound this payment 

with the risk-free government rate to the next pricing day tk. At tk the amount is then 

invested in a risk-free cash account. 

Formally, the price of a corporate zero-coupon bond with rating R is given by 

⎧ 

P d ⎪⎨ 

(tk, T, R, τ) = 

⎪⎩ 

with R d (tk, T, R) = R(tk, T ) + SR(tk, T ). 

exp(−R d (tk, T, R, τ)(T − tk)) , if τ > tk 

RV · exp(R(τ, tk)(tk − τ)) , if tk−1 < τ ≤ tk 

P d (tk−1, T, τ) · exp(R(tk−1, tk)(tk − tk−1)) , if τ ≤ tk−1, 

(5.41) 

For every simulated timestep tk > 0, we determine the total return of the investment, 

45 

See equation (5.1). 

46 

In general, RV is calculated according to RV = L · REC, where L denotes the notional amount. 

According to Remark 5.25, we assume L = 1. 

47 

This assumption is made for reasons of simplicity. According to [Sta06], p. 26, the average recovery 

rate for senior unsecured debt is 42.6%. 

91


R P , by calculating the price change from the previous timestep, tk−1, to the current 

timestep, tk, and relating it to the price in tk−1. Formally, R P can be expressed by 

for a government zero-coupon bond and by 

for a corporate zero-coupon bond. 

5.5.2 Coupon Bonds 

R P (tk−1, tk) = P (tk, T ) 

− 1 (5.42) 

P (tk−1, T ) 

R P (tk−1, tk, R, τ) = P d (tk, T, R, τ) 

P d − 1 (5.43) 

(tk−1, T, R, τ) 

We consider again government and corporate coupon bonds and we assume that a coupon 

bond has n coupon payment dates tc i , with coupon payments amounting to C: 

t c 1 < ... < t c n = T. 

t c 1 denotes the next coupon payment date following the current pricing day tk. At maturity 

T , the notional amount is also paid to the investor. 48 

We illustrate the possible cash flow structure with Figure 5.8. 

Figure 5.8: Coupon payment structure of a coupon bond 

At pricing days tk, we have to discount all future cash flows to determine the price. 

We begin specifying the price of a government coupon bond at simulated timestep 

tk ∈ � 0, ..., T sim� with an initial term of T : 

B(tk, T, C) = 

� 

i∈{j=1...n|t c j >tk} 

C · P (tk, t c i) + 1 · P (tk, T ). (5.44) 

We assume that there is no default, as we consider the government bonds to be non- 

defaultable. P (tk, tc i ) is then calculated by equation (5.40). 

If we price a corporate coupon bond with rating R, we need to consider default times 

τ. The treatment is analogous to that of zero-coupon bonds as described in Section 5.5.1. 

If a default occurs before ˆ T , with ˆ T = min(T, T sim ), the price of the coupon bond at 

τ reduces to the recovery value RV , which is calculated assuming a fixed recovery rate 

REC = 40% of the notional for every coupon bond. We need to compound the recovery 

value to the next simulation date tk. Then, we compound the value of the defaulted bond 

at every remaining pricing day with the 3 month government interest-rate. If no default 

48 For pricing issues, we assume a notional amount of 1 (see Remark 5.25). 

92


occurs, the pricing works analogously to a government bond. Formally, the price of a 

corporate coupon bond with rating R is given by 

⎧ 

⎪⎨ 

B(tk, T, C, R, τ) = 

⎪⎩ 

� 

i∈{j=1...n|t c j >tk} 

C · P d (tk, t c i, R, τ) + 1 · P d (tk, T, R, τ), if τ > tk 

RV · exp(R(τ, tk)(tk − τ)), if tk−1 < τ ≤ tk 

B(tk−1, T, C, R, τ) · exp(R(tk−1, tk)(tk − tk−1)), if τ ≤ tk−1, 

(5.45) 

For every simulated timestep, tk > 0, we determine the total return of the investment, 

R B , by calculating the price change from the previous timestep, tk−1, to the current 

timestep, tk, adding the cash flows in tk, i.e. the coupon payments, 49 and then relating it 

to the price in tk−1. Formally, R B can be expressed by 

for a government coupon bond and by 

R B (tk−1, tk, C) = B(tk, T, C) + C(tk) 

B(tk−1, T, C) 

R B (tk−1, tk, C, R, τ) = B(tk, T, C, R, τ) + C(tk) · 1 {τ>tk} 

B(tk−1, T, C, R, τ) 

− 1, (5.46) 

− 1, (5.47) 

for a corporate coupon-bond. Here C(tk) denotes coupon payments coinciding with pricing 

day tk and 1 {•} denotes the indicator function 50 , which is necessary as the coupon is only 

paid as long as no default has occurred. 

5.5.3 Excursus: Default Intensity Model 

To price CDS and CDS indices 51 , it is necessary to know the expected losses at every 

pricing day for every outstanding spread payment date. For this purpose, we apply the 

constant default intensity model, introduced in this section. 

In contrast to structural models, reduced form models do not tie defaults to funda- 

mental data of a firm, such as the stock market capitalisation or the leverage ratio. 52 

They rather assume defaults to be exogenous events that occur at unknown times τ. Fur- 

thermore, they are not assumed to be directly related to the firm’s balance sheet. These 

models want to assign probabilities to different outcomes of τ. 

Reduced form models characterise the random nature of defaults for an obligor typ- 

ically in terms of the first ”arrival” of defaults over time with a Poisson process. To 

49 

In our framework, the coupon payment dates coincide with simulated timesteps. Therefore, we can 

simply take the coupon payment for the return calculations. If, however, the coupon payment dates deviate 

from the pricing days, we would have to discount the payment at the risk-free rate to the next pricing day. 

50 

The indicator function is defined by 

� 

� 1 , if τ > tk 

1{τ>t k} = 

� 

0 , if τ ≤ tk. 

51 

The pricing procedure of funded CDS can be found in Section 5.5.5 and of funded CDS indices in 

Section 5.5.6. 

52 

For theoretical background regarding credit risk modelling, we refer for example to [DS03] or [Gie04]. 

93


understand these processes, we first need to define the Poisson distribution and the expo- 

nential distribution. 53 

Definition 5.26 (Poisson Distribution) 

A random variable X follows a Poisson distribution with parameter λ > 0 (X ∼ P(λ)), if 

Then, P [X ≤ x] is given by 

P [X = x] = exp(−λ) λx 

, x ∈ N0. 

x! 

P [X ≤ x] = 

x� 

i=0 

exp(−λ) λx 

x! . 

Mean and variance of a Poisson distributed random variable are given by 

E [X] = λ and V [X] = λ. 

In the context of Poisson distributions, we often see an exponential distribution 54 . If 

the occurrence of events follows a Poisson distribution, the time between those events is 

exponentially distributed. An exponential distribution is defined as follows. 

Definition 5.27 (Exponential Distribution) 

A random variable X follows an exponential distribution with parameter λ > 0 (X ∼ 

Exp(λ)), if its density function is of the form 

f(x) = λ exp(−λx), x ≥ 0. 

The corresponding distribution function has the form 

F (x) = 1 − exp(−λx), x ≥ 0. (5.48) 

Mean and variance of an exponential distributed random variable are given by 

E [X] = 1 

λ 

Now, we can define a Poisson process. 55 

Definition 5.28 (Poisson Process) 

and V [X] = 1 

λ 2 . 

Let the random variable X be Poisson distributed (X ∼ P(λ)), where X corresponds to 

the number of events occurring over a continuous time interval [0, ∞). Moreover, let the 

time between the occurrence of these events be independent and exponentially distributed 

with distribution function (5.48) assuming that defaults occur randomly at the mean rate 

(or intensity 56 ) of λ per year, with λ > 0. 

53 

For more background regarding the Poisson distribution, we refer for example to [Har95], pp. 213 or 

to [Kre02], p. 86. 

54 

For more background regarding the exponential distribution, we refer for example to [Har95], pp. 219 

55 

For more background regarding the Poisson processes, we refer for example to [Har95], p. 783 or 

[Kre02], pp. 227. 

56 

That is where the name ”default intensity model” comes from. 

94


Let Xt be the number of events which occurred in the time interval [0, t]. The stochastic 

process {Xt; t ≥ 0} is then called Poisson process with intensity λ. 

The probability function of {Xt} is given by 

P [Xt = x] = exp(−λt) (λt)x 

, x ∈ N0. (5.49) 

x! 

Hence, P [Xt = x] denotes the probability of exactly x defaults within a t-year time 

interval. Mean and variance of a Poisson process are consequently given by 

E [X] = λt and V [X] = λt. 

Let Xt be the stochastic process, corresponding to the number of defaults arriving over 

a continuous time interval and assume that defaults occur randomly at the mean rate of 

λ per year, with λ > 0. We assume Xt to follow a Poisson process. Now, we can easily 

calculate the unconditional probability of no defaults in a t-year time interval by setting 

x = 0 in equation (5.49): 

P [Xt = 0] = exp(−λt). (5.50) 

Analogously, we can calculate the unconditional probability of exactly one default in 

a t-year time interval by setting x = 1 in equation (5.49): 

P [Xt = 1] = λt · exp(−λt). (5.51) 

For pricing credit derivatives, the expected time of the first default is of great inter- 

est. Let τ be the time of the first default, which is a continuous random variable with 

distribution function Q(t). For Q(t) we can write 

⎧ 

⎨ P [τ ≤ t] = 1 − P [τ > t] , if t ≥ 0 

Q(t) = 

⎩ 

0 , if t < 0. 

Note that P [τ > t] is the probability of no defaults by time t, which is given by equation 

(5.50). Thus, for t ≥ 0, we can write 

Q(t) = 1 − exp(−λt). (5.52) 

This can be thought of as the unconditional probability of the first default by time t. 

The unconditional density function of τ is defined as ∂Q(t) 

∂t , and we can write 

q(t) := ∂Q(t) 

∂t 

= λ exp(−λt). (5.53) 

Note that q(t) is the density function and Q(t) the distribution function of an exponen- 

tially distributed random variable. Therefore, the time of first default τ is exponentially 

distributed when defaults follow a Poisson process. 57 

57 See [Bom05], pp. 317. 

95

5.5.4 Floating Rate Notes 


Now, we consider a floating rate note (FRN). We do not want to price FRNs as own 

instruments, but they will play an important role when it comes to pricing funded CDS 

or funded CDS indices, respectively. In our case, a FRN is supposed to be risk-free. It is 

assumed to have quarterly coupon payment dates tc i , with a coupon amounting to 3 month 

LIBOR, always determined at the previous coupon payment date, tc i−1 . At maturity T , 

the nominal amount is paid to the investor, too. The coupon payment dates are given by 

t c 1 < ... < t c n = T. 

t c 1 denotes the next coupon payment date following the current pricing day tk. 58 We denote 

by t c 0 

the previous coupon payment date or settlement date, if the first coupon payment 

has not yet been made. Thus, the coupon payment dates are described relative to pricing 

days and therefore move with pricing days. Figure 5.9 illustrates the relation between the 

pricing days and the coupon payments dates. 

Figure 5.9: Possible cash flow structure of a FRN 

Formally this relation can be expressed by t c 0 ≤ tk < t c 1 . 

In general, the price of a FRN at pricing day tk with a nominal value of 1, future 

coupon payments t c 1 , tc 2 , ..., tc n and not paying a spread over the reference interest-rate 59 is 

given by 60 

F RN(tk, t c 0, T ) = P (tk, t c 1 ) 

P (t c 0 , tc 1 ) − P (tk, T ) + P (tk, T ) 

where P (•, •) is calculated by equation (5.40). 

= P (tk, tc 1 ) 

P (tc 0 , tc , (5.54) 

1 ) 

Funded CDS and funded CDS indices pay LIBOR on the notional amount. Actually, 

LIBOR is a linear interest-rate 61 and has a small spread over the risk-free government 

rate. For the sake of simplicity, however, we use the simulated 3 month government rate 

representing LIBOR, i.e. the continuous rate is used as linear rate and the rate does 

not contain the spread above the risk-free rate, usually observed for the LIBOR. This 

procedure is the same for all CDS and CDS indices, so that the qualitative conclusions 

from our analyses remain unchanged. 

In our simulation and pricing framework, the price of a FRN at tk with a nominal 

58 Hence, we allow coupon payment dates not to coincide with pricing days. 

59 LIBOR can, for example, be such a reference rate. 

60 See [Zag02], pp. 182-184. 


96

value of 1 is then determined according to 62 


F RN(tk, t c 0, T ) = (1 + R(t c 0, t c 0 + ∆t c 1) · ∆t c 1) · P (tk, t c 1), (5.55) 

where (1 + R(tc 0 , tc0 + ∆tc1 ) · ∆tc 1 

1 ) corresponds to 

5.5.5 Funded CDS 

P (tc 0 ,tc 1 ) in equation (5.54). 

We consider CDS from a protection seller’s perspective. Hence we consider CDS as invest- 

ment rather than as a means of hedging. 63 In order to ensure a comparability to bonds, 

we use funded CDS. 

Before explaining the return components of a funded CDS, we briefly describe the 

relation of the relevant timesteps. At every simulated timestep tk ∈ � 0, ..., T sim� we price 

the funded CDS. Furthermore, we assume that 

t c 1 < ... < t c n = T 

denote the spread payment dates, where tc 1 denotes the next spread payment date following 

the current pricing day tk and T denotes the maturity of the CDS. We denote by tc 0 the 

previous spread payment date or settlement date, if the first spread payment has not 

yet been made. Formally this can be expressed by tc 0 ≤ tk < tc 1 . For an illustration of 

the relation of coupon payment dates and pricing days, we refer to Figure 5.9. Spread 

payments are made in arrear – at time t c i for the payment period from tc i−1 to tc i . 

A funded CDS has the following return components: 

• Present value of the CDS for every pricing day tk, P VCDS, resulting from possible 

changes in the default intensity λk at tk, 64 

• present value of the pure, default risk-free FRN at tk, P VF RN, paying LIBOR, 

• coupon payments on the notional amount at tc 0 , comprising LIBOR and the fixed 

spread s, compounded to the following simulated date tk, P Vs (see Figure 5.10), 

• recovery payments at default time τ, compounded to the following simulated date 

tk, P VREC (see Figure 5.11). 65 

For the pricing of a funded CDS, we partly follow [KSW05] adjusting the pricing 

formulas to our needs. 

To determine the present value of the funded CDS at tk, we need to know the expected 

loss at tk up to every spread payment day t c i , i ∈ {1, ..., n}, EL(tk, t c i , λk), the premium 

leg, Premium Leg(tk, T, s, λk), and the protection leg, Protection Leg(tk, T, λk), which are 

described below. 

62 In our case, ∆t c i = 0.25 as we assume quarterly coupon payments. 

63 The functionality of a funded CDS is described in Section 3.2, paragraph ”Product Variations”. 

64 At inception of a funded CDS, the fixed spread is determined so that P VCDS is equal to zero. During 

the term of the funded CDS contract, however, it also can be positive or negative, depending on the default 

intensity λk at tk. 

65 In contrast, the recovery value is already contained in the pricing formulas of the bonds, in the case 

of corporate bonds. (See Sections 5.5.1 and 5.5.2.) 

97


Figure 5.10: Coupon payment structure of a CDS 

Figure 5.11: Compounding of recovery payment 

The calculation of EL(tk, t c i , λk) is done by applying the constant default intensity 

model as introduced in Section 5.5.3, where we assume a constant default intensity λk for 

every pricing day tk. Then, the expected loss at tk up to t c i 

equation (5.56) 

is calculated according to 

EL(tk, t c i, λk) = 1 − exp(−λk(t c i − tk)). (5.56) 

Later in this section, we explain how to determine λk. 

The premium leg is the present value of all expected spread payments. It is calculated 

according to 

Premium Leg(tk, T, s, λk) = 

n� 

∆t c i · s · (1 − EL (tk, t c i, λk)) · P (tk, t c i), (5.57) 

i=1 

where ∆tc i = tci −tci−1 , s is the fixed annual spread of the series and P (tk, tc i ) is the discount 

factor calculated by equation (5.40). EL (tk, tc i , λk) is the expected loss up to tc i . Then, 

(1 − EL(tk, tc i , λk)) = exp(−λk(tc i − tk)) denotes the probability of no default up to time 

t c i . 

Analogously, the protection leg is the present value of all expected protection payments 

made by the protection seller. In case of zero recovery, we can calculate it by 

Protection Leg(tk, T, λk) = 

≈ 

� T 

tk 

P (tk, l)dEL(tk, l, λk) 

n� � 

EL(tk, t c i, λk) − EL(tk, t c i−1, λk) � P (tk, t c i). (5.58) 

i=1 

98


However, if we assume a constant recovery rate of REC = 40% 66 , we calculate the 

protection leg by 


≈ 

� T 

tk 

P (tk, l)dEL(tk, l, λk)(1 − REC) 

n� � 

EL(tk, t c i, λk) − EL(tk, t c i−1, λk) � · 

i=1 

·P (tk, t c i) · (1 − REC). (5.59) 

At issuance of the funded CDS, the fixed annual spread s is determined so that the 

value of the premium leg equals the value of the protection leg. In case of zero recovery 

this can be formally expressed by 

or by 

s = 

�n i=1 

� 

EL(tk, tc i , λk) − EL(tk, tc i−1 , λk) � P (tk, tc i ) 

�n i=1 ∆tc i · (1 − EL (tk, tc i , λk)) · P (tk, tc i ) 

, (5.60) 

�n � 

i=1 EL(t0, t 

s = 

c i , λk) − EL(t0, tc i−1 , λk) � P (t0, tc i )(1 − REC) 

�n i=1 ∆tc i · (1 − EL (tk, tc i , λk)) · P (tk, tc i ) 

, (5.61) 

if we assume a fix recovery rate REC. 

The equality of premium leg and protection leg, however, must also hold for every 

day, when s is substituted by the quoted market spread. Substituting s by the simulated 

spread with rating R, SR(tk, T ) 67 , at every pricing day tk in equations (5.60) or (5.61), 

respectively, we can solve them for λk. This is how we determine the constant default 

intensity at tk, which is necessary to calculate the present value of the funded CDS at 

every pricing day tk. 

The notional amount of a funded CDS at different timesteps tk can either be 1, if the 

reference entity has not defaulted up to tk, or 0, in case of default up to time tk. We 

denote the notional amount at each simulation date by N(tk, τ). Formally, we can express 

the value of the notional amount at tk by 

where 1 {•} denotes the indicator function 68 . 

N(tk, τ) = 1 {τ>tk}, (5.62) 

The pricing of the first return component, P VCDS(tk, T, s, λk, τ), the present value of 

the CDS, is then done by 

P VCDS(tk, T, s, λk, τ) = (Premium Leg(tk, T, s, λk) − Protection Leg(tk, T, λk)) · N(tk, τ), 

(5.63) 

where we calculate the premium leg and the protection leg according to equations (5.57) 

and (5.58), respectively. 

A funded CDS receives besides the fixed quarterly spread, s∆tc i , a quarterly coupon 

payment with a floating interest-rate amounting to 3 month LIBOR at the last coupon 

payment date tc i−1 . The value of this instrument was described in Section 5.5.4. 

66 We also assumed a constant recovery rate of 40% for corporate bonds for every rating class. 

67 We use the same spread for corporate bonds and for funded CDS with the same rating class. 

68 See equation (5.48). 

99


We take this interest-rate from our simulation by regarding the 3 month government 

rate as the corresponding LIBOR. 69 As LIBOR for the first coupon payment we take the 

LIBOR as of 20 th of September, 2006. Then, the present value of the FRN for every tk, 

P VF RN(tk, tc 0 , T, τ), is calculated according to 

P VF RN(tk, t c 0, T, τ) = F RN(tk, t c 0, T ) · N(tk, τ), (5.64) 

i.e. the FRN has a present value of zero, if the reference entity defaulted. 

As we have seen from Figure 5.9, t c 0 

is the relevant coupon payment date for pricing 

day tk. In case of default, we assume a proportional coupon payment as the funded CDS 

contract is assumed to be similar to an insurance contract. During the period between 

the last coupon payment, the protection seller guaranteed to carry a default-related loss. 

Therefore, he is to be compensated for this guarantee by a proportional coupon payment, 

too. For this reason it is necessary to consider the coupon payment date prior to t c 0 : 

t c −1 := 

⎧ 

⎪⎨ 

⎪⎩ 

∅ , if k = 0 

t0 , if k = 1 

t c −1 , if k > 1. 

To allow for a proportional coupon payment, if a default has taken place between t c −1 

and t c 0 , we introduce a time-weighted notional amount at the coupon payment date, tc 0 , 

N t (tk, tc 0 , τ), which is given by 

N t (tk, t c 0, τ) = 

⎧ 

⎪⎨ 

⎪⎩ 

0 , if k = 0 

0 , if τ ≤ t c −1 

τ−tc −1 

∆tc 0 

1 , if τ > tc 0 . 

, if t c −1 < τ ≤ tc 0 

Now we can describe the coupon payments the investor receives at the coupon payment 

date tc 0 , C(tk, tc 0 , s, τ). They consist of 3 month LIBOR and the fixed spread s on the 

weighted notional amount N t (tk, tc 0 , τ). At maturity, the investor additionally receives the 

actual notional amount N(T, τ). Then, the cash flows from the investment at t c 0 

by 

⎧ 

C(tk, t c ⎪⎨ 

0, s, τ) = 

⎪⎩ 

0 , if k = 0 

(s + R(t c −1 , tc −1 + ∆tc 0 )) · ∆tc 0 · N t (tk, t c 0 , τ) , if tk < T 

are given 

(s + R(tc −1 , tc−1 + ∆tc0 )) · ∆tc0 · N t (tk, tc 0 , τ) + N(T, τ) , if tc0 = T 

0 , if tk+1 > T. 

(5.65) 

In order to determine the present value of that payment at the next simulated timestep 

tk, P Vs(tk, tc 0 , s, τ), we need to compound C(tk, tc 0 , s, τ). This principle is shown in Figure 

5.10. Formally, the compounding is calculated as 

P Vs(tk, t c 0, s, τ) = C(tk, t c 0, s, τ) exp(R(t c 0, tk)(tk − t c 0)). (5.66) 

69 This is done for the sake of simplicity. For a brief discussion on implications, we refer to Section 5.5.4. 

100


Recovery payments are the last return component. Regarding a funded CDS, the 

protection seller receives a recovery payment at default time τ. On the other hand, the 

contract terminates. A default occurring before ˆ T (i.e. τ < ˆ T ), with ˆ T = min(T, T sim ), 

has to be treated in the following way: The investor receives the recovery value RV , which 

is calculated as RV = 1 · REC assuming a fixed recovery rate of REC = 40% in the case 

of default. We need to compound the recovery payment to the next simulation date tk. 

Then, the value of the defaulted funded CDS has to be compounded at every remaining 

pricing day. Formally, this is done with the following equation, analogous to the procedure 

already described for corporate (zero-) coupon bonds: 

⎧ 

⎪⎨ 

P VREC(tk, τ) = 

⎪⎩ 

0 , if τ > tk 

RV · exp(R(τ, tk)(tk − τ)) , if tk−1 < τ ≤ tk 

P VREC(tk−1, τ) · exp(R(tk−1, tk)(tk − tk−1)) , if τ ≤ tk−1, 

(5.67) 

We define the value of a CDS investment for every timestep tk, CDS(tk, T, s, λk, τ), 

containing all return components apart from the present value of the spread payments 

P Vs by 70 

CDS(tk, T, s, λk, τ) = P VCDS(tk, T, s, λk, τ)+P VF RN(tk, t c 0, T, τ)+P VREC(tk, τ). (5.68) 

For every simulated timestep tk > 0, we determine the total return of the investment, 

R CDS , by calculating the change in value from the previous timestep, tk−1, to the current 

timestep, tk, adding the compounded coupon payments paid between tk−1 and tk and then 

relating it to the value in tk−1. 71 Formally, R CDS can be expressed by 

R CDS (tk−1, tk) = CDS(tk) + P Vs(tk) 

CDS(tk−1) 

− 1. (5.69) 

For simplicity reasons, we omitted all input parameters for the single pricing components 

apart from the pricing day. 

5.5.6 Funded CDS Indices 

Now, we consider CDS indices. We take again the perspective of a protection seller 

considering funded CDS indices as investment rather than as means of hedging. In order to 

ensure a comparability to government and corporate bonds, we use funded CDS indices. 72 

We think of a CDS index as a large homogeneous portfolio as given in Definition 

5.18. Then, we can assume an average default intensity and thus, we can apply the same 

formulas as for CDS. These formulas were already introduced in Section 5.5.5. 73 We will 

70 

This definition is analogue to the definition of the price of a corporate coupon bond (see equation 

(5.45)), which also contains all return components apart from the coupon payments. 

71 

The return calculation is done analogously to the return calculation of a corporate coupon bond (see 

equation (5.47)). 

72 

We already described the functionality of both funded and unfunded CDS indices in Section 3.9. 

73 

There, we partly followed [KSW05] adjusting the pricing formulas to our needs. 

101

epeat the adjusted formulas here in short. 

Let us assume that 

denote the spread payment dates, where t c 1 


t c 1 < ... < t c n = T 

denotes the next spread payment date following 

the current pricing day tk and T denotes the maturity of the funded CDS index. 74 

At first, we describe the return components of a funded CDS index, which is priced at 

every simulated timestep tk ∈ � 0, ..., T sim� : 

• Present value of the funded CDS index for every pricing day tk, P Vindex, resulting 

from possible changes in the default intensity λk at tk, 75 , 

• present value of the pure, default risk-free FRN at tk, P VF RN, paying LIBOR, 

• coupon payments on the notional amount at tc 0 , comprising LIBOR and the fixed 

spread s76 , compounded to the following simulated date tk, P Vs (see Figure 5.10), 

• recovery payments at default times τ, compounded to the following simulated date 

tk, P VREC (see Figure 5.11). 77 

To determine the present value of the funded CDS index at tk, we need to know the 

expected loss at tk up to every spread payment day t c i , i ∈ {1, ..., n}, EL(tk, t c i , λk), the 

premium leg, Premium Leg(tk, T, s, λk), and the protection leg, Protection Leg(tk, T, λk), 

which are described below. 

Applying the constant default intensity model as introduced in Section 5.5.3, the ex- 

pected loss of a large homogeneous portfolio at tk up to every spread payment day t c i , 

EL(tk, t c i , λk) is given by equation (5.56) 

EL(tk, t c i, λk) = 1 − exp(−λk(t c i − tk)). 

Here, λk is determined in the same way as for funded CDS. 

We calculate the premium leg and the protection leg analogous to CDS. Therefore, the 

premium leg is calculated according to equation (5.57) 

Premium Leg(tk, T, s, λk) = 

n� 

∆t c i · s · (1 − EL (tk, t c i, λk)) · P (tk, t c i), 

i=1 

where ∆tc i = tci − tci−1 , s is the fixed annual spread of the series, P (tk, tc i ) is the discount 

factor and EL (tk, tc i , λk) is the expected loss calculated by equation (5.56). 

Assuming a constant recovery rate REC, we can calculate the protection leg by equa- 

74 

For more information, we refer to Section 5.5.5. 

75 

At inception of a funded CDS index, the fixed spread is determined so that P Vindex is equal to zero. 

During the term of the funded CDS index contract, however, it also can be positive or negative, depending 

on the default intensity λk at tk. 

76 

As explained in Section 3.9, there is one fixed spread for the index, representing a kind of average 

spread for the portfolio of reference entities. 

77 

In the case of corporate bonds, the recovery value is already contained in the pricing formulas of the 

bonds. (See Sections 5.5.1 and 5.5.2.) 

102

tion (5.59) 



≈ 

� T 

tk 

P (tk, l)dEL(tk, l, λk)(1 − REC) 

n� � 

EL(tk, t c i, λk) − EL(tk, t c i−1, λk) � P (tk, t c i)(1 − REC). 

i=1 

For pricing funded CDS indices at different timesteps tk, it is essential to know the 

notional amount of the reference portfolio. At simulation start date, we assume the no- 

tional to be one. Furthermore, we assume a portfolio with 125 equally weighted reference 

entities. Thus, every title in the portfolio has a weight of 0.8%. This is the weight by 

which the notional amount has to be reduced at the time of a default τ. Let τk be a vector 

containing all default times occurred after tk−1 and up to tk with size (l × 1). We denote 

the notional amount at each simulation date by N(tk, τk). Then, we can formally express 

the value of the notional amount at tk by 

N(tk, τk) = N(tk−1, τk−1) − 0.8% · l. (5.70) 

Now, we can price the first return component, the present value of the index, which is 

done analogously to equation (5.63): 

P Vindex(tk, T, s, λk, τk) = (Premium Leg(tk, T, s, λk)−Protection Leg(tk, T, λk))·N(tk, τk), 

(5.71) 

where we calculate the premium leg and the protection leg according to equations (5.57) 

and (5.59), respectively. 

As described in Section 3.9, a funded CDS index receives besides the fixed quarterly 

spread, s∆tc i , a quarterly coupon payment with a floating interest-rate amounting to 3 

month LIBOR at the last coupon payment date tc i−1 . The value of this instrument was 

described in Section 5.5.4. 

We take this interest-rate from our simulations by regarding the 3 month government 

rate as the corresponding LIBOR. As LIBOR for the first coupon payment we take the 

LIBOR as of 20 th of September, 2006. Then, the present value of the FRN for every tk, 

P VF RN(tk, t c 0 , T, τk), is calculated according to equation (5.64) 

P VF RN(tk, t c 0, T, τk) = F RN(tk, t c 0, T ) · N(tk, τk). (5.72) 

Analogous to funded CDS, we introduce a time-weighted notional amount at each 

coupon payment date t c 0 , N t (tk, t c 0 , τ ∗ ) 78 , where τ ∗ is a vector containing all default times. 79 

If defaults have taken place between tc −1 (coupon payment prior to tc0 )80 and tc 0 , we assume 

a proportional coupon payment. 

As already seen, the investor receives a payment, C(tk, t c 0 , s, τ ∗ ), consisting of 3 month 

LIBOR and the fixed spread s on the notional amount N t (tk, t c 0 , τ ∗ ), at the coupon pay- 

78 An explanation for this procedure can be found in Section 5.5.5. The calculation is similar to the 

calculation of the time-weighted notional amount for funded CDS, as described in the previous section. 

79 For one scenario, the vector has the size (125 × 1), as every reference entity in the reference portfolio 

has its own default time. 

80 For a definition, please refer to Section 5.5.5. 

103


ment date tc 0 . At maturity, the investor additionally receives the actual notional amount 

at maturity date, N(T, τ ∗ ). Then, the regular cash flows from the investment at tc 0 are 

similar to equation (5.65) 

C(tk, t c 0, s, τ ∗ ) = 

⎧ 

⎪⎨ 

⎪⎩ 

0 , if k = 0 

(s + R(t c −1 , tc −1 + ∆tc 0 )) · ∆tc 0 · N t (tk, t c 0 , τ ∗ ) , if tk < T 

(s + R(tc −1 , tc−1 + ∆tc0 )) · ∆tc0 · N t (tk, tc 0 , τ ∗ ) + N(T, τ ∗ ) , if tc 0 = T 

0 , if tk+1 > T. 

(5.73) 

In order to determine the present value of that payment at the next simulated timestep 

tk, P Vs(tk, t c i , s, τ ∗ ), we need to compound C(tk, t c 0 , s, τ ∗ ). For an illustration of this prin- 

ciple, we refer to Figure 5.10. 

Formally, the compounding is calculated by equation (5.66) 

P Vs(tk, t c 0, s, τ ∗ ) = C(tk, t c 0, s, τ ∗ ) exp(R(t c 0, tk)(tk − t c 0)). (5.74) 

Recovery payments are the last return component. Regarding a funded CDS index, 

the protection seller receives a recovery payment at default of a reference entity and the 

notional amount is reduced by 0.8% for every default. Each default that occurred before 

ˆT with ˆ T = min(T, T sim ), has to be treated in the following way: The investor receives 

the recovery value RV , which is calculated as RV = 0.8% · REC with a fixed recovery 

rate of REC = 40%. We need to compound each recovery payment received between tk−1 

and tk to the next simulation date tk. This calculation has to be done for every element 

τk,j, j ∈ {1, ..., l}, of τk. The results have to be summed up. Recovery payments received 

before tk−1 have to be compounded to tk and added to the latest recovery payments. This 

principle is illustrated in Figure 5.12. 

Figure 5.12: Compounding of recovery payments of a CDS index 

Formally, we express the present value of the recovery payments by 

104


⎧ 

⎪⎩ 

0, if k = 0 

P VREC(tk, τ ∗ ⎪⎨ 

0, if τ 

) = 

∗ > tk 

P VREC(tk−1, τ ∗ ) · exp(R(tk−1, tk)(tk − tk−1))+ 

+ � l 

j=1 RV · exp(R(τk,j, tk)(tk − τk,j)). 

(5.75) 

Analogous to equation (5.68), we calculate the value of an investment into a funded 

CDS index for every simulated timestep tk, CDSindex(tk, T, s, λk, τk), containing all return 

components apart from the present value of the spread payments P Vs, by 

CDSindex(tk, T, s, λk, τk) = P Vindex(tk, T, s, λk, τk)+P VF RN(tk, t c 0, T, τk)+P VREC(tk, τ ∗ ). 

(5.76) 

For every simulated timestep tk > 0, we determine the total return of the investment 

R CDSindex, by calculating the change in value from the previous timestep, tk−1, to the 

current timestep, tk, adding the compounded coupon payments paid between tk−1 and tk 

and then relating it to the value in tk−1. Formally, R CDSindex can be expressed by 

R CDSindex (tk−1, tk) = CDSindex(tk) + P Vs(tk) 

CDSindex(tk−1) 

− 1. (5.77) 

For simplicity reasons, we also omitted all input parameters for the pricing elements apart 

from the pricing day. 

Now, we have a framework to simulate the necessary risk factors (interest-rates, credit 

spreads, a spread of a CDS index and equity returns) and a framework to price different 

(credit) instruments, even taking into account potential defaults. So, we obtain return 

distributions of the relevant instruments, which are the starting point for portfolio opti- 

misations. Different optimisation criteria are introduced and discussed in the following 

chapter. 

105

Chapter 6 

Portfolio Selection 

After having generated return distributions as described in Chapter 5, we are now inter- 

ested in optimal portfolios containing credit instruments. For this reason, this chapter 

deals with portfolio selection. At first, we introduce the traditional portfolio selection ac- 

cording to Markowitz and illuminate problems due to strict assumptions on the structure 

of asset returns or utility functions. As we have seen in previous chapters, returns from 

credit instruments are not normally distributed. Hence, other optimisation approaches are 

more appropriate to take into account the distribution characteristics of such instruments. 

Therefore, we introduce the conditional value at risk in Section 6.2. This is roughly speak- 

ing the expected value of all losses exceeding a certain significance level. The performance 

measure Ω is presented in Section 6.3. It relates the potential upside of a portfolio’s return 

to its potential downside. Ω is applied to determine the risk-aversion of a certain investor 

if the portfolio selection criteria is the measure score. This is described in Section 6.4. 

We assume the following preconditions for all optimisation frameworks introduced in 

this chapter: 

Remark 6.1 

For all optimisation frameworks, it is assumed that 

• there are neither transaction costs nor taxes, 

• all securities can be divided arbitrarily, 

• the portfolios remain unchanged over time. 

6.1 Traditional Portfolio Selection 

The classical portfolio theory is based on the model of Markowitz. 1 The basic assumption 

is that investors select their portfolios taking into account only the first two moments of 

the asset’s return – mean and variance – and the correlation between the assets. The 

model is described in more detail below. 

6.1.1 Mean-Variance Approach 

The central statements of Markowitz are the following. 2 

1 See [Mar52]. 

2 See [SB00], p. 6. 

106

CHAPTER 6. PORTFOLIO SELECTION 

• Portfolio selection is based on expected returns and variance (as measure of risk). 

• It is sensible to construct portfolios, in order to reduce risk. Correlation is the key 

to risk reduction. 

• Portfolios are denoted as ”efficient” 

– if there is no other portfolio with the same expected return, but with a lower 

risk, or 

– if there is no other portfolio with the same risk, but with a higher expected 

return. 

In order to formally introduce the portfolio selection according to Markowitz, we need 

a few definitions first. We assume n given assets to invest in with returns Ri, i = 1, ..., n, 

and denote the 

• expected return of asset i by µi := E[Ri] and µ := (µ1, ..., µn) T , 

• covariance matrix by C = (cij)i,j=1,...,n, where cij = Cov[Ri, Rj], i, j = 1, ..., n, 3 

• variance V[R i ] = cii =: σ 2 i 

> 0, i, j = 1, ..., n,4 

• correlation matrix Γ = (ρij)i,j=1,...,n. 5 

Let xi be the portfolio weight of asset i with 

n� 

xi = 1. 

i=1 

Then, we denote the portfolio by x := (x1, ..., xn) T . 

If there are n assets to invest in, the expected portfolio return µ(x) is calculated as 

weighted sum of the n expected asset returns µi, with the weight xi 

µ(x) = E[R(x)] = 

n� 

xiµi = µ T x. 

i=1 

The variance of the portfolio return σ(x) 2 is given by 

σ(x) 2 := V[R(x)] = 

= 

n� 

i=1 j=1 

n� 

xixjcij = x T Cx. 

n� 

x 2 i · σ 2 i + 2 

i=1 

n� 

n� 

i=1 j=i+1 

xixjρijσiσj. 

The efficient frontier is the boundary of all µ(x)-σ(x)-combinations, which are not 

dominated by other asset weights xi of the n assets. If µ(x)-σ(x) lies on the efficient 

frontier, then for a given σ(x), there is no portfolio with a higher µ(x), and for a given 

3 The covariance is defined in equation (5.10) in Section 5.2. 

4 The variance is defined in equation (5.7) in Section 5.2. 

5 The correlation is defined in equation (5.9) in Section 5.2. 

107


µ(x) there is no portfolio with a lower σ(x). The efficient frontier, with no short sales 

allowed, is determined according to the following optimisation problem. 

where 1 = (1, . . . , 1) T . 

min 

x 

xT Cx (6.1) 

s.t. µ T x ≥ µ 

1 T x = 1 

x ≥ 0, 

If we solve the optimisation problem (6.1) for every possible µ, we obtain the set of all 

efficient portfolios. Displayed in a µ(x)-σ(x)-diagram, we see the efficient frontier, such as 

in Figure 6.1. 

Figure 6.1: Efficient frontier 

The portfolio on the efficient frontier with the lowest variance is called minimum- 

variance portfolio (MVP). 

In order to come to an optimal asset allocation for a certain investor, further tools 

like utility functions have to be applied. They imply a certain amount of utility to risk- 

return combinations for an investor. Applied to a certain efficient frontier, maximising the 

expected utility leads to a unique optimal asset allocation. 6 

tion. 

There are two main reasons that justify the framework of Markowitz for asset alloca- 

First, if an investor has a quadratic utility function, he decides in accordance with the 

µ(x)-σ(x) criteria, independent from the return distributions. 

Second, if the distribution of security returns and of portfolio returns are completely 

determined by their mean and variance, the traditional portfolio selection is valid for 

arbitrary utility functions. This is the case for normally distributed asset returns. If the 

marginal distributions of asset returns are normal, the resulting portfolio distribution is 

normal, too. 

6 For an introduction into utility theory, we refer for example to [BC04]. 

108


Thus, if either the investor has a quadratic utility function or if the asset returns are 

normally distributed, the portfolio selection according to Markowitz is appropriate. 

Often, the assumptions of the framework of Markowitz are too restrictive. Problems 

that may arise are discussed in the next subsection. Still, the mean-variance approach 

builds the foundation of the modern capital market theory. 

6.1.2 Problems 

Quadratic utility functions only take into account the first and the second moment of 

a return distribution. They do not incorporate higher moments such as skewness and 

kurtosis. 7 In literature, however, one can find empirical evidence on investor’s preference 

towards a positive skewness. 8 Thus, it is not appropriate to justify the application of the 

mean-variance approach by quadratic utility functions. 

Therefore, we now concentrate on the second justification using the mean-variance 

approach – the normal distribution of asset returns which is the standard approach to 

portfolio selection. Often, the hypothesis of normally distributed returns has to be rejected 

for asset returns, particularly for financial instruments which can suffer from defaults such 

as lowly rated bonds. 

In Chapter 4, for example, we showed that neither the US Lehman aggregates nor the 

Euro-Lehman aggregates have normally distritbuted returns. 

To overcome this drawback, we introduce two more optimisation approaches taking 

into account the complete distribution and the tail distribution, respectively. 

6.2 Conditional Value at Risk Optimisation 

Before introducing the concept of the conditional value at risk, we briefly introduce the 

value at risk concept. 

6.2.1 A Note on Value at Risk 

A very popular risk measure is the value at risk (VaR), representing the maximum possible 

loss of a portfolio with respect to a given time horizon and a given significance level. The 

VaR (relative to the distribution’s expected value) can be formally described according to 

the following definition. 

Definition 6.2 (Value at Risk) 

Let (1 − α) be the confidence level for the value at risk with α ∈ (0, 1). Then, the value 

at risk of a portfolio’s return is defined by 

VaR(x, α) = E[R(x)] − sup {y ∈ R : P[R(x) ≤ y] ≤ α} . (6.2) 

In practice, confidence levels of 95% or 99% are usually observed, corresponding to a 

level of α of 5% or 1%, respectively. In the case of a continuous distribution function, the 

7 Skewness and kurtosis are defined in equations (5.28 and (5.29)) in Section 5.4.2. 

8 See for example [HLLM03]. 

109

above definition of the VaR can also be rewritten according to 

where F −1 

R 



VaR(x, α) = E[R(x)] − F −1 

R (α), (6.3) 

denotes the inverse distribution function of portfolio returns R(x).9 

Often, the VaR is measured relative to zero. Then, equations (6.2) and (6.3) reduce to 

VaR0(x, α) = − sup {y ∈ R : P[R(x) ≤ y] ≤ α} , (6.4) 

VaR0(x, α) = −F −1 

R (α). (6.5) 

The VaR as risk measure involves a few problems, such as not taking into account the 

tail distribution and a lack of subadditivity 10 . 

6.2.2 Conditional Value at Risk 

The conditional value at risk (CVaR) is a coherent risk measure, overcoming some defi- 

ciencies of the VaR concept. Artzner, Delbaen, Eber, and Heah ([ADEH99]) presented an 

axiomatic approach building the foundation for a coherent risk measure. 11 

The CVaR 12 represents the expected value of all losses that exceed a certain VaR. 

Formally, we can define the CVaR according to 

Definition 6.3 (Conditional Value at Risk) 

Let (1 − α) be the confidence level for the value at risk with α ∈ (0, 1). Then, the condi- 

tional value at risk of a portfolio’s return is defined by 

CVaR(x, α) = −E[R(x)|R(x) ≤ F −1 

R (α)] 

= −E[R(x)|R(x) ≤ −VaR0(x, α)]. (6.6) 

From the CVaR formulas above, it becomes evident that the CVaR provides informa- 

tion on the negative tail of a return distribution as it is not only focussed on the α-quantile 

9 For more background regarding VaR, we refer for example to [Zag02], pp. 251-253. 

10 The meaning of subadditivity will be explained in the next subsection. 

11 Let X denote the set of all random variables X. Furthermore, a risk measure is defined as measurable 

mapping ρ : X → R, and we call ρ(X) the risk of risk position X. Then, a risk measure is called coherent 

if it satisfies axioms 1 - 4. 

Axiom 1 [Monotonicity] For all X, Y ∈ X with X ≤ Y 

ρ(X) ≥ ρ(Y ). 

Axiom 2 [Translation-Invariance] For all X ∈ X and for all c ∈ R we have 

ρ(X + c) = ρ(X) − c. 

Axiom 3 [Positive Homogenity] For all X ∈ X and for all λ ≥ 0 we have 

ρ(λ · X) = λ · ρ(X). 

Axiom 3 [Subadditivity] For all X, Y ∈ X we have 

ρ(X + Y ) ≤ ρ(X) + ρ(Y ). 

(See for example [Zag02], pp. 254-255.) 

12 CVaR is also known under the names expected shortfall, worst conditional expectation, tail conditional 

expectation, which is for example defined in [Zag02], pp. 262-265. 

110

ut also takes into account the shape of its tail. 

We formulate the portfolio optimisation problem by 


min 

x 

CVaR(x, α) (6.7) 

s.t. µ T x ≥ µ 

1 T x = 1 

x ≥ 0. 

Solving the optimisation problem (6.7) for every possible µ, we obtain the set of all 

efficient portfolios and the corresponding efficient frontier, which can be displayed in a 

µ(x)-CVaR(x, α)-diagram. 

6.3 Performance Measure Omega 

In this section, we introduce the performance measure Ω, developed by Shadwick and 

Keating. 13 It is to overcome the inadequacy of other traditional performance measures 14 

when applied to portfolio selection, particularly if returns are not normally distributed. 

Unlike other performance measures, such as the Sharpe Ratio 15 , it takes into account the 

entire return distribution. 

Definition 6.4 (Performance Measure Omega) 

Let L be a threshold representing the target return, let R = R(x) be the random one-period 

portfolio return and FR(r) the corresponding cumulative distribution function. Then, Ω is 

defined by 

Ω(x, L) = Ω(R, L) = 

∞� 

(1 − FR(r))dr 

L 

L� 

−∞ 

FR(r)dr 

. (6.9) 

From the above definition, we observe that the complete return distribution is con- 

sidered, and so are higher moments. Furthermore, the threshold L can be determined 

individually for every investor. Thus, a better performance measurement can be guaran- 

teed. 

∞� 

L 

The numerator and the denominator in equation (6.9) can be rewritten according to 16 

(1 − FR(r))dr = 

∞� 

L 

(r − L)fR(r)dr = E[max(R(x) − L, 0)] =: E[Upside(x)], (6.10) 

13 See [SK02]. 

14 See for example [SB00], pp. 576-585, for more performance measures. 

15 The Sharpe Ratio S(R(x)) is defined as excess return of a portfolio over the risk-free interest-rate rf , 

relative to the portfolio’s standard deviation σ: 

S(R(x)) = 

µ − rf 

. (6.8) 

σ 

It is a sensible performance measure, if an investor only has preferences for µ and σ, as higher moments 

are not taken into account. 

16 For the derivation, we refer to [KSG03]. 

111

L� 

−∞ 

FR(r)dr = 

L� 

−∞ 


(L − r)fR(r)dr = E[max(L − R(x), 0)] =: E[Downside(x)], (6.11) 

where fR(r) is the density function of the one-period portfolio return. 

It is obvious that the numerator represents the upside potential of an investment, while 

the denominator represents the downside potential. 

We will apply this performance measure for the portfolio selection with the so-called 

score measure, which will be introduced in the next section. 

6.4 Portfolio Selection with Measure Score 

It is our goal to make an asset allocation for two representative investors – one being 

risk-averse, and the other being risk-affine. To find an optimal asset allocation, we need 

to consider the risk-aversion of the corresponding investor. An appropriate way to do so, 

is the measure score. 17 

Definition 6.5 (Score) 

Let E[Upside(x P )], E[Downside(x P )], E[Upside(x B )] and E[Downside(x B )] denote the up- 

side potential and the downside (or loss) potential (as defined in equations (6.10) and 

(6.11), respectively) for a portfolio x P or for a benchmark x B , respectively. λB represents 

the risk-aversion of the investor. It is calculated according to 

λB = E[Upside(xB )] 

E[Downside(xB . (6.12) 

)] 

Then, the score for a portfolio x P with return R(x P ) is calculated as 

score(x P , λB) = E[Upside(x P )] − λB · E[Downside(x P )]. (6.13) 

As we can see from equation (6.12), λB is the same as Ω in equation (6.9). Furthermore, 

we want to remark that λB is determined in such a way that the benchmark score is 0. 

Note, that the higher λB, the higher the risk-aversion of the investor. 

Portfolios can be compared according to their score value – the higher the score, the 

better the portfolio. To find an optimal asset allocation for a given λB, we have to solve 

the following optimisation problem: 

max 

x P 

score(x P , λB) (6.14) 

s.t. 1 T x P = 1 

x P ≥ 0. 

For the portfolio selection we chose L := rf as the target return to calculate the upside 

and downside potential. Before explicitly determining λB, we have a closer look at the 

17 See [Zag03]. 

112


upside potential defined in equation (6.10). We can rewrite it according to 

∞� 

rf 

(1 − FR(r))dr = 

= 

∞� 

rf 

∞� 

−∞ 

(r − rf )fR(r)dr 

(r − rf )fR(r)dr − 

= µ(x) − rf + 

rf � 

−∞ 

rf � 

−∞ 

(rf − r)fR(r)dr 

(r − rf )fR(r)dr 

= µ(x) − rf + E[max(rf − R(x), 0)], (6.15) 

Now, we want to determine λB for the benchmark portfolio x B . Plugging equations 

(6.11) and (6.15) into equation (6.9), we obtain λB 

Ω(x B , rf ) = µ(xB ) − rf + E[max(rf − R(x B ), 0)] 

E[max(rf − R(x B ), 0)] 

= 1 + 

µ(x B ) − rf 

E[max(rf − R(x B ), 0)] = λB. (6.16) 

From equation (6.16) it becomes obvious that the risk-aversion of an investor is measured 

as kind of excess return relative to the downside potential. The latter incorporates the 

complete portfolio return distribution for returns less than rf . 

After having introduced a simulation and pricing framework in Chapter 5, we presented 

a few optimisation criteria for portfolio selection in this chapter, which completed the asset 

allocation framework, so that we are able to apply the asset allocation framework, now. 

We consider an investor in the United States. The corresponding parametrisiation of the 

model and the optimisation results are presented in Chapter 7. 

113

Chapter 7 

Simulation and Simulation Results 

After having introduced the tools for simulating and pricing in Chapter 5 and for portfolio 

optimisation in Chapter 6, we now present the simulation and optimisation results for an 

investor in the United States. First, we describe the data used for the calibration and 

the resulting model parameters. Then, we show the fit of the model interest-rate curve 

to the market curve, as of 30 th of September, 2006 and we briefly present the parameters 

for the simulation of correlated default times. The following subsections are dedicated 

to the simulation results. We analyse the single return characteristics of the financial 

instruments under consideration (government and corporate (zero-) coupon bonds, an 

equity index, CDS and a CDS index). Thereafter, we present the optimisation results 

for the mean-variance approach, the CVaR optimisation and the score optimisation. In 

all cases, we consider different investment universes, different investment horizons and 

different investor types. Then, we describe the main observations, applying a higher 

default correlation between the reference entities. 

7.1 US Economy 

7.1.1 Calibration of the Economic Scenario Generator 

Model Parameters 

To calibrate the ESG, we use parameters estimated by [SZA06] and [Zag06]. We partly 

need to adjust them to better meet the updated historical data. They provide parameters 

for the GDP growth rate, the short-rate and credit spreads for rating classes AA, A2, 

BBB1. 1 

The short-rate parameters were estimated using non-defaultable weekly bond data (US 

Treasury Strips) 2 from 1 st of October, 1993, until 31 st of December, 2004, with maturities 

of 1 year, 2, 3, 4, 5, 7, and 10 years. Figure 7.1 shows the underlying US Treasury Strips 

1 The estimations are done using Kalman filter methodologies. Kalman filters are used if a process is 

to be modelled which is not observable in reality. However, if the connection between the unobservable 

process and observable data is known, Kalman filter methodologies can be applied. Here, for example, 

the short-rate is unobservable, while interest-rates can be observed. This fact is used to estimate the 

short-rate parameters. The parameters for the short spread are also estimated that way. The Kalman 

filter was introduced by R. E. Kalman, see [Kal60]. 

2 Bloomberg ticker symbols are C0791Y Index, C0792Y Index, C0793Y Index, C0794Y Index, C0795Y 

Index, C0797Y Index, C07910Y Index. 

114

CHAPTER 7. SIMULATION AND SIMULATION RESULTS 

from 1 st of October, 1993, until 30 th of September, 2006. 3 

Figure 7.1: US Treasury Strips 

In this figure, we observe a narrowing of interest-rates in recent years. Hence, the US 

interest-rate curve is relatively flat. Most of the time, however, the interest-rate curve 

implied by US Treasury Strips has a normal form, i.e. short-term interest-rates are lower 

than long-term interest-rates. 

In addition to the interest-rate data, quarterly US GDP growth rates were used, ad- 

justed by linear interpolation to weekly data. They were incorporated into the estimation 

process with a three-quarter time lag. Table 7.1 shows the parameters for the short-rate 

and the GDP growth rates. 

ar 0.37867 

âr 0.24782 

br 0.13315 

σr 0.0149553 

aω 1.18532 

âω 0.26847 

θω 0.0158331 

σω 0.0060146 

Table 7.1: Parameter estimation for processes r and ω 

The parameters for the rating classes AA 4 , A2 5 , BBB1 6 were estimated using average 

rates of American Industrials for the corresponding rating classes on a weekly basis from 

3 Note that the estimation period ends at 31 st of December, 2004. 

4 Bloomberg ticker symbols for rating class AA are C0031Y Index, C0032Y Index, C0033Y Index, 

C0034Y Index, C0035Y Index, C0037Y Index, C00310Y Index. 

5 Bloomberg ticker symbols for rating class A are C0061Y Index, C0062Y Index, C0063Y Index, C0064Y 

Index, C0065Y Index, C0067Y Index, C00610Y Index. 

6 Bloomberg ticker symbols for rating class BBB are C0091Y Index, C0092Y Index, C0093Y Index, 

C0094Y Index, C0095Y Index, C0097Y Index, C00910Y Index. 

115


1 st of October, 1993, until 31 st of December, 2004, with maturities of 1 year, 2, 3, 4, 5, 7, 

and 10 years. 

The underlying credit spreads are displayed in Figure 7.2. 7 

From the sub-figures, we can clearly see a term structure in the credit spreads analogous 

to interest-rates. Most of the time, the credit spreads for different maturities have a normal 

form, analogous to the observations regarding interest-rate curves. 

The high level of the credit spreads in 1998 can be attributed to the Asian and Russian 

crises, while the enormous increase in 2001 may be attributed to September 11 and the 

stock market crash in the years 2001 and 2002. 

The credit spread parameters that we use for the simulations are shown in Table 7.2. 

We adjust the estimated parameters âs, âu and bsu to better meet historical data in terms 

of average spreads and spread ranges. 

AA A2 BBB1 

as 2.96099 2.80727 2.41739 

âs 2.96099 2.80727 2.41739 

θs 0.0027474 0.0025139 0.0023778 

σs 0.0038855 0.0030875 0.0028679 

bsω 0.07373 0.07634 0.09369 

au 0.11269 0.11057 0.11048 

âu 0.02180 0.03210 0.04075 

θu 0.12173 0.18566 0.18980 

σu 0.0045920 0.0047604 0.0048862 

bsu 0.98404 0.92214 1.26089 

Table 7.2: Parameter estimation for processes s and u for rating classes AA, A2, BBB1 

CDS Index 

We assume the US CDS index to be composed of the rating classes AA, A, and BBB with 

the weights listed in Table 7.3. 8 

Equity Index 

Rating Weight 

AA 12.28% 

A 39.47% 

BBB 48.25% 

Table 7.3: Composition of US CDS index 

As we have seen from equation (5.14) the process for the continuous stock returns also 

needs an inflation process. We use parameters provided by [Zag06]. They are given in 

Table 7.4. 

7 Note that the estimation period ends at 31 st of December, 2004. 

8 See www.fitchratings.com. As we only have parameters for the rating classes AA, A, and BBB, we 

added the AAA rating class to rating class AA and BB to BBB. Furthermore, we normalised the weights 

to sum up to one. 

116


ai 0.64073 

âi 0.50319 

θi 1.0479 

σi 0.014470 

Table 7.4: Parameter estimation for process i 

Regarding the parameters for the equity index, we only adjust the parameters provided 

by [Zag06] for the level of the returns and for the standard deviation to meet updated 

historical data of a longer period than the estimation period. The parameters are given 

in Table 7.5. 

αE 0.00422 

bER 3.94000 

bEi 5.38436 

bEω 5.10643 

0.16000 

σE 

Table 7.5: Parameters for US equity process 

117


Figure 7.2: US credit spreads, rating classes AA, A2, BBB1 

118

7.1.2 Fit to Market Data 


Zero rates at simulation start date (30 th of September, 2006) have to be fitted to market 

data. We take US zero rates provided by Bloomberg. 9 

We determine the Nelson-Siegel parameters for the log-interest-rate curve. 10 They are 

given in Table 7.6. 

β0 5.2613% 

β1 -0.0130% 

β2 -1.3405% 

β3 246.7524% 

Table 7.6: Nelson-Siegel parameters for US zero rates 

The data and the fit of the curve are shown in Figure 7.3. 

Figure 7.3: Model fit of US zero rates 

7.1.3 Simulation of Correlated Default Times 

We simulate correlated default times according to the migration matrices approach, de- 

scribed in Section 5.4 with the Gaussian and NIG one-factor copula. 

For the US economy we use the transition matrix provided in Table 5.2. The parame- 

ters for the simulation start date are estimated in accordance with the model introduced 

in [KSW05]. In Table 7.7 one can find the parameters of the NIG distribution. 

a 0.37029 

α 0.70138 

β 0 

Table 7.7: Parameters of NIG distribution for US economy 

9 Bloomberg ticker symbols for US zero rates are F08403M Index, F08401Y Index, F08402Y Index, 

F08403Y Index, F08404Y Index, F08405Y Index, F08410Y Index, F08420Y Index, F08430Y Index. 

10 See Section 5.3. 

119


Knowing these parameters, the distributions of the factors M, Xi and Ai are uniquely 

determined by equations (5.35), (5.36) and (5.38). 

To simulate default times with the Gaussian one-factor copula, the only parameter 

needed is a, the correlation of a single reference entity to the common market factor 11 . a 

is the same as in Table 7.7 as we simulate default times with the Gaussian and NIG one- 

factor copula with the same default correlation. The factors M, Xi and Ai are standard 

normally distributed as we have seen in Section 5.4.2. 

The correlation of a single reference entity to the common market factor, a, is equivalent 

to a correlation between two reference entities of a 2 = 0.13711, which is rather low. For 

this reason, we also perform optimisations for a higher correlation. The results can be 

found in Section 7.2. 

Having simulated the government interest-rates, the credit spreads and the correlated 

default times, we can now price various financial instruments. The return characteristics 

are described in the following section. 

7.1.4 Return Characteristics of Financial Instruments 

The simulation outputs from the ESG and the simulated correlated default times are used 

to price the following financial instruments. We price government coupon and zero-coupon 

bonds, corporate coupon and zero-coupon bonds with ratings AA, A, BBB, a CDS index, 

and CDS with ratings AA, A, BBB, according to the formulas provided in Section 5.5. 

All bonds have a maturity of 5 years, while CDS and the CDS index have the maturity 

date 20 th of December, 2011, so that they have a maturity of approximately 5.25 years. 

Furthermore, we consider an equity index, simulated according to equation (5.14) with 

the parameters provided in Table 7.5. 

We consider the instruments and their return characteristics for investment horizons 

of 1 year, 3 years or 5 years. The corresponding key statistics of the financial instruments 

are summarised in Figure 7.4 for Gaussian and in Figure 7.5 for NIG default times. 12 

Of course, the return characteristics of government bonds are identical for Gaussian 

and NIG default times as we consider government bonds to be non-defaultable. The return 

characteristics of the equity index are also identical for both default times, as we simulate 

the index return, assuming that potential defaults of single stocks are already incorporated 

in the index return. Furthermore, the return characteristics for corporate bonds, for CDS 

and the CDS index are the same for Gaussian and NIG default times, if no defaults have 

taken place. 

At first, we start with a remark regarding the statistic CVaR in Figures 7.4 and 7.5. 

We see that the CVaR values can become negative. This is the case, if with the mean of 

11 See Section 5.4.2. 

12 The figures contain, for example, the statistic median, which is defined in the following. Let x1, . . . , xT 

denote a random sample. The order statistic is given by x(1) ≤ x(2) ≤ . . . ≤ x(T ). Then, the median is 

defined as: 

� 

�� 

median = 

�� 

x ( T +1 

2 ) 

x ( T 2 ) + x (1+ T 2 ) 

2 

, if T is odd 

, if T is even. 

Minimum and maximum denote the minimum or the maximum return outcome. The other statistics 

are explained in previous chapters. 

(7.1) 

120


the worst 5% of the instrument’s returns does not get negative. 13 In general, the smaller 

the CVaR, the less risky is the instrument. 

Considering Figures 7.4 and 7.5 we make some general observations. For the one- 

year investment horizon the AA-rated CDS, the A-rated CDS and the CDS index have 

considerably less volatility and a lower CVaR than government and corporate bonds, 

making them relatively save investments. At the same time, CDS and the CDS index 

have a similar expected return to government bonds. Therefore, one can assume that 

CDS and the CDS index are appropriate instruments to partly replace government and 

corporate bonds. This will turn out to be true, as shown in the following sections. 

Generally, we see higher expected returns and higher standard deviations or levels 

of CVaR with increasing risk of the instruments. Besides, we observe that – as soon 

as defaults have occurred – the distributions become strongly non-normal. 14 Though, 

regarding the transition matrix (Table 5.2), it is clear that for 1000 simulations and for a 

one-year time horizon, defaults will not take place for every rating class. For rating class 

AA, for example, the one-year probability of default is 0.01%, i.e. statistically, for 1000 

simulations 0.1 defaults will occur within one year. 

Having a closer look at Figures 7.4 and 7.5, we make a few astonishing observations. 

Sometimes the expected return of a lower rating class is lower than that of a higher rating 

class. For example, the expected return of the BBB-rated corporate coupon bond is lower 

than for the A-rated corporate coupon bond with NIG default times and a 1 year time 

horizon. Regarding the minimum, however, we see that in the simulations of BBB at 

least one default has taken place, while the minimum of A indicates no default. In this 

case, the higher spread for the BBB-rated bond was not able to compensate the number 

defaults. The standard deviation and CVaR behave like expected, they are higher for the 

lower rated bond. Moreover, we observe a higher standard deviation and CVaR-level for 

the AA-rated corporate coupon bond than for the A-rated corporate coupon bond with 

Gaussian default times and a 1 year time horizon. Again, regarding the minimum we see 

that in the simulations of AA at least one default has taken place, while the minimum of 

A indicates no default. In this case, the expected returns, however, behave like one would 

expect: The mean for the higher rated instrument is lower. 

Regarding CVaR, particularly for a 1 year investment horizon and high rated bonds, 

we see that they partly dominate the government bonds, i.e. the expected return of the 

corporate bond is higher than for the government bond, while the CVaR for the corporate 

bond is lower. This implies that with CVaR-optimisation hardly any government bonds 

will occur in optimal asset allocations. 

All in all, one have to keep in mind, that we are dealing with simulations, i.e. for 

a large number of simulations the results should be stable and should behave, how they 

are expected to do. Particularly for default times, however, it is possible, that for a 

short time horizon, such as 1 year, the simulated number of defaults within this period 

is coincidentally higher for a higher rated instrument. Though, for longer time horizons, 

such as 3 or 5 years the results should be more stable, i.e. the number of default times, 

13 

As the CVaR is the mean of the worst 5% of the portfolio return multiplied by −1, it becomes negative, 

if this mean is positive. 

14 

Defaults can be discovered when looking at the minimum returns. If they are smaller than, say, -40%, 

we can assume that this is the consequence of a default. 

121


the mean and standard deviation should be higher for a low rated instrument than for 

an instrument with a higher rating because an investor bearing a higher risk should be 

compensated with a higher expected return. We have already seen this trend in the Figures 

7.4 and 7.5. 

To obtain a better understanding of the return characteristics, we show the resulting 

densities of the instruments. The empirical distribution function can be calculated by 

kernel density estimation, which is a non-parametric method of modelling the univariate 

distribution of a random variable. 15 The results are given in Figures 7.6, 7.7 and 7.8 for 

1 year, and in Figures 7.9, 7.10 and 7.11 for 3 years. 

It is not very meaningful to plot the distributions for a 5 years investment horizon for 

bonds because all bonds have a maturity of exactly 5 years. Therefore, the zero-coupon 

bonds have exactly the same expected returns for every simulated scenario, if no default 

has occurred, since both the term structure of interest-rates for the simulation start date 

and the payout at maturity are deterministic. Then, all changes in interest-rates during 

the holding period are irrelevant. As explained, at maturity a zero-coupon bond with 

a particular rating has the same expected return for all simulations independent from 

the evolution of interest rates, if no defaults have taken place. In the case of default, 

however, the expected return for the corresponding scenario dramatically decreases. This 

is the only driver for the standard deviation of corporate zero-coupon bonds. For coupon 

bonds, however, there are little differences between the expected returns after a 5 years 

holding period, even if no default has occurred, as we assume a reinvestment of the coupon 

payment in the considered bond. Then, the total expected return partly depends on the 

level of interest-rates at reinvestment dates. This is also why the standard deviation, 

particularly for government bonds, decreases for longer investment horizons: The shorter 

the time to maturity, the lower the market risk of bond returns. 16 In contrast to a fixed- 

income security, the return of a funded CDS/CDS index is to a large extent driven by the 

FRN component, i.e. LIBOR. 17 Hence, it nearly excludes the market risk, as the paid 

coupon is adjusted to market interest-rates. Coupon payments, however, are as volatile as 

interest-rates and therefore volatility increases for longer investment horizons, i.e. longer 

simulation horizons. 

In the Figures 7.6 and 7.9 we show the complete distribution of the instruments. 

Though, as defaults make the density plots too compressed, we additionally plot the tails 

and the rest of the distributions in separate Figures (7.7, 7.8, 7.10 and 7.11). 

From these figures we can see a non-normality of the distributions if defaults have 

taken place. Moreover, we see again that for the one-year investment horizon the AA- 

rated CDS, the A-rated CDS and the CDS index have considerably less volatility than 

government and corporate bonds, while having a similar expected return to government 

bonds. 18 

In Figure 7.12, we show the linear correlations 19 between the returns of the financial 

15 

For more background regarding kernel density estimation we refer for example to [Har95], pp. 840-844. 

The figures were made by using the MATLAB function ksdensity with its default parameters. 

16 

The market risk of bonds is explained in Section 2.2. 

17 

The return components of a funded CDS and of a funded CDS index are described in Sections 5.5.5 

and 5.5.6. 

18 

The same observation we already made in the Figures 7.4 and 7.5. 

19 

For a definition of the linear correlation, see equation (5.9). 

122


instruments for the investment horizons 1 year, 3 years and 5 years. This matrices are one 

of the inputs for the mean-variance optimisation conducted in Section 7.1.5. 

Regarding the correlation matrices, we make the following main observations. 

• Bond returns have a high positive correlation for a 1 year and 3 years investment 

horizon. This is reasonable as the main driver of the bond return is the government 

rate, which is the same for all bonds. Differences in bond returns come form different 

credit spreads and different default times. 

• Bond and equity index returns are negatively correlated for a 1 year and 3 years 

investment horizon. A negative correlation between bond and stock returns can also 

be empirically observed. This phenomenon can be explained as follows. In times 

of low interest rates, i.e. low bond returns, companies strongly invest, resulting in 

higher profits of the companies and therefore higher equity returns. 

• There is a small positive or negative correlation between returns of bonds and CDS 

or the CDS index for a 1 year and 3 years investment horizon. We can explain the 

low correlation between bonds and CDS as follows. The bonds are fixed income 

instruments, whose price strongly depends on the level of interest rates: If interest 

rates are falling, the bond returns increase as bond prices increase. For CDS and 

the CDS index its the other way round, since their return strongly depends on the 

floating rate LIBOR and to a smaller part on other components, such as the fixed 

spread. 20 Therefore, returns of CDS and CDS indices tend to decrease with falling 

interest rates. 

• The returns of the equity index and CDS or CDS index are positively correlated. 

This is reasonable since both equity returns and the 3 month government rate are 

to a large proportion driven by the short-rate. 21 

The correlation for the five years investment horizon is not that meaningful for similar 

reasons as explained earlier in this section. 

To conclude, due to appealing risk-return-profiles and low correlations between bonds, 

the equity index and CDS/CDS index, investors should benefit from holding a portfolio 

consisting of several financial instruments. This is examined in the following sections. 

20 

For more information regarding the return components of funded CDS and funded CDS indices we 

refer to Sections 5.5.5 and 5.5.6. 

21 

See equation (5.14) for the dynamics of the equity index and equation (5.19) for the calculation of the 

zero rate. 

123


Figure 7.4: Key statistics US economy, Gaussian defaults 

124


Figure 7.5: Key statistics US economy, NIG defaults 

125


Figure 7.6: Densities of financial instruments, 1 year time horizon, Gaussian defaults 

(left-hand side) and NIG defaults (right-hand side) 

126


Figure 7.7: Left tail densities of financial instruments, 1 year time horizon, Gaussian 

defaults (left-hand side) and NIG defaults (right hand side) 

Figure 7.8: Densities of financial instruments without left tails, 1 year time horizon, 

Gaussian defaults (left-hand side) and NIG defaults (right-hand side) 

127


Figure 7.9: Densities of financial instruments, 3 years time horizon, Gaussian defaults 


128


Figure 7.10: Left tail densities of financial instruments, 3 years time horizon, Gaussian 


Figure 7.11: Densities of financial instruments without left tails, 3 years time horizon, 


129


Figure 7.12: Correlation of financial instruments in US economy, Gaussian defaults and 

NIG defaults 

130



Before presenting the optimisation results, we want to repeat that we aim at finding 

the efficient frontiers for different investment universes described in Remark 7.1 and the 

corresponding asset allocations for every point on the efficient frontiers. We consider three 

different investment horizons, 1 year, 3 years and 5 years. 

Remark 7.1 

The optimisation is conducted for the following three investment universes. 

(a) We assume an initial investment universe consisting of government bonds and stocks. 

(b) Then, we add corporate bonds to the investment universe. 

(c) Finally, we consider the efficient frontier and the respective asset allocations, also 

allowing for a CDS index and CDS. 

Conducting a mean variance-optimisation according to the optimisation problem (6.1) 

and following the steps listed in Remark 7.1, we obtain the results displayed in Figures 

7.14, 7.15 and 7.16. 

These figures are built up in a similar way. On the left-hand side we find the results 

for pricing with Gaussian default times, while on the right-hand side we see the results 

with NIG default times. The sub-figures on top show the efficient frontiers for the three 

investment universes described in Remark 7.1. The following sub-figures show the optimal 

asset allocations in each respective universe, for all resulting levels of risk. Note that for 

reasons of readability we display the weights of the bonds with the same rating class in 

one colour, i. e. we add their weights and display the sum, for example, we add the weight 

of the zero-coupon bond and the coupon bond rated A in the optimal allocation and show 

the sum in a light grey. Analogously, we display the weights of all CDS in one colour. We 

plot the optimal asset allocations for all three investment universes and for all investment 

horizons. 

As already explained in Section 7.1.4, we do not consider defaults for government bonds 

and for equity indices. Therefore, the results are the same for both default times. 

For investors optimising their portfolios according to the mean-variance criterium, we 

can identify some general structures. 

At first, we examine the efficient frontiers 22 : 

• Allowing for corporate bonds in the portfolio optimisation leads to an upward shift 

of the efficient frontier compared to a portfolio only consisting of government bonds 

and an equity index: For the same level of risk, a higher expected return can be 

generated. 

• Allowing additionally for CDS and a CDS index in the portfolio optimisation leads 

to a shift to the left of the efficient frontier, compared to a portfolio consisting of 

government bonds, corporate bonds and an equity index, i.e. the portfolio risk can 

be reduced. The proportion of the potential risk reduction, however, depends on the 

investment horizon: 

22 For more background regarding efficient frontiers, we refer to Section 6.1.1 

131


– For a 1 year horizon, an enormous risk reduction can be achieved. The minimum- 

variance portfolio 23 consisting of government bonds and an equity index has an 

expected return of 6.16% and a standard deviation of 3.12%, while the port- 

folio allowing for all instruments, which has an expected return of 6.16%, has 

a standard deviation of only 0.76% for Gaussian default times and 0.91% for 

NIG default times. 

– For a 3 years investment horizon, a slight risk reduction can be achieved by 

adding CDS and CDS indices. Considering a 5 years investment horizon, the 

benefit of adding CDS is marginal. 

This shift of the efficient frontier by allowing for corporate bonds and CDS/CDS index 

can be attributed to a low correlation between these assets and to the appealing risk-return 

profile of these instruments. 24 

Then, we consider the mean-variance optimal asset allocations: 

• In an efficient asset allocation, government bonds are totally or partially substituted 

with corporate bonds and CDS/CDS index. The reason for that is also the low 

correlation and the attractive risk-return profile for corporate bonds and CDS/CDS 

index compared to government bonds, particularly for the 1 year investment horizon. 

As we have seen from Figures 7.4 and 7.5, CDS have a similar expected return to 

government bonds but a much lower standard deviation for a 1 year horizon. The 

phenomenon of decreasing volatility of bonds and increasing volatility of CDS/CDS 

index over time was already explained in Section 7.1.4. Particularly for low levels of 

risk, corporate bonds are replaced by CDS/CDS index. 

– For levels of standard deviation higher than 5.77% (Gaussian default time) 

and 5.54% (NIG) an optimal portfolio only consists of corporate bonds and an 

equity index for a 1 year horizon. 

– For lower levels of standard deviation, corporate bonds are partially replaced 

by CDS/CDS index, which reach a level of 90% in an optimal allocation with 

a standard deviation of approximately 0.5%. There is only a very small pro- 

portion of government bonds for a very low level of risk, due to the risk-return 

profile of government bonds and CDS. 

– For a 3 years time horizon we see a similar picture: For low levels of standard 

deviation, corporate bonds are partially substituted by CDS/CDS index, reach- 

ing a level of nearly 50%. Now, government bonds are again used in an optimal 

allocation, due to a similar risk-return profile to CDS. 

– For a 5 years horizon, we hardly see any CDS. Now, we observe a large pro- 

portion of government bonds for low levels of risk. The reason is that govern- 

ment bonds have a very low standard deviation, as they have a maturity of 

5 years. Therefore, the payments are received without any uncertainties and 

without reinvestment risk in the case of zero-coupon bonds, while the standard 

23 For an explanation of the minimum-variance portfolio we refer to Section 6.1.1. 

24 See Figures 7.12, 7.4 and 7.5, respectively. 

132


deviation of CDS/CDS index continuously increases with a longer investment 

horizon. 

• Comparing the allocations of instruments with the same rating class, but priced with 

different default times, we make the following observations, which can be attributed 

to the risk-return profile of these instruments: 

– For a 1 year and 3 years investment horizon, instruments which are priced 

using Gaussian default times allocate a larger proportion in A-rated corporate 

bonds, while a pricing with NIG default times leads to a larger proportion 

allocated in BBB-rated corporate bonds. BBB-rated coupon bonds within 

NIG default times are more attractive compared to their counterparts within 

Gaussian default times. In contrast, we make the reverse observation for A- 

rated bonds. 25 The allocation in CDS/CDS index is similar. 

– For a 5 years horizon we see a reverse trend. A larger proportion is allocated 

in BBB-rated corporate bonds with Gaussian default times. Comparing the 

risk-return profiles of these bonds for Gaussian with NIG default times, we now 

see the opposite of what was described for the 1 year horizon. 

Figure 7.13 shows the potential return enhancement for the minimum-variance port- 

folio in the investment universe only allowing for government bonds and an equity index. 

The return enhancements are shown for optimal portfolios with the same level of risk 

as the minimum-variance portfolio, now, however, allowing for other credit risk bearing 

instruments. These portfolios are displayed for the investment horizons of 1 year and 3 

years, and both default times. 

It is not reasonable to show the table for the 5 year investment horizon, because the 

lowest standard deviation is zero in the case of a government zero-coupon bond. Therefore, 

no return improvement is possible for the same level of risk by adding other instruments. 

The first line always shows the minimum-variance portfolio in the investment universe 

only allowing for government bonds and stocks. For the horizons of 1 year and 3 years, we 

see that there are at most marginal differences when additionally allowing for corporate 

bonds, because the portfolio with the lowest standard deviation normally only consists of 

government bonds and stocks. On the contrary, there is an enormous improvement of the 

expected portfolio return, when the investment universe is extended by CDS/CDS index 

for reasons explained earlier. 

From the analyses above, we can draw the following conclusions. An investor applying 

the mean-variance criterium for portfolio selection can add performance to his portfolio for 

a given level of risk, or he can reduce risk for a target return level. This can be achieved due 

to the low correlation and the risk-return profile of the instruments by partially replacing 

his government bond investment by a combination of corporate bonds and CDS/CDS 

index. 

25 Furthermore, the standard deviation of the BBB-rated coupon bond is lower than that of the A-rated 

coupon bond for NIG default times, as already explained in Section 7.1.4. 

133


Figure 7.13: Minimum-variance portfolio with Gaussian defaults (left-hand side) and 

NIG defaults (right-hand side) 

134


Figure 7.14: Results of mean-variance optimisation, 1 year investment horizon with 


135


Figure 7.15: Results of mean-variance optimisation, 3 years investment horizon with 


136




137



Now, we present the results of the CVaR-optimisation according to the optimisation prob- 

lem (6.7) with the investment universes listed in Remark 7.1. α is assumed to be = 5%, 

i.e. we consider the mean of the worst 5% of the portfolio return as risk measure. 26 

The results are displayed in Figures 7.18, 7.19 and 7.20, which are built up in analogy 

to the figures in Section 7.1.5. 

The first striking observation is that the CVaR values can become negative because 

there are portfolios with a mean of the worst 5% of the portfolio returns that does not 

become negative. 27 In general, the smaller the CVaR, the less risky is the portfolio. 

Comparing the results of the CVaR optimisation with the results of the mean-variance 

optimisation, the first impression is very similar. In the following, we present the main 

results from the CVaR optimisation. 

We begin considering the efficient frontiers: 

• Adding corporate bonds to a portfolio consisting of government bonds and an equity 

index leads to an upward shift of the efficient frontier, compared to a portfolio only 

consisting of government bonds and an equity index: For the same level of CVaR, a 

higher expected return can be generated. 

• Adding CDS and a CDS index to the portfolio of government bonds, corporate bonds 

and an equity index, an enormous risk reduction can be achieved for a 1 year horizon. 

The minimum-CVaR portfolio consisting of government bonds and an equity index 

has an expected return of 6.55% and a CVaR of 0.25%, while the portfolio allowing 

for all instruments, which has an expected return of 6.55%, has a CVaR of only 

-3.81% for Gaussian default times, and -3.59% for NIG default times. 

• For a 3 years investment horizon, a slight increase of the efficient frontier can be 

reached by adding CDS/CDS index. 

• Considering a 5 years investment horizon, there is no benefit of adding CDS/CDS 

index to a portfolio. 

The observed shift of the efficient frontier by allowing for corporate bonds and CDS/CDS 

index can be attributed to several facts: A low correlation between these assets, imply- 

ing diversification effects and an attractive risk-return profile of the credit risk bearing 

instruments. 28 

Then, we regard the optimal asset allocations for an investor using the CVaR-criterium: 

• In an efficient asset allocation, government bonds are totally or partially substituted 

with corporate bonds and CDS/CDS index. The reason for that is the low correla- 

tion and the attractive risk-return profile of corporate bonds and CDS/CDS index 

compared to government bonds, particularly for the 1 year investment horizon. As 

26 For more background see Section 6.2. 

27 As the CVaR is the mean of the worst 5% of the portfolio return multiplied by −1, it becomes negative, 

if this mean is positive. 

28 See Figures 7.12, 7.4 and 7.5. 

138


we have seen from Figures 7.4 and 7.5, CDS have a similar expected return to gov- 

ernment bonds, but a lower CVaR for a 1 year horizon. Particularly for low levels 

of risk, corporate bonds are replaced by CDS/CDS index. 

– For levels of CVaR higher than 4.8% (Gaussian default time) and 5.0% (NIG) 

an optimal portfolio only consists of corporate bonds and an equity index for a 

1 year horizon. 

– For lower levels of CVaR, corporate bonds are partially replaced by CDS/CDS 

index, which reach a level of approximately 90% in an optimal allocation with 

a CVaR of approximately 4.7%. There are no government bonds in the optimal 

allocation, due to the risk-return profile of the instruments. 

– For a 3 years time horizon we see a similar picture: For low levels of CVaR, 

corporate bonds are partially substituted by CDS/CDS index, reaching a level 

of at least 40%. Now, government bonds are used in an optimal allocation 

applying Gaussian default times, due to the risk-return profile. 

• We compare the allocations resulting from Gaussian and NIG default times. The 

observed effects can be explained with the risk-return profiles of the instruments: 29 

– For a 1 and 3 years investment horizon, instruments, which are priced using 

Gaussian default times, allocate a larger proportion in A-rated corporate bonds, 

while a pricing with NIG default times leads to a larger proportion allocated in 

BBB-rated corporate bonds. The allocation in CDS/CDS index is similar. 

– For a 5 years horizon, the efficient portfolios priced with Gaussian default times 

partly contain a considerable proportion of government bonds. Efficient portfo- 

lios priced with NIG default times contain more A-rated corporate bonds, since 

these bonds dominate government bonds. 

Figure 7.17 is equivalent to Figure 7.13. 30 We see that there is an enormous improve- 

ment of the expected portfolio return when additionally allowing for corporate bonds and 

CDS/CDS index due to diversification effects. 

Our analyses showed that an investor, optimising his portfolio with the CVaR cri- 

terium, can add performance to his portfolio for a given level of risk, or he can reduce risk 

for a target return level. This can be achieved by partially substituting government bonds 

with corporate bonds and CDS/CDS index. 

29 See Figures 7.4 and 7.5. 

30 For a description, we refer to Section 7.1.7. 

139


Figure 7.17: Minimum-CVaR portfolio with Gaussian defaults (left-hand side) and NIG 

defaults (right-hand side) 

140


Figure 7.18: Results of CVaR optimisation, 1 year investment horizon with Gaussian 

defaults (left-hand side) and NIG defaults (right-hand side) 

141


Figure 7.19: Results of CVaR optimisation, 3 years investment horizon with Gaussian 


142




143

7.1.7 Score 


In this subsection, we describe the results of the score optimisation, which is conducted 

according to optimisation problem (6.14). 

We examine the optimal asset allocation for two representative investors – a risk-averse 

and a risk-affine investor. The composition of the investor’s benchmark portfolio is given 

in Table 7.8. 

Asset Class Risk-Averse Investor Risk-Affine Investor 

Government Bonds 70% 30% 

Equity Index 30% 70% 

Table 7.8: Characterisation of representative investors 

This analysis is done for the investment universes listed in Remark 7.1 for a 1 year, a 

3 and a 5 years investment horizon. 

Some statistics and the results of the optimisation are shown in Figures 7.21, 7.22 

and 7.23. The top sub-figures show the results for the investment universe without CDS 

and CDS index, while the bottom sub-figures show the results for the complete investment 

universe. The statistics are displayed in a light grey, in the first part of the sub-figures. The 

statistic ”risk” denotes Ω as defined in equation (6.9) 31 , the score value is maximised by 

the optimisation. A positive score value indicates that, for a certain investor, the portfolio 

is better than the benchmark portfolio. The optimal allocations can be found in the 

second part of the sub-figures. Analogous to the previous sections, we sum up the weights 

of coupon bonds and zero-coupon bonds with the same credit quality. Additionally, we 

sum up the weights of the CDS. Assets with a proportion higher than 30% are printed 

in bold letters. As before, the results for the Gaussian default times can be found on 

the left-hand side, results for the NIG default times on the right-hand side. Within a 

sub-figure, we present the outcomes for the risk-averse and the risk-affine investor. 

Before analysing the results of the score optimisation, we make a few remarks regarding 

this optimisation procedure. The score value of a portfolio is calculated as expected upside 

potential, less the expected downside potential, which is weighted with λB, the investor’s 

risk-aversion. 32 Hence, the optimisation is done, searching for the best tradeoff between 

upside and downside potential, taking into account the investor’s risk-aversion. Applying 

this optimisation procedure, the complete return distribution of the portfolios is consid- 

ered, i.e. particularly heavy tails of single assets which are implied by defaults. 33 Note 

that the higher the upside potential of an instrument, the higher the downside potential, 

such as for equity indices. In the score optimisation the downside potential is multiplied 

by λB, which is usually considerably higher than 1. 34 Therefore, the downside potential 

has a higher weight than the upside potential when determining the score. Consequently, 

the proportion of rather risky assets will be limited, in order to obtain a positive score. 

For this reason, it will turn out in our optimisations, that the proportion of the equity 

index in an optimal portfolio is at most 36.3%, even for the risk-affine investor. 

31 The higher Ω, the better. 

32 See Section 6.4. 

33 See Section 7.1.4 for distributional properties. 


144


The following observations can all be attributed to a few facts: first, to the character- 

istics of the optimisation procedure as explained earlier; second to the risk-return profiles 

shown in Figures 7.4 and 7.5; third to the diversification effects implied by low correlations 

between the instruments shown in Figure 7.12. 

Figure 7.21: Results of score optimisation, 1 year investment horizon with Gaussian 


For every investment horizon, we can observe that λB is higher for the risk-averse 

investor than for the risk-affine. This confirms what was already stated in Section 6.4: 

The higher the risk-aversion, the higher λB. 

At first, we consider the 1 year investment horizon. We begin with general observations. 

Then, we have a closer look at the different types of investors. 

• For both default times, we see that both investor types have a lower proportion of 

the equity index compared to the initial allocation (see Table 7.8). The optimal 

allocation never contains government bonds. 

• Comparing the level of the benchmark and the expected portfolio return, we see that 

for risk-affine investors the expected portfolio return is smaller than the benchmark 

return. Regarding the risk-averse investors, we cannot observe a clear trend. 

• With Gaussian default times, a risk-averse investor should allocate approximately 

18% into CDS. 

• With NIG default times, a risk-averse investor does not invest into CDS. 

• A risk-affine investor never has CDS in his portfolio. 

145


Figure 7.22: Results of score optimisation, 3 years investment horizon with Gaussian 


• For NIG default times the optimal portfolio consists of only 2 assets, the equity 

index and BBB-rated corporate bonds. 

• Comparing Gaussian and NIG default times, we observe that optimal allocations 

have a higher proportion of the equity index and a much higher proportion of BBB- 

rated corporate bonds for NIG default times, while applying Gaussian default times 

leads to a considerable proportion of A-rated corporate bonds. 

Next, we consider the 3 and the 5 years investment horizon. 

• Now, all resulting allocations have a lower expected portfolio return than their bench- 

mark. The risk-averse investor’s portfolio, however, has similar expected returns as 

the benchmark. 

• For the 3 years horizon, the portfolios never contain government bonds. For a 5 

years investment horizon, the optimal portfolio determined with Gaussian default 

times also allocates into government bonds. 

• Optimising with Gaussian or NIG default times entails different allocations, but 

similar resulting expected returns. The allocation into CDS differs somewhat. A 

risk-averse investor with a 3 years investment horizon should have a CDS proportion 

of 1.27% (Gaussian default time) or 0.37% (NIG default time) in his portfolio. A risk- 

averse investor with a 5 years time horizon should have 8.67% CDS with Gaussian 

146




default times and no CDS with NIG default times. A risk-affine investor does not 

have CDS in his portfolio. 

Our analyses showed that an investor optimising his portfolio with the score measure 

can obtain a better risk-return profile compared to his benchmark portfolio only consisting 

of government bonds and an equity index. This can be achieved by partially substituting 

government bonds and the equity index with corporate bonds and CDS/CDS index. 

147

7.1.8 Comparison of Optimal Portfolios 


We close this section of the simulation results for the US economy with a comparison of 

the optimal portfolios. The optimal allocations are shown for a risk-averse and a risk-affine 

investor having a benchmark portfolio as listed in Table 7.8. For the investment universe 

allowing for all instruments, i.e. government and corporate bonds, CDS/CDS index and 

an equity index, we display the optimal portfolios with the same standard deviation and 

CVaR, respectively, as the benchmark portfolio, as well as the corresponding optimal 

portfolio according to score optimisation. The data for the mean-variance and CVaR- 

optimal portfolios is taken from the optimisation output presented in Sections 7.1.5 and 

7.1.6. The data for the score optimisation can be found in Section 7.1.7. ”Risk” denotes 

the corresponding risk measure used for the optimisation, i.e. the standard deviation, 

CVaR or Ω. The results are shown in the following figures. 

Figure 7.24: Results for risk-averse and risk-affine investor, 1 year investment horizon, 

Gaussian defaults (top) and NIG defaults (bottom) 

Beginning with general observations, we see similar resulting expected portfolio returns 

and portfolio allocations for mean-variance and CVaR optimisation. Having a closer look 

at the results, we see that the resulting expected portfolio returns from CVaR optimisation 

are higher than for the other optimisation procedures. The highest proportion of the equity 

index can also be observed with the CVaR optimisation. The equity index proportion for 

mean-variance optimisation is not substantially lower than with CVaR. In contrast, the 

score optimisation always leads to a considerably lower proportion of the equity index. 35 

CVaR and mean-variance optimisation never allocate in government bonds for the two 

types of investors. This can be attributed to the risk-return profile of the instruments. 36 

For the 1 year horizon, the optimisation according to the CVaR criterium leads to a 

35 The only exception is the result for the 1 year horizon, Ω-optimisation, risk-averse investor. 

36 See Figures 7.4 and 7.5 for the key statistics of the financial instruments under consideration. For a 

detailed discussion of the resulting optimal allocations, we refer to Sections 7.1.5, 7.1.6 and 7.1.7. 

148


Figure 7.25: Results for risk-averse and risk-affine investor, 3 years investment horizon, 


substantial proportion of CDS for both default times. The reason for that is the very 

attractive risk-return profile of CDS/CDS index in the low-risk field compared to bonds. 

For other time horizons, there are at most small proportions of CDS/CDS index, 

because they are primarily attractive for very low levels of risk. Optimal asset allocations, 

however, always contain a considerable proportion of corporate bonds. 

To conclude, comparing the resulting optimal asset allocations for a representative risk- 

averse and risk-affine investor, we see that independent from investment horizon and from 

optimisation criterium an investor always benefits from substituting government bonds by 

corporate bonds and CDS/CDS index. 

149




150


7.2 US Economy with Higher Default Correlation 

In the previous subsections, the correlation between the reference entities is assumed to 

be a 2 = 0.13711, which is rather low. To examine the effects of a higher correlation, 

we conduct the same return calculations and optimisations as in Section 7.1, but now 

applying a default correlation of 0.3. 37 The results are presented in the following. In this 

section, we mainly concentrate on describing the main observations and the differences in 

the simulation results between the actual and the higher default correlation. 

7.2.1 Return Characteristics 

Figures 7.27 and 7.28 show the key statistics of all financial instruments under considera- 

tion for Gaussian and NIG default times. 

From Figures 7.27 and 7.28 we make some general observations. For the one-year in- 

vestment horizon the CDS and the CDS index have considerably less volatility and a lower 

CVaR than government and corporate bonds, making them relatively save investments. 

At the same time, CDS and the CDS index have a similar expected return to govern- 

ment bonds. Therefore, one can assume that CDS and the CDS index are appropriate 

instruments to partly replace government and corporate bonds. Generally, we see higher 

expected returns and higher standard deviations or levels of CVaR with increasing risk of 

the instruments. 

Having a closer look at the statistic CVaR, particularly for a 1 year investment horizon 

and high rated bonds, we see that they partly dominate the government bonds, i.e. the 

expected return of the corporate bond is higher than for the government bond, while the 

CVaR for the corporate bond is lower. 38 This implies that with CVaR-optimisation hardly 

any government bonds will occur in optimal asset allocations. 

We can observe a clear tendency which cannot be observed for the low default correla- 

tion: The lower the credit quality, the more non-normal the return distribution of single 

instruments priced with NIG default time compared to pricing with Gaussian default time. 

This becomes obvious considering the skewness and excess kurtosis, because both statis- 

tics indicate a non-normality, which is the stronger, the bigger the deviation from zero. 

This is the result of what we already have seen in Figure 5.5: The lower the credit rating 

and the higher the default correlation, the more pairwise defaults occur within a short 

investment horizon. 

Comparing the statistics with high default correlation to those with the actual default 

correlation, we also see that within one default time, i.e. only considering the Gaussian or 

NIG default time, the return distributions are more non-normal with default correlation 

of 0.3. 39 Again, this becomes obvious considering the skewness and excess kurtosis. We 

also can explain this observation with the observations we have made in Figure 5.5: For a 

certain rating class and a certain one-factor copula, the joint defaults are the higher, the 

higher default correlation is. 

37 For a detailed description of how the tables and figures are built, we refer to Section 4.2. 

38 See, for example, AA-rated coupon bond vs. government bond, 1 year investment horizon. This is 

particularly the case if no defaults have taken place, which can be attributed to the number of simulations. 

For a discussion, we refer to Section 7.1.4. 

39 See Figures 7.4, 7.5, 7.27 and 7.28. 

151


To obtain a deeper insight into the return distributions, we show the empirical distri- 

bution functions and the distribution functions split into tail and rest of the distribution 

in Figures 7.29, 7.30 and 7.31 for the 1 year investment horizon. The corresponding dis- 

tribution functions for the 3 years investment horizon are shown in Figures 7.32, 7.33 and 

7.34. 

From these figures, we can see a non-normality of the distributions, if defaults have 

taken place. Moreover, we recognise again that for the 1 year investment horizon, the 

CDS and the CDS index have considerably less volatility than government and corporate 

bonds, while having a similar expected return to government bonds. 

The linear correlations between the returns of the financial instruments for the invest- 

ment horizons 1 year, 3 years and 5 years are displayed in Figure 7.35. 

Regarding the correlation matrices, we make the similar observations as for low default 

correlation between the reference entities. 40 

• Bond returns have a high positive correlation for a 1 year and 3 years investment 

horizon. 

• Bond and equity index returns are negatively correlated for a 1 year and 3 years 

investment horizon. 

• There is a small positive or negative correlation between returns of bonds and CDS 

or the CDS index for a 1 year and 3 years investment horizon. 

• The returns of the equity index and CDS or CDS index are positively correlated. 

40 For an interpretation of the correlations, we refer to Section 7.1.4. 

152


Figure 7.27: Key statistics US economy, Gaussian defaults 

153


Figure 7.28: Key statistics US economy, NIG defaults 

154


Figure 7.29: Densities of financial instruments, 1 year time horizon, Gaussian defaults 


155


Figure 7.30: Left tail densities of financial instruments, 1 year time horizon, Gaussian 


Figure 7.31: Densities of financial instruments without left tails, 1 year time horizon, 


156


Figure 7.32: Densities of financial instruments, 3 years time horizon, Gaussian defaults 


157


Figure 7.33: Left tail densities of financial instruments, 3 years time horizon, Gaussian 


Figure 7.34: Densities of financial instruments without left tails, 3 years time horizon, 


158


Figure 7.35: Correlation of financial instruments in US economy, Gaussian defaults and 

NIG defaults 

159



Conducting a mean variance-optimisation according to the optimisation problem (6.1) with 

the different investment universes listed in Remark 7.1, we obtain the efficient frontiers 

and optimal asset allocations shown in the figures below. 

All in all, the observations are similar to those in Section 7.1.5. Considering the 

efficient frontiers, we see again an upward shift of the efficient frontier when allowing for 

corporate bonds and a shift to the left, when allowing additionally for CDS/CDS index, 

i.e. a potential to reduce risk. These shifts can be attributed to a low correlation between 

the financial assets and to the appealing risk-return profile of these instruments. 41 

Now, we analyse the optimal asset allocations according to the mean-variance cri- 

terium. The following results can mainly be explained by the risk-return profile of the 

instruments and the correlation structure between them. 

• The main observation is that government bonds are totally or partially substituted 

with corporate bonds and CDS/CDS index. 

• In contrast to the lower default correlation, now optimal allocations contain CDS 

up to a higher level of standard deviation: For levels of standard deviation higher 

than 6.91% (Gaussian default time) and 6.74% (NIG) an optimal portfolio only 

consists of corporate bonds and an equity index for a 1 year horizon. These limits 

are approximately 1% lower for the lower default correlation. We make a similar 

observation for the 3 years time horizon and NIG default time. This result can 

be explained by the risk-return profile of the BBB-rated CDS. 42 The BBB-rated 

CDS, for example, generates a rather high expected return compared to its standard 

deviation. 43 Compared to the BBB-rated CDS return for low default correlation 44 

it has a more attractive risk-return profile. Its expected return is higher for higher 

default correlation, while its standard deviation is lower. 

Figure 7.36 shows the potential return enhancement for different investment universes, 

based on the same level of risk as the minimum-variance portfolio in the initial investment 

universe. The results are very similar to those in Section 7.1.5. 

After having analysed the results of the higher default correlation and compared to 

the results of the lower default correlation, we can draw the following conclusions. An 

investor applying the mean-variance criterium for portfolio selection can add performance 

to his portfolio for a given level of risk, or he can reduce risk for a target return level. This 

can be achieved due to the low correlation and the risk-return profile of the instruments 

by partially replacing his government bond investment by a combination of corporate 

bonds and CDS/CDS index. The optimal allocations, however, vary somewhat between 

the different default correlations, particularly for low levels of risk. 


42 Having a closer look at the simulation results, it turns out, that for higher levels of standard deviation 

only BBB-rated CDS are contained in the optimal allocations. 

43 BBB-rated CDS, for example, dominates the government bonds, i.e. it has a higher expected return, 

but a lower standard deviation. 

44 See Figures 7.4 and 7.5. 

160


Figure 7.36: Minimum-variance portfolio with Gaussian defaults (left hand side) and 

NIG defaults (right-hand side) 

161


Figure 7.37: Results of mean-variance optimisation, 1 year investment horizon with 


162




163




164



In analogy to Section 7.1.6, we conduct a CVaR-optimisation according to the optimisation 

problem (6.7) with the investment universes listed in Remark 7.1. The resulting efficient 

frontiers and optimal asset allocations are displayed in the figures below. 

The main observations are similar to those in Section 7.1.6. When we regard the 

efficient frontiers, we see again an upward shift of the efficient frontier if the investment 

universe is extended by corporate bonds and a shift to the left, if the investment universe 

is additionally extended by CDS/CDS index. The latter extension particularly means a 

potential to reduce risk. These shifts can be attributed to a low correlation between the 

financial assets and to the appealing risk-return profile of these instruments. 45 

• The main observation is that government bonds are totally substituted with corpo- 

rate bonds and CDS/CDS index. For the lower default correlation in contrast, we 

see a high proportion of government bonds for the 5 years investment horizon with 

Gaussian default times, due to the risk-return profile of the instruments. 

• Now, for the higher default correlation between reference entities, optimal allocations 

contain CDS up to a higher level of CVaR, in contrast to the lower default correlation: 

For levels of CVaR higher than 6.91% (Gaussian default time) and 6.35% (NIG) an 

optimal portfolio only consists of corporate bonds and an equity index for a 1 year 

horizon. These limits are more than 1% lower for the lower default correlation. 46 

This result can be explained by the risk-return profile of the BBB-rated CDS. 47 The 

BBB-rated CDS, for example, generates a rather high expected return compared 

to its CVaR. 48 Compared to the BBB-rated CDS return for low default correlation 

it has a more attractive risk-return profile. Its expected return is higher for higher 

default correlation, while its CVaR is lower. 

• For the 3 years investment horizon and for Gaussian default times, the proportion 

of CDS is considerably lower for the higher default correlation than for the lower 

one. Having a closer look at the optimal allocations reveals a very large proportion 

of A-rated CDS in the case of low default correlation. This can be explained by two 

facts: First, the more attractive risk-return profile of A-rated CDS within the 3 year 

time horizon for Gaussian default times and low default correlation. Second, the 

more attractive risk-return profile of A-rated CDS compared to the same instrument 

for high default correlation. 

Figure 7.40 shows the potential expected return enhancement for different investment 

universes, based on the same level of CVaR as the minimum-CVaR portfolio in the initial 

investment universe. 

The results slightly differ from those in Section 7.1.6. This can be attributed to different 

risk-return profiles of the financial instruments for different default correlations and the 

45 

See Figures 7.35, 7.27 and 7.28. 

46 

See Section 7.1.6. 

47 

A closer look at the simulation results reveals that for higher levels of CVaR only BBB-rated CDS 

are contained in the optimal allocations. 

48 

The BBB-rated CDS, for example, dominates the government bonds, i.e. it has a higher expected 

return, but a considerably lower CVaR. 

165


Figure 7.40: Minimum-CVaR portfolio with Gaussian defaults (left-hand side) and NIG 

defaults (right-hand side) 

resulting optimal asset allocations. Though, it becomes obvious that an enormous return 

enhancement can be achieved by extending the investment universe by credit-risk bearing 

instruments. 

To sum up, an investor applying the CVaR criterium for portfolio selection, can add 

performance to his portfolio for a given level of risk, or he can reduce risk for a target 

return level. This can be achieved due to the low correlation and the risk-return profile of 

the instruments by substituting government bonds by a combination of corporate bonds 

and CDS/CDS index. The optimal allocations between the two analysed levels of default 

correlation, however, partly differ from each other, particularly for low levels of CVaR. 

166


Figure 7.41: Results of CVaR optimisation, 1 year investment horizon with Gaussian 


167




168




169

7.2.4 Score 


Finally, we also conduct the score optimisation according to optimisation problem (6.14), 

analogous to Section 7.1.7. Again, we consider two representative investors with bench- 

mark portfolios as given in Table 7.8. The figures below show the optimisation results. 

Figure 7.44: Results of score optimisation, 1 year investment horizon with Gaussian 


The main observations are similar to those in Section 7.1.7. All observations and 

differences to the low default correlation can be attributed to the specific characteristics of 

the optimisation procedure 49 , the risk-return profiles of the instruments and the correlation 

between them. 50 

• All investors have a lower proportion of the equity index in their optimal portfolio 

than the benchmark portfolio. The optimal allocation never contains government 

bonds. In contrast to that, the optimal allocation for the lower default correlation 

contains government bonds for the risk-averse investor with a 5 years time horizon 

with Gaussian default times. 

• The proportion of the equity index is rather similar for both default correlations in 

most of the cases. There is one major exception: Applying NIG default times, an 

investor with a 1 year time horizon should invest a considerably lower proportion 

into the equity index and a considerably larger proportion into CDS (BBB-rated). 51 

49 For a discussion, we refer to Section 7.1.7. 

50 See Figures 7.27 and 7.28 and 7.35. 

51 This phenomenon was already explained in Sections 7.2.2 and 7.2.3. 

170




• Risk-averse investors for the 3 years investment horizon should also have a slightly 

higher proportion of CDS in their portfolio for the higher default correlation. 

• Regarding the resulting expected portfolio returns there is no clear trend between 

both default correlations. Sometimes, it is higher with the higher default correlation 

than for the lower one and sometimes it is the other way round. We make the same 

observation for score values. 

The results show that an investor optimising his portfolio with the score measure can 

obtain a better risk-return profile compared to his benchmark portfolio. This can be 

achieved by substituting government bonds with corporate bonds and CDS/CDS index. 

171




172

7.2.5 Comparison of Optimal Portfolios 


Now we show the optimal portfolio allocations for a risk-averse and a risk-affine investor 

having a benchmark portfolio listed in Table 7.8, analogous to Section 7.1.8. For the 

investment universe allowing for all instruments, we display the optimal portfolios with 

the same standard deviation and CVaR, respectively, as the benchmark portfolio. Besides, 

we display the corresponding optimal portfolio according to score optimisation. The data 

for the mean-variance and CVaR-optimal portfolios is taken from the optimisation output 

presented in Sections 7.2.2 and 7.2.3. The data for the score optimisation can be found in 

Section 7.2.4. The results are shown in the following figures. 

Figure 7.47: Results for risk-averse and risk-affine investor, 1 year investment horizon, 


The main results are similar to those in Section 7.1.8. We see similar resulting ex- 

pected portfolio returns and portfolio allocations for mean-variance and CVaR optimisa- 

tion. Again, the portfolio returns from CVaR optimisation are higher than for the other 

optimisation procedures. 

Comparing resulting expected portfolio returns, we cannot reveal a clear trend between 

both default correlations. In some cases, the expected returns are higher with the higher 

default correlation than with the lower one, and vice versa. 

In contrast to the lower default correlation, we observe here a considerable proportion 

of CDS in an optimal allocation for a risk-averse investor with a 1 year time-horizon. 52 

To conclude, comparing the resulting optimal asset allocations for a representative 

risk-averse and risk-affine investor, we see that independent from investment horizon, op- 

timisation criterium and default correlation between reference entities, an investor always 

benefits from (partially) substituting government bonds by corporate bonds and CDS/CDS 

index. 

52 We already explained this phenomenon earlier in Sections 7.2.2 and 7.2.3. 

173






174

Chapter 8 

Summary and Outlook 

In this thesis, we aimed at giving a deep understanding of credit instruments and at 

analysing these instruments in a portfolio context with a consistent asset allocation frame- 

work, which comprises a simulation, pricing and optimisation framework. 

Therefore, Chapter 2 gave a deep and comprehensive insight into credit risk transfer. 

After having explained some regulatory aspects regarding credit risk, we described the 

traditional credit related instruments, loans and bonds. They often build the underlyings 

for credit risk transfer. Then, the main vehicles to transfer credit risk were introduced, 

which are credit derivatives and securitisations. We presented the functionality of these 

instruments, the reasons for using them, and both, chances and risks related to these 

markets. This chapter closed with an overview on market evolution and size. 

Chapter 3 was dedicated to single credit risk transfer instruments. After a classifica- 

tion of credit risk transfer instruments according to the dimensions unfunded/funded, i.e. 

without or with initial investment of the protection seller, and single-name/multi-name, 

i.e. one or several underlying reference entities, we explained the most important instru- 

ments. In particular, these instruments are CDS, total return swaps, credit spread options, 

basket CDS, portfolio CDS, credit linked notes, collateralised debt obligations and CDS 

indices. We mainly focussed on CDS and CDS indices because they are the most liquid 

instruments in the market. For this reason they were also used in the asset allocation 

framework to represent credit derivatives. 

In Chapter 4, we analysed Lehman aggregate indices, which are fixed income indices, 

to learn more about distributional properties of returns of credit instruments. Statistical 

tests for normality for several US- and Euro-aggregate indices clearly rejected that the 

returns were normally distributed. We saw heavily tailed distributions, particularly for 

negative returns, which potentially indicated defaults. As we wanted to develop an asset 

allocation framework that is able to map reality as good as possible, it was necessary to 

set up a simulation and pricing system that can create return distributions accounting for 

the distinct properties of credit instruments. 

Chapter 5 presented a cascade model allowing to simulate interest-rates, credit spreads 

and the spread of a CDS index. To realistically price credit instruments, we needed to 

take into account potential defaults. Therefore, this chapter introduced the popular one- 

factor copula approach for modelling correlated default times. We used this approach not 

only for a Gaussian one-factor copula, but also for a Normal-Inverse-Gaussian (NIG) one- 

175

CHAPTER 8. SUMMARY AND OUTLOOK 

factor copula, because the latter is able to produce default times with a higher probability 

of joint defaults between reference entities than the Gaussian version. This is a more 

realistic assumption, since, for example, during a recession more firms default than during 

a booming period. We calculated the distribution function for default times with migration 

matrices provided by rating agencies. Finally, we introduced the pricing formulas for 

the credit instruments in the relevant investment universe: Zero-coupon bonds, coupon 

bonds, funded CDS and funded CDS indices. In order to price CDS and CDS indices, 

it was necessary to know expected losses, which were determined by the constant default 

intensity model. 

Chapter 6 completed the asset allocation framework. In order to find optimal portfo- 

lios, we first considered the mean-variance approach according to Markowitz. The follow- 

ing discussion showed some problems, either driven by strict assumptions on the structure 

of asset returns, such as normal distributions, or driven by strict assumptions on utility 

functions. As seen earlier, returns of credit instruments were not normally distributed. 

Therefore, other optimisation procedures accounting for the distinct distributional prop- 

erties of credit instruments were presented. One appropriate risk measure for the optimi- 

sation is the conditional value at risk, which is roughly speaking the expected value of all 

losses exceeding a certain significance level. With this measure, the complete tail distri- 

bution of financial instruments is considered. The last section was dedicated to portfolio 

selection with the measure score. This measure takes into account an investor’s risk- 

aversion and the entire upside and downside potential of a portfolio distribution related 

to a fixed target return. 

We applied the asset allocation framework to investors in the United States. In Chapter 

7, we presented the simulation and optimisation results. On the one hand, the model 

parameters for the GDP growth rate, the inflation rate, the short-rate, the credit spreads 

and the equity index were presented. In addition, the starting curve for the US zero 

rates was fitted to market data as of 30 th of September, 2006 (simulation start date). On 

the other hand, the parameters for the correlated default times were also provided as of 

simulation start date. At that day, the default correlation between two reference entities 

was only at 0.13771. 

After having calculated the risk-return profiles of all instruments in the investment 

universe and the optimal allocations for the investment horizons 1 year, 3 years and 5 

years, we found that the risk-return profiles of the credit instruments were very attrac- 

tive compared to government bonds. Particularly in a 1 year investment horizon, CDS 

(AA- or A-rated) and the CDS index had a considerably lower risk (measured in stan- 

dard deviation or CVaR) than government bonds, while having nearly the same expected 

return. However, it also became obvious that the returns could be strongly non-normal 

for credit instruments, which was caused by defaults. Besides, the correlations between 

the returns of different instruments were rather low due to distinct characteristics of the 

instruments under consideration. So, investors could benefit from diversification effects in 

their portfolios by adding credit instruments. 

Independent of the applied investment horizon, default time or optimisation criterium, 

a shift of the efficient frontier compared to the efficient frontier of the initial investment 

universe (government bonds and equity index) was observed. After extending the invest- 

176

CHAPTER 8. SUMMARY AND OUTLOOK 

ment universe by corporate bonds, CDS and a CDS index, we saw a shift of the efficient 

frontier – either an upward shift or a shift to the left. Consequently, an investor could en- 

hance the expected portfolio return for a given level of risk, or he could reduce portfolio risk 

for a target return level by partially replacing the government bond investment by credit 

instruments. The optimal asset allocations always contained a considerable proportion of 

corporate bonds, CDS or a CDS index. 

As explained earlier, the actual default correlation between the reference entities was 

rather low at simulation start date. For this reason, we also performed the simulation 

and optimisation with a default correlation of 0.3. Regarding the risk-return profiles 

of the credit instruments, we detected that with a decreasing credit quality the return 

distribution of single instruments priced with NIG default times became even more non- 

normal compared to pricing with Gaussian default times. This could be attributed to 

the realistic ability of the NIG distribution to generate more joint defaults for relatively 

small time horizons. We also saw that within one default time, i.e. only considering 

the Gaussian or NIG default time, the return distributions were more non-normal with 

default correlation of 0.3. This could be attributed to the ability of the one-factor copula 

to generate more joint defaults within a relatively small time horizon for a higher default 

correlation, given a certain rating class and a certain one-factor copula. 

The general observations, however, were similar to a low default correlation. The 

efficient frontier of the initial investment universe also experienced a shift by extending the 

investment universe by credit instruments. Moreover, optimal allocations always contained 

a considerable proportion of corporate bonds, CDS or a CDS index, too. 

All in all, with the presented asset allocation framework, the following central results 

were found: Credit instruments have an appealing risk-return profile, allowing to enhance 

the expected portfolio return compared to a portfolio of traditional asset classes, such as 

equity indices and government bonds. Due to the correlation structure between returns of 

credit instruments and those of traditional asset classes, credit instruments offer enormous 

diversification potential. 

Based on the findings described above, we can actually consider credit instruments 

as a single asset class. Every investor, independent of his risk-aversion and optimisation 

criterium, can profit from adding credit instruments to his portfolio. 

The framework presented in this thesis can be the starting point for further research: 

At the moment, CDS and CDS indices have only been considered as means of investment. 

One field of research can be the analysis of CDS and CDS indices as means of hedging, 

too. Then, the investor would act as protection buyer, rather than as protection seller, by 

shorting funded CDS or a CDS index. 

Another subject for further research can be the extension of the investment universe 

by other credit instruments, such as CDOs or other structured credit products, in general. 

So, there might be more potential to diversify one’s portfolio by low-correlated returns. 

From this research, we should learn even more about the implications on optimal asset 

allocations for different types of investors. 

177

Appendix A 

Mathematical Preliminaries and 

Definitions 

In this chapter we want to provide the mathematical preliminaries relevant for this thesis. 

We use [Zag02] and [BK04], for the description of the theory. 

A.1 Probability Spaces and Stochastic Processes 

Definition A.1 (σ-Algebra) 

If Ω is a given set, then a σ-algebra F on Ω is a family F of subsets of Ω with the following 

properties: 

(a) Ω ∈ F, 

(b) A ∈ F ⇒ A C = Ω\A ∈ F, 

(c) A1, A2, . . . ∈ F ⇒ A := ∞� 

Ai ∈ F. 

i=1 

Definition A.2 (Measurable Space) 

The pair (Ω, F) is called a measurable space. 

Definition A.3 (Probability Measure) 

A probability measure Q on a measurable space (Ω, F) is a function P : F −→ [0, 1] 

such that 

(a) Q(∅) = 0, Q(Ω) = 1. 

� 

(b) If A1, A2, . . . ∈ F are disjoint (i.e. Ai Aj = ∅ if i �= j) then 

Definition A.4 (Probability Space) 

∞� 

Q( Ai) = 

i=1 

∞� 

Q(Ai). 

The triple (Ω, F, Q) is called a probability space. A set A ∈ F with Q(A) = 0 is called 

a (Q−) null set. (Ω, F, Q) is called a complete probability space if F contains all subsets 

of the (Q−) null sets. 

178 

i=1

APPENDIX A. MATHEMATICAL PRELIMINARIES AND DEFINITIONS 

To define stochastic processes we additionally introduce filtered probability spaces. 

Definition A.5 (Filtration) 

A filtration F is a non-decreasing family of sub-sigma-algebras (Ft)t≥0 with Ft ⊂ F and 

Fs ⊂ Ft for all 0 ≤ s < t < ∞. We call (Ω, F, Q, F) a filtered probability space, and 

require that 

(a) F0 contains all subsets of the (Q−) null sets of F, 

(b) F is right-continuous, i.e. Ft = Ft+ := ∩s>tFs. 

(Ω, F, Q, F) is a complete filtered probability space, if F and each Ft, 0 ≤ s < t < ∞, 

is complete. One can think of Ft as the information available at time t, and F = (Ft)t≥0 

describes the complete flow of information over time assuming that no information is lost 

in the course of time. 

To describe the behavior of the financial instruments, their volatility and correlation, 

we use stochastic processes. 

Definition A.6 (Stochastic Process) 

A stochastic process is a family X = (Xt)t≥0 = (X(t))t≥0 of random variables Xt de- 

fined on the filtered probability space (Ω, F, Q, F). The stochastic process X is called 

(a) adapted to the filtration F if Xt = X(t) is Ft− measurable for all t ≥ 0, 

(b) measurable if the mapping X : [0, ∞) × Ω −→ R k , k ∈ N, is (B([0, ∞)) ⊗ F − 

B(R k )−) measurable with B([0, ∞)) ⊗ F denoting the product sigma-algebra created 

by B([0, ∞)) and F, 

(c) progressively measurable if the mapping X : [0, t]×Ω −→ R k , k ∈ N, is (B([0, t])⊗ 

Ft − B(R k )) measurable for each t ≥ 0. 

Note that for each t fixed, we have a random variable 

with ω ∈ Ω. 

ω −→ Xt(ω), 

When fixing ω ∈ Ω, we have a function in t, i.e. 

called a path of Xt. 

t −→ Xt(ω), 

An important example for a stochastic process is the Wiener process, denoted by 

W = (Wt)t≥0 = (W (t))t≥0. Sometimes it is also called Brownian motion. 

Definition A.7 (Wiener Process) 

Let (Ω, F, Q, F) be a filtered probability space. The stochastic process W = (Wt)t≥0 = 

(W (t))t≥0 is called a (Q−) Brownian motion or (Q−) Wiener process if 

(a) W (0) = 0 Q− a.s., 

179


(b) W has independent increments, i.e. W (t) − W (s) is independent of W (t ′ 

) − W (s ′ 

) 

for all 0 ≤ s ′ 

≤ t ′ 

≤ s ≤ t < ∞, 

(c) W has stationary increments, i.e. the distribution of W (t + u) − W (t) only depends 

on u for u ≥ 0, 

(d) Under Q, W has Gaussian increments, i.e. W (t + u) − W (t) ∼ N(0, u), 

(e) W has continuous paths Q− a.s.. 

We call W , W = (W1, . . . , Wm) = (W1(t), . . . , Wm(t))t≥0 a m-dimensional Wiener 

process, m ∈ N, if its components Wj, j = 1, . . . , m, m ∈ N, are independent Wiener 

processes. 

One basic concept for modelling in finance, is the so-called martingale. 

Definition A.8 (Martingale) 

Let (Ω, F, Q, F) be a filtered probability space. A stochastic process X = (X(t))t≥0 is called 

a martingale relative to (Q, F) if X is adapted, EQ[|X(t)|] < ∞ for all t ≥ 0, and 

EQ = [X(t)|Fs] = X(s) Q − a.s. for all 0 ≤ s ≤ t < ∞. 

A.2 Stochastic Differential Equations 

A tool to describe the behaviour of financial assets and derivatives is the Itô Process. 

Definition A.9 (Itô Process) 

Let Wt be a m-dimensional Wiener process, m ∈ N. A stochastic process X = (X(t))t≥0 

is called an Itô process if for all t ≥ 0 we have 

X(t) = X(0) + 

= X(0) + 

� t 

0 

� t 

0 

µ(s)ds + 

µ(s)ds + 

� t 

0 

m� 

j=1 

σ(s)dW (s) (A.1) 

� t 

0 

σj(s)dWj(s), 

where X(0) is (F0−) measurable and µ = (µ(t))t≥0 and σ = (σ(t))t≥0 are m-dimensional 

progressively measurable stochastic processes with 

� t 

0 

� t 

0 

|µ(s)| ds < ∞ (A.2) 

σ 2 j (s)ds < ∞ Q − a.s. for all t ≥ 0, j = 1, . . . , m. (A.3) 

A n-dimensional Itô process is given by a vector X = (X1, . . . , Xn), n ∈ N, with each Xi 

being an Itô process, i = 1, . . . , n. 

Remark A.10 

For convenience we write (A.1) symbolically 

dX(t) = µ(t)dt + σ(t)dW (t) = µ(t)dt + 

m� 

σj(t)dWj(t), (A.4) 

j=1 

180


and call this stochastic differential equation (SDE) with drift parameter µ and diffu- 

sion parameter σ. 

To use Itô’s Lemma, we have to define the quadratic covariance process. 

Definition A.11 (Quadratic Covariance Process) 

Let m ∈ N and W = (W1(t), . . . , Wm(t))t≥0 and X2 = (X2(t))t≥0 be two Itô processes with 

dXi(t) = µi(t)dt + σi(t)dW (t) = µi(t)dt + 

m� 

σij(t)dWj(t), i = 1, 2. 

Then we call the stochastic process 〈X1, X2〉 = (〈X1, X2〉 t )t≥0 defined by 

〈X1, X2〉 t := 

m� 

j=1 

� t 

0 

j=1 

σ1j(s) · σ2j(s)ds 

the quadratic covariance (process) of X1 and X2. If X1 = X2 =: X we call the stochastic 

process 〈X〉 := 〈X, X〉 the quadratic variation (process) of X, i.e. 

where �σ(t)� := 

〈X, X〉 t := 

m� 

� t 

σ 

j=1 

0 

2 � t 

j (s)ds = 

0 

�σ(x)� 2 ds, 

� �m 

j=1 σ2 j , t ∈ [0, ∞) denotes the Euclidean norm in Rm and σ := σ1. 

Theorem A.12 (Itô’s Lemma) 

Let W = (W (t))t≥0 be a m-dimensional Wiener process, m ∈ N, and X = (X(t))t≥0 be 

an Itô process with 

dX(t) = µ(t)dt + σ(t)dW (t) = µ(t)dt + 

m� 

σj(t)dWj(t). (A.5) 

Furthermore, let G : R × [0, ∞) −→ R be twice continuously differentiable in the first 

variable, with the derivatives denoted by Gx and Gxx, and once continuously differentiable 

in the second, with the derivative denoted by Gt. Then we have for all t ∈ [0, ∞) 

or briefly 

G(X(t), t) = G(X(0), 0) + 

+ 

+ 1 

2 

� t 

0 

� t 

0 

� t 

0 

j=1 

Gx(X(s), s)dX(s) 

Gt(X(s), s)ds 

Gxx(X(s), s)d 〈X〉 (s) 

dG(X(t), t) = (Gt(X(t), t) + Gx(X(t), t)µ(t) 

+ 1 

2 Gxx(X(t), t) �σ(t)� 2 )dt 

+Gx(X(t), t)σ(t)dW (t). 

181


A.3 Equivalent Measure 

Definition A.13 (Equivalent Measure) 

Let Q and ˜ Q be two measures defined on the same measurable space (Ω, F). We say ˜ Q is 

absolutely continuous with respect to Q, written ˜ Q ≪ Q,if ˜ Q(A) = 0 whenever Q(A) = 0, 

A ∈ F. If both ˜ Q ≪ Q and Q ≪ ˜ Q, we call Q and ˜ Q equivalent measures and denote this 

by ˜ Q ∼ Q. 

The definition of equivalent measures states that two measures are equivalent if and 

only if they have same null sets. 

Definition A.14 (Radon Nikodym Derivative) 

Let Q be a sigma-finite measure and ˜ Q be a measure on the measurable space (Ω, F) with 

˜Q < ∞. Then ˜ Q ≪ Q if and only if there exists an integrable function f ≥ 0 Q−a.s. such 

that 

� 

˜Q(A) = 

A 

fdQ ∀A ∈ F. 

f is called the Radon-Nikodym derivative of ˜ Q with respect to Q and is also written as 

f = d ˜ Q 

dQ . 

Let γ = (γ(t))t≥0 be a m-dimensional progressively measurable stochastic process, 

m ∈ N, with 

� t 

0 

γ 2 j (s)ds < ∞ Q − a.s. ∀t ≥ 0, j = 1, . . . , m 

Let the stochastic process L(γ) = (L(γ, t))t≥0 = (L(γ(t), t))t≥0, ∀t ≥ 0 be defined by 

L(γ, t) = e − � t 

0 γ(s)′ dW (s)− 1 � t 

2 0 ||γ(s)||ds 

Note that the stochastic process X(γ) = (X(γ, t))t≥0 = (X(γ(t), t))t≥0 with 

or 

X(γ, t) := 

� t 

0 

γ(s) ′ dW (s) + 1 

2 

� t 

0 

||γ(s)|| 2 ds 

dX(γ, t) := 1 

2 ||γ(t)||2 dt + γ(t) ′ dW (t) 

is ∀t ∈ [0, ∞) an Itô process with µ(γ(t), t) = 1 

2 ||γ(t)||2 = 1 

2 

�m j=1 γ2 j (t) and σ(γ(t), t) = 

γ(t) ′ . Thus, using the transformation G : R × [0, ∞) ↦→ R with G(x, t) = e −x and Itô’s 

Lemma (Theorem A.12) with G(X(γ, t), t) = e −X(γ,t) = L(γ, t) we obtain: 1 

Lemma A.15 (Novikov Condition) 

dL(γ, t) = −L(γ, t)γ(t) ′ dW (t) 

Let γ and L(γ) be as defined above. Then L(γ) = (L(γ, t)) t∈[0,T ] is a continuous (Q -) 

martingale if 

EQ 

1 For a detailed calculation see [Zag02], p. 33. 

� 

e 1 � t 

2 0 ||γ(s)||2ds � 

< ∞. 

182

Remark A.16 


Under Novikov’s condition 

� T 

0 

�γ(s)� 2 ds < ∞ Q − a.s. 

and � T 

�γ(s)� 2 ds < ∞ Q − a.s. for all t ∈ [0, T ] 

Remark A.17 

0 

For each T ≥ 0 we define the measure ˜ Q = Q L(γ,T ) on the measure space (Ω, FT ) by 

� 

˜Q(A) := EQ[1A · L(γ, T )] = 

A 

L(γ, T )dQ for all A ∈ FT , 

which is a probability measure if L(γ, T ) is a Q-martingale. In this case, L(γ, T ) is the 

Q-density of ˜ Q, i.e. L(γ, T ) = d ˜ Q 

dQ on (Ω, FT ). 

In the following, we provide the Girsanov Theorem, which shows how a ( ˜ Q−) Wiener 

process ˜ W = ( ˜ W (t)) t∈[0,T ] starting with a (Q−) Wiener process W = (W (t))t≥0 can be 

constructed. 

Theorem A.18 (Girsanov) 

Let W = (W1(t), . . . , Wm(t))t≥0 be a m-dimensional (Q−) Wiener process, m ∈ N, γ, L(γ), ˜ Q, 

and T ∈ [0, ∞) be as defined above, and the m-dimensional stochastic process ˜ W = 

( ˜ W1, . . . , ˜ Wm) = ( ˜ W1(t), . . . , ˜ Wm(t)) t∈[0,T ] be defined by 

i.e. 

˜Wj(t) := Wj(t) + 

� t 

0 

γj(s)ds, t ∈ [0, T ], j = 1, . . . , m, 

d ˜ W (t) := γ(t)dt + dW (t), t ∈ [0, T ]. 

If the stochastic process L(γ) = (L(γ, t)) t∈[0,T ] is a (Q−) martingale, then the stochastic 

process ˜ W is a m-dimensional ( ˜ Q−) Wiener process on the measure space (Ω, FT ). 

For γ(t) constant the change of measure corresponds to a change of drift from µ to µ − γ. 

183

Appendix B 

Ratings 

Rating Description 

AA Very strong capacity to meet financial commitments 

A Strong capacity to meet financial commitments, but somewhat susceptible to 

adverse economic - conditions and changes in circumstances 

BBB Adequate capacity to meet financial commitments, but more subject to adverse 

economic conditions 

BB Less vulnerable in the near-term but faces major ongoing uncertainties to 

adverse business, financial and economic conditions 

B More vulnerable to adverse business, financial and economic conditions but 

currently has the capacity to meet financial commitments 

CCC Currently vulnerable and dependent on favorable business, financial and economic 

conditions to meet financial commitments 

CC Currently highly vulnerable 

C A bankruptcy petition has been filed or similar action taken but payments or 

financial commitments are continued 

D Payment default on financial commitments 

Table B.1: Standard & Poor’s credit ratings 

Ratings in the AAA, AA, A and BBB categories are regarded by the market as 

investment grade. 

Ratings in the BB, B, CCC, CC and C categories are regarded as having significant 

speculative characteristics. 

Ratings from AA to CCC may be modified by the addition of a plus (+) or minus (-) 

sign to show relative standing within the major rating categories. 1 

1 See www.standardandpoors.com. 

184

Appendix C 

Calculation of the Dynamics of the 

CDS Index 

With Itô’s Lemma 1 we can calculate the dynamics of the CDS Index. We consider the 

CDS index spread as weighted spread of n different rating classes. The dynamics of the 

short spread is given by equation (5.13). Then, the dynamics X of n rating classes is given 

by 

with 

⎛ ⎞ 

X1(t) 

⎜ ⎟ 

⎜ X2(t) ⎟ 

X(t) = ⎜ ⎟ 

⎜ ⎟ 

⎝ . ⎠ , θs 

⎛ 

⎜ 

= ⎜ 

⎝ 

Xn(t) 

⎛ ⎞ 

W1(t) 

⎜ ⎟ 

⎜ W2(t) ⎟ 

W (t) = ⎜ ⎟ 

⎜ ⎟ 

⎝ . ⎠ . 

dX(t) = [θs + bsuu(t) − bsωω(t) − ass(t)] dt + ΣdW (t), (C.1) 

θs1 

θs2 

. 

θsn 

⎞ 

⎛ 

⎟ 

⎠ , bsu 

⎜ 

= ⎜ 

⎝ 

bsu1 

bsu2 

. 

bsun 

⎞ 

⎛ 

⎟ 

⎠ , bsω 

⎜ 

= ⎜ 

⎝ 

bsω1 

bsω2 

. 

bsωn 

⎞ 

⎛ 

⎟ 

⎠ , as 

⎜ 

= ⎜ 

⎝ 

Wn(t) 

Let wT = (w1, w2, ..., wn) be a vector with portfolio weights. We define G(X(t), t) as 

The dynamics of dG(X(t), t) is then given by 

dG(X(t), t) = w T dX(t) 

as1 

as2 

. 

asn 

⎞ 

⎟ 

⎠ , 

G(X(t), t) = w T X(t). (C.2) 

= w T [θs + bsuu(t) − bsωω(t) − ass(t)] dt + w T ΣdW (t). (C.3) 

The last term of equation (C.3) yields to a scalar. We want to express it as product of 

two scalars and therefore, we equate: 

1 See Theorem A.12. 

σ ∗ dW ∗ (t) = w T Σ dW (t) 

185

APPENDIX C. CALCULATION OF THE DYNAMICS OF THE CDS INDEX 

Comparing the volatilities, we get: 

σ ∗2 dt = w T ΣΣ T w dt 

⇔ σ ∗2 = w T ΣΣ T w 

⇔ σ ∗2 = w T COV [X(t)]w 

⇔ σ ∗2 = w T 

⎛ 

⎞ 

σ1 0 0 . . . 0 

⎜ 

⎟ 

⎜ 0 σ2 0 . . . 0 ⎟ 

⎜ 

⎟ 

⎜ . 

⎝ . .. 

⎟ 

. ⎠ Γ[X(t)] 

⎛ 

σ1 0 0 . . . 0 

⎜ 0 σ2 0 . . . 0 

⎜ . 

⎝ . .. . 

0 0 . . . σn 

0 0 . . . σn 

⎞ 

⎟ 

⎠ w, 

(C.4) 

where Γ[X(t)] is the correlation matrix of X(t). Then, we rewrite equation (C.3) according 

to 

dG(X(t), t) = w T [θs + bsuu(t) − bsωω(t) − ass(t)] dt + σ ∗ dW ∗ (t). (C.5) 

In this form, we use the equation to simulate the evolution of the CDS index spread. 

186

Appendix D 

Transition Matrix for European 

Union 

The transition matrix 1 was adjusted by issuers who withdrew their rating, by normalising 

the row sums so that they sum up to one. 


AAA 89.92% 9.26% 0.66% 0.16% 0.00% 0.00% 0.00% 0.00% 

AA 0.22% 89.63% 9.63% 0.48% 0.01% 0.01% 0.01% 0.01% 

A 0.00% 2.31% 92.27% 5.16% 0.20% 0.03% 0.01% 0.01% 

BBB 0.00% 0.22% 5.07% 90.79% 2.79% 0.71% 0.14% 0.29% 

BB 0.00% 0.00% 0.00% 3.56% 85.18% 9.88% 0.60% 0.79% 

B 0.00% 0.00% 0.32% 0.64% 7.37% 79.50% 6.09% 6.09% 

CCC/C 0.00% 0.00% 0.00% 0.00% 0.00% 14.28% 31.43% 54.29% 

D 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 0.00% 100.00% 

Table D.1: European Union average one-year transition rates, 1981 to 2005 

1 The transition matrix is taken from [Sta06]. 

187

Appendix E 

List of Abbreviations 

ABS Asset-Backed Security 

bp basis point 

CVaR Conditional Value at Risk 

CBO Collateralised Bond Obligation 

CDO Collateralised Debt Obligation 

CDS Credit Default Swap 

CLN Credit Linked Note 

CLO Collateralised Loan Obligation 

CPI Consumer Price Index 

CVaR Conditional Value at Risk 

EEMEA Eastern Europe, Middle East & Africa 

ESG Economic Scenario Generator 

etc. et cetera 

EURIBOR European Interbank Offered Rate 

FRN Floating Rate Note 

GDP Gross Domestic Product 

IIC International Index Company 

ISDA International Swaps and Derivatives Association 

LIBOR London Interbank Offered Rate 

MRL Mean Reverting Level 

MVP Minimum Variance Portfolio 

NIG Normal Inverse Gaussian 

OTC Over-the-counter 

SDE Stochastic Differential Equation 

SPE Special Purpose Entity 

SPV Special Purpose Vehicle 

TRS Total Return Swap 

TMT Telecom, Media, Technology 

VaR Value at Risk 

188

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