The Hidden Correlation of Collateralized Debt Obligations N. N. Kellogg College University of Oxford A thesis submitted in partial fulfillment of the MSc in Mathematical Finance April 13, 2009 Acknowledgments I would like to thank my supervisor Dr Alonso Pe~nafor his advice and help and for encouraging me to follow the approaches taken in this thesis. I want to thank my wife Blanca for her direct (by reading and correcting) and indirect (by motivating me) support of this work. I would like to thank my former employer d-fine for the possibility to attend the MSc Programme in Mathematical Finance at the University of Oxford. For their engage- ment in this programme, I particularly would like to thank Dr Christoph Reisinger, the course director, and Prof Dr Sam Howison. Thank also goes to the other lecturers of the Mathematical Finance Programme. Finally, I thank my friend Ari Pankiewicz for the effort to read and correct this thesis. ii Abstract We propose a model for the correlation structure of reference portfolios of collater- alized debt obligations. The model is capable of exhibiting typical characteristics of the implied correlation smile (skew, respectively) observed in the market. Moreover, it features a simple economic interpretation and is computationally inexpensive as it naturally integrates into the factor model framework. iii Contents List of Figures v List of Tables vi 1 Introduction to Collateralized Debt Obligations 1 1.1 The CDO Market . .1 1.2 Valuation of STCDOs . .4 1.3 Outline . .5 2 Modeling of Multivariate Default Risk 7 2.1 Structural Models . .8 2.2 Copula Functions . .9 2.3 Factor Models . 11 3 Default Correlation 18 3.1 Implied Correlation . 19 3.2 Correlation Smile . 22 4 Modeling the Correlation Matrix 25 4.1 Empirical Correlations and RMT . 26 4.2 Group Models . 28 4.3 Attainable Correlation Smiles . 31 5 Numerical Implementation 40 5.1 Integration Schemes . 40 5.2 Existence and Uniqueness of Correlation . 48 5.3 Performance . 49 Conclusions 51 iv CONTENTS CONTENTS A Inverse Transform Sampling 53 B Correlation in Student-t Factor Model 54 C Used Parameters 56 Bibliography 59 v List of Figures 3.1 Example of correlation smile/skew observed in the market. 22 3.2 Loss distribution, un-skewed vs. skewed. 23 4.1 Example of implied correlations for non-flat correlation matrix. 25 4.2 Implied corr. for varying µ in the two-layer hierarchical model. 32 4.3 Implied corr. for varying ∆ in the two-layer hierarchical model. 32 4.4 Implied corr. for varying n in the two-layer hierarchical model. 33 4.5 Implied corr. for varying ν in the two-layer hierarchical model. 34 4.6 Implied corr. for varying ∆n in the two-layer hierarchical model. 35 4.7 Implied corr. for varying µρ in the two-layer hierarchical model. 36 4.8 Implied corr. for varying ∆ρ in the two-layer hierarchical model. 36 4.9 Implied corr. for varying ∆n in the block model. 37 4.10 Implied corr. for varying ∆ρ in the block model. 38 5.1 Integration schemes for the 1-factor Gaussian copula. 46 5.2 Integration schemes for the 2-factor Student-t copula. 47 5.3 Integration schemes for the 2-factor Gaussian copula. 48 5.4 Uniqueness and existence of implied compound correlations. 49 vi List of Tables 4.1 Types of group models. 29 5.1 Test cases for the validation of the developed program code. 41 5.2 Tranche spreads: LHP vs. numerical results in factor model framework. 49 5.3 CPU times for CDO pricing in different parameter regimes. 50 C.1 Used parameters for the validation of integration schemes. 56 C.2 Used parameters for the implied correlations of group models. 57 vii Chapter 1 Introduction to Collateralized Debt Obligations 1.1 The CDO Market After a rapid growth in the recent ten years, the market for collateralized debt obli- gations (CDOs) has seen drastic and devastating changes in the months prior and during the creation of this thesis. Being in the center of the subprime mortgage crisis, the value of existing CDO positions has sharply dropped {if a price can be assigned at all, given the almost non-existing market{ and the issuing of new CDOs has slowed down by almost 90% in 2008 compared to 2007 [1]. For the riskiest CDO portions on subprime mortgages, the term "toxic waste" has been coined and the underlying mortgage-backed securities (MBS) are the main focus of the US$ 700 bil- lion "Troubled Assets Relief Program" (commonly referred to as "bailout") of the US government [2]. We will later come to the involvement of CDOs in these recent developments, but first lay out the basic ideas behind CDOs. CDOs are structured finance products that transfer and distribute credit risk on a reference portfolio of assets into tranches of increasing seniority. The risk is transferred from the arranger of the CDO, who acts as buyer of protection, to the investor, who acts as seller of protection. The common feature of all CDOs is the splitting of the underlying reference portfolio into different tranches, which defines a sequential allocation of the losses that occur in the portfolio. The lowest-lying tranche (equity tranche) is the first one to absorb losses, with the losses resulting in a payment from the investor to the arranger (and a decrease of the premiums paid to the investor, see below), up to a coverage limit which defines the size of the tranche (e.g. 3% of the CDO notional). If the cumulative losses exceed this limit, the next tranche (mezzanine tranche, e.g. 3%-6%) is affected, and so on. The CDO investors thus 1 1.1 The CDO Market 1 Introduction to Collateralized Debt Obligations take on exposure to a particular tranche, where the higher risk of the lower (junior) tranches is naturally rewarded with a higher premium, whereas the higher (senior) tranches are protected by the junior tranches, hence yielding a much lower spread. In essence, CDOs are based on the idea that the idiosyncratic risk of the names in the portfolio can be diversified out and that the diversified pool of risky assets has a relatively predictable loss pattern, where losses hardly ever touch the senior tranches. This allows the senior tranches to achieve a higher rating than the average rating of the underlying portfolio. However, the residual systematic risk crucially depends on the dependency structure (loosely termed "correlation" in the following, see Chapters 2 and 3 for details) of the portfolio and gives rise to correlation risk and correlation trading. Importantly, the correlation risk varies among the tranches: a high correlation benefits equity investors, whereas the price of senior tranches drops as correlation rises, see Chapter 3. Originally, the aim of the first CDO deals was the management of the balance sheet of the originating banks by making use of the possibility to transfer credit risk and free up regulatory capital (balance-sheet CDOs), with the underlying assets typically being cash assets like bank loans, bonds, or asset-backed securities (ABS, MBS). The ownership of these assets then needs to be transferred to a separate legal entity (SPV, special purpose entity) which acts as arranger of the CDO and issues the tranches (cash CDO). In contrast, synthetic CDOs are backed by credit default swaps (CDS); the arranger typically does not hold these CDS positions beforehand but enters into them only in order to set up the CDO. This is done to take advantage of a possible spread difference between the average yield of the underlying CDS portfolio and the spread paid on the tranches (arbitrage CDO) 1. The spread difference can be seen as service charge for bundling the portfolio and reflects the benefit for the investor, who otherwise would have to assemble the portfolio himself. As further step in the product development of synthetic CDOs, single-tranche CDOs (STCDOs) entered the market in 2003 and in the same year already accounted for 90% of the issued synthetic CDOs [12]. In a STCDO, only one isolated tranche is sold to a investor, thereby carving out a piece of the loss profile. From the investor's point of view, the popularity of STCDOs originates from the flexibility in the risk structure (the underlying credit portfolio, level of subordination, and tranche size can be chosen by the investor according to his needs) while providing a substantially higher yield than similarly rated investments, whereas the arrangers appreciate the comparably 1An arbitrage CDO can also be set up as cash CDO. 2 1 Introduction to Collateralized Debt Obligations 1.1 The CDO Market easy set-up of the transaction. However, as the arranger sells only part of the CDO structure, he exposes himself to the "remaining" credit risk and needs to hedge that risk. STCDOs are often written on standardized CDS indices like CDX and iTraxx which facilitates this hedging for the arranger. Obviously, STCDOs are leveraged products with respect to the spreads of the underlying names: if the portfolio losses would increase from e.g. 3% to 6% due to a change in the credit spreads of the same order of magnitude, a 3%-6% STCDO would suffer a total loss. In the course of the product evolution, also more sophisticated products have appeared on the market of which we only want to give two examples: CDO2 are CDOs backed by tranches of other, underlying CDOs (and ABS, typically), where additional leverage is achieved by the overlap of names in the reference portfolios of the underlying CDOs (can be further enhanced to CDOn).