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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
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1
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Structured credit products: a business review
1.1 Introduction
1.2 Elementary building blocks
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4
4
4
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1.2.1
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Credit Default Swaps
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4
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1.2.2
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Credit indices
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6
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1.2.3
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Collateralized Synthetic Obligations
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6
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1.2.4
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Structured Non-Correlation Products
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9
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2
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Modelling and pricing CSO tranches
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12
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2.1
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Modelling a CSO tranche payoff
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12
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2.2
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Default and premium legs of a CSO tranche
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12
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2.2.1
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Default leg
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13
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2.2.2
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Premium leg
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13
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2.2.3
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Fair premium
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13
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2.3
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The semi-analytic approach: one factor Gaussian Copula model . .
.
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14
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2.3.1
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Copula functions: basic properties
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14
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2.3.2
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The one factor gaussian copula model
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14
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2.4
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Back to CSO tranche pricing: computing the expected portfolio
loss
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16
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2.4.1
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Monte-Carlo simulations
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16
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2.4.2
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Evaluating the loss characteristic function
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16
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2.4.3
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Andersen's recursive formula
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17
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3 Modelling and Rating Dynamic Proportion Portfolio Insurance
prod-
ucts 19
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3.1
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Moody's approach to rating CPDO and
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CPPI/DPPI products
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19
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3.1.1
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Historical vs risk-neutral probability measures
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19
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3.1.2
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Moody's Metric and coherent risk measures
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20
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3.2
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Modelling risk factors
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23
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3.2.1
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Credit spread processes influenced by defaults and ratings .
.
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23
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3.2.2
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Rating migrations and default events
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24
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3.2.3
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Interest rates process and other parametres
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26
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3.3
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Portfolio investment rules
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27
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3.3.1
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Dynamic leverage function
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27
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3.3.2
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Deferred coupons
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29
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3.3.3
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Other key structural features
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30
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3.4
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A study of the DPPI's sensitivities
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30
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3.4.1
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Tailor-made structural features to achieve target rating. . .
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31
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3.4.2
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Hypothetical stress-scenarios
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35
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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
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1.1
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Cash flows of a Credit Default Swap with physical delivery
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5
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1.2
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Structuring of a single-tranch CSO
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7
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1.3
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Structuring of a first-generation CPDO, referencing credit
indices . .
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9
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3.1
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Moody's rating conversion table
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21
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3.2
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Moody's idealized EL values by rating category and tenor
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22
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3.3
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DPPI base-case loss distribution conditional on the structure
not cash-
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ingin
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31
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3.4
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Distribution of cash-in times
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31
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3.5
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Estimated expected loss as a function of S.
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32
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3.6
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Estimated expected loss as a function of è (2000
simulations per
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coupon level)
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33
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3.7
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Loss and Moody's Metric as a function of OL and TM, 1000
simula-
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tions per couple of parametres
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34
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8
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Parametres of Moody's CDS spread processes
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39
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9
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Correlation matrix of Moody's CDS spread processes
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39
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10
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Moody's 10Y corporate rating transition matrix
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39
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11
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DPPI optimized structural features
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40
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12
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DPPI reference portfolio
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41
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