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From pricing to rating structured credit products and vice-versa

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par Quentin Lintzer
Université Pierre et Marie Curie - Paris VI - Master 2 2007
  

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From pricing to rating structured credit products and
vice-versa

Quentin Lintzer
Université Paris VI - Pierre & Marie Curie

Thesis submitted for the degree of:
Master M2 in Probability Theory and Applications

· September 2007 ·

Abstract

Credit risk area is one of the most rapidly developing areas in finance. A wide range of synthetic structured credit products builds on liquid credit instruments such as Credit Default Swaps, credit indices or Credit Synthetic Obligations (CSO) referenced on the latter. In this document, we first recall the principles of CSO pricing in a one factor gaussian copula model and outline two numerical procedures aimed at mapping joint loss distributions. We also formalize the theoretical modelling framework of Moody's, a rating agency, for rating Constant Proportion Debt Obligation (CPDO) and Constant Proportion Portfolio Insurance (CPPI) products. We then present our conclusions regarding an innovative Dynamic Proportion Portfolio Insurance (DPPI) product and raise some risk management issues.

Contents

1

Structured credit products: a business review

1.1 Introduction

1.2 Elementary building blocks

4
4
4

 
 

1.2.1

Credit Default Swaps

4

 
 

1.2.2

Credit indices

6

 
 

1.2.3

Collateralized Synthetic Obligations

6

 
 

1.2.4

Structured Non-Correlation Products

9

2

Modelling and pricing CSO tranches

12

 

2.1

Modelling a CSO tranche payoff

12

 

2.2

Default and premium legs of a CSO tranche

12

 
 

2.2.1

Default leg

13

 
 

2.2.2

Premium leg

13

 
 

2.2.3

Fair premium

13

 

2.3

The semi-analytic approach: one factor Gaussian Copula model . . .

14

 
 

2.3.1

Copula functions: basic properties

14

 
 

2.3.2

The one factor gaussian copula model

14

 

2.4

Back to CSO tranche pricing: computing the expected portfolio loss

16

 
 

2.4.1

Monte-Carlo simulations

16

 
 

2.4.2

Evaluating the loss characteristic function

16

 
 

2.4.3

Andersen's recursive formula

17

3 Modelling and Rating Dynamic Proportion Portfolio Insurance prod-

ucts 19

3.1

Moody's approach to rating CPDO and

 
 

CPPI/DPPI products

19

 

3.1.1

Historical vs risk-neutral probability measures

19

 

3.1.2

Moody's Metric and coherent risk measures

20

3.2

Modelling risk factors

23

 

3.2.1

Credit spread processes influenced by defaults and ratings . .

23

 

3.2.2

Rating migrations and default events

24

 

3.2.3

Interest rates process and other parametres

26

3.3

Portfolio investment rules

27

 

3.3.1

Dynamic leverage function

27

 

3.3.2

Deferred coupons

29

 

3.3.3

Other key structural features

30

3.4

A study of the DPPI's sensitivities

30

 

3.4.1

Tailor-made structural features to achieve target rating. . . .

31

 

3.4.2

Hypothetical stress-scenarios

35

3.5 DPPI: any hidden pricing issue? 36

3.5.1 From the investor's perspective 36

3.5.2 From the investment bank's perspective 36

Conclusion 37

Appendix 38

Bibliography 42

List of Figures

1.1

Cash flows of a Credit Default Swap with physical delivery

5

1.2

Structuring of a single-tranch CSO

7

1.3

Structuring of a first-generation CPDO, referencing credit indices . .

9

3.1

Moody's rating conversion table

21

3.2

Moody's idealized EL values by rating category and tenor

22

3.3

DPPI base-case loss distribution conditional on the structure not cash-

 
 

ingin

31

3.4

Distribution of cash-in times

31

3.5

Estimated expected loss as a function of S.

32

3.6

Estimated expected loss as a function of è (2000 simulations per

 
 

coupon level)

33

3.7

Loss and Moody's Metric as a function of OL and TM, 1000 simula-

 
 

tions per couple of parametres

34

8

Parametres of Moody's CDS spread processes

39

9

Correlation matrix of Moody's CDS spread processes

39

10

Moody's 10Y corporate rating transition matrix

39

11

DPPI optimized structural features

40

12

DPPI reference portfolio

41

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